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FARMER RISK MANAGEMENT BEHAVIOR AND WELFARE UNDER
ALTERNATIVE PORTFOLIOS OF RISK INSTRUMENTS

By

Hong Wang

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Agricultural Economics

1996

ABSTRACT
FARMER RISK MANAGEMENT BEHAVIOR AND WELFARE UNDER
ALTERNATIVE PORTFOLIOS OF RISK INSTRUMENTS
By

Hong Wang

Agricultural returns are risky because production is highly sensitive to weather,
disease, and pests. Production risk contributes to price variability because of biological
time lags, inelastic demand, and asset fixity. Farmer income risk comes from the joint
effect of price and yield risks. Futures, options, crop insurance, and government
programs can be used to manage the overall income risk.

Existing research tends to study particular risk management instruments in
isolation from others. This dissertation studies farmers’ optimal use of alternative
instruments in a portfolio setting and the resulting welfare. Current political debates are
ooncemed with the budgetary cost of supporting crop insurance and government
programs. Potential policy changes in the crop insurance and government program may
significantly impact a farmer’s risk management behavior and welfare. Some of these
effects are analyzed in this dissertation.

A bivariate ARCH model with seasonality is used to capture the time-varying
volatilities and excess kurtosis prOperties of commodity cash and futures prices. A
deterministic trend model with nonnorrnal errors from a hyperbolic sine transformation
is used to characterize yields based on the limited sample size and asymmetric

distribution. Because the distributions and income Structure are complicated, numerical

multivariate distributions for prices and yields are simulated, and numerical results are
analyzed in an expected utility framework.

The results suggest crop insurance is usually more valuable than the futures and
options because yield is generally more volatile than price. Futures and options may be
used to "cross hedge" yield risk if price and yield are correlated.

With futures and options, the value of the government program is derived
primarily from the implicit subsidy, so it decreases as the subsidy is reduced and its risk
reducing effect is not important when futures and options are available.

There is a tradeoff between the yield basis risk using area yield crop insurance
(AYCI) and the individual yield crop insurance (IYCI) premium. AYCI may provide
higher welfare gain than IYCI while significantly reducing the government cost, if the
IYCI premium is above the actuarially fair level, yield basis risk is low, or AYCI trigger

yield restrictions are above lYCI levels.

To my husband Hao Zhang,

and my parents Dongxiong & Fengsheng Wang

iv

ACKNOWLEDGMENTS

I’m very grateful to many faculty, staff, and my fellow graduate students in the
Department of Agricultural Economics, from whom I have received constant support and
encouragement which made my graduate study at Michigan State University pleasant and
fruitful.

I’m greatly in debt to my dissertation supervisor, Dr. Steve Hanson, who has
continually encouraged me through his kindness and patience. His guidance has made
the completion of this dissertation a smooth and enjoyable process, and his advice has
helped me develop professionally in the discipline. Furthermore, his personal care and
deep concern for his students has kept me in high spirit, without which this dissertation
may never have been accomplished. It is impossible to fully express in words my respect
and gratitude toward him.

I extend gratitude to my major professor, Dr. Roy Black, for his insightful
guidance to my academic program, comments on my dissertation, scholarly challenge to
my professional progress, and years of financial support. I also thank Dr. Robert Myers
for his active service in my dissertation committee, and the valuable inputs he has
contributed to my dissertation. He has not only helped me to widen the scope of my

academic vision, but also inspired me with the power of economic theory. I’m also

benefited from Dr. Jack Meyer, another committee member, through his excellent
teaching in economic theory and consultation on risk analysis.

My gratitude also goes to Dr. Lindon Robison who taught me important concepts
in risk analysis; to Dr. Scott Swinton who has helped me develop teaching skills; to
Elizabeth Bartilson who helped me format this dissertation; and to Rose Greenman,
Nicole Alderman, and Sherry Rich who also provided a lot of help that might go beyond
their responsibilities.

I give great thanks to my parents and my sister who would sacrifice everything
for my education and have encouraged me since my early childhood to try my best on
everything I start and to develop perseverance. My deepest thanks go to my husband,
Hao Zhang, to whom I owe the most. His unconditional love, constant encouragement,
aeademic inspiration, and sacrifice are the indispensable factors contributing to my Ph.D

degree and this dissertation.

vi

TABLE OF CONTENTS

LIST OF TABLES ...................................... xi
LIST OF FIGURES ..................................... xiii
Chapter I
INTRODUCTION ....................................... 1
Chapter 11
DECISION MAKING UNDER RISK IN THE AGRICULTURAL SECTOR . . . 7
2.1 Risk and Risk Management .......................... 8
2.2 Cash, Futures, Options, Government Program, and Crop Insurance . 10
Cash Contracts ............................... 10
Futures Contracts .......................... '. . . . 11
Options Contracts ............................. 12
Risk Management Using Futures and Options ............. 13
Government Deficiency Payments .................... l4
Crop Insurance ............................... 16
Individual Yield Crop Insurance (IYCI)---MPCI ...... 19
Area Yield Crop Insurance (AYCI)---GRP .......... 22
2.3 Decision Making Under Risk ........................ 24
Expected Utility Model .......................... 25
Application of the EU Model .................. 26
Risk Preference Measures .................... 28
Alternative Models for Decision Making Under Risk ........ 30
SD Model .............................. 31
MV Model ............................. 32
Choosing a Decision Model ....................... 32
2.4 Risk Management with Price and Yield Instruments ........... 33
Futures and Options as Risk Management Instruments ....... 35
Government Deficiency Payment Program .............. 38
Crop Insurance ............................... 38
Price and Yield Instrument Portfolios ................. 42
Chapter III
THE FARMER DECISION MAKING MODEL .................... 44

vii

3.1 Model ......................................
3.2 Characteristics of Risk Management Instruments .............
Futures ....................................
Options ....................................
Government Deficiency Payment Program ..............
Crop Insurance ...............................
3.3 Methodology ..................................
3.4 Model Validation ................................

Chapter IV
THE JOINT PRICE AND YIELD DISTRIBUTION .................
4.1 The Joint Cash and Futures Price Distribution ..............
Mean Price Levels .............................
Properties of Com Price Data ......................
Analysis of Price Level Data ..................
Analysis of Higher Moments ..................
Price Model Specification and Estimation ...............
Empirical Price Distributions .......................
4.2 The Joint Farm and Area Yield Distribution ...............
Properties of Yield Data .........................
Yield Model Specification and Estimation ...............
Empirical County-level Yield Distribution ...............
Farm-level Yield Distributions ......................
Bivariate Farm-level and County-level Yield Distributions .....
Correlation Between the Farm- and County-level Yields . .
Imposing Correlation Between Farm and County Yields . .
4.3 Multivariate Price and Yield Distribution .................
Correlation Between Cash Price and Area Yield ...........
Imposing Correlation Between Price and Yield ............

Chapter V
OPTIMAL RISK MANAGEMENT CHOICES UNDER ALTERNATIVE
PORTFOLIOS OF RISK INSTRUMENTS .......................
5.1 Price Instruments ................................
No Risk Management Instrument ....................
Futures Only ................................
Futures and Options ............................
Government Deficiency Payment Program Only ...........
Futures and Government Program ....................
Futures, Options and Government Program ..............
Summary ..................................
5.2 Price Instruments With Individual Yield Crop Insurance ........
IYCI Only ..................................
Futures and IYCI ..............................

viii

47
47
48
49
49
52
54

58
59
64
68
74

90
93
101
101

111
116
116
119
121
122
123
124
125
127
127
131

Futures, Options and IYCI ........................ 132

Government Program and IYCI ..................... 133

Futures, Government Program and IYCI ............... 134

Futures, Options, Government Program, and IYCI ......... 135

Summary .................................. 136

5.3 Price Instruments With Area Yield Crop Insurance ........... 138

AYCIOnly...................... ........... 138

Futures Market and AYCI ........................ 140

Futures, Options and AYCI ....................... 142

Government Program and AYCI .................... 143

Futures, Government Program and AYCI ............... 145

Futures, Options, Government Program and AYCI ......... 146

Summary .................................. 147

5.4 Conclusions ................................... 149
Chapter VI

ECONOMIC IMPACTS OF ALTERNATIVE RISK STRUCTURES AND

PREFERENCES ....................................... 152

6.1 Price-Yield Correlation ............................ 154

6.2 Risk Preferences ................................ 158

6.3 Yield Risk .................................... 163

Farm-level Yield Risk ........................... 164

Yield Basis Risk .............................. 168

6.4 Summary and Conclusion ........................... 169
Chapter VII

ECONOMIC EVALUATION OF CROP INSURANCE CONTRACT DESIGN . 172

7.1 Introduction ................................... 172

Feasibility of Current Crop Insurance Programs ........... 173

Crop Insurance Contract Design Parameters .............. 175

7.2 Crop Insurance ................................. 178

Economic Impacts of Coverage Restrictions .............. 178

Economic Impacts of Yield Basis Risk ................. 184

No Trigger Yield Restriction .................. 185

Current Restrictions ........................ 188

7.3 Futures and Options .............................. 190

Economic Impacts of Coverage Restrictions .............. 191

Economic Impacts of Yield Basis Risk ................. 194

No Trigger Yield Restriction .................. 195

Current Restrictions ........................ 196

7.4 Government Program ............................. 198

Economic Impacts of Coverage Restrictions .............. 198

Economic Impacts of Yield Basis Risk ................. 201

No Trigger Yield Restriction .................. 201

ix

Current Restrictions ........................ 203

7.5 Other Contract Design Parameters ..................... 203
7.6 Summary and Conclusion ........................... 205

Chapter VIII

ECONOMIC EVALUATION OF THE GOVERNMENT DEFICIENCY PAYMENT

PROGRAM .......................................... 208
8.1 Change in Target Price ............................ 210
8.2 Change in Flexible Acreage ......................... 214
8.3 Change in Acreage Reductions ....................... 218
8.4 Summary .................................... 223

Chapter IX

CONCLUSION ....................................... 225

APPENDDI A

STATISTICAL HYPOTHESIS TESTING ....................... 231

APPENDIX B

SIMULATING BIVARIATE NORMAL DISTRIBUTIONS ............. 234

APPENDIX C

HYPERBOLIC SINE TRANSFORMATION ...................... 236

APPENDIX D

CORRELATION MATRICES WITH ZERO PRICE-YIELD CORRELATION . 242

BIBLIOGRAPHY ...................................... 243

LIST OF TABLES

Table 2.1 Classification of Risk in Agriculture ...................... 8
Table 2.2 Classification of Risk Management Strategies and Representation . . . . 9
Table 4.1 Phillips-Perron Unit Root Test for Prices .................. 61
Table 4.2 Bivariate Futures and Cash Price Process .................. 73
Table 4.3 Sample Moments of the Simulated Harvesting Prices ........... 78
Table 4.4 Phillips-Perron Unit Root Test for Yield .................. 86
Table 4.5 County-level Yield Model Estimates ..................... 89
Table 4.6 Statistics of the Simulated Yield Distribution .......... ’ ...... 91
Table 4.7 Maximum,Minimum and Average Standard Deviation of Farm Yield . 97
Table 4.8 Transformation Parameters for Mean-preservin g Spread Yield ..... 99
Table 4.9 Statistics of the Simulated 1994 Farm-level Yield Distributions . . . . 99
Table 4.10 Statistics of the Estimated Yield Correlations ............... 102
Table 4.11 Correlation Matrices of Simulated Prices and Yields .......... 109
Table 5.1 Model Parameters for the Base Solution .................. 114
Table 5.2 Optimal Positions, Income Distribution, and WTP without Crop

Insurance ..................................... 1 17
Table 5 .3 Optimal Positions, Income Distribution, and WTP with IYCI ..... 129
Table 5.4 Optimal Positions, Income Distribution, and WTP with AYCI ..... 139
Table 6.1 Risk Structure and Risk Preference Parameters .............. 153
Table 6.2 Optimal Positions and WTP with Zero Price-Yield Correlation ..... 156
Table 6.3 Futures and Options Positions for Alternative Risk Preferences . . . . 159
Table 6.4 Trigger Yield Levels for Alternative Risk Preference ........... 160
Table 6.5 Willingness-to-Pay for Alternative Risk Preferences ........... 161
Table 6.6 Futures and Options Positions for Alternative Yield Risks ........ 165
Table 6.7 Trigger Yield Levels for Alternative Yield Risks ............. 166
Table 6.8 Willingness-to-Pay for Alternative Yield Risks .............. 167
Table 7 .1 Participation and Willingness-to-pay with Coverage Restrictions . . . . 180
Table 7.2 Welfare-Equivalent IYCI Premiums for Alternative AYCI Basis

Levels ....................................... 185

Table 7 .3 Participation and Willingness-to-pay with Futures and Options ..... 192

xi

Table 7.4 Welfare-Equivalent IYCI Premiums for Alternative AYCI Basis

Levels with Futures and Options ....................... 195
Table 7.5 Participation and Willingness-to-pay with the Government Program . . 199
Table 7.6 Welfare-Equivalent IYCI Premiums for Alternative AYCI Basis

levels with Government Program ...................... 202
Table 8.1 Alternative Government Program Design Parameters ........... 209
Table 8.2 Participation Levels for alternative Target Price Levels ......... 211
Table 8.3 Willingness-to—Pay for Alternative Target Price Levels .......... 212
Table 8.4 Participation Levels for Alternative Flexible Acreage Levels ...... 215
Table 8.5 Willingness-to—Pay for Alternative Flexible Acreage Levels ....... 216
Table 8.6 Participation Levels for Alternative ARP Levels ............. 219
Table 8.7 Willingness-to—Pay for Alternative ARP Levels .............. 220
Table A.1 LR Tests for Degrees of Frequencies in Seasonality ........... 232
Table C.1 Distribution Statistics of HST of a Standard Normal Variable (6:0) . 238
Table C.2 Distribution Statistics of the HST of a Standard Normal Variable . . . 239

Table D.1 Correlation Matrices of Simulated Prices and Yields with ppy=0 . . . 242

xii

LIST OF FIGURES

Figure 4.1 Logarithm of December Corn Futures Prices from May 1989 to

April 1994 .............................. ' ...... 62
Figure 4.2 Logarithm of Iowa Corn Cash Prices from May 1989 to April 1994 . 63
Figure 4.3 Squared Logarithm of December Corn Futures Prices from May

1989 to April 1994 .............................. 65
Figure 4.4 Squared Logarithm of Iowa Corn Cash Prices from May 1989 to

April 1994 .................................... 66
Figure 4.5 Histograms of the Simulated Futures and Cash Price Distributions . . 77
Figure 4.6 Iowa County Average Corn Yield levels During 1928-1993 for

Adair, Adams and Case County Farms .................. 82
Figure 4.7 Iowa County Average Corn Yields in Logarithms During 1928-

1993 for Adair, Adams, and Case County Farms ............ 84
Figure 4.8 Simulated 1994 Corn Yield Distributions for Adair, Adams and

Case County .................................. 92
Figure 4.9 Histograms of Farm-level Yield Standard Deviation Distributions

for Adair, Adams and Case County .................... 96
Figure 4.10 Histograms of Simulated 1994 Farm-level Yield Distributions for

Low, Medium and High Risk Farms ................... 100
Figure 4.11 Histogram of Farm- and County-level Yield Correlations in Adair
County ..................................... 103

Figure 4.12 Illustration of the Taylor Transformation ................. 106
Figure 5.1 Histograms of the Income Distribution without Crop Insurance . . . . 118
Figure 5.2 Histograms of Income Distribution with IYCI .............. 130
Figure 5.3 Histograms of Income Distribution with AYCI .............. 141

Figure 7.1 Tradeoff Curve Between the AYCI Basis Risk and the IYCI
Premium without Trigger Yield Restriction ................ 186
Figure 7.2 Tradeoff Curve between the AYCI Basis Risk and the IYCI

Premium under the Current Trigger Yield Restrictions ......... 189

xiii

Chapter I

INTRODUCTION

Farm incomes are characterized by instability and risk that stem from several
sources. First, agricultural production is biological in nature so yields depend on natural
conditions, such as weather (temperature, moisture and sunshine), disease, insects and
other catastrophes (flood, drought, wind and fire). Second, many agricultural products
are perishable and difficult to store, such as dairy products, eggs, poultry, and livestock
products. Production processes also tend to be longer than most manufacturing
production processes, and investment is often characterized by asset fixity, causing
agricultural production to be inflexible in adjusting to market signals. In addition, the
market demand for agricultural products is price inelastic, so small fluctuations in supply
can cause large fluctuations in price. To help manage these risks, a variety of risk
management instruments are available to farmers.

Yield and price are the two main sources of risk in agriculture. These risks can
be managed through both market mechanisms and government programs. Commodity
futures markets have been available since the 1800s as a means of price risk
management. Options on futures, available since 1984, provide an additional way to
manage price risk for some agricultural commodities. Government commodity price

support and deficiency payment programs also protect some producers from unfavorable

l

2

prices. In addition, government sponsored crop insurance provides a tool to help manage
yield risk. Available since the late 19303, crop insurance now exists for most major
crops and in most regions throughout the US.

However, the crop insurance program imposes a heavy budgetary burden on the
government. To encourage participation, the government has subsidized up to 30 percent
of estimated premium costs and provided indirect subsidies in the form of free delivery
and administrative costs. The aggregate program loss ratio (indemnities paid / premiums
earned) during the 1981-1990 period was 1.42‘. Between 1980 and 1988, government
outlays for the federal crop insurance program exceeded 4.2 billion dollars, accounting
for over 80 percent of the total indemnities paid to farmers. In fact, indemnities
exceeded premiums in every year during the ten year period.

The deficiency payment program reduces price risk and provides an implicit
government subsidy to farmers by guaranteeing a minimum target price. The average
annual cost to the government has been about $6 billion in the past five years.

Recent political debates have focused on reducing and/or eliminating many farm
programs, including deficiency payments and crop insurance programs, because their
effectiveness has been questioned and they tend to impose a large cost on the government
budget. Current federal budget difficulties have increased this pressure.

Altering or eliminating these farm programs will influence the risk position of

farms, and farm risk management strategies and welfare may change correspondingly.

 

‘ This computation of the loss ratio uses total premiums in the denominator. Total
premiums are the sum of producer premiums and premium subsidy, Thus, if only the
producer premiums are included the loss ratio would be even higher.

3

The farmer’s use of risk management instruments depends not only on the properties of
the individual instrument, but also on the properties of other instruments available in the
portfolio. Therefore, it is important to study the impacts of change in farm programs on
risk management in a portfolio setting.

Considerable research has been done on the behavior of competitive firms using
futures and options markets. Danthine (1978); Holthausen (1979); and Feder, Just and
Schmitz (1980) investigated production and hedging behavior when producers face only
output price risk, and hedge using futures markets. Lapan, Moschini and Hanson (1991)
extended this approach to model production and hedging decisions when both futures and
options markets are available simultaneously.

Many studies on the optimal use of futures and options to manage income risk
assume yields are deterministic. However, as discussed earlier, agricultural commodity
yields tend to be stochastic. Sakong, Hayes and Hallam study futures and options under
both price and yield risks without crop insurance. A number of studies have examined
income risk management in the presence of stochastic yields using crop insurance.
Ahsan, Ali and Kurian (1982) showed that, in the absence of price uncertainty,
actuarially fair crop insurance can completely eliminate income risk. Chambers (1989)
studied crop insurance in the presence of moral hazard. Coble, Knight, Pope and
Williams (1993) tested for the existence and level of moral hazard and adverse selection
in the use of individual yield crop insurance. Miranda (1991) showed that area yield
insurance provides better risk protection for most producers than individual yield

insurance and significantly improves the actuarial performance of the insurance program.

4

Despite the rich set of research in the risk management area, most studies focus
on a particular risk management instrument studied in isolation, without acknowledging
that the instrument may be part of an overall portfolio of risk management instruments.
This dissertation explores the optimal behavior and welfare of agricultural producers,
when futures markets, options markets, government deficiency payment programs and
crop insurance are simultaneously available to manage price and yield risks. This is one
of the first comprehensive studies of farm risk management behavior which allows for
such an extensive portfolio of risk management instruments. More Specifically, the
objectives of this research are:

1. Develop a simulation model which can be used to evaluate farmer risk
management behavior and welfare in alternative policy and risk settings.

2. Evaluate the behavior of risk averse farmers when futures, options, crop
insurance and a government deficiency payment program are included in the risk
management portfolio.

3. Compare and evaluate alternative crop insurance designs and examine the relative
performance and tradeoffs between individual yield crop insurance (IYCI) and
area yield crop insurance (AYCI).

4. Investigate the impact of alternative designs for the government deficiency
payment program on farmer behavior and welfare.

It is necessary to have a model flexible enough so that the impacts of changes in
the availability and design of these instruments can be studied. In particular, crop
insurance and other government programs are subject to change, and so it is important
to be able to evaluate the impact of potential changes on a farmer’s risk management

behavior and welfare.

5

The dissertation is organized into nine chapters. Chapter 11 provides background
information on risk and risk management instruments, decision making models under
risk, and previous studies on risk management in agriculture. The general model is then
developed in Chapter III and the function of each risk management instrument is
explained. Because of the portfolio nature of the model it can not be solved analytically
except under very restrictive conditions. As a result, numerical optimization and
simulation methods are used to solve the model.

Chapter IV explains the simulation model parameterization for stochastic prices
and yields. A bivariate ARCH model with seasonal components is used to simulate the
eash and futures price distribution, and a deterministic trend model with nonnormal
random errors is used to simulate the yield distribution. The correlation between prices
and yields is imposed using a technique which constructs a multivariate distribution from
univariate distributions.

Chapter V analyzes a representative farmer’s optimal behavior and welfare for
different combinations of risk management instruments. The optimal combination of risk
instruments used by the representative farmer is explored, and the substitute or
complement relationships among the instruments is also investigated.

Chapter VI studies the impacts of changes in yield risk and risk preferences on
the optimal use of the risk management instruments and welfare. The price-yield
correlation, individual farm yield risk, yield basis risk and risk aversion level may vary

by individual farm and the geographical location. The results for the representative farm

6

with specific values of these parameters are generalized by examining the changes in
farmer behavior and welfare as these parameters change.

Chapter VII compares IYCI with AYCI in an attempt to evaluate the relative
performance of the alternative yield indices. Performance under alternative contract
designs are evaluated in order to explore the potential use and benefit of each type of
indices. The results of this chapter will provide insights into the effectiveness of AYCI
as a risk management alternative to IYCI, and the conditions that allow AYCI to
outperform IYCI.

Chapter VIII studies the impacts of changes in the government deficiency payment
program on participation and farmer welfare. The current policy discussions related to
reducing the government subsidy in an attempt to phase out the program in the future
may impact farmers’ income levels and risk exposure. The use of other risk management
instruments in the portfolio may also be affected by changes in the design of the
government program.

Chapter IX concludes the dissertation by summarizing the major results of the
study and discussing policy implications. The strengths and weakness of the study are

also discussed, along with suggestions for future research.

Chapter 11

DECISION MAKING UNDER RISK IN THE AGRICULTURAL SECTOR

In agricultural production, risk may have an important impact on decision making,
and the existence of risk management instruments and strategies enriches the decision set.
The concepts of risk, risk management, and important risk management instruments are
introduced in this chapter.

There is a vast literature on risk management strategies for individuals who
operate in the agricultural sector. In this chapter, important studies on decision making
under risk and the use of risk management instruments are reviewed. These studies are
used to develop the theoretical framework used in this research effort.

The expected utility model has been widely used to model decision making under
risk. Its advantages and disadvantages in the current research application are explored
in this chapter. In this study, utility is a function of net income which is composed of
eash income from production and income from risk management instruments. Existing
studies of a farmer’s decision making under risk using alternative risk management
instruments such as futures, options, government programs and crop insurance, are also
reviewed in this chapter. Examination of the existing literature suggests a need for

additional emphasis of risk analysis in a portfolio setting.

2.1 Risk and Risk Management

When an event has more than one possible outcome, it is uncertain. Robison and
Barry define risk as a subset of uncertainty where the outcome of an event can alter the
decision maker’s well being. Risk faced by a firm can be broken into business risk and
financial risk. Business risk is the variability in net returns to total assets as a result of
such things as production and price risks. Financial risk is the variability added to the
firrn’s net returns to equity holders that results from fixed financial claims against the
firm. Thus, business risk is the risk associated with profit independent of the financial
structure of the firm. Table 2.1 lists the general types of business and financial risks

faced by a farm business. In this study we focus primarily on business risk.

Table 2.1 Classification of Risk in Agriculture

 

 

Type of Risks Example
Business risk
production risk yield variability
market risk input and output price variability
technological risk technological change
legal and social risk policy change
human risk labor or ownership change
Financial risk high leverage

 

A firrn’s response to risk generally focuses on reducing the likelihood that it is
exposed to unfavorable outcomes, e.g. , selecting crops that are more disease resistant;

transferring risk to other agents, e. g., buying insurance; and increasing its risk bearing

9

ability, e.g., setting up irrigation systems. Risk management strategies can be classified

into production, marketing and financial strategies, as outlined in Table 2.2.

Table 2.2 Classification of Risk Management Strategies and Representation

 

Risk Management Strategies

Example

 

Production

Marketing

Finance

selection of stable enterprises; diversification; and
operating flexibility

inventory management; hedging, forward
contracting; sequential marketing; participating in
public programs, vertical integration; and crop
insurance

self-liquidating loans; liquid assets; liquid credit
reserves; purchase of life, disability, and property
insurance; and credit reserves

 

This study focuses on a firm’s marketing responses to price and yield risks when

a portfolio of risk management instruments is available to the farmer. Evaluating the

farmer’s behavior and welfare by focusing on each risk instrument individually can

provide misleading results so it is important to examine the problem in a portfolio

setting. For exarnple, when options are the only instruments, the optimal position may

be different than when the government program is added into the portfolio, because the

two instruments are substitutes. The risk management portfolio studied here includes

futures, options, crop insurance, and a government deficiency payment program.

10
2.2 Cash, Futures, Options, Government Program, and Crop Insurance

There are three basic types of pricing contracts available to the firm: cash, futures
and options. Using these contracts, a firm not only realizes the value of its production
but may capture more profit opportunities and/or avoid price risk by shifting it to others
willing to accept the risk.

In addition to the market driven pricing instruments, the US. Government has
provided pricing programs and sponsored crop insurance. The pricing programs are
intended to help farmers manage their price risk while the crop insurance program helps
farmers manage their yield risk. Both programs are subsidized and result in an income

transfer to farmers who participate.

W

A cash contract is an agreement negotiated individually between a buyer and a
seller that may be outside the rules and guarantees of an organized exchange. Cash
contracts allow buyers and sellers the maximum flexibility in specifying contract terms.
Commodity price, quality, quantity, delivery period and delivery location etc. can all be
negotiated and specified in the contract. Because of their flexibility, cash contracts are
not normally traded on an exchange but are settled by delivery.

Delivery of the commodity may be immediate or at some specified date in the
future. Cash contracts for immediate delivery of the physical commodity are referred
to as spot transactions and are said to take place on the spot market. The price specified

in the spot market is referred to as spot price, or cash price. Cash contracts for future

11

delivery of the physical commodity are known as forward contracts and are said to take

place on the forward market.

W

A futures contract is a standardized legal contract to make or take delivery of a
commodity during a specified future period for an agrwd upon price. However, unlike
forward contracts, futures contracts are normally traded on an exchange. To make
trading possible, the exchange specifies certain standardized features in the contracts.
Sellers and buyers trade through brokers, and all contracts for a specific commodity and
delivery period are identical except for the trading price on a given exchange. One party
to the contracts agrees to buy the physical commodity in the future and is said to hold
a long position; the party who agrees to sell the commodity holds a short position. The
contractor who is long (short) futures can either take (make) delivery at the specified
time or sell (buy) back the futures contract at the current trading price before delivery
is due. The futures price is the price that results in an equal number of short and long
holders, and is regularly reported in the financial press.

A futures contract is different from a forward contract in that an exact delivery
date is not usually specified, but can be at any time during a pre-specified period, say,
the last month of the contract. The holder of the short position has the right to choose
the delivery date during that period.

Another difference is that futures contracts are marked to market or settled each

day. Both buyers and sellers are required to maintain a margin account with their

12
brokers. At the end of each day when that day’s futures price is revealed, adjustment

is made to the margin accounts by subtracting (adding) the loss (gain). The initial
margin is usually a small portion of the total value of the commodity but is set large
enough to cover the maximum daily price movement allowed by the exchange. If the
margin drops to a level lower than the "maintenance" margin, which is somewhere below
the initial margin, a margin call will require the trader to restore his margin account to
its original level. The cost to the trader to maintain the margin account is small because
Treasury Bills can be used to satisfy the account requirement.

The performance of each party to honor the futures contract terms is guaranmd
by the exchange clearinghouse. Brokers are required to settle accounts daily and
maintain margin accounts with the clearinghouse just as their clients are required to do

so with them.

9211mm

Options contracts give the buyer of the option the right but not the obligation to
buy or sell the commodity during some period at a specified price. The primary
difference between futures and options contracts is that the owner of a futures contract
is legally obligated to make or take the delivery, or an equivalent cash settlement, while
the owner of an option only agrees to make or take the delivery when it is favorable to
do so, otherwise the contract just expires. While there is no cost to obtain a futures

contract, buyers of an option must pay a premium to the sellers for the right to only

13

”exercise“ the option when it is profitable. The price specified in the options contract
at which delivery is made is known as the exercise price or striking price.

There are two types of options. A call option gives the owner the right to buy
the underlying commodity, and a put option gives the owner the right to sell the
commodity. The date on which the option can no longer be exercised is known as the
expiration or maturity date. American options can be exercised at any- time up to the
expiration date, while European options can only be exercised on the expiration date.

Options can be thought of as price insurance. If the price level is lower (higher)
than the exercise price, the owner of the put (call) option will exercise it at maturity; if
the price is higher (lower), the owner chooses not to exercise the option. Similar to

insurance, the premium is related to the expected value of the option.

'n F r d ' II

There are three basic behavior motives associated with the use of different
contracts: arbitrage, hedging, and speculating. Arbitrage involves locking in a riskless
profit by simultaneously entering into transactions in two or more different markets.
Speculating is a risky investment made in an effort to achieve a financial profit. Hedging
is a technique of establishing an approximate price for a cash commodity, or in some
eases, ensuring that adequate supplies of some asset are available. The hedging motive
is most closely associated with risk management behavior.

A hedger can be a producer, processor, marketing intermediary or a financial

intermediary. He takes a position in the futures and/or options market which is opposite

14
his actual or expected cash position. Because cash prices and futures prices are

positively correlated, the gain or loss in one market will be offset by the loss or gain in
the other. A ”perfect hedge” refers to the perfect offset of a gain in one market by a loss
in the other markets and rarely occurs because of basis risk. Basis is the difference
between futures and cash prices. The basis level is the result of transportation cost,
storage cost, variable quality and product characteristics, and the change in supply and
demand conditions over time. The basis level changes over time and reduces the ability
to hedge price risk.

Cross hedging is a concept used to denote the use of futures and/or options to
hedge a risk associated with an asset other than the commodity underlying the futures and
options contracts. Cross hedging typically occurs when there is no futures contract on
a commodity/product, but its price is correlated with another commodity/product that
does have an associated futures contracts (e. g., hedge slaughtered cows with beef
futures). It is also possible to use futures and/or options to cross hedge yield risk when

the commodity yield is correlated with futures price.

P m n
In many countries, governments provide deficiency payments to farmers to protect
them from income risks, to help increase farm income, and/or to help maintain a stable
food supply. Farmers who wish to participate in the program for specified commodities
enter into contracts with the government that often include a cost. However, the cost is

not always a direct payment; for example, a particular program may require a percentage

15
of total acreage to be set aside in order to receive the benefits the program offers. This

allows the government to use the program to help control supply of particular
commodities. The Commodity Credit Corporation, established in 1933 has administered
deficiency payment programs in the US.

The specific benefits and obligations of the program vary by year, but the basic
format is the same. For the Wheat and Feed Grain Programs provided by the United
States Department of Agriculture (USDA), a target price is set, such that when the
market price level (or some index related to the market price) falls below the target
price, the difference will be paid to the farmer by the government. The amount of
deficiency payment is limited to the difference between the target price and a critical
price, known as the loan rate. If the price falls below the loan rate, the farmer is eligible
to receive a nonrecourse loan up to an amount equal to the difference between the loan
rate and the market price. This essentially provides a participating farmer a minimum
cash price equal to the target price.

The program payment for each acre is the product of the price difference and a
prespecified yield level. In recent years, the farm’s 1981-1985 five-year—average farm
program payment yields have been used to establish the base yield levels upon which the
payments are based. A participant’s obligation is to set a specified percentage of their
acreage aside; no crop is allowed to be produced and no benefits are paid on this
acreage. Besides the set aside acreage, another portion of acreage may be excluded from
the benefits and is classified as flexible acres. Unlike the set aside land, flexible acres

may be used to produce any crop, although no program benefits will be received for that

16
portion of production. The total program payment for each crop, called the deficiency

payment, is the product of the per acre payment and the covered acreage. The base
acreage in each year is the previous five year average of planted acreage plus diverted

conservation acreage for that crop.

W

Insurance is a risk instrument which protects property, life, one’s person, etc,.
against loss or harm, in consideration for a payment proportionate to the risk involved.
Buying insurance enables an individual or firm to transfer part of its risk to an insuring
party. Insurance companies can lower the cost of risk bearing by diversifying over a
large number of clients, assuming the risky outcomes facing the clients do not exhibit a
high level of positive correlation. The insurance company accepts the risk in return for
a premium which exceeds the certainty equivalent of the loss.2 In turn, the insurance
buyer improves his or her welfare by paying a premium that is less than the loss of
certainty equivalent income created by the risk. In this way both the insurance seller and
buyer gain from the exchange. So, by pooling individual risk, insurance leads to Pareto-
prefcrred states. Insurance enables individuals to engage in risky activities which they
would not otherwise undertake.

There are two main types of insurance, discrete—disaster insurance and insurance

with continuous outcomes. Discrete-disaster insurance covers the case in which there

 

2 Certainty equivalent is a certain value that makes an agent as well off, if he accepts
this certain return, as if he faces the risky return.

17

exist only two possible outcomes: 1) the disaster occurs or 2) it doesn’t occur. Fire
insurance is a good example. If the disaster (fire) occurs, the insurance company will
pay the coverage according to contract, otherwise nothing is paid. The insurance
premium is calculated based on both the coverage and the probability of disaster
occurrence.

Insurance with continuous outcomes provide payoffs scaled to the realization of
continuous outcome risky event. Crop insurance is an example of insurance with a
continuous set of outcomes. In the crop insurance contract, a level of output is specified
as the trigger point at which loss starts to occur (e. g. corn yield at 80 bu/acre), and the
loss level is the difference between realized output and this level. The insurance payment
is based on the loss level, and could be anywhere from 0 to 100% loss. For example,
if the trigger yield is 80 bu/acre and the realized corn yield is 70 bu/acre, a 10 bu/acre
loss occurs and the insurance holder would receive compensation equivalent to the value
of 10 bu/acre.

Moral hazard is an important insurance concept. Without insurance, a firm
realizes costs in an effort to reduce the likelihood of undesirable outcomes. However,
after purchasing insurance, the firm loses part of the incentive to reduce the probability
of adverse outcomes and may alter its behavior accordingly. This altered behavior is
called moral hazard, and can lead to a breakdown in an insurance market if the insurance
company can’t observe and price the moral hazard of its clients. When moral hazard
occurs, the ex ante probability function used by the insurance company to calculate

premiums is different than the ex post one after the insurance is purchased. This results

18
in a premium below the level required by the insurance company to bear the actual level

of risk. In the presence of moral hazard, insurance companies may experience excess
losses and become economically inviable.

There are measures that can be undertaken by insurance companies to avoid or
reduce moral hazard, such as charging deductible payments, coinsurance and/or adjusting
the premium rate. A deductible payment is a fixed amount that the insured has to pay
first in order to get the insurance company to pay the rest of the loss. Examples of
deductibles ean be found in auto insurance. Coinsurance requires the insurance holder
and the insurance company each to pay a percentage of the total loss. Health insurance
often uses coinsurance. Premium rate adjustment is an adjustment by the insurance
company for a particular insured agent based on a periodically documented standard.
Once a claim is paid, the insurance company may view the insured as a higher risk and
raise his premium. Auto insurance providers frequently use this technique. The idea
underlying each of these methods is to make the insured bear some of the losses so that
they have an incentive to avoid loss, which will hopefully reduce or eliminate moral
hazard. Unfortunately, it has been found that sometimes none of these techniques
adequately corrects for the problems associated with moral hazard.

Adverse selection is another difficulty that faces insurance providers. Adverse
selection arises when insured individuals have better information about the insured risk
than the insurance company setting the premium rate. When the insurance purchaser and
provider possess asymmetric information, only those whose certainty equivalent loss is

larger than the insurance premium would agree to buy the insurance. Thus, the premium

19

calculated based on the information of all potential clients tends to be too low to cover
the insurance company’s cost of indemnity payments. As a result, insurance companies
have to conduct intensive investigations to get accurate information on the likelihood of
alternative outcomes, increasing cost and raising premiums, in order to avoid adverse

selection problems.

Individual Yield Crop Insurance (IYCI)---MPCI

Crop growers face high yield risk due to uncertain natural conditions. Crop
insurance can be used to help manage the risk associated with yield fluctuations. Unlike
most types of insurance, agricultural crop insurance is seldom provided by the private
sector because providers of crop insurance face a number of special difficulties. One
difficulty is the yield risk of crop growers across large geographic areas tends to be
positively correlated because of Similar weather and soil types. Most small companies
have difficulty in pooling these risks in an economical way.

The Federal Crop Insurance Corporation (FCIC), founded in 1938, has provided
crop insurance to US crop growers. During the late 1930’s and 1940’s, the program
was criticized for its high costs and low participation. Legislation, passed in 1947,
limited the scope of the program, and from 1948 to 1980 the federal crop insurance
program was run on an experimental basis. Major revisions were embodied in the

Federal Crop Insurance Act of 1980, which set up Multiple Peril Crop Insurance

(MPCI).

20
MPCI provides indemnity payments based on a farmer’s individual yield

realizations, so it is a special form of IYCI. Under the current MPCI program, corn
producers may select one of three yield levels up to a maximum of 75% of their
insurable yield’, and one of the three price levels determined from FCIC forecasts of
expected prices with the top price election level set close to the expected market price.
If the producer’s real yield falls below the elected trigger yield, he/she receives an
indemnity payment equal to the product of the elected price and the yield shortfall. This
yield shortfall is determined by the amount that actual yield falls below the elected
trigger yield. The crop insurance premium varies depending on the elected price and
trigger yield levels.

The government has historically subsidized this program in an effort to encourage
participation. However, the FCIC has realized billions of dollars in losses over and
above the subsidy. Symptomatic of the problem is the fact that the recent indemnity
payout is 40% higher than the premium collected. Also troublesome is that in spite of
the favorable indemnities-received/premiums-paid ratio for participants, producer
participation in the MPCI program fell short of expectations. Not until 1989 did the

participation rate exceed 25 percent of insurable acres (Coble, et al, 1993).

 

3 Insurable yield is determined by FCIC. Prior to 1985, the insurable yield for a
particular farm was determined as the average yield in the farm’s geographic area. But
this exacerbated the problem of adverse selection as farmers with above average loss
risks comprised an ever-increasing proportion of the insured pool (Skees and Reed,
1986). After 1985, actual production history (APH), the average of ten years of the
farm’s actual production, is used as the insurable yield.

21

In principle, the insurer will break even if they charge an actuarially fair premium
to everyone. Approximately, an actuarially fair premium level equals the expected
indemnity paid to all the participants, plus a small amount to cover transaction costs.
This requires that 1) the insurance agency has reasonably objective knowledge about the
risks involved, 2) a large number of similarly exposed individuals are present, 3) the
incidence of risk is independently distributed over individuals, 4) individuals cannot
influence the nature and occurrence of the risky incident, and 5) nor can they influence
the indemnity receivable once they buy a policy. However, it is possible that none of
the above criteria is satisfied for crop yield insurance. The failure of MPCI can be at
least partially explained by adverse selection and moral hazard.

Moral hazard occurs when behavior changes as the result of holding insurance.
Beeause crop insurance contracts are Specified in terms of ”result states” (yield
realizations) rather than the ”state of nature" (weather), even though individuals have no
control over the state of nature, they are able to influence the indemnities received
through influencing their yields. The insured will have less incentives to take care of the
crop when participating in the insurance program and may take steps to reduce
production cost (Chambers 1989). AS a result, yields tend to be reduced and indemnity
payments increase.

Adverse selection occurs when asymmetric information exists between the insurer
and the insured. Farmers often have superior information about their own farm yields
relative to the insurance company, so those whose yield loss occurs more than the

insurance company expects tend to participate at the predetermined premium rate.

 

”u

4.4

{4%

WM

to:

LT

22

The level of transaction cost associated with supplying insurance also impacts its
attractiveness to users and the financial competence of insurers. The administrative cost
is high for MPCI because the information related to yield shortfall has to be obtained and

maintained for each individual farm.

Area Yield Crop Insurance (AYCI)-«GRP

Many of the difficulties experienced with MPCI are related to the individual farm
yield index used to calculate indemnity payments. Currently, testing is being undertaken
on an alternative form of crop insurance which uses an area yield index to calculate
indemnity payments. The idea is that both indemnities and premiums would be based
not on a producer’s individual yield but on the aggregate yield of a surrounding
geographical area, usually the county.

This idea was first put forth as a Group Risk Plan (GRP) by Halcrow in 1949.
Under GRP, a special form of AYCI, a participating producer would receive an
indemnity equal to the difference, if positive, between the trigger yield level chosen by
farmers and the realized area yield. All participating farmers in a given area would
receive the same indemnity per insured acre if they purchase the same policy, regardless
of each one’s own farm yield. Each farmer therefore would pay the same premium rate
for the same contract. GRP is based on the premise that when a county’s average yield
is low, then most farmers in that county will have a low yield. Therefore, GRP pays
only when the yield of the entire county drops below the trigger yield. This may serve

the risk management needs of farmers if the major sources of yield risk are common

23

within a small enough area. In addition, because individual farmers will generally have
little impact on area yields and are unlikely to have superior information about the area
yield distribution, GRP has the potential to avoid adverse selection and moral hazard
problems associated with MPCI.

GRP is a new program the Federal Crop Insurance Program established in the
United States during 1994 (Baquet and Skees, 1994). The GRP indemnity payment is
based on the county yield shortfall below the trigger yield and the coverage level the
individual farmer purchases. Both trigger yield and coverage level can be chosen by the
farmer within FCIC specified ranges. The price index used to calculate the indemnity
payment is specified by FCIC. The insurance payment per insured acre is the product
of the price index, coverage level, and the area yield shortfall.

GRP offers a number of advantages over individual yield crop insurance. Because
information regarding the distribution of the area yields is generally more available and
reliable than that of individual yields, insurers could more accurately assess the actuarial
fairness of premiums under GRP, thereby significantly reducing adverse selection
problems. Moreover, because the indemnities would be based on the area yield rather
than the farmer’s individual yield, a farmer could not significantly increase his indemnity
by unilaterally altering his production practices. Thus, under the GRP program, moral
hazard essentially would be eliminated. Administrative costs would also be substantially
reduced under the GRP program because claims would not have to be adjusted
individually. GRP has been implemented on an experimental basis in selected areas by

the Commission for the Improvement of the Federal Crop Insurance Program.

24

The use of GRP to manage yield risk has one glaring weakness, it does not fully
cover the yield risk faced by farmers because individual yield is not perfectly correlated
with area yield. The difference between area yield and individual yield is called yield
basis risk. When an individual realizes higher loss than the area average, i.e., basis is
positive, he won’t be completely reimbursed under GRP. The relative performance and
tradeoff between IYCI and AYCI schemes are important policy issues and will receive

significant attention in this study.

2.3 Decision Making Under Risk

The idea that the expected value of an outcome cannot fully describe a risky event
was raised by researchers some two hundred years ago. Bernoulli used a logarithmic
function to convert dollar values to ”moral expectation values", later called utilities, in
the 1700s. However, it wasn’t until the 19405 that the expected utility model was
carefully deduced and became the cornerstone of risk analysis (Von Neumann and
Morgenstem, 1944). In this framework the characteristics of the decision maker’s utility
function reflect risk preferences, and the decision maker maximizes expected utility when
facing a risky situation.

Expected utility (EU) is not the only technique which can be used to order risky
choices. The stochastic dominance (SD) model first introduced by Hadar and Russell
(1971), and the mean variance (MV) model originated by Markowitz (1952) and extended
by Tobin (1958), are also widely used in risk analysis. The EU, SD and MV models are

introduced in the next two subsections.

25
mm
The expected utility theory of decision making under uncertainty is based on the
decision maker’s (DM) personal beliefs about the likelihood of uncertain outcomes and
his personal valuation of the possible outcomes. The axioms that guarantee the existence
of a utility function are now well known (Anderson, Dillon and Hardaker, 1977; Robison
and Barry, 1987), and are listed in the following:

1. M: For any two action choices, the DM either prefers one
action choice to the other or is indifferent between the two action choices. For
choices A and B, either A «’3, A >B, or A ~ B. This says that the DM must be able
to make a choice between any two distributions of outcomes or else be indifferent
between them.

2. W: If the DM prefers action choice A to action choice B and action
choice B to action choice C, then A must be preferred to C. If the DM is
indifferent between A and B and indifferent between B and C, then A and C must
be indifferent. For A >B, B>C, then A >C; and for A ~B, B~ C, then A ~ C.

3. Emit—5221113 If the DM prefers A to B and B to C, then there exists some
probability p between 0 and 1, for which, the DM is indifferent between B and
a lottery pA+(1-p)C where 0<p< I. This indicates that a DM facing a risky
situation involving a favorable and an unfavorable outcome will take the risk, if
the probability of the unfavorable outcome is small enough. If A>B >C, then 3

0<p<1, stB ~ pA+(1-p)C.

26
4- WW1 If A is preferred to B, and C is some

other action choice, then a lottery involving A and C must be preferred to a
lottery involving B and C if the probability of receiving A and B is the same in

each lottery. IfA>B, for v 0<p<I, pA+(I-p)C >pB+(I-p)C.

The expected utility theorem, also known as Bemoulli’s principle, says if a DM’S
preferences do not violate the preference axioms of ordering, transitivity, continuity, and
independence of irrelevant alternatives, then there exists a utility function that: 1) can
transform each possible outcome to a real number, so that the ranking of the real number
represents the preference ranking of the corresponding outcome; and 2) in the case of
uncertainty, can generate an expected utility value under the subjective probability for
each action choice, such that the ranking of the value represents the preference ranking
of the corresponding choice. Positive linear transformation of a utility function does not

change its ranking property.

Application of the EU Model
The expected utility model allows either the choice variable or the stochastic
factors to be discrete or continuous. If the stochastic factor in the model, Y, is discrete

with n possible outcomes, and the probability, p,, associated with each outcome, y,, is

known, the expected utility for each action, It, is defined as: Egg) = 21,313,”). On

in]

the other hand, if the stochastic factor, Y, is continuous, and the probability density

27

function associated with it, fly). is known, the expected utility for any action, 1:, is
defined as= Evrx) = [U(x.y)fly)dy-’

The continuity of the expected utility depends on the choice set. When the choice
set is discrete, e.g., the decision on whether to participate in the government program
or not, the expected utility function is also discrete. The decision can then be made by

ranking all the possible expected utility levels, and selecting the one that provides the

maximum expected utility. For any action set {xiv choose x], so that EU(x}) =

M ,
max{EU(x,), EU(x,),..., EU(X,..)}.

When the choice set is continuous, the expected utility function» is also usually
continuous, and the optimal decision is made by maximizing the expected utility function.
There are convenient ways to maximize a continuous function either analytically or
numerieally.

One advantage of the EU model is that it provides a functional form associated
with the probability density function, so that precise optimal solutions can be obtained
either analytically or numerically. Another important advantage is that the EU model can
be used to measure the relative welfare levels of any two choices. Two measures of
welfare change, equivalent variation (EV) and compensating variation (CV) are

commonly used.

 

‘ There may be more than one stochastic factors and/or choice variables, in which
case x and/or Y are vectors. When Y is a vector, the summation in the discrete case or
the integral in the continuous case is multiple, and there is a probability p associated with
each possible outcome combination of the stochastic factors and the density function f is
a joint density function.

28

Equivalent variation is the certain income adjustment to the current income

distribution which provides the DM with the same expected utility level as would be

derived from an alternative income distribution: EU(M + EV) = EU(M), where M and Ma

represent the stochastic current and alternative income, respectively. EV represents the
DM’s willingness-to—pay in certain dollars to obtain the alternative income distribution.
Compensating variation is the income adjustment to compensate the agents under

the alternative income distribution, so that they will be as well off as under the current
income distribution: EU(M) .-. Evan“ _ CV) . CV represents the DM’s willingness-to-pay

in certain dollars to keep the alternative income distribution from moving back to the
base income distribution.

EV is used in this study because it measures DM’s willingness-to-pay based on
the current income distribution which is invariant with the change of risk instruments in
alternative portfolios. Thus, the current income distribution provides a constant base
from which to evaluate alternative income distributions.

A major disadvantage of the EU model is that the form of the utility function is
sometimes hard to obtain. However, various methods for inferring information about
farmer risk attitudes have been developed (Saha, et a], 1994). This makes the EU

framework a viable option to evaluate the farm decision making and welfare.

Risk Preference Measures
Decision makers’ risk preferences can be classified into three general categories:

risk averse, risk neutral and risk preferring. Within each category, there still exists

29

differences in the degree of risk preference. The curvature of a utility function reflects
the risk attitude of the DM. The more risk averse the DM, the more concave the utility
function.

A measure of the curvature of a utility function, or risk attitudes, was introduced
by Pratt and Arrow independently and is called absolute risk aversion:

.. ‘11” (Y)
AR - (2.1)
(Y) (1,0,)

 

AR(Y) is always positive for risk averse individuals; zero for risk neutral individuals;

and negative for risk preferring individuals. Preferences are said to exhibit decreasing

absolute risk averse (DARA) if 2i; < o; constant absolute risk averse (CARA) if

93A): = ; and increasing absolute risk averse (IARA) if 98g); > 0-

Another measure of the curvature of a utility function is relative risk aversion,

defined as:

_U// (11)); =

__ AR Y (2.2)
0,") (Y)

RRO’) =

Preferences are said to exhibit decreasing relative risk averse (DRRA) (constant relative

risk averse (CRRA)) (increasing relative risk averse (IRRA)), when %; is < (=) (>)

3O

0. A commonly used CRRA function takes the form 9:, for RR = a > I; or mm,
for RR = 1. Utility functions with DRRA and CRRA also exhibit DARA.

Saha, Shumway and Talpaz (1994) have reviewed existing empirical research on
risk preferences. They claim CARA is generally rejected in favor of DARA, while the
nature of relative risk aversion remains ambiguous. Chavas and Holt (1990) also provide
the evidence that the risk preference of corn and soybean growers is DARA rather than
CARA. Pope and Just (1991) conducted econometric tests for risk preferences of Idaho
potato growers, and found CARA was not supported by the data, and that the CRRA
hypothesis could not be rejected.

Specific functional forms for utility have also been investigated in some studies
which allow the coefficients of absolute and relative risk aversion to be estimated. Using
aggregate data, Hansen and Singleton (1982) studied CRRA model with a utility function
U(W) = W‘la, and estimated a relative risk aversion coefficient between 0 and 2 for
consumers. Szpiro (1986) showed that US. 1951—1975 insurance data support the CRRA
hypothesis and that the relative risk aversion coefficient is between 1.2 and 1.8. Love
and Buccola’s (1991) results for data from Iowa corn show the coefficient of relative risk

aversion is between 2.4 and 19.

' i i n M 'n nd r Ri k
In addition to the EU model, stochastic dominance (SD) and mean variance (MV)

models are also commonly used in the studies of risk and uncertainty. They both have

31
advantages and disadvantages compared to the EU model, which are discussed in the

following.

SD Model

The SD model provides a general way to study the DM’S choice of risky events
(I-Iadar and Russell). It compares the cumulated density functions associated with each
choice, and tells which choice is preferred by certain types of DMS when the underlying
cumulated density functions exhibit Specific relationships. For example, first degree
stochastic dominance (FSD) says choice A is always preferred to choice B by DMS who
prefer more to less if the cumulated density function associated with B is not less than
that with A for any outcome level and is greater than A for at least one outcome level.
Second degree stochastic dominance (SSD) reveals decision making principles for DMS
who prefer more to less and are risk averse. Higher degree stochastic dominance rules
ean be defined for more specific types of DM preferences.

The advantage of the SD model is that it does not require a specific form for the
DM’s utility function. Instead, it only places constraints on the DM’S risk preferences.
However, there are several difficulties with SD rules: 1) they only apply to finite discrete
choice sets, because each possible choice generates one outcome distribution to which all
other possible distributions must be compared to find the optimal decision; 2) the optimal
solution may contain a set of choices so that it may not be possible to identify a single
optimum choice; and 3) SD model does not provide a way of measuring the welfare level

associated with each choice.

32
MV Model

The MV model’, originated by Markowitz, is also widely used in risk analysis.
The model selects the action that can maximize the certainty equivalent, Ya, of the
stochastic outcome, which makes the DM as well off as if he faces the random returns,
Y, ie. U(Yag = EU(Y). The expression of certainty equivalent as a function of its mean
and variance is generally derived from the expected utility model using Taylor series, so
the MV model can be viewed as an approximation of the EU model.

The major advantage of the MV model is that it is often more tractable than EU,
and analytieal solutions are sometimes obtainable in a MV model in cases where the EU
model provides no closed form solutions. However, the MV model is only consistent
with the EU model when probability distribution and/or the DM’S risk preferences are
restricted. Sufficient restrictions are either 1) the DM’s utility function is quadratic
(Tobin, 1958); 2) the random attribute is normally distributed and the DM has a concave
utility function (Samuelson, 1970); or 3) the random attribute is a monotonic linear

transformation of a single random variable (Meyer, 1987).

Cl . I! . . 1 I 1 1

Among the three major decision making models under risk, the expected utility
model is chosen as the framework for the analysis conducted in this study for several
reasons. In the farm decision model which includes alternative risk instruments in the

portfolio, we hope to obtain solutions of the optimal position for each instrument and

 

5 It is also referred as mean standard deviation model (MS).

33

measure the welfare changes associated with alternative portfolios and design changes for
the various instruments. Thus, the SD model is immediately eliminated because it
generally will not provide unique optimal solutions nor allow welfare measurement.

While the MV model can provide unique solutions as well as welfare measures,
the sufficient conditions for consistency with the EU model are generally not satisfied
under the complete multivariate distribution faced by the farmer. Furthermore, the
income distribution is so complex that it is generally not possible to derive analytical
results even in a MV setting.

Based on existing studies of risk preferences, the CRRA function is selected as
the utility function for this research, and the relative risk aversion coefficient is set at 2
for the base model. Comparative static analysis is done to explore the impacts of

alternative levels for the relative risk aversion coefficient.

2.4 Rkk Management with Price and Yield Instruments

Under the expected utility model, the farmer maximizes the expected utility of the
farm return which is composed of stochastic returns from the cash market as well as
alternative risk management instruments. Sandmo (1971) studied the competitive firms’
production decision under price uncertainty without any risk management instruments.
In his EU model, the deterministic output level was chosen to maximize the expected
utility of net income. In the absence of any risk management instruments, the output

level depends on the distribution of price and the farmer’s risk attitude; and risk averse

34

farmers produce less when facing stochastic prices than in the case of a certain price set
at the mean of the random price.

A number of risk management instruments are now available to help farmers
manage the risk they face from price changes. Numerous studies have focused on the
use of futures contracts, forward contracts, options contracts and the government
deficiency payment program (Danthine, 1978; Iapan et al, 1991; and Turvey et al,
1990). Yield risk also contributes to farm income risk, and may alter the optimal use
of risk instruments designed to manage price risk. If prices and yields are correlated,
the pricing instruments may be able to help manage the risk associated with yields. Even
so, the pricing instruments are not likely to be efficient at managing yield risk. Crop
insurance is available for many agricultural commodities and can be used to directly
manage yield risk. A number of studies have addressed farmers’ use of. crop insurance
(Ahsan, et. al., 1982; and Chambers, 1989).

Several studies allow for both price and yield risks where several risk
management instruments are included simultaneously in the portfolio (Myers, 1988; and
Poitras, 1993). Before developing the model used in this research, relevant studies in
which farmer portfolios include futures, options, government program, and/or crop
insurance are reviewed. Most of these existing studies investigate pricing instruments
or crop insurance in isolation. Studies on futures and options tend to either assume
yields are deterministic or fail to include crop insurance in the portfolio. Likewise, most
studies on crop insurance assume prices are deterministic or fail to include futures and

options in the portfolio. With a few notable exceptions, most studies on futures and

35

options and/or crop insurance do not allow for a government deficiency payment program

in the portfolio.

m n In m n

Research on the use of futures contracts started to appear two decades ago.
Danthine (1978) developed an EU model where the production decision and hedging
position on the futures market are made simultaneously in a setting where output is
nonstochastic, and the cash price is the only source of risk. He found that when the
possibility of trading futures contracts exists, farmers’ output depends on the
deterministic futures price and the input price only, not on the risk associated with cash
price or farmers’ risk attitudes.

Holthausen (1979) examined the competitive firm’s behavior under cash price
uncertainty while allowing the firm to select output and hedge to maximize expected
utility of farm income. He drew similar results to Danthine: 1) production decisions are
based on the forward price but not on the price the firm expects to prevail or the degree
of risk aversion; 2) the optimal hedge depends on the forward price and the expected
eash price. If the forward price is higher (equal) (lower) than the expected cash price,
the farmer will hedge more (the same) (less) than he produces; and 3) risk aversion and
the level of price risk affect the firm’s optimal hedge if the forward price differs from
the expected cash price.

Feder, Just, and Schmitz ( 1980) modified the model to account for opportunity

cost across time by assuming the return from the futures contract occurs at planting and

36

the return from spot market occurs at harvest. In addition to the Danthine and
Holthausen’s results, they found the volume the farmer hedges (or speculates) in the
futures market is affected by the discounted expected cash price, the futures price, and
the risk of cash price if the discounted expected cash price differs from the futures price.

In these early studies, the "futures contact” was actually a cash forward contract,
because the ”futures” price at harvest was assumed at harvest cash price level, implying
the contract would be settled by delivery. This specification ignores basis which is
generally present when futures contracts are used. Recent studies introduce basis and
basis risk in the decision problem and find some of the above results are invalid, such
as the three studies reviewed in the following.

More recently, a number of studies have addressed the use of commodity options
as a risk instrument. Lapan, Moschini and Hanson (1991) examined the use of both
options and futures in a farmer’s portfolio allowing for both cash price and basis risk.
Based on expected utility maximization, they drew several analytical results: 1) when
futures and options prices are unbiased and cash price is a linear transformation of
futures price plus a white noise, then a fraction of production is hedged in the futures
market, options are not used as hedging instruments, and the portion of non-diversifiable
basis risk affects the nonstochastic production level; 2) under CARA, a marginal change
in the futures price leads to a change in futures and options sold in the same direction;
3) under risk aversion and a symmetric distribution of prices, the qualitative optimal
speculative futures position depends only upon the bias in the futures price; and 4) under

non-increasing absolute risk aversion (DARA or CARA), an unbiased straddle price and

37
a symmetric distribution of the cash price, the optimal straddle position will always be

long when there is an open speculative futures position; also, the straddle position will
always be smaller than the open futures position.

A recent study by Vercammen (1995) uses the model of Lapan, Moschini, and
Hanson, and allows the stochastic price to be nonsymmetric. He shows that with a
biased futures price, options are relatively more valuable for reducing the skewness of
a nonsymmetric price distribution than in the symmetric price distribution case. If the
market price distribution is skewed to the right, hedgers who perceive a downward bias
in the futures price take a long position in the futures market and write put options.
Again, deterministic production is implicitly assumed in the model.

Sakong, Hayes and Hallam (1993) extended the model by Lapan, Moschini, and
Hanson to include yield risk. In Lapan, Moschini and Hanson’s paper,lcash price risk
can be diversified into a component attributable to changes in the futures price and an
orthogonal component reflecting undiversifiable basis risk. Because the diversifiable risk
is linear in futures price, futures contracts dominate options contracts, so that when both
prices are unbiased options are not used. When yield risk is introduced, it is almost
always optimal for the producer to purchase put options and to hedge less on the futures
market than expected production. These results are strengthened if the producer expects
local production to influence national prices (price and yield are negatively correlated)

and ifhe is DARA.

38
WW

Robison and Barry (1987) use the MV model to study the use of a government
deficiency payment program to manage price risk. They find that even though the
production per acre increases in the presence of the government program, total output
may or may not increase, because a portion of acreage has to be set aside to participate
in the program.

Turvey and Baker (1990) use an EU model to study farmers’ use of futures,
options, and farm programs in the presence of price risk, financial risk, and liquidity
constraints. Their results suggest the farmer’s use of futures and options decreases in
the presence of loan rates and target prices. These results indicate that the government
program may act as a substitute for futures and options.

These studies lend many insights into a competitive firms’ behavior under
uncertainty and the use of instruments designed to manage price risk. However, with the
exception of Sakong, Hayes and Hallam, each study assumes output to be deterministic
and controllable. Allowing yield risk and yield risk management instruments into the

analysis may alter the implications of these studies.

W

Ahsan, Ali and Kurian (1982) developed a model of crop insurance and examined
the implications of adverse selection. Their work, one of the earliest studies in this area,
used a discrete EU model (disaster occurs or does not occur), where producers allocate

resources between risky and riskless assets, and select the level of output (yield) to

39

insure. The intent was to find analytical evidence to support the observation that even
though there are clear advantages for crop insurance over alternative institutional
arrangements, the market generally has failed to provide such a mechanism.

Their results suggest risk-averse farmers choose full insurance coverage if
insurance is offered at actuarially fair odds and the optimal allocation of risky resources
is chosen by setting the expected marginal product of the resource equal to its
opportunity cost, the return on a riskless investment. They also find that it is difficult
for the insurance agency to separate the high risk farmers from the low risk farmers so
that, in equilibrium, each kind of farmer will buy different contracts. In addition, they
find farmers invest less on risky farming in the absence of crop insurance. In the
presence of crop insurance, farmers’ risky investment will decrease when 1) the premium
increases; 2) the endowment decreases; 3) the random yield mean decreases without
changing the risk; and 4) the yield risk is increased by a mean-preserving spread. This
work provides a theoretical foundation for studying adverse selection.

Nelson and Loehman (1987) studied crop insurance with symmetric information
between farmers and insurers. They derived a Pareto optimal result which requires full
information, an actuarially fair premium, full coverage, and a different contract for each
individual, so that every individual acts risk neutral and there is no moral hazard or
adverse selection problems. They also developed a model with moral hazard and adverse
selection, in which the insurer chooses the production input after the insurance choice is
made other than choosing them simultaneously. They conclude that a Pareto optimal

agricultural insurance program would provide full coverage individualized insurance.

40

Such an insurance program would eliminate income variability and cause farmers to
behave as if they were risk neutral. However, they suggest the problems caused by lack
of information and incompatible incentives prevent this theoretical optimum from being
attained.

Chambers (1989) extended the Nelson and Loehman model to allow asymmetric
information in order to consider moral hazard and positive costs of providing insurance.
He finds full insurance contracts are dominated by contracts involving coinsurance and
deductibles for risk averse insurers. Chambers also considered other alternatives to deal
with moral hazard. For example, insurance agencies may collect more information to
detect ”poor farming practices" and refuse to indemnify avoidable losses, or write multi-
year insurance contracts to detect consistent cheating. However, neither alternative is
efficient because the former induces high transaction cost and the later can’t prevent
cheating as long as insurance contracts are finite lived.

Coble, Knight, Pope and Williams (1993) conducted an empirical test for moral
hazard and adverse selection in Multiple Peril Crop Insurance (MPCI). First, they
separated adverse selection and moral hazard theoretically by "hidden knowledge" or
”hidden action”. They then set up an econometric model to compare the yield
distributions of those who have participated in MPCI to those who have not participated;
as well as the yield distributions of farmers before and after participating in MPCI. They
concluded that for the crops and districts they studied, moral hazard is always present,

but there is less evidence of adverse selection.

41

Vercammen and van Kooten (1994) study a dynamic model for a risk neutral
farmer using individual yield crop insurance with premium surcharges and discounts,
depending on the historical ratio of indemnities to premiums. They find even when the
insurance contract punishes moral hazard in previous periods through higher premium
in the current and future periods, the farmer still has incentive to practice moral hazard.

Concerned with the fact that, even with a 25% deductible, the FCIC still loses
money providing MPCI, Miranda (1991) examines the area yield insurance plan.
Modelling an individual producer’s total yield risk as a systematic component that is
explained by factors affecting all producers in his area and a nonsystematic residual
component, he concludes that l) for a given critical yield level, the risk reduction from
area yield insurance is determined by the correlation between the farmer’s individual
yield and the average area yield; 2) the higher the correlation is, the greater the risk
reduction he gets from the area-yield insurance; and 3) the more risky one’s yield is, the
greater the risk reduction he can obtain from area yield insurance.

Using farm-level data from 102 western Kentucky soybean farms, Miranda
concludes that for most producers, area yield insurance would provide better overall risk
protection than individual yield insurance due to lower deductibles and higher coverage;
the reduction of adverse selection and the virtual elimination of moral hazard would
significantly improve the actuarial performance of the federal crop insurance program;
and area-yield insurance would be less expensive to administer because verification of

individual production histories and adjustment of individual yield-loss claims would not

be necessary.

42
n f li

Myers (1988) studied a model with a futures market for trading in contracts to
deliver or take delivery of farm output at a future date and a crop insurance market
allowing farmers to insure against low yield realizations at the cost of an actuarially fair
insurance premium. The futures market can eliminate price risk and the crop insurance
market provides insurance against unfavorable yield realizations. By assuming risk
averse farmers maximize their expected utilities of net farm income, Myers finds: l) in
the absence of production uncertainty, an unbiased futures market induces complete
hedging and expected profit maximizing behavior by risk averse farmers --- the result of
Holthausen; 2) in the absence of price uncertainty, actuarially fair crop insurance induces
the farmer to completely insure yields and maximize expected profit --- the result of
Ahsan, Ali, and Kurian; and 3) if both price and production uncertainty are present, both
futures and crop insurance markets are required to complete the market structure and ean
induce expected profit maximizing behavior. Again, this research does not differentiate
futures from forward contracts in that there is no basis risk involved. Also, the stylistic
crop insurance design restrains the flexibility in modelling actuarially unfair premium
and/or trigger yield restrictions.

Poitras (1993) modeled both price and yield uncertainty with futures and "crop
insurance” in the portfolio. ”Crop insurance", as defined by Poitras, includes three types
of instruments: quantity insurance (yield insurance), price insurance (put options), and
revenue insurance. He used mean, variance and skewness to approximate the expected

utility model and finds that when yield insurance is included in the portfolio, hedging in

43

futures is increased. However, when put options are included in the portfolio, the use
of futures is decreased. Studying futures plus one form of ”crop insurance” at a time,
the portfolio including futures and options in the presence of yield insurance is left
uninvestigated.

These studies provide many insights into farmer behavior under risk. However,
the price and yield distributions and/or the portfolio of assets available to the farmer have
typically been restricted. In the model used in this study, cash price, futures price and
yield are stochastic and their joint distribution is consistent with historical data. The
farmer is allowed to participate in the spot market, futures, options, crop insurance, and
a government deficiency payment program. Decisions will be made on the level of
participation in futures, options, crop insurance and the government program in an effort
to maximize expected utility.

This portfolio model will allow us to investigate the optimal risk management
behavior given the currently available risk management instruments and a realistic
representation of the joint price and yield distribution faced by the farmer. The model
provides the flexibility to add (remove) any existing or potential risk management
instrument into (from) the portfolio to study the marginal effects of each instrument. The
model can also be used to study alternative contract designs, compare welfare levels, and
do sensitivity analysis, so that potential changes in crop insurance and government

program policy can be evaluated.

Chapter III

THE FARNIER DECISION MAKING MODEL

This chapter develops an expected utility model for a risk averse farmer who
makes decisions on the use of risk management instruments in a portfolio setting. The
farmer maximizes the expected utility of income which is derived from selecting positions
in the futures market, options market, government deficiency payment program and crop

insurance. The model assumptions, specification and calibration are introduced in this

chapter.

3.1 Model

A two-period model is developed for a representative farmer who produces a
single crop. In the first period (planting) the farmer makes hedging decisions subject to
price and yield distributions conditional on information known at planting. In the second
period (harvest) prices and yields are realized. Profits are specified on a dollar per acre
basis, because the farm production is assumed to exhibit constant return to scale such that
the number of acres in production does not impact the farmer’s optimal decisions. The
farmer exhibits CRRA which, together with the constant return to scale assumption,
allows maximizing the expected utility of per acre income to generate the same position

in risk management instruments in proportion to output as the expected utility of the

44

45

whole farm income, when the planted acreage is constant. This is determined by the
properties of CRRA functions.

Production cost is assumed constant because the production decision is assumed
to be made prior to the risk management decisions. In other words, the price and yield
distributions faced by the farmer are known and fixed at the time the risk instruments are
selected. The production decision could be modeled explicitly but this is not necessary
here because our interest lies in investigating risk management behavior conditional on
given input choices. This allows us to focus directly on the use of the risk management
instruments.

No discount factor for time is included in the model because, again, the emphasis
is on risk management. Actually, most of the cash transactions occur in the harvest
period. Option and crop insurance premiums are paid in the initial period, but futures
transactions involve no up-front cost other than a margin account which can be satisfied
with interest bearing securities. Options are the European type which means no early
exercise is allowed, and only one exercise price is investigated.

The farmer’s objective function is defined as:

MAX EU[i'r]
x,z,£.n
wherei-=CR+F'I+(5]+G'S+CI-F
C73 = ”’3’ (3.1)
F] = x(f-fo)

01 = z[max(0,s-])-h]

(75 = ElmaX(0,Pr'fi)Yb(1-q-r)-r(C7?+CDl
CI = IP - map)

[F = m max{0.nE(i’)-Y’J,

46
where E is the expectation operator; 00 is the CRRA utility function; 1 denotes the

farmer’s net income per acre; and the tilde above a variable denotes it is stochastic. The
component CR is the cash income from the spot market, where cash price is denoted by
p and the farm-level yield by Y'. F is the fixed production cost.

F1 is the income from the futures market, where 1}, is the future price at planting;
f is realized future price at harvest; and x is the hedging amount purchased (sold if
negative). 01 is the cash income from the options market, where s the strike price; h is
option premium required to own the option; and z is the quantity of put options
purchased (sold if negative).

GS is the income from the government program. The binary indicator, .2, signals
government program participation, with E = 1 (0) indicating that the farmer participates
(does not participate) in the program. Y, is base yield and p, is target price used to
calculate the deficiency payment. The proportion of acreage reduction under the program
is denoted by r, and the proportion of flexible acres is denoted by q.

CI is the income from crop insurance, where [P is the insurance indemnification
payment; 3' is the proportion of acreage to be covered by the insurance; Y is the yield
index used by insurance agents (farm yield 1" for IYCI and county yield 1“ for AYCI);
p, is the price index used to calculate the crop insurance indemnity payment; 11 is the
trigger yield level Specified as a proportion of the expected yield index; and A denotes

the crop insurance premium relative to actuarially fair premium level.

47
3.2 Characterktics of Risk Management Instruments

The farmer produces corn and makes his marketing decision at the planting time
based on the information available then. The realized cash return is the cash price, p,
times the farm-level yield, 1". Besides the cash market, the farmer is allowed to
participate in the futures market, options market, government deficiency payment
program and crop insurance. The remainder of this section provides a more detailed
discussion of the model specification of each instrument included in the portfolio, where
the emphasis is on the base model. Modifications to the base model will be made in later

chapters when alternative contract designs are studied.

Futures

If the quantity hedged on the futures market, x, is negative (positive), the farmer
is short (long) futures. The farmer with a short (long) futures position enters a contract
at planting time to sell (buy) the commodity for the current futures price fi,, and buys
(sells) the contract back at the time of harvest for the realized futures price f which is
unknown when the hedging decision is made. If the futures price at harvest is above the
futures price when the contract was established at planting, the holder of a short (long)
position experiences a loss (gain) of 0%). On the other hand, if the futures price at
harvest is below the price at planting, the holder of a short (long) position experiences
a gain (loss) of 031‘). The farmer is assumed to face no transaction costs from using the
futures market. Thus any brokerage cost or opportunity costs from margin accounts are

assumed to be negligible in order to focus on the risk management functions of the

,,.\

.d
. .‘i

131“"

er

I;

lit

48

instrument. The net income from the futures market in the model is accordingly denoted

Jr(fife)-

Quinn:

Only put options are included in the model, because puts are traded more
frequently in the agricultural commodity markets than calls, and the role of calls can be
represented by combining futures and puts. So adding futures, puts and calls into the
model would result in a redundant instrument. Producers usually hold long cash
positions and hedge it with short futures positions. Likewise, put options can be used
directly to hedge the long cash position.

If 2 is positive the farmer buys put options. The cost to buy a put option at
planting is the premium, h. The buyer of a put option exercises the contract at the
maturity if the strike price s is higher than the futures price f in order to capture the gain
of (s-f). If the Strike price is lower than futures price the holder of the option lets the
contract expire by simply declining to sell at the exercise price and thus avoids the loss
01s . If the options contract is fair, the premium, h, equals the expected options value
which is E[max(0,s-f)].

If 2 is negative, the farmer sells the put option to collect premium h. If the option
is exercised, the farmer must pay to the owner of the option s while the current price is
f resulting in a payout at maturity of (s-j). Again, the transaction cost in the options
market is assumed zero. The farmer’s net income from the use of the option contract

is therefore defined in the model as z[max(0,s-f)—h].

49
W
The deficiency payment program specified in the model is a stylized version of

the 1994 USDA Wheat and Feed Grain Programs. Total deficiency payment is the
product of eligible acreage, base yield and price shortfall. The same crop is also
assumed to be planted on the flexible acreage, q (on which any crop can be planted but
is not eligible for deficiency payments). Acreage, r, must be removed from production
completely for the farmer to be eligible for the deficiency payments. The farm’s
historical yield moving average from 1981-1985 is used as the base yield in the program.
In this stylized model, the expected farm yield for the current year is assumed to be the
yield base, Yb. In the program, price shortfall is defined as the difference between target
price, p1, and the greater of the harvest cash price, p, and the loan rate, if the difference
is positive. For simplicity, the realized cash price is assumed to be at or above the loan
rate. Thus, the deficiency payment is (I-q-r) meax(0,prp) per base acre.

The cost to the farmer to participate in the program is the loss on the acreage
removed from production. The loss comes from the lost income both from lower
production and crop insurance (the farmer cannot buy crop insurance for the land set
aside). Therefore, the net income from participating in the deficiency program is (I—q-

r)Y,,max(0,p,«-p)—r(CR+CI) per base acre.

lensutance

Currently, two versions of crop insurance are offered to growers on a range of

crops. Multiple Peril Crop Insurance (MPCI) is the major individual yield crop

50
insurance program provided by FCIC. The indemnity payment per insured acre is the

product of the price index, p,, and yield shortfall, where the price index can be chosen
by the insured farmer. For example, for corn in 1990, farmers could choose a price of
$1.45/bu, $1.65/bu, or $2.30/bu. The yield shortfall is the amount the farmer’s
individual yield realization falls below a pre-specified yield level chosen by the farmer.
Because this pre-specified yield level triggers the indemnity, it is called the trigger yield.
In the current MPCI program, farmers can select 50%, 65%, or 75% of an FCIC
determined insurable yield, 1”, which is the farmer’s historical yield moving average

fi'om the preceding ten years. Thus, the indemnity payment can be specified as:

IP = p, max(0, nY‘ -Y’) (3.2)
where the trigger yield factor, n, is chosen by the farmer from {0.50, 0.65, 0. 75}.
Restricting trigger yield to below 75 % of the insurable yield implies that the contract
features a deductible.

Because of moral hazard, adverse selection, and transaction costs, insurance
companies charge a premium which is greater than the expected indemnification, E(IP),
even when the insurance companies are subsidized by the federal government. In our
stylized IYCI model, we allow 27 to be selected by the farmer continuously in the range
[0, 0.75], and the predetermined insurable yield, 1", is set at expected yield index E(Y).
The yield index is the farmer’s individual realized yield, price is fixed at the expected
cash price at harvest, Em), and the premium is the expected indemnification payment
multiplied by a coefficient A. If A is 1, the premium is actuarially fair; if it is larger

than 1, the insurance company charges a premium loading to the farmer; and if it is

51

smaller than 1, the insurance contains a subsidy. The value of A is set at 1.35 in the
base model.6 The level of insured acreage, 5“, denotes the number of acres the farmer
elects to cover under crop insurance, which is set at 1 in the base model because it is
restricted to 100% of the base acreage under MPCI.

Recently, the Group Risk Plan (GRP), an alternative to MPCI, has been offered
to farmers by the FCIC. GRP reduces moral hazard, adverse selection, and transaction
costs by basing indemnity payments on an area yield index. Its indemnification payment
is based on a coverage level parameter, tp, selected by the farmer from 0 to 1.5; the
FCIC determined price index, p,; and a yield shortfall which is specified in a more
complicated fashion than in MPCI. The yield shortfall in GRP is specified as the product
of the expected county yield and the percentage of county yield realization falling below
the trigger yield. The trigger yield can also be selected by the farmer at 65%, 70%,
75%, 80%, 85%, or 90% of the expected county yield. The indemnity payment per acre
can be represented by:

IP = 0, E Y "E(Yc) ‘ Ye (3.3)
ptmaXI ‘P ( 6) 11E(YC) ]

 

where Itc is the county yield; 11 is the indemnification trigger yield factor. Restricting

trigger yield to lie below 90% of the expected county yield also implies a deductible.

 

‘ The break-even level of crop insurance premium is difficult to determine from
existing research, however, it is estimated that the reimbursement cost the FCIC pays
insurance companies to deliver MPCI is currently between 30% to 50% of the total
premiums paid by farmers and the subsidy. Here, the transaction cost of 35% of the
actuarially fair premium is used as the premium loading.

52

Because the indemnification payment in equation (3.3) can be rewritten as
[p g («P/n)P,maX[0,flE(Y‘) _ ye] , it is obvious that if the coverage level, (p, is smaller

than n, a co-insurance is implied in the contract; and if it is greater than n, the loss is
inflated. Compared with model (3.1), go/n plays the same role as 1’.

In our stylized AYCI model, the predetermined price level is set at the expected
cash price at harvest, Em), and n is allowed to be selected within the range of [0, 0.9].
Beeause of the lower transaction costs and reduced moral hazard and adverse selection
problems, the premium is set at the actuarially fair level, E(IP). Notice that why plays
the same role as the insured acreage parameter, f, so restricting r to 100% of the base
acreage in the base model implies (p/n = I . These restrictions assure the indemnification
payment equation (3.3) can be expressed by the same equation as in (3.2) in the base

model except now Y“ is used as the yield index.

3.3 Methodology

Farm income is derived from the spot market, futures market, options market,
government deficiency payment program, and crop insurance, each of which is
stochastic. Thus, income depends on the joint distribution of the cash price, futures
price, farm-level yield, and county-level yield (when AYCI is included in the portfolio).

The farmer maximizes the expected utility of income which can be written as:

Eve) = l l I U(1r)8(f,P.Y'.YC|0)dfdde’dYC (3.4)

ha

53

where gfl 0) is a joint density function for futures prices, cash prices, individual farm
yield and county-level yield conditional on 0, a set of information available when the
decision is made. When AYCI is replaced by IYCI, the area yield variable does not
enter the model explicitly.

The analytical form of the joint distribution of yields and prices is not available.
Furthermore, options, government deficiency payment program and crop insurance
truncate the distribution of farm income, making it difficult to find an analytical form of
the solution to maximize expected utility. However, using market data, the joint
distribution of price and yield can be estimated by econometric and simulation
techniques. Numerical Optimization can then be used to solve the farmer’s optimization
problem.

The procedure begins by using historical and current market data to estimate the
joint price and yield generating process. Simulation techniques are then used to generate
a discrete joint distribution of prices and yields. This price-yield distribution is used to
obtain a discrete distribution of net income for a given set of possible choice variables.
The corresponding level of expected utility is then calculated by substituting the income
values into the utility function and taking the expectation. Finally, the maximum value
can be found using standard optimization algorithms. The values of choice variables
corresponding to the maximum expected utility is the optimal solution. All estimation,
simulation, and optimization is done using GAUSS.

In order to compare the effects of that these risk management instruments on

farmer welfare, the Equivalent Variation (EV) measure of the decision maker’s

54

willingness-to-pay for the instruments is used. The base income distribution is the spot
market income distribution pl”. The alternative income distribution may be impacted by
the inclusion of futures, options, government program, and/or crop insurance in the
portfolio. Letting 1' represent the income from the alternative portfolio with the risk

instruments, the EV is calculated such that:

EU(pY’ + EV) = EU(rr‘). (3.5)

When there is no analytical form of expected utility, numerical techniques can be used
to find the EV by minimizing the squared difference between the two sides of equation
(3.5) as:

1” I, _ 1” - 2 (3.6)
MEIVN [fig U(p,.Y. EV) WE U(1r.- )l,

i-l
where N is the sample size and i indexes the realizations of random variables.

3.4 Model Validation

Because both the multivariate distribution for prices and yields and the income
structure are very complicated, a numerical model is used to solve the representative
farmer’s portfolio problem. In order to validate the numerical model, analytical results
from previous studies can be used by replicating the results with the numerical model and

the appropriate restrictions.

55

In the study of Lapan, Moschini, and Hanson, where yield is deterministic,
analytieal results are drawn from expected utility model. When cash price is assumed
to be a linear transformation of the stochastic futures price plus an independent white
noise, p = a + bf + r, and when unbiased futures and options are included in the
portfolio, a fraction b of production is hedged in the futures market and no options are
used. Same results are obtained from the numerical model when the yield is restricted
to be deterministic and the linear relationship between cash and futures price is imposed.

In the portfolio study of Myers, he derives analytical results under the
assumptions that the farmer has constant relative risk averse preferences I and there is no
basis risk. When an unbiased futures market and an actuarially fair crop insurance are
available, a full level of crop insurance is purchased and the insured output is fully
hedged on the futures market. The numerical analysis in this study provides the same
result that, without basis risk the farmer selects full level of IYCI trigger yield as the
premium is reduced to the actuarially fair level, and the maximum yield is hedged at the
same time on the unbiased futures market. Thus, the model faithfully reproduces the

analytical results supporting the model’s validity.

Chapter IV

THE JOINT PRICE AND YIELD DISTRIBUTION

In this chapter, the price and yield generating process for the representative farm
is outlined. The stochastic returns to the farm depend on jointly distributed prices and
yields. When a farmer uses futures or options contracts, participates in crop insurance
programs, or utilizes government deficiency payment programs, his returns may depend
on the futures price and/or "area" yield as well as the cash price and his own farm-level
yield. Therefore, a joint distribution of futures price, cash price, farm-level yield and
area-level yield is necessary in order to study the farmer’s risk management behavior and
welfare in a portfolio setting. However, directly estimating and/or generating the joint
distribution of commodity prices and yields is difficult because empirical evidence
suggests the random variates have complicated distributions that may not have closed-
form density functions. As a result, solving the maximum expected utility problem will
require numerical techniques and an empirical joint distribution.

The procedure used in this study is to first generate the price and yield
distributions independently, and then impose a correlation between them by using the
numerical algorithm introduced by Taylor (1990). Section 4.1 explains the development
of the joint cash and futures price distributions. Section 4.2 describes the development

of the marginal farm-level and county-level yield distributions, and the technique used

56

57

to derive the bivariate distribution of farm and county yields. Section 4.3 explains the
technique used to join the simulated price distributions and yield distributions to obtain

a multivariate distribution of prices and yields.

4.1 The Joint Cash and Futures Price Distribution

In this section, the techniques to model the mean and the higher moments of the
bivariate cash and futures price distribution are introduced separately. While the mean
price levels are fairly straight forward to model using information available at planting,
the higher moments are more difficult to characterize. We begin by examining the
stochastic properties of historical corn futures and cash prices, and reviewing literature
on modelling commodity prices. The price generating process and parameter estimation
used in this research are then presented, along with the procedure for simulating the
empirical distribution of futures and cash prices at harvest.

The data used to estimate the price distribution in this study are weekly corn
futures and cash prices from the first week of May 1989 to the last week of April 1994.
The futures prices are Thursday closing prices of each week for the December contract
at the Chicago Board of Trade, and the cash prices are Thursday prices for cash markets
in the Southwestern Crop Reporting District of Iowa.7 When the futures contract expires

in December the data series switches to next year’s December contract. The contract

 

7 Thanks go to the Agricultural Extension Service in the Department of Economics
at Iowa State University for supplying these data.

58

switching date is the first trading day in December. December futures are the new crop

futures contracts which farmers would most likely use to hedge their harvest.

W15

Cash and futures prices are stochastic and may have a complicated joint
distribution. Generally, the futures market is believed efficient in the sense that there is
no expected gain from trade in futures contracts. This suggests that the expected futures
price at harvest is equal to the futures price at planting for the same contract. Thus, the
farmer’s best forecast of the futures price at harvest is the futures price at planting.

Similarly, forward contract prices combine information from both historical prices
and current market conditions to estimate forward cash price levels and provide market
beliefs about mean cash price levels in the future. The forward contract price at planting
provides the farmer with the market’s forecast of the cash price at harvest. Therefore,
the futures and forward prices at planting are used as the farmer’s expectation of the

mean futures and cash prices at harvest.

E . E C rn .13 . E

Previous studies have found stochastic trends and time-varying volatility in
commodity prices (Myers, 1994). These properties are determined by the characteristics
of the markets in which prices are generated. For example, arbitrage opportunities in
futures markets generally bid the current price up or down until the price in next period

only deviates from the current price by some unpredictable random amount, which is

59

consistent with a stochastic trend, or unit root, model. Although the cash price series
could be mean-reverting, there is still evidence suggesting some cash price series may
contain a unit root (Ardeni, 1989). Seasonal production and supply of agricultural
commodities may cause the prices to show seasonal patterns in either their levels, or
volatilities. Also, the volatility of commodity price changes often appears to vary over
time, moving between high volatility periods where large price changes tend to be
followed by other large changes, and tranquil periods where small price changes tend to
be followed by other small changes. Here the stochastic properties of cash and futures
price movements for corn are examined first, and results are then used to develop the

joint cash and futures price distribution faced by the representative farmer.

Analysis of Price Level Data
A stochastic trend increases or decreases from its value in the previous period by
some fixed amount on average, but in any given period the trend deviates from the

average by a random error with mean zero. The process can be modeled as:

Pl = Pl'l + F, + 61’ (4.1)

where P, is the price at time t; u is a constant drift; and e, is a mean-zero random error
which is initially assumed identically and independently distributed (i.i.d) (Bachelier,
1900).

It is common practice in studies of commodity prices to work in the logarithm of

the price level. The natural logarithm of the prices can eliminate trends in levels and

60

variances, and reduce the impact of outliers; and has been widely used in previous
research on commodity pricing models (Baillie and Myers; Yang and Brorsen).

A time series with a stochastic trend, such as in (4.1), is "non-stationary” because
the statistical properties of the random variable can change over time and therefore
conventional econometric methods are not generally applicable. The econometric
difficulties of this particular specification are easily handled by rewriting the model and

working in terms of price changes, or first differences, such as:
AP, = u + 6,,
where A is the difference operator, A P. = pl .. pm. When the price level is replaced by

the logarithm of price, the first difference of the logarithms becomes the approximate
percentage change in prices.

The price series used in this research are examined first to determine if the
hypothesis of a stochastic trend can be rejected. Phillips (1987) and Perron (1988) have
developed unit root tests which are more robust than the traditional Dickey-Fuller tests
to autocorrelation and heterogeneity in the distribution of the residuals, which are
common problems exhibited by commodity prices. Here, Phillips-Perron unit root tests
are undertaken orghelogarithmof futures and cash prices.

Table 4.1 shows the Phillips-Perron unit root tests for the logarithms of the
futures and cash price series. The hypothesis of a unit root cannot be rejected for either
futures or cash prices. The first difference model with a drift appears to be appropriate
for the mean levels of both futures and cash prices, although the errors may not be i.i.d.

The Phillips-Perron test is explained in appendix A. Figures 4.1 and 4.2 show the

61
Table 4.1 Phillips-Perron Unit Root Test for Prices

 

 

 

Pr = '31P” + E, (a)

P. = u’ + v‘P-, + e.’ (b)

P. = ii + Zia-g) + 3P.-. + e, (c)
Parameter _Esrimate_

matte: cash
4, 0.9995 0.9996
u’ 0.0674 0.0235
7— 0.9253 0.9705
,1 0.0678 0.0237
3 1.095e—5 1.99e-5
.7 0.9249 0.9702
Statistic Mie— CriticaLYaluL_ He
10% 5% 2.5% 1%

20,-) -19.421 -7.669 -18.3 -21.8 -25.1 -29.5 7 1
Z(t1.) -3.200 -1.966 -3.13 —3.43 -3.69 -3.99 7 1
20’.) 3.186 1.953 2.16 2.53 2.84 3.19 M 0
205,) 5.123 1.933 3.81 4.63 5.45 6.52 7 =1,,— =0
2(5) -19.519 -7.737 -18.3 -21.8 -25.1 -29.5 '7 —1
2(15) -3.217 -1.987 -3.13 -3.43 -3.69 -3.99 .~, —1
Z(;i) 3.202 1.973 2.72 3.08 3.38 3.72 ,1 =
2(5) 0.513 1.018 2.38 2.79 3.12 3.49 B =
2m) 3.507 1.639 4.07 4.75 5.40 6.22 3 = [,= 7:1
Z(<I>,) 5.258 2.459 5.39 6.34 7.28 8.43 [3' =09 =

 

Note: Z, t, and ¢I> are statistics defined in Perron’s (1988) paper and discussed in
Appendix A. The critical values are from the tables composed by Dickey and Fuller

(1981).

62

nus-.3

32 E2 9 $2 a: 58 80E chasm 58 S588 E aefimfi 2. 93E

not: 33.. out... out:

 

63

3...:

32 ie.. a $2 a: see SE and 58 «as E 2558.. a... 2.9...

«use: :32 our.) on;

 

64
changes in the logarithm of futures and cash prices from the first week of May 1989 to

the last week of April 1994.

Analysis of Higher Moments

Previous studies suggest high frequency commodity price movements have time-
varying volatility and excess kurtosis (Gordon, 1985; Hall, Brorsen and Irwin, 1985).
These stochastic time-varying volatility and excess kurtosis properties of commodity price
changes have been successfully modeled using the Autoregressive Conditional
Heteroskedasticity (ARCH) model, developed first by Engle(1982) and later generalized
by Bollerslev(1987) into the Generalized ARCH (GARCH) model. Baillie and Myers
(1991) applied univariate GARCH-t models to daily cash and futures prices for corn,
soybeans and other agricultural commodities, assuming the conditional error term in the
process is distributed as a student-t random variate. Bivariate GARCH-normal models
were also used to describe daily cash and futures prices jointly, assuming the conditional
error terms are distributed as bivariate normal random variates (Myers, 1991). Though
there is some evidence indicating the t-distribution is the appropriate conditional
distribution for both cash and futures price errors, modeling a bivariate t-distribution
remains technically infeasible because the joint density function is undefined (Yang and
Brorsen, 1992; Baillie and Myers, 1991).

Time-varying volatility features can also be modeled with deterministic seasonal
variables in the variance equation. A time dependent seasonal function can be useful in

capturing volatility changes that occur at regular intervals. Figures 4.3 and 4.4 Show

65

 

:41 .

 

32 E3 a $2 a: 58 88F. 85d 58 S583 E 555»...— ESS m... 2.5a

no;

. 2.14....

NOE!

 

 

SE

.95. one:

$33.4 , 2a 1.4—.

 

 

 

on;

 

 

 

row

ION

10*

Ion

10h

 

 

66

 

32 ie.. a $2 a: 82.. 82E =5 68 .32 E 5.233 Beam 4... 85E

not: «at: 3...: one: one:

a: E a]: fag . 411;: 31. .- a .

IO—

 

 

ION

IOV

 

 

 

 

 

 

ran

row

105

TOO

 

 

 

 

Oa

67
squared changes in the logarithm of futures and cash prices from the first week of May

1989 to the last week of April 1994. A cyclic pattern appears present indicating the
existence of a seasonal effect where higher volatility is found during the planting and
growing months and lower volatility found in the winter months, which is consistent with
Anderson’s (1985) observation.

Unfortunately, the seasonal and stochastic components of volatility are often
related and difficult to identify separately. Fackler (1986) developed a process
represented by both a deterministic seasonal component and a GARCH component, which
did a good job of characterizing the variance process for corn prices. The price model
used in this study follows that of Fackler, combining GARCH with seasonality.

There are significant and important differences between deterministic seasonality
and a GARCH representation of the variance process. First, the time-varying variance
of price changes is deterministic in the seasonality model and stochastic in the GARCH
model. Second, the volatility prediction of the seasonality model is conditional on the
time period for which the forecast is made, while the volatility prediction of the GARCH
model converges to a constant unconditional variance that does not depend on the time
period. The GARCH specification is best suited to model volatility when volatile and
tranquil periods emerge randomly, while the deterministic seasonality model is best suited
to model volatility that appears cyclically. Thus, using the GARCH specification by
itself can lead to an erroneous long-term variance forecast if the true process contains a

strong seasonal component.

68

Sine and cosine functions are commonly used seasonal functions because they are
cyclical and can be combined in different frequencies to provide a flexible representation
of many types of cycles (Yang and Brorsen, 1992; Kang and Brorsen, 1993). The
frequencies of the seasonal variables and the specification of ARCH/GARCH process
interact in characterizing the variance process, and the correct specification is determined
by the data. In this study the variance process for cash and futures prices is represented
by a bivariate model which includes both a deterministic seasonal component and a
stochastic ARCH component. The bivariate models for futures and cash prices in this
study assume the conditional errors are bivariate normal. The next section introduces

the empirical model of the price process and presents estimates of the process

parameters.

Futures and cash prices are not independent, so a bivariate mOdel is used to
characterize the price process. Before presenting the bivariate model, the concepts of
the deterministic seasonal component and stochastic ARCH component are first
introduced in a univariate model which is used to conduct preliminary analysis of the
variance process.

The general univariate price generating process can be modeled as:

AP. = u + .,, 6.0.-. ~ N(0.a.’) “-2)

69

where AP, is the change in logarithm of price at time t; 0’, is the time-varying variance
at time t for the random error, 5,, conditional on information available at time H, 0”;
and N(.,.) is a normal distribution with the first argument representing the mean and the
second argument signifying the variance.

Examination of Figures 4.3 and 4.4 suggest the presence of a seasonal pattern in

the volatility of price changes for both cash and futures prices. A deterministic seasonal

variance model can be specified as:
I
a? = w + E [¢,cos(2wid,/52) + ¢1Sln(21ld,/52)] (4.3)
i-l

where I is the degree of frequency in seasonality variables, and d, is the current week
number. The parameters, 6:, (1),-and (h, fori = 1,2,...1, represent the constant term and
the coefficients of the sine and cosine functions respectively. The parameters are
restricted so that the variance is positive at each time t.

This model allows the variance to evolve over time periodically moving from high
levels to low levels with 52/] weeks as a cycle, where the timing of high and low
variance levels are determined by the combined values of ¢.~ and 41,-.

Likelihood ratio tests on the univariate deterministic volatility model suggest
seasonality can be well represented by setting I =1 (See appendix A). It should be noted
that the frequency degree and parameter estimates may change if a stochastic component
is introduced and/or the univariate models are specified as a bivariate model. However,

these results provide insight regarding the frequency level with which to initialize the

It 1'

till.

she

 

70

bivariate model. The suggested low frequency level is consistent with results in previous
research (Fackler, 1986; Yang and Brorsen, 1992; and Kang and Brorsen, 1993).
The stochastic component of the time-varying variance is described by an ARCH

model. The variance equation of a univariate ARCH(J) model is defined as:
J
0,2 = w + 20:]. 5,27- 6, 92-1 ~ N(0,o',2) (4-4)
j-l

I
where all the parameters, to, 0‘1" j = I,...,J, are non-negative, and 20,] < 1, which

1'1

ensures the unconditional variance, w/(1—ZJ: 05-) , is positive (Engle, 1982).
j-l

The ARCH(J) model allows the volatility to evolve over time driven by the
shocks from the previous J periods; where a previous large shock, either positive or
negative, contributes to the current variance level by adding a relatively large positive
value to it; and correspondingly, a small shock contributes a relatively small value.
Because the shocks exhibit some degree of correlation, a large (small) current variance
level means large (small) current shocks, either positive or negative, are likely to occur,
which is consistent with the behavior of many commodity prices where large shocks tend
to be followed by other large shocks and small shocks are often followed by other small
shocks.

A combined univariate ARCH(J) with I degree seasonality model has a variance

equation specified as (4.5):

71

a? = w + 209’ e.., + 21¢ cos(21rid/52) + t s1n(21rrd /52)] (4.5)

i-l

where the deterministic component should be positive at each time, t, that is,
I J
cor-‘2‘;[¢pos(21'id,/52)+\l/,sin(2‘rid,/52)] > 0, and once again. 12a, < 1.

This model allows the volatility to evolve over time driven by both a seasonal
cycle and previous shocks. The contribution of each component to the variance level is
determined by ARCH parameters 0,, j = I,...,J, and seasonal parameters 45, and #1,, i =
1,. . . ,I.

The bivariate specification is similar to the univariate specification where the
variance process for cash and futures price shocks is characterized by a process with both
stochastic and seasonal components. In addition, the error terms are assumed to be
bivariate normal, and the covariance term is also characterized by both a stochastic and
seasonal component. The bivariate ARCH(J) model with I degree seasonality in the

logarithm of futures and cash prices is specified as:
AP, = p + 6,, ~,N(0 H,)

”1713:,“ =[’:
l 9
np‘ (4.6)

I
Vech(H,) = W + £A,Vech(e,_,.e,’.j) + z [¢,cos(21rid,/52) + ‘It,sin(2wid,/52)],

1.1 1.1

where P, =

 

 

72

where P, is the vector of natural logarithms of futures and cash prices at time t; p. is the
mean vector of the differenced log prices; the error vector, 6,, is conditionally distributed
as bivariate normal with zero mean, and variance-covariance matrix H,; Vech(.) is an
operator which stacks the lower triangle of a symmetric matrix; W, <I>,, and ‘15, are (3x1)
parameter vectors for the constant, sine and cosine coefficients respectively; and A}, are
(3x3) parameter matrices for the ARCH coefficients. A diagonal parameterization serves
to reduce the number of parameters in this model to make the estimation feasible (Baillie
and Myers, 1991). As a result, A,, for j = I,2,...,J, are specified as diagonal matrices.
Various specifications of model (4.6) were estimated using the maximum
likelihood approach and subjected to a number of diagnostic tests. The specification
which seemed to do the best job of characterizing the behavior of the price series was
a process with a restricted ARCH(9) with first degree seasonality (See appendix A).
The parameter estimates for the model and diagnostic statistics are shown in Table 4.2.
As expected, the constant mean for futures price is not significantly different than
zero, suggesting there are no constant price changes expected in the futures market which
is consistent with weak-form market efficiency. There is a small positive drift in the
eash price equation, suggesting an upward trend in cash prices over time. This may be
partly attributed to the upward pressure on cash prices during the 1993 flood, which was
the last year of the data. The ARCH coefficients at lag 1, except in the futures equation,
are statistically significant at the 10% level. Removal of any of the ARCH coefficients

in the futures price equation results in invalid diagnostic Q tests which suggest the

73

Table 4.2 Bivariate Futures and Cash Price Process

 

 

AP, = u + 6,, e,|D,_1 ~ N(O,II,)
In}, "I V:
where P, - u = e, -
lap, 11 u,
“11"1-1 a], 0 0 :-3 (1,, 0 0 ”H i.
”.th ' 02 T 0 “21 vt-lut-l + 0 0 0 vt-Jut-S + 0 0 0 Vt—Qut-s + *2 mug/52)
O O a,l ”:1 0 0 0 “:3 O 0 0 “:9 t,
Coefficient Estimate t-Value p-Value
p, 0.0586 0.526 0.300
,1, 0.1949 1.583 0.057
a, 4.7732 8.639 0.000
a, 4.2835 8.727 0.000
63, 5.1533 9.208 0. 000
a" 0.0499 1.114 0.133
a2, 0.0538 1.424 0.078
a,, 0.0697 1.412 0.080
01,, 0.0179 0.867 0.193
01,, 0.0252 1.249 0.106
1’1 3.7641 6.481 0.000
l; 3.3444 6.172 0.000
4:, 3.4265 5.353 0.000
Futures Residual Cash Residual Cross Product
Statistics Statistics ResidualStatistics
M, = 0.178 M3 = -0.266
M, = 2.934 M = 5.626

Q(1) = 0.056 02(1) = 0.808 Q(1) = 0.368 Q2(1) = 0.620 Q’(1) =0.068
Q(5) = 5.348 02(5) = 8.388 Q(5) = 5.824 Q2(5) = 5.286 Q3(5) =5.071

Q(10) = 12.68 Q2(10) = 16.68 Q(10) =11. 28 Q2(10) = 10.25 Q’(10)=
Q(15) = 15.29 Q2(15) = 17.50 Q(15) = 14.44 Q2(15) = 15.62 ‘Q’(15) = 11.66
Q(20) = 18.55 Q2(20) = 24.84 Q(20)= 15. 92 Q2(20) = 18.33 Q’(20) = 14.55

 

Note: M, is sample skewness; M4 is sample kurtosis; Q(dj) is a Q test for df degree
autocorrelation in the residuals; Q2(df) is a Q test for df degree autocorrelation in the
squared standardized residuals; and Q’(dj) is a Q test for the standardized cross product
of the residuals from the futures price and cash price equations.

74
presence of unexplained ARCH effects in the squared standardized residuals. All the

seasonality coefficients for sine functions are significant and positive, which results in
cyclical volatility levels high in the summer time and low in the winter time.8 All the
insignificant coefficients, such as cosine terms and other ARCH terms, are excluded from
the model. The Ljung-Box Q statistics for the errors and standard squared errors of all
futures, cash and cross errors suggest the model is well specified in the sense that the

model does a good job of capturing all the autocorrelations.

E ”12.13,.

Unfortunately, the estimated price process does not have a closed form solution
and so it is necessary to generate the joint distribution of futures and cash prices
numerically in order to solve the farmer’s optimization problem. Using the estimators
of the joint price process from Table 4.2, we can use simulation techniques to obtain a
terminal distribution of the cash and futures prices, conditional on information available
at the time of the decision problem.

Let p0 and f}, be the initial cash and futures price respectively on the last week of
April 1994, to, which is assumed to be the planting period. The terminal date, t,, is the
first week of December 1994, which is the assumed harvest period. Starting from t =

to, the simulation steps are:

 

' The data series begin from the first week of May and the sine function reaches its
peak in August. Similarly, the lowest level of the sine function is at three quarters of
a period, which corresponds to February.

«.4 75
l. Simulate Hu1+11 Hm” and H22”, by iterating the variance equation in (4.6)
forward one period.

2. Generate two independent standard normal variates, (emu cu”), so that

(«HmM evrol’1/H22ul e, 1+1) are two independently distributed normal variates

with mean zero and variance H, 1.1+ , and H22”, respectively.

3. Calculate the errors of joint normal distribution as:

1+1 = VH11.1+1 ev,t+l

V

u =p ‘/H e +J(1-p2)H e

1+1 1+1 22,1+1 v,1+1 1+1 22,1+1 u,1+r (4.7)
H

21,t+l

\[H 11.1+1H22.1+1

(See appendix B for the deduction of the above equation.)

 

where 11M =

 

 

4. Generate one pair of log prices at t+I by:

lnfM = 11, + lnf, + v“, (4 8)

lan = 11,, + 1np, + u“1
5. Repeat step 1 - 4 for t, - to = 31 weeks until one realization of (19611".01)’ is

obtained, and take the exponential of the realized log prices to restore them to
price levels. This is used as one random sample of the bivariate price distribution
at harvest conditional on information available at planting.

6. Repeat step 1 - 5 m times to generate an m-size sample of the bivariate price

distribution at harvest, where m is 1000 in this study.

76

The above process generates a bivariate distribution of cash and futures prices
based on the information available in historical futures and cash prices at planting, which
provides important information on higher moments. The mean of each price is then
adjusted to the market determined mean levels (current futures and forward prices).

Futures and cash forward prices contain not only the information in historical
prices but also other currently available market information. As a result, current futures
and forward prices are used as the mean levels of futures and cash prices at harvest. The
December cash forward contract price during the last week of April 1994 was $2.48/bu
in the southwest Iowa,9 and the December futures price during the same week was
$2.56/bu at Chicago Board of Trade. The 8 cents difference between these two prices
at planting is the markets’ expected basis at harvest, representing the cost to transport the
corn from Iowa to Chicago. A linear transformation of the simulated price distribution
is used to shift cash and futures means at harvest to 2.48 and 2.56 respectively, in order
to eliminate the mean increasing effect caused by the positive mean estimates. This
adjustment does not affect the higher moments of the distribution, but ensures the means
correspond to the designated levels.

The histograms of the simulated marginal distributions of futures and cash prices
for the first week of December 1994 are shown in Figure 4.5. The first four moments
of the bivariate distribution are calculated and listed in Table 4.3. As expected from the

estimated price process, the joint sample distribution of futures and cash prices at harvest

 

9 This is the average of the high and low forward prices for April 28, 1994, obtained
from the Agricultural Extension Service in the Department of Economics at Iowa State
University.

77

 

Froquonoy
.0 70 no no 100

20304-080

10

 

u 2.2 2.6 an 3.4 311 41
Future: t/bu

A. A A A A

 

 

100

Frequency
30 40 50 00 70

20

10

 

u 2.2 2.6 3.0 3.4 3.8 4.2
Cash S/bu

Figure 4.5 Histograms of the Simulated Futures and Cash Price Distributions

 

78
Table 4.3 Sample Moments of the Simulated Harvesting Prices

 

Mean Standard Deviation Skewness Kurtosis Correlation

 

futures 2.5626 0.39237 0.41368 3.5744 0.8433
cash 2.4826 0.38532 0.43658 3.5529

 

is not normally distributed. The marginal distributions have slight positive skewness and
excess kurtosis, while the normal distribution is symmetric and has a kurtosis level of
about 3. This result is consistent with previous observations regarding the distribution
of commodity prices (Gordon, 1985). The two prices are highly correlated at harvest,

with a conditional correlation of 0.84.

4.2 The Joint Farm and Area Yield Distribution

Although the normal distribution has been used to model yield distributions in past
studies of crop insurance, it may not be an appropriate model. It is generally argued
from a conceptual standpoint that farm-level yield distributions have a thin long tail on
the left side because of the occasional severe catastrophes that can drastically reduce, or
totally eliminate, yield; and the distribution’s right tail is short because there is a
maximum yield limit given current technology.

Day (1965) conducted statistical tests on yields of several major field crops in the
Mississippi Delta Region, and found the yields are nonnormal and nonlognormal, and
have negative skewness. Recently, Nelson (1990) concluded that beta distribution can

better characterize the skewness of yield distributions based on an empirical analysis

79

using data from seven Iowa counties. Nelson and Preckel (1989) studied a conditional
beta distribution and drew similar conclusions. The beta distribution is a bounded
distribution with the two bounds characterized by two of the distribution’s parameters,
and it provides the flexibility for fitting skewness by changing two shape parameters.

However, there are a number of difficulties in estimating the parameters of the
beta distribution for farm-level yields. In order to estimate the parameters describing the
yield generating process for a particular farm, historical yield data from the farm are
required. Estimating the central tendency and higher moments of the yield distribution
typieally places heavy demands on the data and consequently a large set of data is needed
to obtain reliable estimates of the yield generating process.lo Unfortunately, farm-level
yield data are usually not well recorded; for example, the Actual Production History
(APH) yield data available for individual farms in southwest Iowa contain only ten years
of yield observations, which makes it unrealistic to attempt to estimate the beta
distribution directly for individual farms.

It is natural to think about using cross sectional data to reduce the difficulties in
working with time series and increase the amount of data available for parameter
estimation. For example, Nelson (1990) pools the cross sectional data from neighboring
farms each of which has several years of time series data on yield levels and estimates

the beta distribution. Unfortunately, cross sectional data from the farmer’s neighboring

 

‘° The Beta distribution has four parameters. They are the upper bound, lower
bound, left skewness and right skewness. As technology improves over time, the upper
bound and the two skewness parameters of yield all change, and perhaps the lower bound
as well. This increases the number of parameters that must be estimated in order to
characterize the yield process.

80
region are generally highly correlated during the same time period because of local

production factors, such as weather, which can lead to biased parameter estimates. On
the other hand, cross sectional data from randomly selected remote regions are difficult
to use because the variation in factors such as soil fertility, rainfall, and temperature in
the different regions must be taken into account. As a result, many studies have resorted
to using county-level yield data to estimate representative yield distributions because
these data are usually available for longer historical periods, often more than 50 years,
and the geographical conditions in a county may be similar for most farms, which would
imply that county-level yield distribution may reveal information about the farm-level
yield distributions in that county. Hence, in this study county-level time series data are
used to estimate the farm-level yield distribution.

Evidence has been found that the historical yield data exhibit time trends due to
technological change which needs to be accounted for in yield distribution estimation
(Skees and Reed, 1986; Swinton and King, 1991). Recent work by Moss and Shonkwiler
(1993) uses an inverse hyperbolic sine function to model yield distributions around a
measure of central tendency. The yield process might be thought of as having stochastic
shocks, which are due to factors such as weather, insects and disease, around a mean-
yield level which increases at a deterministic rate as technology is made available. The
residuals around the time-varying mean level are corrected for skewness and kurtosis by
using a transformation based on the inverse hyperbolic sine function. This model
simultaneously shrinks large residuals toward zero and parameterizes the skewness and

kurtosis of the distribution. Equally important, the model characterizes both the

81

deterministic time trend and the stochastic component of yields in a way that easily
accommodates the use of time series data. The model used in this study to describe the
yield generating process is a variation of the model used by Moss and Shonkwiler.
The remainder of this section introduces the data characteristics and presents the
model specification. Next, the simulation procedure for county-level yields and farm-
level yields are discussed. Finally, the technique used to generate the bivariate

distribution for both county-level and farm-level yields is explained.

E . [1,. 11 L

Due to the lack of farm-level yield data, the method used in this study is to
estimate parameters describing the county-level yield distribution, whichitself is needed
in the analysis of the area yield crop insurance program, and then use available data on
farm-level yields to make adjustments to the county-level yield generating process that
will reflect the characteristics of farm-level yield distributions. Empirical county-level
and farm-level yield distributions are then generated using numerical methods. Taylor’s
(1990) method is used to impose correlation between the farm-level and county-level
distributions and obtain a bivariate yield distribution.

Figure 4.6 shows county-level corn yields for Adair, Adams and Case counties
in southwest Iowa from 1928 to 1993. Yields in each county increase over time at a
nonlinear rate with a slightly upward bending curve, suggesting a nonlinear trend is
present in the process. In addition, yield volatility is increasing as the central tendency

goes up. Conceptually, this form of heteroskedasticity is expected because the larger the

82

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83
central tendency of the yield, the larger the disturbance can be because there is more

room for yields to drop due to negative shocks and there is always the possibility that a
major catastrophe cause yields to fall to zero."

It is important to account for the nonlinear time trend and heteroskedastic
properties of the yield data. One method would be to incorporate these features directly
in the model describing the yield process and estimate parameters using historical data.
For example, a quadratic time trend can be used instead of the linear trend to reflect the
nonlinear central tendency. The disadvantage of this approach is that the process
generating the time trend and heteroskedasticity must be explicitly defined and then
estimated, which imposes additional demands on the data set. An alternative approach
is to impose a transformation that corrects the data for these general characteristics and
then use the transformed data in the estimation.

One common transformation is the logarithm function which adjusts both the
convexity of the central tendency and the heteroskedasticity across time. The details of
how it accommodates the heteroskedasticity across time will be explained after the model
is introduced. Figure 4.7 shows the logarithm of historical yield levels in each county,

which appears to have a linear trend and is homoskedastic across time except for one

 

" To test for a trend in variance, the errors are obtained as observed deviations from
the central tendency. The model used to estimate the central tendency is important to
this test, here the inverse hyperbolic sine transformation model, which will be later
explained in detail is used. A X2 test on the residuals (Harvey, 1990, p158) is
undertaken.

The values of the statistic for Adair, Adams, and Case are 54.6, 65.0 and 66.9
respectively, and their probabilities are all less than 0.1%. The null hypothesis of
homoskedasticity is rejected essentially at any significance level.

84

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85
large negative shock in the beginning of each series. The null hypothesis of

homoskedasticity across time cannot be rejected for the log yield data. ‘2

In order to determine whether differencing of the yield is necessary to specify the
model, the Phillips-Perron unit root tests are undertaken on the logarithm of the yield
levels to check if the log yields have a unit root. The results are listed in Table 4.4.
Unlike the price series, the hypothesis of unit root in the yield series can be rejected and

a linear time trend hypothesis cannot be rejected.13

ifi ' n im ti n
Yields are assumed to be governed by the following process:
y,=a+by,_,m+ct+e,

(4.9)

 

 

where y, is the logarithm of yield at time t; e, is the nonnormal disturbance; sinh(- ) is

the hyperbolic sine transformation (HST); v, is an i.i.d normal disturbance with mean

 

‘2 The same )8 test for a trend in variance, as specified in footnote 11, is undertaken
on the logged yield data with the same model. The statistics for Adair, Adams and Case
are 5.174, 6.463 and 5.571 respectively, and the probabilities are all larger than 99.95 %.
The null hypothesis of homoskedasticity can’t be rejected at the 5% significance level.

‘3 Again, following Perron’s suggestion, the equation with the linear time trend is

tested first. Z(<Ir3) is large enough to reject the hypothesis that E=0 and 5 = l at 5%
significant level, but it doesn’t tell which of the above two equations is violated.
However, the small 2(5) and Z( :5) tell us that 5 =1 can be rejected. Therefore, the unit
root hypothesis can be rejected for the yield series. Also, 2(111) is large enough to reject
that a =0.

86

Table 4.4 Phillips-Perron Unit Root Test for Yield

 

2(5)
20,)
Z( 1,)
Z( 1,)
201’s)

 

 

 

 

y; = 0.4” b.y:—1 + e!
y, = d + 5y,_, + era-g) + e,
Estimate
Ada'rL m m
2.6158 1.7377 1.8462
0.35597 0.57624 0.54968
2.7596 1.8927 1.9598
0.31944 0.53755 0.52141
0.0071465 0.0036892 0.0032305
Estimate M" __1114_
10% 5% 1%
-44.237 -30.060 -3l.108 -18.3 -21.8 -29.5 5=l
-5.7718 -4.3080 -4.5346 -3.13 -3.43 ~3.99 5=1
5.6984 4.3326 4.5195 2.72 3.08 3.72 a=0
1.4170 1.3856 1.0138 2.38 2.79 3.49 c=0
16.669 9.3155 10.299 5.39 6.34 8.43 c=0 5:1

 

Note: Z, t, and <I> are statistics defined in Perron’s ( 1988) paper. The critical values are
from the tables composed by Dickey and Fuller ( 1981).

87

zero and variance (:2; 15 and 0 are parameters measuring skewness and kurtosis
respectively. Generally, when 5 is positive e, is skewed to the right, if it is negative e,
is skewed to the left, and if it is zero e, is symmetric. When 0 is zero, e, is as kurtotic

as the normal distribution (the limit of e, is v,+5 as 0 approaches zero, lime, = v, + 5) and
M

e, becomes more and more kurtotic as the magnitude of 0 increases either positively or
negatively. The lagged term y,_, captures autocorrelation, so that e, is i.i.d. The
functional form of the HST is outlined in appendix C.

The first equation in (4.9) is the linear trend equation of log yield, which
describes the central tendency of the current yield as being partially determined by time
and last period’s yield“. The transformed stochastic variable, e,, has a non-zero mean
which is determined by the three parameters of the transformation. Although still
deterministic, the central tendency of y, is the sum of the first three terms in that equation
and E(e). The realized yield is a combination of the deterministic central tendency and
a stochastic nonnormal shock. The second equation in (4.9) transforms the normal
shocks into nonnormal shocks by the modified HST. Appendix C also discusses some

of the properties of the HST. In order to estimate the parameters of model (4.9), the

 

“ It is useful at this point to notice that the log yield described in (4.9) implies the
errors are multiplicative to the time trend and lagged yield levels rather than additive,

Y, = Yt,exp(a +ct+e,) , where Y, is the yield level, and the variance of yield levels equals
to the variance of the exponential stochastic shocks multiplied by a trend factor. The

conditional mean and conditional variance of yield are: E,_,[Y,] = Y flexp(a +ct)EIexp(e,)] ,
and V,_,[Y,] = Yzjexpaa +2ct) V[exp(e,)] respectively, and both of them are'increasing with

time even if the mean and variance of the errors are constant over time, which is
consistent with the yield heteroskedasticity observed in Figure 4.6.

88

maximum log likelihood technique is used. Equation (4.10) gives the maximum log

likelihood function (details of this equation are explained in appendix C):

2
1 T ”1 2
MAX L=—_ ln 2+_+ln02e,+1
d,b,c,c’,0,5 2 g [ C c2 ( )1
a "l '
u, = 8230193.). _ a (4.10)

%ln(0e, + (0e,)2+1) — 6

e,=y,-a—by,-,—ct

The model is estimated for county-level yield using annual corn yield data from
Adair, Adams and Case county from 1928 to 1993 as obtained from the University of
Kentucky. The log yield is multiplied by 100 for computational convenience. The
estimated parameters and their t-statistics are shown in Table 4.5. The second, third and
fourth moments and Ljung-Box Q statistics for the nonnormal errors are listed in the
table as well. All the parameters are significantly different than zero at the 5% level
except the variance for Case County.

The estimated yield processes for all the three Counties show the same pattern.
The positive b and c estimates indicate expected yields are increasing over time, and
positively related with the yields in the previous period. The negative 0 and 5 estimates
indicate errors are skewed to the left and are more kurtotic than the normal distribution,
which is consistent with the calculated third and fourth sample moments listed in Table

4.5. Adair and Adams county have similar estimates for the transformation parameters,

89

Table 4.5 County-level Yield Model Estimates

 

 

 

 

 

County
Adair Adams Case
Coefficient Estimates t-Value Estimates t-Value Estimates t-Value

a 360.5 918.5 355.91 908.3 325.28 44.6

b 0.052 9.8 0.050 6.4 0.142 6.8

c 1.86 35.0 1.93 24.5 1.72 21.0
_ c 90.0032 3.5 0.0083 3.5 20.00 1.2
T -16.21 -41.3 -11.27 -28.8 -0.2375 -2.3

6 -0.41 -31.4 -0.55 —25.2 -8.80 -3.8

M2 1987.47 2319.31 1699.44

M3 -4.3532 -3.5274 -3.9091

M4 26.9436 19.0132 22.6256

Q(1) 0.388 0.293 1.341

Q(5) 4.483 6.775 5.911

Q(10) 5.384 8.259 7.085

 

Note: M2 is sample variance; M3 is sample skewness; M4 is sample kurtosis; and Q(dj)

is a Q test for df degree autocorrelation in the residuals.

90

while Case county clearly has different parameter estimators. However, the
transformation parameter estimates combine to produce quite similar distributions for all
three counties (see appendix C for further discussion of the relationships among the three
parameters). The Ljung-Box Q test suggests there is no remaining autocorrelation in the

residuals from this model.

- i 1 i ' ' 11
Because the time series data on yields are annual observations, we can predict the
yield process for the upcoming year by specifying the current year’s yield and iterating
the model forward one period. There is no closed form distribution for the yield
process, but it is straightforward to simulate the yield distribution empirically. The
simulation starts by generating a sample of n normal variates, and then plugging each
variate into the recursive equation specified by (4.9), to get n realizations of yield. The
detailed simulation procedure for n = 1,000 is:
1. Generate normally distributed random variates, ”1.1+“ with mean zero and
variance (:2, fori = I,...,IOOO.
2. Transform the simulated random variates, ”1.1+ 1+ into nonnormal disturbance using
the HST:
em, = sinh(0(v,.,,+,+6)) /0, fori = I, 1000.
3. Simulate log yields using the yield trend equation
ym, = a + by, + c(t+I) + em, , fori = I,...,1000.

4. Take exponential of the log yields to obtain yield levels:

91
11",, = expOrmfl) , fori = I,...,IOOO.

This provides a sample of 1,000 yield observations conditional on information
available at planting. To illustrate, a simulation was conducted for each of the three
counties using the parameters estimated in Table 4.5. Histograms for each simulated
distribution are given in Figure 4. 8 and Table 4.6 lists the statistics for these

distributions.

Table 4.6 Statistics of the Simulated Yield Distribution

 

 

Standard
County Mean Deviation Skewness Kurtosis Minimum Maximum
Adair 117.51 30.40 -1.203 4.072 .6749 156.9
Adams 117.38 32.84 -1.209 3.921 1.562 157.5
Case 124.22 31.00 -1.317 4.567 5.400 164.3

 

For 1994, each simulated yield distribution for the three counties has a mean yield
of approximately 120bu/acre, a maximum possible yield of about 160bu/acre, is skewed
to the left, is more kurtotic than the normal distribution, and has a possibility, though
low, of a yield realization near zero. Note, again, the similarities in each of the

simulated distributions despite the difference in the estimated values of the transformation

parameters.

92

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Because data on farm-level yields are generally available for only relatively short
time horizons, for example, less then ten years, it is difficult to directly estimate a farm-
level yield generating process using historical farm-level data. Although several years
of farm-level yield data are generally not sufficient to estimate the higher moments of the
distribution, such as skewness and kurtosis, they may provide useful information
regarding the distribution’s central tendency and variance. In order to proceed, it is
assumed that the farm—level yields in each county have the same central tendency as the
county-level yield and that the distributions are similar except for differences in the
variances.“ The reasoning is that farmers in a given county face similar technology,
soil, and weather conditions, so that average yield levels will be similar across farms.
However, even if farm-level yields are i.i.d. , the county average yield distribution will
have a smaller variance due to the averaging effect. Hence, the farm-level yield
variation would be expected to differ from the county-level yield variation, and in general
to be larger.

A linear mean-preserving spread technique is used to generate farm-level yield

distributions from the county-level yield distributions.“ The mean-preserving spread

 

‘5 The constant central tendency assumption can be relaxed but the emphasis of this
study will be on the dispersion of the distribution around the central tendency.

'6 Suppose X is a random variable with mean, 11, and variance 0’, then, Y is called
a mean-preserving spread from X when Y = l+kX, and l and k are chosen so that the
mean of Y is also 11, and the variance of Y is kzo".
E(Y) = E(I+kX) = l+kE(X) = l+k11, V(Y) = k2V(X) = 1120’, so thatl = (I-k),1.
If 0<k<1, it is a mean preserving shrink; if k> I, it is a real mean-preserving
spread; ifk = 1, Y = X. The case k<0, or k = 0, is beyond our study.

94
technique provides the flexibility to alter the spread of the distribution, so that alternative

yield distributions with different risk levels can be generated. First, we use historical
farm-level data to estimate reasonable values for the variance of the representative farm
yield, and then the mean-preserving spread is used to transform the simulated county
yield distribution to the farm-level yield distribution.

The available farm-level yield data consist of ten years of APH data for Adair,
Adams and Case county of Iowa from 1983 to 1992.‘7 There are 218 farms in the
counties: 113 in Adair, 66 in Adams, and 39 in Case. Each farm may have more than
one plot of land. The yield and acreage are recorded each year for each of the plots in
the data set. A whole farm average yield is obtained each year by calculating a weighted
average of all yields for each farm, weighted by the relative acreage for each plot.
Because the simulated county-level yield distribution is for 1994, we estimate the 1994
yield variance of each farm.

Because the APH are time series data and not homoskedastic, the yield for each
year has a different variance, and so the sample variance cannot be used to forecast the
yield variance. Therefore, a procedure has to be used to transform the time series data
into 1994 farm-level yield data so that a variance which is compatible with the 1994
county yield ean be calculated.

The farm-level APH data from 1983 to 1992 are detrended using the first equation

in model (4.9), where the parameters a, b, and c for farms in a given county are those

 

‘7 Thanks go to Jerry Skees in the Agricultural Economics Department at the
University of Kentucky for supplying these data.

95
estimated for county yield in Table 4.5 and t=1 is corresponding to the year 1928, to

obtain a set of residuals for each farm that are independent of time. The detrended
residuals provide a small sample of the time-independent, nonzero-mean, and nonnormal
random errors, 1: = (e,,, e58--. e,,)’, which can be imposed on the time trend in 1994
to obtain a sample of farm-level yield realizations for 1994. The time trend for yield in
any year depends on the yield of the previous year and, unfortunately, the 1993 farm-
level yield is not available. The expected farm-level yield for 1993 to be predicted first

as:

E(ym) = a = by, + c(t+1) + E(e). (4-11)

A small sample of farm-level yield for 1994 can then be generated by iterating the

equation forward again with the random error imposed as:

y... = a + b130,...) + mm + e. ' (4.12)

Noting that equation (4.12) is still in logarithms, the exponential transformation
is taken to restore the log yield sample to yield levels. Then, the sample standard
deviation of each farm’s 1994 yield sample can be calculated and will be directly
comparable with the standard deviation of simulated 1994 county-level yields. The
distribution of the farm-level yield standard deviations for each county is graphed in
Figure 4.9, and the maximum, average and minimum standard deviations are listed in

Table 4.7.

96

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97
Table 4.7 Maximum,Minimum and Average Standard Deviation of Farm Yield

 

 

County Maximum Average Minimum County Yield

Adair 52.18 37.02 19.30 30.40
(1.72) (1.22) (0.63)

Adams 47.51 35.96 26.89 32.84
(1.45) (1.10) (0.82)

Case 54.57 36.11 21.25 31.00
(1.76) (1.16) (0.69)

 

Note: The numbers in parentheses are the ratios of the individual yield standard
deviations to the county yield standard deviations.

The standard deviations of the farm-level yields range from 63 % to 176% of the
county-level standard deviation with an average of 122 % , 110% and 116% of the county-
level standard deviation for Adair, Adams and Case county respectively.

Because all three county-level yield distributions are similar and Adair county has
113 farms (more than half of the farms in all the three counties), Adair is used as the
representative county in southwest Iowa. Three representative farm-level yield
distributions are then simulated, representing the yield distributions for low risk, medium
risk and high risk farms. A linear "mean-preserving" spread on the logarithm of the
county-level yield distribution is used to generate these distributions for two reasons: 1)
the yield model has a deterministic central tendency on the logarithm of yield; and 2) the
mean-preserving spread on yield levels may result in negative yields, an impossible case

for crop yields. By performing the transformation on log yield, the models remain

98

consistent and a negative yield will never occur because the exponential function restores
all negative log yields to nonnegative yield levels.

The two parameters of the linear transformation are estimated by numerical
methods because the linear transformation is taken on the log yields while the mean-
preserving spread criteria are specified with respect to yield levels, so there is no
apparent way to explicitly find analytical expressions for the two linear transformation

parameters. Formally, the transformation parameters were estimated by solving (4.13):

Min oar = [E(Y‘)-E(Y’)]2 + w[S,-S(Y’)]2
M

where I = y’ — kyc (4.13)
y’ = 1n(Y’)
y“ = ln(Y‘)

where Yc and Y' are the county-level and farm-level yields respectively; y" and y' are the
logarithm of the two yields; S (. ) is the standard deviation operator; ST is the target
standard deviation for the representative farms; and w is a weight parameter set at 10,
to normalize the magnitudes of the mean and standard deviation.

The distribution of farm-level yield standard deviation in Figure 4.9 for Adair
shows that the majority of the standard deviations fall into the category of 30 to 43
bu/acre, which corresponds to a range of about 100% to 140% of the county-level yield
standard deviation, with a mean at about 120%. Actually, only 15% of the standard
deviations fall outside this range. Therefore, 100%, 120%, and 140% of the county-
level yield standard deviation are used as the target standard deviations in the mean

preserving spread process to calculate the parameters for low risk, medium risk and high

99
risk farm-level yield, respectively. Table 4. 8 shows the mean preserving spread

parameter estimates for the representative farms in each county.

Table 4.8 Transformation Parameters for Mean-preserving Spread Yield

 

 

__1&111_Risk_ M i m Ri M51;—
1 k l k I k
0.000 1.000 -0.1113 4.083 -0.2087 19.639

 

One thousand farm-level yields are simulated for each representative farm using
the linear transformation on the logarithm of yields as in equation (4.14), where the

transformation parameters 1 and k are taken from Table 4.8 for each type of farms:

y: = 1 .1 kyc (4.14)

The histogram of each simulated distribution is shown in Figure 4.10, and Table

4.9 lists statistics describing the simulated distributions.

Table 4.9 Statistics of the Simulated 1994 Farm-level Yield Distributions

 

 

Risk Mean Standard Skewness Kurtosis Minimum Maximum
Ranking Deviation

Low 117.51 30.40 -1.203 4.072 0.674 156.9
Medium 117.51 36.48 -0.966 3.280 0.147 168.4

High 117.51 42.56 -0.759 2.744 0.027 181.2

 

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101
Bixanate' Egan-me] and County-level Yield Distributions

Area yield crop insurance uses an average yield measure for an area as the index
value from which to determine indemnity payments. In this study, county-level yields
are used as the index for area yield crop insurance programs. Thus, the county-level
yield distribution developed earlier provides the necessary distribution to calculate farm
income for area yield crop insurance participants. However, farm-level and county-level
yields are not independent because they are affected by similar factors such as weather.
Accordingly, county- and farm-level yields must be generated simultaneously by a joint
process which imposes the appropriate level of correlation between the yields. This
correlation may impact risk management decisions so it is important to take it into

account.

Correlation Between the Farm- and County-level fields

The correlation between area yield and farm-level yield is very important to
farmers participating in an area yield crop insurance program. Miranda finds that the
higher the correlation, the more protection a farmer receives from the program. The
univariate distributions of 1994 farm-level and county-level yield have already been
simulated. It is now necessary to generate the joint distribution between farm- and
county-level yields by imposing a correlation between the two yield distributions.

The sample correlation between farm-level yield, 1", and county-level yield, YC,

is defined as:

102

Cov(Y’, Y ‘)
J(Var(Y’) Var(Y°)
Historical data can be used to estimate the correlation. However, as discussed

pm = corr(Y’,Y‘) = (4.15)

 

 

earlier, it is important to detrend the mean and variance, and account for any
autocorrelation in the data, in order to obtain an unbiased estimate of the correlation.
Using a procedure similar to the one used to estimate the standard deviation of farm-level
yields, we can detrend the both farm-level and county-level yields using the log-linear
model (4.9). Remember, e= (e57, e58” . . , e65) ' are the residuals from farm-level log yield
process which are homoskedastic and trend free. By taking the exponential, we have
farm-level "residuals” in yield levels, D, = exp(e,), for t = 5 7,...,65 , that are trend free,
uncorrelated, homoskedastic and consistent with yield levels. Using the same procedure
on the county-level yields, we can obtain county-level ”residuals”, DC,, for the same
period. Now we can use 0,, DC” and equation (4.15) to obtain sample correlations.
The statistics and a histogram of the sample distribution of the correlations are

shown in Table 4.10 and Figure 4.11 respectively.

Table 4.10 Statistics of the Estimated Yield Correlations

 

Mean Standard Deviation Maximum Minimum

 

0.8549 0.0575 0.9603 0.6489

 

103

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0.68 0.72 0.76 0.80 0.84 0.88 0.92 0.96

Figure 4.11 Histogram of Farm- and County-level Yield Correlations in Adair County

The sample correlation between the farm-level yields and the county-level yields
from 1983 to 1992 ranges from 0.653096 for the 113 farms in Adair with an average
of about 0.85. These estimates reflect the fact that farms in the same area are affected
by the similar conditions, such as weather, so that their yields tend to exhibit strong
positive correlations.

In order to study the optimal use of area yield crop insurance by farms, it will be
useful to study the farms whose yield distributions exhibit different kinds of correlations
with the county-level yield, which is used as the yield index in the area yield crop
insurance. In this study, we will consider correlation levels of 0.65, 0.85 and 0.95 to

represent low, medium, and high correlation levels, respectively.

104
Imposing Correlation Between Farm and County Yields

Although a multivariate nonnormal random variable generator is not commonly
available in most computer programming languages, most do include an independent or
multivariate standard normal random number generator. In any event, correlated normal
variables can be easily obtained by algebraic transformation of the uncorrelated standard
normal variables, and appendix B has outlined this transformation process. Taylor
(1990) developed a method to generate multivariate nonnormal random variables by
transforming correlated normal random variables. This can be used to impose correlation
between the simulated univariate distributions for the farm-level and county-level yields.

Using Taylor’s method, the following steps will convert the simulated farm-level
and county-level yield distributions into a joint distribution with a correlation of pm:

1. Generate the empirical cumulated density functions F(Y') and 60‘) for farm-level
and county-level yields respectively, using the 1,000 yields observations simulated
from the univariate yield processes.

2. Generate 1,000 jointly distributed bivariate standard normal variates (X’,,X2. ,
i=1 ,...,1000 from iid normal variates, with correlation pm using the linear
transformation method in equation (B7) in appendix B, where pm is the estimated
correlation between farm-level and county-level yield.

3. Empirically generate the marginal distributions for each of the correlated standard
normal variables fN(X’) and gN(X2). Empirically generate the two cumulated
density functions, FN(X’) and GN(X2 , corresponding to the standard normal

marginal distributions.

105
4. For each of the normal variates obtained from step 2, (X’,, X2 , find their

cumulated function values FN(X’,) and CNN). Then, find the closest cumulated
function values from the empirical distributions for farm- and county-level yields
so that 1704,) = FN(X’,), and 00”,) = GN(X2,). Then, locate the farm- and
county-level yields corresponding to these function values, (1", .Yc,). The pair
(1”,, 1”,) provide one sample observation of jointly distributed farm- and county-

level yields which will have a correlation level of p ,C.

An example using the Taylor transformation to project (X’o, X20) on (1'0, 1”,) is

illustrated in Figure 4.12.

4.3 Multivariate Price and Yield Distribution

Because the spot market is competitive, the cash price acts to clear the market by
equating demand and supply. Given the generally inelastic demand for crop
commodities, cash prices tend to exhibit negative correlation with aggregate production.
Because of the generally fixed nature of land available for production, acreage doesn’t
change dramatically from year to year, so that the total production is determined mainly
by the yield levels of land in production. The level of correlation between prices and
yields can have significant impacts on the risk position of a farm and, consequently, on
the demand for different types of risk management instruments. As a result, it is

important to consider the joint process that generates prices and yields.

106

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107

The previous sections have developed empirical bivariate distributions for both
cash and futures prices and farm- and county—level yields. This section concludes the
chapter by using Taylor’s method again to impose correlation between prices and yields
and generate empirically a multivariate distribution of cash price, futures price, farm-

level yield and county-level yield.

h Pri d Ar Yi 1

The first step is to obtain an estimate of the correlation between the cash price at
maturity and county-level yield conditional on information available at planting. Because
we are interested in the conditional correlation based on information at planting, the ideal
procedure would be to find the correlation between the residuals of forward price and
yields that are trend free. Unfortunately, a historical forward contract price series was
not available, and so historical futures prices, which are highly correlated with forward
contract prices, are used as an approximation. The county-level yield residuals were .
obtained using the same procedure as was used to estimate the correlation between
county- and farm-level yields: detrending the log yields, and taking the exponential. The
price changes between harvest and planting are used as price residuals.

The weekly futures price data from April 1975 to December 1995 and the Adair
county yield data from 1974 to 1993 are used to estimate the price-yield correlation.
Matching price and yield, we have 19 samples from 1975 to 1993. The sample
correlation for Adair County are calculated as -O.50. The estimated negative correlation

between price and yield is consistent with the market demand/supply mechanism. That

108
is when yield is high, the increased grain supply tends to reduce the price; and when

yield is low, the market demand bids the price up due to the lack of supply.

n Pri Yi 1

Once again, Taylor’s method is used to impose the correlation between the
independently simulated bivariate price and yield distributions. To impose a fixed
correlation between cash price and county yield, the same steps are followed as when
imposing the correlation between farm- and county-level yields, except the variables (Y',
Ye) are replaced by (p, l") and the correlation pm is replaced by p,,, which is set at —0.5.
Because futures and cash price are jointly simulated through the bivariate ARCH model,
they are kept pairwise during this procedure to guarantee the correlation between them
is not affected. Likewise, farm- and county-level yields are kept pairwise to maintain
their imposed correlation level. After repeating those steps, a set of 1,000 jointly
determined realizations, {fip,Y',YC}, are obtained.

There are three sets of farm-level yields representing the low risk, medium risk
and high risk farms, and three levels of correlations between county-level yield and farm-
level yield, so that the process is repeated nine times. Because of simulation error, the
simulated correlations among the variables of the joint distribution are sightly different
than the designated values. The correlation matrices of the four variables are listed
separately for each distribution in Table 4.11. The marginal distributions are not affected
by Taylor’s transformation, so that the moments of the marginal price and yield

distributions remain unchanged.

109
Table 4.11 Correlation Matrices of Simulated Prices and Yields

 

MWWiinl 1mm
fp Y’I‘f p Y'Y‘fp YY‘

 

Low Risk

f 1.00 .843 -.212 -.370 1.00 .843 -.313 -.370 1.(X) .843 -.340 -.370
p 1.00 -.277 -.459 1.00 -.378 -.459 1.00 -.434 -.459
1" 1.00 .623 1.00 .830 1.0) .941
2" 1.00 1.00 1.00
Medium Risk

f 1.00 .843 -.215 -.370 1.00 .843 -.310 -.370 1.0) .843 -.338 -.370
p 1.00 -.281 -.459 1.00 -.378 -.459 1.(X) -.436-.459
Y’ 1.00 .628 1.00 .831 III) .941
Yc 1.00 1.00 1.00
High Risk

f 1.00 .843 -.218 -.370 1.00 .843 -.307 -.370 1.(X) .843 -.337 -.370
p 1.00 -.284 —.459 1.00 -.378 -.459 1.(X) -.436 -.459
Y’ 1.00 .628 1.00 .830 M!) .937

r 1.00 1.00 _ 1.00

 

110
The realized correlation is 0.843 between futures and cash prices, and 0.459

between cash price and county-level yield for all the distributions. The realized
correlations between farm-level yield and county-level yield, are about 0.63 for low
correlation, 0.83 for medium correlation, and 0.94 for the high correlation.

Because cash price is correlated with county-level yield and county-level yield is
correlated with farm-level yield, the correlation between cash price and farm-level yield
is implicitly determined by the joint distribution, and depends on the two directly
imposed correlations. The same reasoning holds for the correlations between futures
price and farm-level yield. When the correlation between county- and farm-level yields
is low (0.63), the correlation between cash price and farm-level yield is at about —0.28,
and the correlation between futures price and farm-level yield is at about —0.22. When
the correlation between county-level and farm-level yields is medium (0.83), the
correlations with farm-level yield are -0.38 and -0.31 for cash and futures, respectively.
When the correlation between county-level and farm-level yields is high (0.94), the farm-
level yield correlations with cash and futures are -0.44, and -0.34 respectively. The
correlation between futures price and county-level yield is also implicitly determined by
the joint distribution. It depends on the correlation between futures and cash prices and
the correlation between cash price and county-level yield, and is at -0.37 for all the

distributions.

Chapter V
OPTIMAL RISK MANAGEMENT CHOICES UNDER ALTERNATIVE

PORTFOLIOS OF RISK INSTRUMENTS

In this chapter, we study the optimal use of different risk management instruments
in an expected utility framework using numerical methods and the multivariate price and
yield distributions simulated in the previous chapter. Risk instruments available in the
farmer’s portfolio include futures contracts, options contracts, crop insurance, and a
government deficiency payment program. The objectives are to study the demand for
the various risk management instruments by risk averse farmers and to estimate their
welfare effects. Additional insights into the impact of various factors on the demand for
the alternative risk instruments will be provided in the comparative static analysis of the
next chapter.

The demand for a particular risk management instrument may typically change
when another instrument is added or dropped from the portfolio. In some cases the
instruments may partial substitutes and, in other cases, they may be complements. When
instruments are substitutes, they tend to play the same role in reducing the impact of a
given type of risk, so adding a substitute instrument will tend to reduce the demand
and/or value of the existing instrument in the portfolio. For example, put options and

the government deficiency payment program may be substitutes because the mechanics

111

112

of the two instruments are similar and both act to reduce the same type of price risk.
Complementary instruments work together to manage a particular type of risk more
efficiently than when used in isolation. Thus, adding a complementary instrument will
tend to increase the demand and/or value of the existing instrument(s). For example,
price instruments such as futures contracts, and yield instruments, such as crop
insurance, may be complements if they can combine to reduce income risk more
efficiently than if either was used in isolation. These substitution and complement effects
are explored in detail in this chapter.

The two forms of crop insurance, individual yield crop insurance (IYCI) and area
yield crop insurance (AYCI), are also compared in this chapter, each of which is studied
in alternative portfolios.

For convenience, the expected utility model introduced in Chapter III is presented

here again:

MAX EUfi']

x.z.£,n
sti=C7€+Fl+07+G~S+C7—F

C1? = pi”

F7 = x(f-fo)

07 = z[max(0,s-f) -h]

GS = Elmax(0,pT-fl)12(1-q-r)-KCF+CD]
CI = 1P - x5010)

17> = m max{0,nE(YO-7],

(5.1)

where p is the cash prices at harvest; 9’ is the farm-level yield per acre; f is the futures

price at harvest; fo is the futures price at planting; x is the futures position; 2 is the

113

option position; It is the option premium; 3 is the option strike price; £ is a binary
indieator for participation in the government deficiency payment program; Y, is base
yield; r is the percentage of acreage reduction program (ARP); q is the percentage of
flexible acres; pT is target price; 3’ is the insured acreage; A is the crop insurance
premium proportion relative to the actuarially fair level; p, is the price index used to

calculate the crop insurance indemnity payment; 17 is the trigger level of crop insurance;

and f! is the yield index used to determine crop insurance payouts. )7 is specified as Y’ ,

the farm-level yield for IYCI and YC, the county-level yield, for AYCI.

Initially, the maximization problem in (5.1) is solved for a set of parameters
representative of those faced by southwest Iowa corn growers in 1994 as listed in Table
5 .1. After analyzing this base solution in detail in the remainder of this chapter, a
comparative static analysis is conducted in next chapter to study cases where some of the
model parameters deviate from their base values, e. g. , the relative risk aversion increases
or decreases.

The base model is characterized by a futures market that is perceived unbiased,
which implies that the futures price at planting is also the mean of the stochastic futures
price at harvest. This ensures hedging with futures is done to manage risk with no
speculative profit motive in the base solution. Likewise, the options market is also

perceived unbiased in that the premium equals the expected value of the option."

 

“ We have not explicitly accounted for the difference in the timing of the cash flow
to pay the option and the crop insurance premiums at planting and the other cash flows
which occur at harvest. Using standard option pricing arbitrage arguments, the implicit
assumption in the model is that all cash flows are discounted at the risk-free interest rate.

114
Table 5.1 Model Parameters for the Base Solution

 

 

Parameter Notation Value
Futures price at planting f0 $2.56/bu
Option strike price 5 $2.60/bu
Option premium h 51;,“me —])]
Target price p, $2.75/bu
Acreage reduction program r 0
Flexible acreage 15 %
Base yield Y» 1507')
Insured acreage r 1.00
IYCI premium rate A 1.35
AYCI premium rate A 1.00
Crop insurance indemnification price p, E(p)
Ratio of farm-level yield standard deviation R 1.20

to county-level yield standard deviation
Correlation between farm and county yield p ,C 0.83
Correlation between cash price and county yield pp, -0.46
Relative risk aversion RR 2

 

115

The strike price for the option contract in the model is set at the level closest to the
current futures price. The government deficiency payment program is specified
according to the 1994 USDA Wheat and Feed Grain Programs, setting target price at
$2.75/bu, requiring no set-aside acreage, and imposing flexible acreage of 15% . The
expected farm-level yield is used as the predetermined yield base.

The insured acreage is set at 100% of the base acreage. The IYCI premium is
set at a level 35 % above the actuarially fair premium to reflect transaction costs and the
impacts of moral hazard and adverse selection of farmers being modeled, and the AYCI
premium is set at the actuarially fair premium. The farmer can select the indemnification
trigger yield up to 75 % of his expected yield for IYCI and 90% for AYCI. It is assumed
that both IYCI and AYCI pay the indemnification at a price equal to the expected cash
price at harvest.

The joint price and yield distributions were developed in Chapter IV. The farm-
level yield distribution for the base solution is set at the medium risk farm whose yield
standard deviation is 120% of the county yield standard deviation, and is positively
correlated with area yield at a medium correlation level, 0.83. The county-level yield
is negatively correlated with cash price at a correlation level of -0.46 in the base
solution. The utility function used in the base model is a constant relative risk aversion
function with the relative risk aversion parameter set at 2.

The remainder of the chapter is divided into four sections. The first section
analyzes equation (5 . 1) when only price instruments are used to manage income risk.

The second section studies equation (5. 1) when both price instruments and IYCI are used

116

to manage risk. The third section explores equation (5. 1) when price instruments and
AYCI are used to manage risk. In each case, the optimal levels of participation in the
risk instruments, the first four moments of the income distribution, and willingness-to-
pay for alternative instruments are examined. Histograms of the income distribution are
also provided for some cases, in order to illustrate the effects of various instruments on
the income distribution. The fourth and final section summarizes the results and gives

conclusions.

5.1 Price Instruments

Six portfolios are studied in this section, which are: 1) no risk management; 2)
futures only; 3) futures and options; 4) government program only; 5) futures, and
government program; and 6) futures, options, and government program. The Optimal
participation levels; resulting mean, standard deviation, skewness and kurtosis of income;
and willingness-to-pay for the various instruments are listed in Table 5 .2 for all six

portfolios. Each portfolio is discussed in turn.

m n In men
When no risk management instruments are used, income risk comes from the joint
effect of yield and cash price risk. Model (5. 1) is restricted by setting futures, options,
government program and crop insurance positions to be zero (x=z=£=n=0). The

histogram of the resulting income distribution is shown in Figure 5.1(a). The distribution

117

Table 5.2 Optimal Positions, Income Distribution, and WTP without Crop Insurance

 

 

 

Portfolio
Futures,
Futures Futures, Options,
No Risk and and and
Instr. Futures Options Govt. Govt. Govt.
Futures 0 0.219 0.276 0 0.649 0.472
Options 0 0 0. 131 0 0 -0.497
Govt. 0 0 0 1 1 1
13(1) 286.43 286.43 286.43 319.11 319.11 319.11
Stdv(r) 89.20 90.54 90.25 90.29 89.90 90.93
Skew(w) -0.617 -0.437 -0.432 -1.024 -0.532 -0.509
Kurt(r) 3.324 3.169 3.201 3.696 3.379 3.199
Willingness-to—pay 0 5. 87 5.95 0.35 32. 1 l 32.53

 

Note: The participation levels for futures and options are reported as the percent of the
expected yield; the income mean and standard deviation and willingness-to-pay

levels are reported in 8.

has a mean of $286.43 per acre, a standard deviation of $89.20 per acre, and is skewed

to the lefi.

The coefficient of variation, defined as the ratio of standard deviation to mean,

measures the relative standard deviation of a stochastic variable and is sometimes used

as a measure of the standardized risk of the stochastic event. The coefficient of variation

is 0.311 for income when no risk management instruments are used, and the cash price

and farm-level yield coefficients of variation are 0.155 and 0.310 respectively. The

relative standard deviation of income is similar to that of yield because the negative

118

 

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correlation between price and yield provides an implicit form of risk protection so that
income, the product of price and yield, is not more volatile. Also, the large coefficient
of variation for yield relative to price suggests yield is relatively more volatile, and it

contributes more to the income variability than does price.

menu

In this case, futures contracts are included as the only risk management instrument
in the portfolio. Equation (5. 1) is restricted by setting options, government program, and
crop insurance positions to be zero (z=£ =n=0). A negative (positive) 1: value means
the farmer sells (purchases) futures at planting and buys (sells) the equivalent amount
back at harvest, i.e., the farmer takes a short (long) position in the futures market which
is offset at harvest.

The optimal futures position in the base case is to go long 22% of the expected
yield. This behavior, at first, appears in conflict with traditional notion of hedging in
which the position in the futures market is opposite to the cash market position. The
traditional notion of hedging is that by taking a short position in the futures market, price
changes in the eash market (the producer’s long position) tend to be offset by price
changes in the futures market. Indeed, most hedging studies have found the optimal
position in the futures market is to go short futures.

The optimal long position in the futures market found here is a result of the strong
negative correlation between cash price and farm-level yield, combined with the

relatively large yield variability. The benefit of offsetting yield variability through the

120

negatively correlated price variability tends to outweigh the benefit of reducing price
variability by taking a short position in the futures market. Put another way, taking a
long position in the futures market allows the farmer to ”cross hedge” a portion of the
yield risk using futures.

The mean of income stays the same as when no risk management instruments are
available because the futures market is assumed to be unbiased. However, the standard
deviation of income increases slightly as a result of futures, though skewness and kurtosis
both decrease. It is apparent that the long position in futures helps reduce the negative
skewness of the income distribution, making it more symmetric, because of the cross
hedging effect between the futures contract and negatively correlated yield. A histogram
of the income distribution with futures is shown in Figure 5.1(b).

Although a risk averse decision maker (DM) usually prefers the stochastic return
with a small variance to a large one when the mean is the same, higher moments also
affect the DM’s preference. Tsiang (1972) found that a risk-averse individual would
have a preference for positive skewness and an aversion for variance of the distribution.
However, a DM often has to evaluate the trade off between skewness and variance. In
this case, the DM prefers the income distribution with the smaller negative skewness but
slightly larger variance, which gives the farmer a higher expected utility level.

Willingness-to-pay for the opportunity to trade futures is $5.87 per acre. Because
the futures market is unbiased, the $5.87 is the amount the farmer is willing to pay for
the reduction in the negative skewness and kurtosis of the income distribution that futures

provide.

121

W

Ifboth futures and options are allowed in the portfolio, equation (5.1) is restricted
by setting the government program and crop insurance positions to zero (5 =n=0). We
only allow put options and define 2 such that a positive (negative) value means the farmer
buys (writes) the put option. ‘9

The optimal futures and options positions are to go long futures 28% of the
expected yield and buy put options equivalent to 13 % of expected yield. The put options
position reduces some of the price variability, exposing the farmer to additional yield
variability because of the negative price-yield correlation. As a result, the long position
in the futures market increases slightly to cross hedge the increased yield risk exposure.
This result is consistent with the analysis of Sakong, Hayes and Hallam (1993), who find
that with yield risk and no crop insurance it is almost always optimal for the producer
to purchase put options and to underhedge on the futures market under CARA. They
also claimed these results are strengthened further if price and yield are negatively
correlated and if the farmer has DARA preference.

The mean of income remains as in the previous two cases because both futures
and options are unbiased. The other three moments of the income distribution are similar
to the case of futures only. This is not surprising given the relatively small position in

the options market and the small change in the optimal futures position.

 

‘9 Synthetic call options can be constructed from combinations of futures and puts,
so call options are included implicitly in the analysis.

122

The farmer’s willingness-to-pay is increased slightly to $5.95 over the case where
only futures are available. The marginal value to the farmer of adding the put options
into the portfolio is only $0.08 per acre, indicating options have little value in this

setting.

n Pr

In this case, the farmer is assumed to participate in the government deficiency
payment program, but does not use any other risk management instruments in the
portfolio. Equation (5.1) is restricted by setting futures, options and crop insurance
positions to be zero, and the government program position to one (5 =1 , x=z=n=0).

The government program is similar to a put option but provided 'at no cost with
zero ARP. Because willingness-to-pay for the government program is positive, the
farmer chooses to participate and f = I is the optimal position.

The mean of income when participating in the program increases by about $33
per acre over the no risk instrument case, which is the amount of the expected subsidy
implicit in the program in this setting. However, participation also causes the standard
deviation of income to increase, and the distribution of income to skew more to the left
relative to the no risk management case. This is because the elimination of the downside
price variability resulting from participation in the government program reduces the
implicit income risk protection provided by the negative correlation between cash price
and yield, and therefore the farmer is exposed to more yield variability. The resulting

income distribution is illustrated in Figure 5.1(c).

123
Despite the high level of expected subsidy, the willingness-to—pay measure is only

$0.35 per acre. The cost to the farmer to bear the additional income risk (higher
standard deviation and larger leftward skewness) that results from participation is high
enough to offset much of the benefit from the higher mean income resulting from the

implicit subsidy.

W

In this case, the farmer participates in the government deficiency payment
program and the futures market, but does not use any other risk management instruments
in the portfolio. Equation (5. 1) is restricted by setting the options and crop insurance
positions equal to zero, and the government program position equal to one (£=I,

=n=0). Again, expected utility increases from participation in the program, so that the
government program is in the farmer’s optimal portfolio.

The farmer’s optimal futures position is long 65% of expected yield. As in the
case when put options were included in the portfolio, the government program eliminates
downside price variability so as to reduce the implicit income risk reduction from the
negative price-yield correlation. A larger long futures position is then taken to help cross
hedge the increased yield risk exposure.

As expected, mean income with futures and the government program is the same
as the government program only case, $33 above the no risk management instrument
ease. However, the standard deviation, negative skewness and kurtosis are reduced

below the levels in the government program only case because the futures position helps

124

to cross hedge the increased yield risk exposure that results from participation in the
government program. Nevertheless, the skewness and kurtosis are increased from the
futures only case because, again, participating in the government program increases the
farmer’s yield risk exposure.

The willingness-to-pay increases significantly to $32 per acre, close to the level
of the expected subsidy built into the government program. The ability to cross hedge
the additional yield risk exposure resulting from participation in the government program
by taking a larger long position in the futures market allows the farmer to capture most

of the expected subsidy value without bearing the additional risk exposure.

v n Pr m

In this case, futures, options and the government program are all allowed in the
farmer’s portfolio. Equation (5. 1) is restricted by specifying participation in government
program and zero crop insurance (5 =1, 77 =0). Once again, it is optimal for the farmer
to participate in the government program, so setting 5:] is not restrictive.

The farmer again takes a long position in futures, in this case equal to 47% of
expected yield. Unlike in earlier cases, the presence of the government program causes
the farmer to write put options equal to 50% of expected yield. The reason for this is
that the government deficiency payment program eliminates much of the downside price
variability faced by the farmer, and the long position in the futures market and the
writing of put options combine to help cross hedge the additional yield risk exposure

which results from participation in the government program. This result is also

125
consistent with the result from Sakong, Hayes and Hallam (1993), who find that when

a government program is used, farmer tend to write put options.

As before, the expected income level is not changed by adding options into the
portfolio because the options are fairly priced. However, the standard deviation of
income increases relative to the case without options in the portfolio, but the skewness
and kurtosis levels decrease slightly. The income distribution histogram is graphed in
Figure 5.1(d).

The value of adding options to the portfolio that contains the government program
and futures is small, increasing willingness-to-pay only $0.42 per acre above the case
with no options in the portfolio. So, despite the significant size of the options position,

the gain to the farmer of adding options to the portfolio is again relatively small.

Summary

In the base case where price and yield are negatively correlated and yield
variability is relatively higher than price variability, price instruments are used to cross
hedge yield risk rather than to hedge price risk. This may lead to market positions which
appear contrary to conventional wisdom, in which they actually accomplish more in the
reduction of yield risk than they do in the reduction of price risk. In addition, low
willingness-to-pay measures suggest that price instruments do not work very effectively
to manage income risk in the absence of crop insurance.

In the base case, the government deficiency payment program provides an implicit

subsidy but does not reduce income risk, because the reduction in downside price risk

126

resulting from participating in the program reduces the implicit income risk protection
of the negatively correlated price and yield. In fact, the program actually increases the
farmer’s income risk exposure to the point that the costs of bearing the additional risk
are almost as great as the benefit of the expected subsidy provided by the program
(willingness-to-pay for the government program is very low despite the high value of the
implicit subsidy).

The use of futures contracts allows the farmer to capture most of the benefits of
the implicit subsidy by cross hedging the additional risk exposure resulting from
participation in the program. In the base case, when crop insurance is not allowed in the
portfolio, futures contracts and the government program act as compliments to each
other.

In the presence of the futures market, options contracts seem to have little
additional value to the farmer as indicated by the small increase in willingness-to-pay
when options are included in the portfolio. When the government program is also
included in the portfolio, a significant number of options are written by the farmer to
offset the extra risk exposure caused by the government program. However, the
marginal welfare gain to the farmer from including options in the portfolio with futures
and the government program is still relatively small.

In several cases, the standard deviation of income actually increases under optimal
use of risk management instruments while the mean is kept unchanged. In these cases,
welfare gains are attained through changes in higher moments of the income distribution,

mainly from reducing leftward skewness. This suggests that a mean-variance modeling

127

approach would provide quite different results to the expected utility maximization
approach used here.

The above discussion suggests the combined effect of price risk and yield risk on
income have important implications for the farmer’s optimal use of price risk
management instruments. In the next section, we allow the use of a yield risk
management instrument in the portfolio and examine implications for all risk management

instruments when both price and yield risk can be directly managed.

5.2 Price Instruments With Individual Yield Crop Insurance

In this section, the risk management role of individual yield crop insurance (IYCI)
is studied. The effectiveness of crop insurance in the presence of price instruments is
compared with the cases in last section which only included price instruments in the
portfolio. Six portfolios are explored in this section: 1) IYCI only; 2) futures and IYCI;
3) futures, options and IYCI; 4) government program and IYCI; 5) futures, government
program and IYCI; and 6) futures, options, government program and IYCI. The optimal
market participation levels, the first four moments of the income distribution, and
willingness-to—pay for the optimal portfolios are each listed in Table 5.3. The major

results from each portfolio are discussed in the following subsections.

112112111!
In this case, only IYCI is included in the portfolio. Equation (5 .1) is restricted

by setting the futures, options and government program positions to be zero (x=z=£ =0).

128
The farmer chooses the trigger yield level to be the maximum of 75 % of expected

yield, despite the fact that the premium charged is 35 % above the actuarially fair level.
Participation in IYCI partly eliminates the downside yield variability, which increases
price risk exposure due to the negative correlation between price and yield. However,
because the variance of price is relatively lower than yield, partly eliminating downside
yield variability reduces the income standard deviation.

Mean income is $5.28 per acre less than with no risk management instruments.
This represents the cost of the premium being set 35% above the expected indemnity
payment. The standard deviation of income is decreased significantly and the income
distribution becomes skewed to the right. The histogram of the income distribution is
shown in Figure 5 .2(a). Comparing this histogram to Figure 5.1, it is obvious that the
income distribution is more compressed when using the crop insurance than under any
alternative portfolio in the absence of crop insurance.

The farmer’s willingness-to—pay for the IYCI is $25.66 per acre, which is much
higher than the value of futures, Options, or even the government program when they are
used in isolation, despite the fact the farmer has to pay an actuarially unfair premium.
This again suggests that IYCI is an effective risk management instrument in the base

model where yield variability contributes more to income risk than price variability.

129

Table 5.3 Optimal Positions, Income Distribution, and WTP with IYCI

 

 

 

Portfolio
Futures, Futures, Futures,
Futures, Options, Govt. Govt. Options,
No Risk and and and and Govt.
Instr. IYCI IYCI IYCI IYCI IYCI and IYCI
Futures 0 0 0.265 -0.023 0 0.214 0.208
Options 0 0 0 0.507 0 0 -0.012
IYCI 0 0.750 0.750 0.750 0.750 0.750 0.750
Govt. 0 0 0 0 l 1 1
E(w) 286.43 281.15 281.15 281.15 313.83 313.83 313.83
SIdV(‘l’) 89.20 65.12 63.00 62.66 62.61 62.51 62.50
Skew(t) -0.617 0.324 0. 195 0.228 -0.077 0.086 0.085
Kurt(r) 3.324 2.578 2.257 2.342 1.985 2.247 2.242
Willingness- 0 25.66 26.20 26.38 51.45 51.76 51.76
tO-Pa)’

 

Note: The participation levels for futures and options, and the selected trigger yield for
IYCI are reported as the percent of the expected yield; the income and standard

deviation and willingness-to-pay levels are reported in $.

130

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131
W

In this case, IYCI and futures contracts are both included in the portfolio.
Equation (5. 1) is restricted by setting the options and government program positions to
be zero (z=£ =0).

These two instruments allow the farmer to manage both price and yield risks
directly. The optimal futures position is to go short 27% of the expected yield, and the
selected trigger yield for IYCI is again the maximum 75% of expected yield. Because
the yield risk is now managed by the IYCI, futures can be used to mange price risk
directly by taking a position opposite of the farmer’s cash position. However, because
the 75% trigger yield restriction prevents IYCI from fully reducing the yield risk, part
of the price variability is still necessary to cross hedge the remaining yield variability
through the negative price-yield correlation. This is what prevents the futures position
from going beyond short 27 % to manage price risk.

Because the futures market is unbiased and the IYCI trigger yield level stays the
same as without futures, the mean of income is the same as with IYCI only. The
standard deviation of income is reduced relative to the IYCI only case, and the
distribution is more symmetric and less kurtotic.

The willingness-to-pay for the two instruments is only about $0.50 per acre more
than the IYCI only case, indicating that futures market is not very valuable to the farmer
once the yield risk is reduced. The reason for the small short position and the small
value associated with futures is, again, that even though the IYCI eliminates the

downside yield risk associated with yield shortfalls below 75 % of expected yield, there

132
is still a large proportion of yield risk present which is at least partially offset by the

negative correlation between price and yield.

W

In this case, IYCI, futures, and options are all available in the farmer’s portfolio.
The restriction on equation (5. 1) is to set the government program position to be (5 =0).

The futures market position is close to zero and the farmer takes a'relatively large
options position by buying put options equivalent to 51% of expected yield. When yield
is effectively managed by IYCI, the income distribution depends primarily on the price
distribution. As indicated by Vercammen (1995), options are valuable to risk averse
farmers because they can increase the skewness of income distribution. The trigger yield
is still selected at the maximum possible level, 75% of expected yield.

The income distribution has same mean as the previous two models, and the
standard deviation is slightly smaller. Its skewness is larger than the case without options
which is consistent with the above analysis. It is also slightly skewed to the left and less
kurtotic than a normal distribution, similar to the previous case with futures and IYCI.
The income distribution is graphed in Figure 5.2(b), which is similar to the IYCI alone
case, but has the left tail truncated.

The farmer’s willingness-to-pay for this portfolio is only $0.18 per acre higher
than the portfolio without options, suggesting futures and Options are substitutes in the
presence of IYCI, and that the value of options when IYCI and futures are in the

portfolio is also not high. However, the almost zero futures position and the additional

133

$0.18 per acre willingness-to—pay show that Options can manage price risk more
effectively that futures when IYCI is in the portfolio. When either options or futures are

included in the portfolio with IYCI, the resulting income distribution is similar.

W

In this case, the farmer’s portfolio includes IYCI and the government deficiency
payment program. Equation (5.1) is restricted by setting the futures and options
positions to zero, and the government program position to one (é =1 , x=z=0).

The optimal trigger yield selection is still the maximum 75% of expected yield
and it is optimal for the farmer to participate in the government program. The IYCI
reduces the downside yield risk, and the government program eliminates the downside
cash price risk.

The mean Of income is nearly $33 higher than the previous cases because of the
subsidy implicit in the government program; the standard deviation of income is further
reduced from the cases without the government program; and the distribution is skewed
to the right under IYCI but symmetric in this case. The histogram is graphed in Figure
5.2(c). Comparing Figures 5 .2(c) to (a), the income distribution is shifted to the right
and its left tail is truncated, showing a more preferable shape to the farmer.

Willingness-to—pay increases by about $26 per acre over the IYCI only case,
which is slightly below the amount of the expected government subsidy. The value of
adding the government program into the portfolio is relatively low considering its subsidy

value. Part of the reason is that even though IYCI reduces downside yield risk, there

134

is still yield risk left unprotected, and participating in the government program eliminates
price risk and increases the farmer’s exposure to the remaining yield risk due to the
negative price-yield correlation.

Notice that the farmer’s absolute risk aversion is decreasing as his/her wealth
level increases under CRRA preferences, and when the subsidy inherent in the
government program increases the mean income, the farmer values the IYCI less than
before. As a result, the true value Of government program may be more than the
additional $26 per acre, which is not separable from the total willingness-to-pay.
Therefore, an experiment is conducted by adding an ”actuarially fair" government
program, Obtained by subtracting the expected subsidy, $33 per acre, from the
government program, into the portfolio containing IYCI only, and calculating the
willingness-to—pay to this portfolio which is $25.83 per acre, $0.17 higher than the value
of IYCI. This result suggests the government program slightly reduces the income risk
by eliminating part of the downside price variability when IYCI is in the portfolio, but
its risk management value is much smaller than IYCI and even smaller than futures and
options. The resulting income distribution is the same as the case with a subsidized

government program, except the mean is smaller.

W

In this case, the farmer uses the government deficiency payment program, crop
insurance, and futures contracts in the portfolio. Restrictions in equation (5. 1) set the

options position to zero and government program position to one (z=0, £=I).

135
It is optimal for the farmer to select 75 % of expected yield as IYCI trigger yield,

participate in the government program, and change the futures position from short 27%
to long 21% of expected yield. The elimination of the downside price variability by
participating in government program results in additional exposure to the yield risk
remained from using IYCI under 75 % trigger yield restriction. The long futures position
allows the farmer to cross hedge the additional yield risk.

As before, adding futures to the portfolio does not change the mean of income,
but the standard deviation decreases slightly. The distribution remains nearly symmetric
as in the case without futures, but the skewness measure increases from a small negative
value to a small positive value.

The additional willingness-to-pay when adding futures to a portfolio containing
both government program and crop insurance is only $0.30 per acre, suggesting that the
futures market has a little additional risk management value when IYCI and the

government program are in the portfolio.

m n m Y
In this case, the farmer’s portfolio includes the government program, crop
insurance, futures contracts and options contracts. The restriction on equation (5.1) is
to set the government program participation level to be one (5 = I).
In the presence of the government program and IYCI, the put options position
changes from buying 52% to writing 1% Of expected yield, supporting the conclusion in

the previous section that the government program and options are substitutes. Again,

136

because the government program reduces downside price variability which exposes the
farmer to additional yield risk when the maximum IYCI trigger yield is restricted to 75 %
of expected yield, the long futures position helps to cross hedge this yield risk. The
futures position remains nearly the same as in the last case because the options position
is close to zero, and the IYCI trigger yield is again selected to be 75 % of expected yield.

The income distribution is almost identical to the previous one, and willingness-to-
pay stays the same because the options position is very small and the Options play almost
no role in the portfolio. The histogram of the income distribution is shown in Figure

5.2(d).

Summary

In the base model where yield is relatively more volatile than price, adding crop
insurance into the farmer’s risk management portfolio provides direct risk protection for
yield risk which is more effective than the ”cross hedging" through the negatively
correlated cash price. Crop insurance frees the price risk management instruments to
deal more directly with price risk.

In the base model, the farmer selects the indemnification trigger yield at the
maximum permitted level, 75 % of his expected yield, despite the fact the crop insurance
premium is set 35 % above the actuarially fair premium. The willingness-to-pay for IYCI
is relatively high, over $25 per acre for the IYCI only case, even with the restriction on
trigger yield and the premium loading. The standard deviation of income is greatly

reduced whenever IYCI is allowed in the portfolio. These facts suggest crop insurance

137

can manage income risk effectively, and the farmer’s potential demand for crop insurance
may go beyond the current restriction even in the absence of a moral hazard motivation.

When IYCI is used to manage yield risk, the farmer goes short futures. When
the government program is included in the portfolio the farmer goes long futures;
however, the magnitude of long futures is decreased from the cases when IYCI was not
available. In the base model, the size of the futures position in each case is about 20%,
long or short, and the marginal willingness-to—pay is never more than about $0.50 per
acre, indicating it is not a very valuable risk management instrument in the presence of
IYCI.

The marginal willingness-tO-pay for the government deficiency payment program
is significant in the presence of IYCI, at around $25 per acre. This suggests that, in the
base model, the government program’s value to the farmer is a result Of the subsidy to
income. The program does not contribute much towards the management of income risk,
because it eliminates downside price variability which increases the farmer’s exposure
to yield risk because Of the negative price-yield correlation.

The farmer tends to buy more (or write fewer) put Options when crop insurance
is introduced into the risk management portfolio, even though the trigger yield level is
restricted to 75%. This suggests that, in the presence of IYCI, options contracts can be
used to manage price risk directly. The value to the farmer of including options in the
portfolio is generally small when IYCI and futures are already available, suggesting that

the options and futures are substitutes in the presence Of IYCI.

138
5.3 Price Instruments With Area Yield Crop Insurance

In this section, the role of area yield crop insurance (AYCI) as Opposed to IYCI
is studied as a risk management instrument. In the AYCI scheme, the indemnity index
is the county-level yield as opposed to the individual farm-level yield. AYCI is assumed
to charge an actuarially fair premium and has a maximum trigger yield of 90% Of the
expected yield. Correlation between farm- and county-level yields is assumed to be 0.83.
Each ease in this section is the same as those in section 5 .2 except that IYCI is replaced
by AYCI. The optimal market participation levels, the first four moments of the income
distribution, and the willingness-to—pay levels are all listed in Table 5.4. The results for

each portfolio are discussed in the following subsections.

AICLQDI!
In this case, only AYCI is included in the farmer’s portfolio. Equation (5.1) is

restricted by setting futures, options and government program positions to be zero
(x=z=£ =0)-

The farmer chooses the maximum trigger yield level of 90% of expected yield.
Using AYCI to manage yield risk instead of IYCI introduces yield basis risk as a result
of the imperfect correlation between farm- and the county-level yield indices used for
crop insurance. In the base model, the farm-county yield correlation is 0.83, indicating
that county-level yields are highly correlated with farm-level yields. Furthermore, some
of the cost of yield basis risk can be offset by the benefit from being allowed to select

higher trigger yield levels and only paying an actuarially fair premium.

139

Table 5.4 Optimal Positions, Income Distribution, and WTP with AYCI

 

 

 

Portfolio
Futures, Futures, Futures,
Futures Options Govt. Govt. Options,
No Risk and and and and Govt.
Instr. AYCI AYCI AYCI AYCI AYCI and AYCI
Futures 0 0 -0.095 0.010 0 0.325 0.192
Options 0 0 0 0.206 0 0 —0.266
AYCI 0 0.900 0.900 0.900 0.900 0.900 0.900
Govt. 0 0 0 0 1 1 1
13(1) 286.43 286.43 286.43 286.43 319.11 319.11 319.11
Stdv(1r) 89.20 71.44 70.32 70.19 67.35 68.44 68.42
Skew“) —0.617 -0.303 -0.377 -0.359 -0.785 -0.483 -0.507
Kurt(r) 3.324 3.476 3.534 3.598 3.920 3.840 3.730
Willingness- 0 22.38 22.47 22.52 49.23 50.07 50.13
tO‘PaY

 

Note: The participation levels for futures and options, and the selected trigger yield for
AYCI are reported as the percent of the expected yield; the income mean and
standard deviation and the willingness-tO-pay levels are reported in $.

140

Optimal mean income is $286.43 per acre which is same as when no risk
management instruments are available because AYCI is actuarially fair. The standard
deviation of income is smaller than the case of no risk management but larger than the
case when IYCI is the only risk management instrument available, even though the
trigger yield for IYCI is only 75 % as opposed to 90% for AYCI. Unlike the IYCI only
case, the income distribution is skewed to the left but more symmetric than the no risk
management case. The higher standard deviation and negative skewness under AYCI
results from yield basis risk, which is zero in the IYCI only case. The income
distribution is graphed in Figure 5.3(a). This distribution is more "spread-out" than the
distribution using IYCI only in Figure 5.2(a) but more compressed than the no risk
management case in Figure 5.1(a).

The willingness-tO-pay for AYCI alone is $22.38 per acre, $3.28 less than for
IYCI, which suggests that AYCI works slightly less effectively than IYCI by itself to
manage the income risk, despite the fact that trigger yield is higher and the premium is
actuarially fair for AYCI. On the other hand, this result also suggests that AYCI
actually works quite well in managing the farmer’s yield risk, while providing with FCIC

an inexpensive way to administer crop insurance at the same time.

W

In this case, AYCI and futures contracts are both in the farmer’s portfolio. The
restrictions on equation (5 . 1) are to set the options position and government program

participation level to zero (z=£ =0).

141

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142

The farmer selects the maximum trigger yield level of 90% of the expected yield
for AYCI and goes short futures 10% of expected yield. Similar to the case with IYCI,
the yield risk is partially managed by the AYCI, so futures contracts are then used to
manage price risk. However, because Of the restricted trigger yield level and yield basis
risk, AYCI leaves a certain amount of yield risk unprotected. Despite the higher trigger
yield level, the existence of yield basis risk exposes the farmer to additional yield risk
which reduces the short futures position below the position in the IYCI only case, in
order to reduce the exposure to the remaining yield risk which is negatively correlated
with price.

The mean of income does not change because the futures market is unbiased. The
standard deviation of income is reduced, and the distribution is skewed more to the left
than in the case without futures contracts.

The marginal willingness-tO-pay when adding futures to the portfolio is relatively
small, $0.09 per acre, which is not surprising because of the small futures position. The
small short position and the small marginal willingness-tO-pay suggeSt that the risk

management value of futures is limited when the portfolio contains AYCI.

W
In this case, AYCI, futures contracts and options contracts are included in the

farmer’s risk management portfolio. The restriction on equation (5.1) is to set

government program participation level to zero (E = 0).

143

The optimal futures position in this case is essentially zero. The farmer buys put
options equivalent to 21% of expected yield and selects the trigger yield to be the
maximum 90% of expected yield. As in the IYCI case, the options contract reduces
downside price variability and AYCI reduces downside yield variability. The Options
position is smaller than in the IYCI case, because the AYCI is not as efficient at reducing
yield risk, so there is more yield risk left unprotected which requires more price
variability to offset it.

The distribution is graphed in Figure 5.3(b). The mean of income stays the same
as in the cash only case because all risk instruments are perceived to be priced at a level
equal to their expected payoff. The standard deviation of income is smaller than the
previous two cases with AYCI, and is more symmetric than the case without options in
the portfolio.

The willingness—to-pay for including Options in the portfolio is increased by only
$0.04 per acre. The relatively large options position combined with the decrease in the
use of futures in the portfolio suggesting that Options are substitutes for futures in the
presence of AYCI. Similar to the IYCI case, the almost zero futures position and the
additional $0.04 per acre willingness-to-pay suggest that Options can manage price risk

at least as effectively as futures in the presence of AYCI.

W

In this case, the farmer participates in AYCI and the government deficiency

payment program to manage income risk. In equation (5.1), the futures and options

144

positions are restricted to zero, and the government program participation level is
restricted to be one (£=1, x=z=0).

Participating in the government program is again the farmer’s Optimal choice and
the trigger yield is again chosen at the maximum 90% of expected yield. When yield is
the major source of income risk, the farmer uses crop insurance to reduce yield risk.
When the trigger yield is restricted and yield basis risk exists, buying the maximum level
of AYCI still leaves a certain amount of yield risk unprotected. The government
program reduces the downside price variability but exposes the farmer to more yield risk
that is left after using AYCI. The government program may increase income risk in this
case. However, the implicit subsidy in the government program still makes it attractive
to the farmer.

The mean of income is increased by about $33 per acre, which is the expected
level of subsidy from participating in the government program. The standard deviation
of income is reduced, but the negative skewness is larger in magnitude than without the
program. The histogram of the income distribution is shown in Figure 5.3(c).

The willingness-tO-pay is increased by $27 per acre when the government program
is added to the portfolio, which is below the expected subsidy value, indicating the
government program is not very effective at managing risk in the base model but
provides benefit to the farmer through the implicit subsidy.

An experiment where the "actuarially fair" government program is replaced by
a subsidized government program by subtracting the expected subsidy, $33 per acre.

The resulting income distribution is the same except the mean is $33 lower. Willingness-

145
to—pay for this portfolio, unlike the case with IYCI, is only $20.08 per acre which is

lower than the value of AYCI alone, suggesting the government program actually

increases income risk when AYCI is used.

W

In this ease, the farmer’s portfolio includes the government deficiency payment
program, AYCI, and futures. Restrictions on equation (5. 1) are to set the options
position to zero and the government program position to one (z=0, £=I).

Participating in the government program is optimal and the maximum trigger yield
of 90% of expected yield is chosen. The farmer now chooses to go long futures 33%
of expected yield. As in the IYCI case, participation in the government program exposes
the farmer to additional yield risk when the price and yield are negatively correlated,
yield risk is not fully eliminated by AYCI, and the price variability is reduced. Futures
are then used to cross hedge this additional yield risk. The long futures position is larger
than in the IYCI case but smaller than the no crop insurance case, implying that AYCI
can manage yield risk, but not as effectively as IYCI in the base model.

The mean of income stays the same when futures are added to the portfolio,
however, the standard deviation of income is larger and the negative skewness is smaller
than that in the previous cases without futures. Compared to IYCI, the income
distribution in this case has a larger standard deviation and a larger negative skewness.

The marginal willingness-to-pay to include futures in the portfolio is $0.85 per

acre. Though still small, it is larger than the value when including futures without the

146

government program. The role of futures contracts to manage income risk with AYCI
is more important when the government program is available. In this case, the value Of
using futures to cross hedge the additional yield risk exposure caused by the government
program is greater than the value of using futures to hedge price risk in the absence Of

the government program.

1311mm, Options, Government Program @d AYCI

In this case, the government deficiency payment program, crop insurance, futures
contracts and options contracts are all allowed in the farmer’s risk management portfolio.
The restriction on equation (5.1) is tO set the position in the government program to one
(5 = 1)-

It is again optimal for the farmer to participate in the government program. The
optimal portfolio is to go long futures 19% of expected yield, write put options at 27%
of expected yield, and select the trigger yield at its maximum level of 90%. Futures and
options combine to cross hedge the increased residual yield risk exposure due to the
reduced downside price variability resulting from participation in the government
program. Compared to the case with IYCI instead Of AYCI, the futures position is at
a similar level while more put Options are written. This suggests that AYCI is not as
effective as IYCI at managing yield risk, so that more put Options are written to cross
hedge the remaining yield risk.

The mean of income stays the same as when futures, government program and

AYCI are in the portfolio because the options are priced at their expected payout. Both

147

the standard deviation and skewness of income are slightly decreased. The income
distribution histogram is shown in Figure 5.3(d), and is quite similar to the case with
government program and AYCI (no futures or options).

The marginal willingness-to-pay to include options in the portfolio is only $0.05
per acre, which, although small, is above the value in the IYCI case. When options are
added into the portfolio, the farmer takes a relatively large options position and reduces
the futures position, and the resulting willing-tO-pay increases very little. This suggests
that futures and options are acting as substitutes.

Willingness-to—pay for AYCI is close to the IYCI case, which suggests that when
all the price instruments are available, AYCI provides almost as same welfare gain as
IYCI at managing income risk given the specific IYCI premium loading and the current

trigger yield restrictions.

Summary
The farmer always selects the AYCI trigger yield at the maximum 90% of

expected yield. Even though the trigger yield level is higher and the AYCI premium is
actuarially fair, willingness-to—pay for AYCI is still lower than the corresponding IYCI
cases, suggesting that AYCI is generally not as effective in reducing the farmer’s income
risk as individual yield crop insurance, given the level of basis risk in the base model.

Mean income is consistently larger than in the corresponding cases with IYCI and
equal to mean income without any crop insurance, because the insurance participant pays

an actuarially fair premium for AYCI as Opposed to a premium set 35% above the

148
actuarially fair level for IYCI. However, the willingness-to-pay for AYCI is smaller

than for IYCI, with the magnitude of the difference ranging from $1.65 to $3.86 per
acre. This suggests that AYCI can be a reasonable substitute for IYCI. When all the
price instruments are included in the portfolio, the difference between willingness-to—pay
using AYCI and IYCI is only about 3%.

Because of yield basis risk, AYCI leaves more yield risk unprotected than the
IYCI, despite the farmer’s ability to choose a higher trigger yield level using AYCI. As
a result, futures and Options are used less to manage price risk directly and more to
manage yield risk through cross hedging when AYCI is used in the portfolio rather than
IYCI. This also results in smaller willingness-to-pay values for futures and options in
cases where they are used to hedge price risk as opposed to cross hedging yield risk.
Regardless, the willingness-to-pay value for futures and options are negligible in
comparison to the willingness-to-pay values for crop insurance.

Again, the willingness-to—pay for participating in the government deficiency
payment program is significant but always smaller than the expected subsidy from
participation. This suggests the government program’s value to the farmer is the income
subsidy and that, with negative price-yield correlation, participation actually exposes the
farmer to additional income risk because of the additional exposure to yield risk.

When participating in the government program, the farmer goes long futures,
writes put options, and sets the AYCI trigger yield to the maximum level. AYCI
provides similar risk protection as IYCI, but IYCI is generally more effective in the base

model.

149
5.4 Conclusiom

In the base model, yield risk is a more important component of income risk than
price risk. In addition, prices and yields are negatively correlated which provides a built-
in form of income risk management.

In the absence of crop insurance, futures and Options are generally used to cross
hedge yield risk rather than hedge price risk, in contrast to the conventional hedging
results. This result rests on a strong negative price-yield correlation. However, the
willingness-to-pay for futures and Options are relatively low, under $6 per acre, and the
resulting income distributions are similar to the no risk management case, implying that
futures and Options are not effective income risk management instruments in the absence
of crop insurance.

When IYCI is included in the portfolio, the income risk is significantly reduced.
The farmer always selects the maximum trigger yield level. The willingness-to—pay for
IYCI is significant (approximately $26 per acre), despite the fact that the trigger yield
is restricted and the premium is 35 % above the actuarially fair level. The optimal trigger
yield level and willingness-to-pay for IYCI suggest it is an effective instrument for
managing income risk. When IYCI is in the portfolio, futures and optiOns are used to
manage price risk in the absence Of the government program; and they are used to cross
hedge the additional yield risk exposure caused by the government program when the
program is in the portfolio. However, the additional willingness-to-pay for including

futures and options in the portfolio is less than $1 per acre.

150
When AYCI is included in the portfolio, income risk is reduced in a manner

similar to when IYCI is included. The maximum trigger yield level is always chosen by
the farmer. The willingness-to—pay for AYCI is much larger than the price instruments
at about $22 per acre. However, as result Of the yield basis risk, willingness-to—pay for
AYCI is slightly below the value of IYCI, even though the AYCI premium is actuarially
fair and its maximum trigger yield is higher than IYCI. This indicates AYCI is also an
effective risk management instrument, but somewhat less effective than IYCI. When
AYCI is included in the portfolio, futures and options are again used to hedge price risk
in the absence of the government program, but their positions are generally smaller than
in the ease of IYCI, because more yield risk is left unprotected. Once again, the value
of futures and options is quite small relative to the value Of AYCI at less than $1 per
acre.

Participating in the government deficiency payment program can eliminate
downside price variability, but it also increases the farmer’s income risk through
additional exposure to yield risk given the negative price-yield correlation and the
relatively large yield risk. The risk management impact of the government program
depends on the level of unprotected yield risk in the sense that if a lot of yield risk is left
unprotected, the elimination Of the downside price variability through program
participation actually increases income risk. However, when there is little yield risk left
unprotected, the program can reduce income risk through reducing price risk.

The government program provides a high level of expected subsidy through the

deficiency payment, and the willingness-to-pay to include it in the portfolio is nearly as

151

high as the expected subsidy value when at least one other risk management instrument
is present to help manage the income risk. This implies that the risk management value
of the government program is relatively small. Indeed, when the government program
is the only risk management instrument in the portfolio, its value is near zero despite the
subsidy. This is because of the increased yield risk exposure from the elimination of the
negatively correlated price movements.

When all instruments are included in the portfolio, the farmer’s optimal choice
is to participate in the government program, take a small long futures position, write a
small number of put Options, and choose the IYCI trigger yield at the maximum level.

These results are based on the particular set of parameters specified for the base
model. Some of the parameters are very specific to the representative farmer,
geographical region, and crop year, such as the farmer’s risk preference, yield basis risk
level, relative variability of price and yield, and price-yield correlation. These
parameters may differ across farmers, regions and time, and the results may change
accordingly. In the next chapter, a comparative static analysis is conducted to investigate

results when the parameters deviate from their base values.

Chapter VI
ECONOMIC IMPACTS OF ALTERNATIVE RISK STRUCTURES AND

PREFERENCES

The use of different risk management instruments and their welfare effects were
studied in Chapter V for a farmer in southwest Iowa, whose farm-level yield variance,
yield basis risk, and price-yield correlation were representative of medium levels in that
region. In addition, the farmer’s risk preference is represented by a constant relative risk
averse (CRRA) utility with R = 2. In this chapter, the implication of alternative risk
structures and risk preferences on the use of alternative risk management instruments are
explored along with the resulting welfare effects.

The use and value of the risk management instruments may differ significantly as
a result of changes in risk preferences, and/or the joint distribution Of prices and yields
faced by the farmer. The alternative parameter values studied in this chapter are listed
in Table 6.1. I

Price-yield correlation can have a significant impact on income risk and, as a
result, affect the use and value Of alternative risk management instruments. Based on
historical data for southwest Iowa, the price-yield correlation for the representative farm
is set at -0.46. However, the price-yield correlation for other com growing areas may

be much smaller than southwest Iowa which is located in the heart of the corn belt where

152

15 3
Table 6.1 Risk Structure and Risk Preference Parameters

 

 

Base Alternative
Parameter Notation Value Values
Price-yield correlation pp, -0.46 0
Relative risk aversion RR 2 1, 3
Farm/county yield standard deviation R 1.20 1.40, 1.00
Farm-county yield correlation p“: 0.83 0.94, 0.63

 

production levels can significantly impact price levels. In order to study the impacts of
lower price-yield correlations on the use and value of the risk instruments, the model is
evaluated with price-yield correlation set at zero.

Risk preferences for the representative farm exhibit CRRA. In the base model,
CRRA=2 and the utility function is of the form u(1r) = -1r". The level of risk aversion
may impact the use of the risk management instruments as well as the associated welfare
effects. To explore the impacts of the level of risk aversion on the use and value of the
risk management instruments, the base model is evaluated for different levels of risk
aversion. A more risk averse farmer’s utility function is represented by setting RR= 3,
so that u(1r) = -0.51r'2; and a less risk averse farmer’s utility function is represented by
setting RR = I, so that u(1r) = 111(1).

The volatility of yields may differ greatly across farms even in close geographic
proximity. The standard deviation of yields for the representative farm is set at 1.2 times

the county yield standard deviation. In the southwest Iowa study area, high risk farms

154

have a standard deviations about 1.4 times the county level; and low risk farms exhibit
yield standard deviations about the same level as the county yield. To examine the
impacts of the level of yield volatility on the performance of the risk instruments, the
individual yield standard deviation is set at 1.0 and 1.4 times the county yield standard
deviation and the changes in the optimal use and welfare of the instruments are
examined.

Finally, the correlation between farm- and county-level yields (yield basis risk)
affects decisions when AYCI is included in the portfolio instead of IYCI. To explore the
effects of yield basis risk on the performance of the instruments, the farm-county yield
correlation is decreased to 0.63 (an increase in yield basis risk) and increased to 0.94 (a
decrease in yield basis risk), and the change in the use Of the instruments and farmer
welfare are examined. These levels reflect the range of yield basis risk for farms located
in the study area in southwest Iowa.

The remainder of this chapter is divided into four sections. In the first section,
we study the impact of the price-yield correlation level. Next, we examine the impact
of changes in the risk preferences. In the third section, the impact of different yield risk
structures, including both farm-level yield volatility and yield basis risk are examined.

The chapter concludes with a section summarizing the analysis.

6.1 Price-Yield Correlation
Price-yield correlation plays a key role in determining the use Of risk management

instruments and farmer welfare. The analysis in Chapter V assumed cash price and

155

county yield were negatively correlated. To determine the impacts of the price-yield
correlation on participation levels and farmer welfare, the analysis is repeated here for
the case of zero price-yield correlation.

The joint price-yield distributions are generated using the procedure outlined in
Chapter IV with the same parameter values, except that the correlation imposed between
the eash price and the county yield is set to zero. The marginal distributions of each
random variable remain unchanged. The sample correlations for the simulated joint
distribution are shown in appendix D. Although the price-yield correlation is specified
to be zero, simulation error results in a very small level Of positive correlation.

The Optimal market positions and willingness-to-pay are reported in Table 6.2.
The zero price-yield correlation results in some significant changes compared to the
negative price-yield correlation case. As in the negative correlation case, the farmer
continues to choose the maximum crop insurance trigger yield and always participates
in the government program when available. However, with no price-yield correlation,
the farmer generally takes a significant short position in the futures market because
futures are no longer used to cross hedge yield risk and are used to hedge price risk
directly. The options position is highly sensitive to the inclusion of crop insurance and
government program in the portfolio. Adding crop insurance causes the farmer to switch
from selling to buying put Options; while adding the government program causes the
farmer to once again sell puts. This is as expected because when yield risk is managed
by crop insurance, put options are bought to manage the price risk; and the government

program is a substitute of puts, so the use of which is accompanied by selling puts.

156

Table 6.2 Optimal Positions and WTP with Zero Price-Yield Correlation

 

 

 

Portfolio
Futures Futures Futures,

No Price and and Options

Instrument Futures Options Govt. Govt. and Govt.
Without Crop Insurance
Futures 0 -0.086 -0. 893 0 —0. 129 —0.585
Options 0 0 -0.796 0 0 -1.019
Govt. 0 0 0 l 1 1
E(rr) 292.24 292.24 292.24 324.93 324.93 324.93
Stdv(1r) 101.98 103.65 96.11 93.07 92.05 92.97
Willingness-to-pay 0.00 0.13 7.34 33.76 33.96 36.21
With IYCI
Futures 0 -0.658 -0.577 0 -0.194 -0.366
Options 0 0 0. 169 0 0 0350
Govt. 0 0 0 1 l l
IYCI 0.75 0.75 0.75 0.75 0.75 0.75
E(t) 286.96 286.96 286.96 319.64 319.64 319.64
Stdv(1r) 78.59 68. 82 68.76 67.68 66. 12 65.92
Willingness-to—pay 28.96 31.75 31.77 57.58 57. 80 57.87
With AYCI
Futures 0 -0.684 -0.520 0 -0.230 -0.325
Options 0 0 0.316 0 0 0183
Govt. 0 0 0 l 1 l
AYCI 0.90 0.90 0.90 0.90 0.90 0.90
E(rr) 292.24 292.24 292.24 324.93 324.93 324.93
Stdv(r) 82.25 72.04 72.08 71.10 69.18 69.02
Willingness-to-pay 24.14 29.91 30.00 57.11 57.50 57.53

 

Note: The participation levels for futures and options and the trigger yield level of crop
insurance are all reported as a percent Of the expected yield; and income mean,
standard deviation, and willingness-tO-pay levels are reported in dollars.

157

When both price and yield management instruments are included in the portfolio,
the price-yield correlation has less effect on the market positions than the case when
pricing instruments or crop insurance are used in isolation. The willingness-to-pay for
risk management instruments is generally higher with zero price-yield correlation than
with negative correlation. With zero price-yield correlation, the cash market income is
more volatile with a standard deviation of $101.98 in contrast to $89.20 in the negative
price-yield correlation case.

There are two cases where the willingness-tO-pay for alternative instruments are
signifieantly different when the price-yield correlation is zero compared to the base case.
First, when futures are the only risk management instrument in the portfolio, willingness-
to—pay is less than in the negative correlation case. The reason is that yield is more
volatile than price, so that the risk management value of using futures, to cross hedge
yield risk in the negative price-yield correlation case is higher than the value from
hedging price risk when there is no price-yield correlation.

Second, when the government program is the only instrument in the portfolio, its
value is higher than the expected subsidy when there is no price-yield correlation,
because it now reduces risk as well as providing a subsidy to the farmer. In the negative
price-yield correlation case, the value Of the program is almost zero because its subsidy
value is offset by the increased income variability which results from the elimination Of
price variability by the program and increasing yield risk exposure.

The optimal portfolio with no price-yield correlation is to sell futures and put

options at 37% and 35% of expected yield, participate in the government program, and

158
use IYCI with a trigger yield at the maximum 75% of expected yield level. The

willingness-to-pay measure is $57.87 per acre.

6.2 R'Bk Preferences

The use of futures, Options, or AYCI impacts the risk associated with the income
distribution while the government program and IYCI impact both mean income and the
risk associated with the income distribution. Farmers with different risk preferences will
value mean income and income variability with different weights. As a result, farmer
risk preferences impact both the use of risk instruments and the resulting welfare for
every portfolio.

The Optimal futures and Options positions, the Optimal trigger yields for crop
insurance, and the willingness-tO-pay for farmers with alternative risk preferences are
reported in Tables 6.3, 6.4, and 6.5, respectively. Futures and Options are not reported
separately in order to simplify the presentation of results. For convenience, the base
values when R is 2 are again provided in the tables.

The direction of change in futures and options positions depends on the portfolio.
The magnitudes of the changes are large in a number Of instances, which suggests the
level of participation in futures and Options can be very sensitive to risk attitudes.

A ceteris paribus decrease in RR tends to make the IYCI trigger levels go below
the maximum level, while the AYCI trigger yield remains at the maximum level. The
reduction in risk aversion reduces the benefits Of risk reduction enough that the premium

loading in IYCI causes the farmer to select a trigger level below the 75% level.

159

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162

Beeause there is no premium loading in AYCI, it is still optimal to select a trigger yield
at the maximum level even though the value of the risk reduction is lower.

The government program provides a subsidy value to the farmer but, as discussed
before, actually increases income variability in the base model where price and yield are
negatively correlated. As a result, when the farmer becomes more risk averse and no
other instrument is available to manage the income risk, the farmer no longer participates
in the government program because the cost of the increased variability exceeds the
subsidy benefit. However, the more risk averse farmer does participate in the
government program when the price-yield correlation is zero or when at least one other
instrument is in the portfolio.

Willingness-to-pay generally decreases as the level of risk aversion decreases,
because the benefits of risk reduction decline as risk aversion falls. Once again,
exceptions occur when the government program is included in the portfolio without any
crop insurance. In this case, willingness-to-pay actually increases (decreases) as risk
aversion decreases (increases). This is because participation in the program, in the
absence of crop insurance, results in an increase in income variability to the farmer and
the decrease (increase) in risk aversion reduces (increases) the cost of the additional risk
the farmer must bear to capture the subsidy. The change in willingness-to-pay ranges
from $0. 16 to $28.53 per acre as the relative risk aversion level increases or decreases
by l.

The optimal risk management strategy for farmers with high and medium levels

of risk aversion is to participate in futures, options, the government program and IYCI.

163

Farmers with low level of risk aversion substitute AYCI in the portfolio for IYCI.
Although IYCI provides better yield risk management effect than AYCI, it imposes an
actuarially unfair premium loading. The low risk averse farmers care more about mean

income and less about income risk less which makes AYCI more valuable.

6.3 Yield Risk

Even in the same geographic region, farmers may have different individual yield
distributions due to different soil characteristics, management practices, and weather
related factors, such as rainfall, hail, or flood. A farmer’s individual yield distribution
may impact the use of risk instruments and their resulting welfare.

Farm-level yield risk, represented by the standard deviation of farm-level yield,
and yield basis risk, represented by the correlation between farm- and county—level
yields, represent two important characteristics of the farmer’s yield distribution. Changes
in the farm yield risk will affect income risk and so the use and value of crop insurance
may be impacted. In addition, changes in farm yield risk will affect the relative
variability of price and yield, which may affect the use and value of price instruments
when the price and yield are negatively correlated.

Change in yield basis risk can affect the use of risk management instruments in
several ways. As the correlation between farm and county yields decreases (increases),
AYCI is less (more) effective at controlling yield risk, which can affect the use and value
of AYCI as well as the pricing instruments in the portfolio. In addition, as the

correlation between farm and county yields declines (increases), the correlation between

164

price and farm yield will decline (increase) in the negative price-yield correlation case,
which may affect the optimal use of the instruments and the resulting welfare.

The impacts of alternative yield distributions on the futures and options positions,
the crop insurance trigger yield selection, and the willingness-to-pay are reported in

Tables 6.6, 6.7, and 6.8 respectively.

E _] I]? ”3.1

In the case of negative price-yield correlation, the farmer continues to select the
maximum trigger yield level for different levels of yield variability. The futures and
options positions change as the yield variability increases, but the direction of the change
depends on the risk instruments included in the portfolio. The magnitudes of the changes
are significant in some portfolios suggesting participation levels in futures and options
can be highly sensitive to the level of yield variability.

Willingness-to—pay increases with yield variability when crop insurance is included
in the portfolio because the crop insurance becomes more valuable. However,
willingness-to-pay decreases as yield variability increases when crop insurance is not
included in the portfolio, because the pricing instruments become less effective in
managing the increased yield risk.

When yield variability is low, IYCI is replaced in the optimal portfolio by AYCI
so that the optimal portfolio consists of AYCI, futures, options and the government

program. When the yield is not very risky, the farmer is not willing to' pay the

165

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168
actuarially unfair premium in order to use IYCI when he can use AYCI paying only the

actuarially fair premium.

The change in positions and willingness-to-pay in the zero price-yield correlation
case is similar to the negative correlation case. One exception is that the futures position
consistently increases as the yield variability decreases for all portfolios; because without
the negative correlation, futures are used to hedge price risk directly and the increase in

price variability relative to yield variability increases the demand for futures.

Kim '3']

The farmer chooses the maximum IYCI trigger yield regardless of the level of
yield basis risk. The AYCI trigger yield is also generally at the maximum level, except
for a few cases where the farm-county yield correlation is low and the government
program is not included in the portfolio.

The change in optimal futures and options positions as the yield basis changes
depends on the portfolio. Changes in futures and options positions in portfolios that
don’t include AYCI are the result of changes in the correlation between price and farm
yield. When AYCI is included in the portfolio, the resulting changes occur because of
the combined impact of changes in price-yield correlation and the farm-county yield
correlation. Not surprisingly, the changes in the AYCI portfolio tend to be of larger
magnitude than in portfolios with IYCI.

The willingness-to-pay measures with IYCI in the portfolio changes insignificantly

with the basis risk. When AYCI is included in the portfolio, the willingness-to-pay

169

measure declines (increases) as the yield basis increases (decreases), and the magnitudes
of the changes are significant relative to the changes in the IYCI portfolios. In the AYCI
portfolios, the higher farm-county yield correlation improves the ability of AYCI to
manage yield risk which significantly increases the value of the risk management
instrument. As a result, when the basis risk is low, portfolios containing AYCI
outperform those with IYCI, and the optimal portfolio for a low basis risk farmer
includes AYCI, futures, options and the government program.

In case of zero price-yield correlation, the changes in positions and willingness-to-
pay are generally similar to the negative correlation case. However, there is little change
in the value of the pricing instruments when crop insurance is not in the portfolio. When
the price-yield correlation is zero, changes in the farm-county yield correlation have little
impact on the correlation between price and farm yield, therefore on the pricing

instruments’ ability to manage risk.

6.4 Summary and Conclusion

Two potentially important factors that affect the optimal use of risk management
instruments and the resulting farmer welfare are: (l) the relative variability of price and
yield because income risk is a function of the product of price and yield and the relative
importance of reducing price and yield risks depend on the relative level of the two types
of risks; and (2) the correlation between price and yield, which impacts the level of
income risks and also may provide cross-hedging opportunities for price and yield

instruments.

170

When the price-yield correlation is zero, the optimal futures pOsition is to go
short, in contrast to the long position in the negative correlation case. When both price
and yield instruments are included in the portfolio, the optimal positions are less affected
by the price-yield correlation, but the welfare level is higher for the zero correlation case
because the cash market income is more volatile, allowing larger gains from reducing
income risk.

Changes in risk preferences affect the relative importance of mean income and
income risk in determining farmer welfare. Because IYCI has an actuarially unfair
premium loading, and the government program has an implicit subsidy, portfolios
including IYCI and/or the government program are more sensitive to a change in risk
preferences.

Increases in yield variability increase the value associated with both types of crop
insurance and decrease the value of pricing instruments. Changes in yield basis risk
affect the cross-hedging ability of the instruments as a result of the change in correlation
between cash price and farm—level yield whenever cash price and county yield are
correlated. However, this impact is small. An increase in yield basis risk significantly
decreases the value of AYCI regardless of the price-yield correlation. These results are
consistent with Miranda’s study.

IYCI generally provides better risk protection to farmers than AYCI because yield
basis risk reduces AYCI’s effectiveness in managing yield risk. However, in cases when
l) the farmer has a low level risk aversion, 2) yield variability is low, and/or 3) yield

basis risk is low, AYCI enters the optimal portfolio instead of IYCI. The reason is the

171

actuarially unfair premium loading imposes cost to the farmer to use IYCI, and when
AYCI can provide similar risk management gains or the importance of yield risk declines

enough, the farmer prefers AYCI.

Chapter VII

ECONOMIC EVALUATION OF CROP INSURANCE CONTRACT DESIGN

7.1 Introduction

It is widely recognized that farmers are subject to high income risk in the

production of many agricultural commodities, and a major component of the income risk
originates with crop yields. Futures and options can be used to manage price risk, and
the government deficiency payment program can also manage price risk as well as
subsidize farmer income. A more urgent need facing farmers is the lack of a well
functioning instrument to manage yield risk.
Crop insurance has the potential to help farmers manage yield risk. However, the
current crop insurance program established in 1980, has never been financially sound.
Many operational and structural modifications have been discussed in an effort to
improve the program. Examples include differentiating high risk farmers from low risk
farmers, modifying reinsurance mechanisms, and altering contract designs.

The objective of this chapter is to study the effectiveness of the two currently
available yield index designs for crop insurance: IYCI and AYCI. The performance of
each yield index will be evaluated for a number of alternative contract design
specifications. The tradeoff between the premium loadings inherent to the IYCI index

and the yield basis risk associated with the AYCI index will also be evaluated.

172

173
n n Pr

Beeause of the level of yield volatility and the significant probability of very small
yield realizations, crop insurance may play an important role in reducing income risk and
increasing farmer welfare. The analysis reported in the previous two chapters shows the
value of crop insurance is significantly higher than futures and options for some farmers.

Crop insurance has been provided since the 1930s, and has been widely used
across the US for most major crops since 1980 in the form of MPCI. However, for a
number of reasons, including problems associated with moral hazard and adverse
selection, the FCIC has accumulated a deficit of about $3.3 billion from 1980 to 1994,
in addition to direct government subsidization of the program. From 1981 through 1994,
the ratio of indemnifications paid to premiums collected was 1.4. Congress has required
the program to reduce its projected indemnification/premium ratio to 1.1, so that at least
91% of anticipated claims can be covered by premiums collected, the so called ”91-
percent adequacy” requirement. The urgency of finding more efficient ways to deliver
economically feasible crop insurance programs has increased in recent years.

The indemnity payment of IYCI, a stylized form of MPCI, is determined by each
individual farmer’s yield. Because each farmer can manipulate his yield distribution
through management decisions after purchasing crop insurance (moral hazard), and
because each individual has better information about the yield distribution than the
insurance companies (adverse selection), IYCI is difficult to implement. Besides moral
hazard and adverse selection, the indemnity for IYCI is different for each individual

farmer based on his realized yield, which increases operating costs. In addition,

174

weather-related, and in some cases insect- and disease-related hazards ean cause
signifieant yield reduction over large geographically contiguous areas, thereby increasing
the chance that a substantial number of farmers in specific regions claim large
indemnifieation payments in the same year. This geographic correlation between
indemnity payouts can subject crop insurance agencies to catastrophic risk. Moral
hazard, adverse selection, high operating cost, and catastrophic risk have each
contributed to the financial difficulties surround the current form of IYCI.

New alternative forms of crop insurance have been proposed by economists and
politicians in an attempt to improve the financial situation of FCIC. The Group Risk
Plan (GRP), a form of AYCI, is one alternative design which has been developed and
is currently being tested in the field.

AYCI has the potential to reduce moral hazard and adverse selection problems
that have hindered IYCI because indemnities do not depend on the farmer’s yield.
Rather, indemnities depend on area yield which the farmer has little ability to influence.
Furthermore, the farmer will have little, if any informational advantage over the
insurance provider regarding area yields which reduces the adverse selection problem.
In addition, it is easier to administer a program indexed by an area yield than a program
which uses a separate index for each individual farm, which should result in lower
transaction costs for AYCI.

A disadvantage of the AYCI index is that the indemnity payment is based on an

area yield index which may not be highly correlated with the farmer’s yield. Using an

175

area yield index makes it possible for a farmer to have a large yield shortfall and receive
no indemnity payment if there is no short fall in the area yield index.

The risk management potential and cost of any form of crop insurance depends
on design parameters of the contract, the income distribution faced by the farmer, and
the availability of other risk management instruments. Next we discuss the major design

features that characterize the currently available crop insurance instruments.

W

The design features of current forms of crop insurance contracts have evolved
over time. Various features, such as the maximum trigger yield level, have resulted
from difficulties in administering a financially feasible program that, at the same time,
offers an attractive risk management instrument to farmers.

One important contract parameter is the yield index used to calculate the
indemnity payment. The individual yield index (IYCI) and area yield index (AYCI) offer
two alternative options which have significantly different implications for performance,
cost, and administration of the insurance contracts. The relative performance of the two
indices also depends on a number of key design features.

Because the indemnification payout is based directly on the farmers yield, IYCI
has the potential to provide a very efficient mechanism to manage yield risk.
Unfortunately, moral hazard, adverse selection, catastrophic risk, and administrative
difficulties have resulted in operating costs that exceed actuarially fair premium levels

by a significant amount for some farmers. To counteract these difficulties, maximum

176

trigger yield restrictions and/or increased premium rates have been instituted. Restricting
the maximum trigger yield level (currently 75% of expected yield in IYCI) can reduce
the moral hazard incentive thus limiting potential losses to insurance providers. At high
trigger yield levels, the farmer has a high probability of making a claim so he may make
inferior management decisions which lower expected yields and/or increase yield risks.
However, at low trigger yield levels there is a lower probability of claiming the
indemnification and the farmer is less likely to alter management decisions. The
disadvantage of restricting the trigger yield level is that the risk management capability
of the insurance instrument is reduced, which may lead to reduced farmer welfare.

In the area yield plan, GRP restricts the maximum trigger yield to 90% of
expected yield. However, moral hazard and adverse selection are not likely to be
significant problems for AYCI because the indemnity payment is unlikely to be affected
by individual farmers’ management decisions, and individual farmers will generally have
little informational advantage regarding the distribution of area yields. Therefore, the
reasons for placing restrictions on trigger yield levels are less clear for an insurance
contract that uses an area yield index.

In contrast to IYCI, low levels of moral hazard, adverse selection, and
administrative cost should allow AYCI to be offered with few, if any, trigger yield
restrictions at premium rates that are close to being actuarially fair. The disadvantage
to farmers, relative to IYCI, is that the area yield index introduces yield basis risk, i.e. ,
less than perfect correlation between farm yield and the area yield index. The yield basis

risk associated with AYCI reduces the effectiveness of the insurance contract in managing

177

individual yield risk. The economic implications of these tradeoffs is an empirical issue
we will address shortly.

The current farms of AYCI and IYCI contracts restiict the amOunt of acreage
insured in the program to a farmer’s base acreage, which corresponds roughly to 100%
of planted acres. An alternative crop insurance contract design would allow farmers to
select the number of acres covered by crop insurance. Under this design, a farmer may
select to insure a number of acres greater or less than the level of planted acreage.
Choosing the number of acres to insure is analogous to selecting the number of bushels
to buy or sell in the futures and options markets.

Once again, moral hazard and adverse selection difficulties may make it
impractical to allow farmers to choose IYCI acreage coverage levels greater than planted
acreage. However, as discussed above, these issues are less of a concern in AYCI and
allowing a flexible level of acreage coverage is a feasible design alternative which may
increase the‘ability to manage risk.

The remainder of the chapter evaluates the participation incentives and welfare
levels for IYCI and AYCI designs. The effects of trigger yield restrictions and acreage
coverage restrictions are evaluated for different levels of price—yield correlation and
different risk management portfolios. The next section explores the participation and
welfare effects of IYCI and AYCI in the absence of price instruments under alternative
coverage restrictions and yield basis risk. Changes in the crop insurance participation
and welfare levels as a result of including futures and options contracts in the portfolio

are evaluated in section 7.3. Section 7.4 then considers the implicatiOns of adding a

178

government deficiency payment program to the portfolio containing crop insurance. A
number of remaining design issues are discussed in section 7.5; and finally, section 7 .6

concludes the chapter by highlighting the major policy implications.

7.2 Crop Insurance

The risk management performance of IYCI and AYCI depends on the
specification of contract designs, such as coverage restrictions, and the characteristics of
the farmer’s yield distribution, such as yield basis risk. In this section, the impacts of
coverage restrictions and yield basis risk on the relative performance of IYCI and AYCI

are studied.

Current crop insurance contract designs specify the maximum trigger yield to be
75 % of expected yield for IYCI and 90% of expected yield for AYCI. These restrictions
may impact each instrument’s ability to manage risk, as well as farmer welfare, because
they limit the portion of the farmer’s yield distribution that can be protected by the
insurance instrument.

Under the current contract designs, the insured acreage is restricted to be roughly
100% of the planted acreage. Eliminating the 100% restriction and allowing farmers to
choose the acreage coverage at any level may also improve contract performance and

farmer welfare.

179

The trigger yield and acreage coverage restrictions in the current contract designs
place an opportunity cost on the farmer relative to the unrestricted case. In addition,
the relative performance, in terms of welfare effects, may differ significantly under the
restrictions. The opportunity costs of trigger yield levels and acreage coverage
restrictions in terms of farmer welfare are evaluated in this section.

The optimal trigger yield and willingness-to-pay measures are shown in the top
half of Table 7.1 for a variety of trigger yield restrictions ranging from 75 % of expected
yield to the maximum possible yield level. The maximum trigger yields in the current
IYCI and AYCI program are 75 % and 90% of expected yield respectively. The
maximum possible yield levels from the simulated yield distributions are 143% of the
expected yield for IYCI and 134% of the expected yield for AYCI. Crop insurance is
the only risk management instrument held in the portfolio, the premium for IYCI is 35 %
above the actuarially fair premium and the AYCI premium is actuarially fair, and the
insured acreage is restricted to 100% of planted acreage.

If the trigger yield is selected at the maximum yield, yield risk is completely
eliminated by IYCI. When the premium cost is set at the actuarially fair level or higher,
the farmer would not choose any trigger yield above the maximum yield level, because
further increasing the trigger yield will result in increasing the indemnity by a constant,
while the corresponding premium will increase by an equivalent, or larger, constant.
Therefore, setting the maximum trigger level at the maximum yield essentially places no

restriction on the trigger yield. A similar argument holds for AYCI.

180

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181
With negative price-yield correlation, the optimum IYCI trigger yield is 86.8%

of expected yield, which leaves a large portion of yield variability left unprotected. This
is partly a result of the premium loading. If the trigger yield is selected at a level higher
than 86.8% of expected yield, the marginal increase in expected utility from reducing
income risk will be lower than the marginal decrease in expected utility from lowering
the income level as a result of the higher premium cost. In this case, the trigger yield
only becomes a constraint when it set below 86.8%. Trigger yield restrictions set above
86.8% of expected yield impose no opportunity cost to this farmer. The current 75%
restriction imposes an opportunity cost of only $0.71 per acre.

The optimum AYCI trigger yield is 121.6% of expected yield which is close to
the maximum level. The AYCI premium is actuarially fair, so the farmer can use the
instrument to eliminate risk without any reduction in expected income. The optimal
trigger yield still remains lower than the maximum yield because, with the negative
price-yield correlation, income risk is actually reduced by leaving a portion of yield
variability. Any trigger yield restriction below 121% of expected yield imposes an
opportunity cost to the farmer. The current 90% trigger yield restriction imposes an
opportunity cost of $4.36 per acre.

Under the current trigger yield restrictions IYCI is the preferred instrument,
providing a willingness-to-pay measure of $3.28 per acre more than AYCI. In contrast,
when the trigger yield is not restricted, AYCI is the preferred instrument with a

wfllingness-to-pay measure $0.37 per acre higher than IYCI. This suggests that if the

182

trigger yield restrictions are removed, or at least increased, the AYCI can be an
acceptable or even preferable alternative to IYCI.

As discussed earlier, eliminating the trigger yield restriction from IYCI may be
difficult given the transaction costs inherent to an individual yield index. However,
eliminating the trigger yield restriction in a contract that uses an area yield index may
be feasible. When the IYCI trigger yield restriction is left at the 75 % level, but the
AYCI trigger yield is left unrestricted, AYCI is the preferred instrument and provides
a willingness-to-pay measure $1.08 per acre above IYCI.

With no price-yield correlation, the optimum trigger yield increases to 92% of
expected yield for IYCI and to the maximum yield for AYCI. The willingness-to-pay
measures are higher than with negative price-yield correlation because the cash market
income is more volatile, increasing the value of the crop insurance instruments.

With no price-yield correlation, the willingness-to-pay measure for AYCI with no
trigger yield restriction is $3.02 per acre higher than IYCI, in contrast to the negative
price-yield correlation case where it was only $0.37 per acre higher. Under the current
trigger yield restrictions, the willingness-to-pay for IYCI is $4.82 per acre above AYCI.
In both the zero and negative price-yield correlation cases, removing the trigger yield
restrictions for AYCI results in higher farmer welfare than the actuarially unfair IYCI
with or without trigger yield restrictions.

The optimal insured acreage and trigger yield levels, and the corresponding
willingness-to—pay levels are listed in the bottom half of Table 7.1 for both IYCI and

AYCI, when the number of acres covered by crop insurance (insured acreage) is also

183

allowed to be chosen by the farmer.20 The maximum trigger yield restrictions range
from 75 % of expected yield to the maximum yield level.

When price and yield are negatively correlated, the optimal trigger yield for IYCI
is 87.9% of expected yield and the optimal insured acreage is 95.1% of planted acreage.
Once again, the actuarially unfair premium, along with the negative price-yield
correlation, contributes to the relatively low trigger yield and is an incentive to insure
less than 100% of planted acreage.

For AYCI, the optimum trigger yield is 132.4% of expected yield and the optimal
insured acreage is 85.3% of planted acreage. The trigger yield is slightly below the
maximum yield and insured acreage is below the planted acreage as a result of the
negative price-yield correlation. When there is no instrument available to manage price
risk, the farmer has an incentive to leave a portion of the yield variability uninsured to
offset the risk associated with the negatively correlated price. Without trigger yield or
insured acreage restrictions, the willingness-to—pay for AYCI is $0.65 above IYCI.

When trigger yield restrictions are imposed, the insured acreage level increases.
The increased insured acreage level is a substitute for the reduced trigger yield level.
At the current level of trigger yield restrictions, the farmer’s optimal insured acreage is

larger than 100% of base acreage and IYCI is preferred to AYCI.

 

2° From the general model (5.1), we can see that the position can also be
interpreted as the percentage of selected price index to the predetermined price level (17,),
ie. expected cash price. In this research, the position is interpreted as the insured

acreage.

184

Imposing the insured acreage restriction results in an opportunity cost to the
farmer. However, the opportunity costs appear to be relatively small. With no trigger
yield restrictions, the decrease in willingness-to-pay from restricting insured acreage to
equal planted acreage is only $0.02 per acre for IYCI and $0.30 per acre for AYCI.
Under the current trigger yield restrictions, the decrease in willingness-to—pay is $0.22
per acre for IYCI and $0.16 per acre for AYCI. These results also suggest that adjusting
the level of insured acreage is not as effective as relaxing trigger yield restrictions to
improve farmer welfare.

When the price and yield are not correlated, the insured acreage tends to increase
along with the willingness-to-pay from the negative correlation case. As in negative
price-yield correlation case, IYCI is preferred to AYCI under current trigger yield

restrictions even if the insured acreage is not restricted.

i l i Ri k

The transaction costs associated with IYCI have resulted in a number of
restrictions on contract design. In an effort to cover these costs, the IYCI premium is
set at a level greater than the expected indemnification level for the typical farmer. The
result is a decrease in participation and welfare for many farmers.

The low level of transaction costs associated with contracts using an area yield
index allow the AYCI premium to be set close to the actuarially fair level. However,
the existence of yield basis risk reduces the ability of AYCI to manage yield risk.

The tradeoff between higher IYCI premiums and the reduced ability of AYCI to

185
manage yield risk as a result of higher yield basis risk is evaluated in this section. For

each level of yield basis risk in AYCI, there exists a IYCI premium level which makes

the farmer equally well off participating in IYCI as in the actuarially fair AYCI.

No Trigger Held Restriction

The AYCI basis risk, welfare-equivalent IYCI premium and corresponding
willingness-to-pay are listed in the t0p half of Table 7 .2 for both negative and zero price-
yield correlation cases. Crop insurance is the only risk management instrument allowed
in the portfolio and there are no restrictions on the level of trigger yield. The tradeoff

between yield basis and IYCI premium is show graphically in Figure 7.1.

Table 7.2 Welfare-Equivalent IYCI Premiums for Alternative AYCI Basis Levels

 

Negativeflmelation Zero Correlation

Wm WTP fiasis Risk IYQI Premium WTP M
No Trigger Yield Restrictions

 

 

4.336 $13.16 0.63 2.415 $18.92 0.63
1.329 26.74 0.83 1.220 33.45 0.83
1.066 32.06 0.94 1.048 38.89 0.94
Under Current Restrictions
22.142 $ 5.70 0.63 17.491 $ 6.50 0.63
1.663 22.38 0.83 1.816 24.14 0.83
1.094 28.46 0.94 1.049 31.62 0.94

 

Note: Yield basis risk is measured by the farm- and county-level yield correlation.
Premium levels are expressed as a percent of the actuarially fair premium. WTP
is willingness-to-pay in dollars.

186

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In the case of negative price-yield correlation, the IYCI premium that produces
the same welfare level as AYCI ranges from 107% of the actuarially fair premium at a
low level of basis risk (farm-county yield correlation of 0.94) to 434% at high level of
basis risk (farm-county yield correlation of 0.63).

Notice that, when the yield basis risk is at the average level (farm-county yield
correlation of 0.83), the corresponding IYCI premium is 133% of the actuarially fair
premium, close to the 135 % base premium level assumed in the previous analysis. This
suggests that, with no trigger yield restrictions, a farmer with the average level of yield
basis risk would be nearly indifferent between actuarially fair AYCI and IYCI under the
135% IYCI premium. The cost to a farmer of the basis risk in AYCI is roughly the
same as the 35% IYCI premium loading.

A farmer in the high yield basis risk class (0.63) would prefer IYCI until the
premium rises above 434% of the actuarially fair level; while a farmer in the low basis
risk class (0.94) would prefer AYCI unless IYCI can charge a premium close to the
actuarially fair level. Thus, without trigger yield restrictions, farmers with high levels
of yield basis risk would prefer IYCI, while farmers with low basis risk will prefer
AYCI.

When price and yield are not correlated, the welfareequivalent IYCI premium
decreases, suggesting that AYCI becomes a relatively more attractive instrument than in
the negative price-yield correlation case. Farmers with the medium level of basis risk

would prefer AYCI when the IYCI premium is above 122% of the actuarially fair level.

188
Thus, at the base 135% IYCI premium rate, AYCI would be the preferred instrument

with no price-yield correlation.

Current Restrictions

The relative performance of IYCI and AYCI are evaluated again for different
levels of AYCI basis risk under the existing trigger yield and insured acreage restrictions.
The previous analysis found that the relative performance of AYCI compared to IYCI
becomes poorer as restrictions were placed on trigger yield and insured acreage. The
welfare—equivalent IYCI premiums for different levels of basis risk under current trigger
yield and insured acreage restrictions are reported at the bottom of Table 7 .2. Figure
7.2 shows this tradeoff between the IYCI premium and the level of basis risk.

The IYCI premium levels are higher than the corresponding values when no
trigger yield restrictions are present, which is consistent with the previous results. When
the yield basis risk is high, AYCI with a trigger yield restriction of 90% of expected
yield provides the same welfare as IYCI with a premium 22 times the actuarially fair
level, in contrast to 4 times the actuarially fair level with no trigger yield restriction.

At the average level of yield basis risk, IYCI is preferred to AYCI at the current
IYCI 35% premium loading for both zero and negative price-yield correlation. This is
in contrast to the case with no trigger yield restrictions where AYCI is slightly preferred
to IYCI. One implication of the results is that constraints on the trigger yield level

reduce the attractiveness of AYCI relative to IYCI.

189

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This section has examined the relative performance of IYCI and AYCI under a
number of alternative contract designs. The results indicate that, because of the high
costs associated with IYCI, AYCI has the potential to be a preferred instrument, despite
the yield basis risk associated AYCI.

Restricting the maximum trigger yield reduces the effectiveness of AYCI relative
to IYCI. Under current trigger yield restrictions, IYCI is generally preferred to AYCI
by the farmer with average yield basis risk. AYCI eliminates most of the moral hazard
and adverse selection problems that have plagued IYCI and there appears to be little
argument for restricting AYCI trigger yield levels. Removing AYCI trigger yield
restrictions results in a contract design that, at average levels of yield basis, is generally
preferred to the IYCI with a 35% premium loading and also avoids some of the
transaction costs associated with IYCI.

The current insured acreage restriction (100% of planted acres) imposes little cost
on the farmer and has little impact on the relative performance of AYCI to IYCI. There
is a substitution effect between insured acres and the trigger yield because allowing
farmers to select the insured acres above (below) 100% of the base acreage increases

(decreases) the indemnity payment proportionally, but the substitution effect is weak.

7.3 Futures and Options
The previous section examined the performance of IYCI and AYCI when crop

insurance was the only instrument available to manage income risk. When price and

191

yield are correlated, crop insurance can be used to manage the price risk in addition to
the yield risk faced by the farmer.

Adding instruments into the portfolio which are designed to manage price risk
changes the role of crop insurance in the portfolio to one which focuses more directly
on managing yield risk. This may change the performance of AYCI and IYCI. In this
section, the economic performance of IYCI and AYCI are evaluated when futures and

options are included in the farmer’s portfolio.

W

The selected trigger yields and willingness-to-pay measures under alternative
trigger yield restrictions are reported in the top half of Table 7.3 for portfolios containing
crop insurance, futures and options. The amount of insured acreage is set at the base
acreage level (planted acres), the AYCI premium is actuarially fair, and the IYCI
premium is set 35% above the actuarially fair premium.

With no trigger yield restrictions and negative price yield correlation, the optimal
trigger yield for both IYCI and AYCI increase slightly when futures and options are
added to the portfolio. Futures and options are used to reduce the price risk faced by the
farmer, reducing the need to cross hedge the price risk with the crop insurance
instrument. This allows the farmer to increase the trigger yield level and reduce the
level of yield risk.

The addition of futures and options also increases the willingness-to-pay at each

trigger yield level although the increases are relatively small ranging from $0.15 to $2.15

192

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193

per acre with negative price-yield correlation. The increase in willingness-to—pay is
larger when the trigger yield is higher for both IYCI and AYCI, suggesting that price
instruments and crop insurance compliment each other.

The increase in willingness-to-pay for AYCI is higher than for IYCI when the
trigger yield is high. However, when the trigger yield is low, the increase in
willingness-to-pay for IYCI is higher than for AYCI. These results suggest including the
pricing instruments in the portfolio further improves the performance of AYCI relative
to IYCI, when the trigger yield restrictions are removed.

When price and yield are not correlated, the optimal trigger yield changes very
little when futures and options are added to the portfolio. With no price-yield
correlation, crop insurance could not be used to cross hedge price risk and so the
addition of the pricing instruments to the portfolio has little impact on the use of the crop
insurance instrument.

In contrast to the negative price-yield correlation case, the increase in willingness-
to-pay when future and options are added to the portfolio is consistently higher for AYCI
than IYCI. The reason is that the added value to the AYCI portfolio is decreasing with
the decrease in trigger yield when prices and yields are negatively correlated but
increasing in the no correlation case. This suggest adding futures and options to the
portfolio improves the relative performance of AYCI to IYCI regardless of the trigger
yield restrictions when there is no price-yield correlation.

The optimal insured acreage position, the optimal trigger yield, and willingness-

to—pay for alternative maximum trigger yield restriction levels are reported in the bottom

194
half of Table 7.3. The level of insured acreage increases slightly when futures and

options are added to the portfolio. The size of the increase is larger when price and
yield are negatively correlated than in the zero correlation case. Once again, when price
and yield are negatively correlated, adding the price instruments to the portfolio reduces
the cross-hedging role of crop insurance. This allows additional yield variability to be
eliminated with the crop insurance instrument.

The unrestricted trigger yield selections increase insignificantly when adding
futures and options while the insured acreage increases more than those in Table 7.1,
suggesting that increasing insured acreage improves the risk management effectiveness
of the crop insurance when futures and options are available.

While in many cases the optimal insured acreage differs significantly from base
acreage, the opportunity cost of restricting the level of insured acres to the base acreage
level remains small even when futures and options are added to the portfolio, ranging
from $0.00 to $1.38 per acre. Thus, restricting insured acreage to base acreage imposes

little cost on the farmer.

i l i Ri k
The addition of futures and options into the portfolio may affect the relative
performance of IYCI and AYCI under alternative basis risk levels. These effects are

studied in this section.

195
No Trigger Yield Restriction

The top half of Table 7.4 shows the welfare—equivalent IYCI premium for alter-
native levels of AYCI yield basis for both negative and zero price-yield correlations when

there is no trigger yield restriction and futures and options are included in the portfolio.

Table 7 .4 Welfare-Equivalent IYCI Premiums for Alternative AYCI Basis Levels
with Futures and Options

 

 

 

 

 

 

Nemmlmion Mn
IXCI Emm’nm WTP AXQI flasis IYQI Eremium WTP A AXQI Basis
No Trigger Yield Restriction
3.444 $15.16 0.63 2.879 $19.27 0.63
1.278 28. 89 0.83 1.185 37.49 0.83
1.057 34.50 0.94 1.052 42.01 0.94
Under Current Restrictions
5.946 $11.25 0.63 7.110 $11.47 0.63
1.717 22.52 0.83 1.514 30.00 0.83
1.115 28.96 0.94 1.100 34.51 0.94

 

Note: Futures and options are included in the portfolio with crop insurance. Yield basis
risk is expressed by the farm- and county-level yield correlation; WTP is the willingness-
to-pay in dollars.

Naturally, the willingness-to-pay for price and yield risk instruments in Table 7 .4
is higher than the willingness-to-pay for crop insurance alone in Table 7.2. For the

average level of yield basis risk (.83), adding futures and options to the portfolio

196

increases willingness-to-pay by $2.15 and $3.99 per acre for the negative and zero price-
yield correlation cases, respectively.

When price and yield are negatively correlated, the welfare-equivalent IYCI
premium is lower with futures and options in the portfolio. This indicates that, when the
price-yield correlation is negative, the relative performance of AYCI improves when the
price instruments are included in the portfolio. With price instruments available, AYCI
can be used to manage yield risk more directly which allows the farmer to select a larger
trigger yield and eliminate a larger portion of the yield risk. Adding the price
instruments to the portfolio also allows IYCI to be used to manage yield risk more
directly; however, the transaction costs built into the IYCI premium impose an additional
cost as the farmer increases the trigger yield, providing a relative advantage to AYCI.

With no price-yield correlation, adding futures and options to the portfolio results
in a small change in the welfare-equivalent IYCI premium. At the average level of basis
risk, the welfare-equivalent premium decreases by 3.5 percent of the actuarially fair

premium with futures and options added to the portfolio.

Current Restrictions

The welfare-equivalent IYCI premiums and willingness-to—pay for alternative
levels of yield basis risk when futures and options are included in the portfolio, and both
IYCI and AYCI are under current maximum trigger yield restrictions, are shown at the
bottom of Table 7 .4. The willingness-to—pay measures increase when futures and options

are added to the portfolio. In general, the increases are in the $3 to $6 ”per acre range.

197

However, for average and low levels of yield basis risk, the increases are less than $0.50
per acre when price and yield are negatively correlated. Because AYCI does not manage
yield risk efficiently for high basis risk farmers, futures and options are again used to
cross hedge yield risk, and the risk management value of futures and options is relatively
high when combined with AYCI for high basis risk farmers.

At high levels of yield basis risk (0.63), adding futures and options to the
portfolio causes a significant decrease in the welfare-equivalent IYCI premium, because,
futures and options are used to manage price risk in the portfolio with IYCI, and their
added risk management value is less than that with AYCI. Therefore, a lower IYCI
premium is needed to obtain the equivalent welfare level. Thus, adding futures and
options improves the relative performance AYCI when the yield basis risk is high.
However, IYCI still significantly outperforms AYCI when the yield basis risk is high,
as evidenced by the level of welfare-equivalent IYCI premium (5.9 and 7.1 with negative
and zero price-yield correlation, respectively).

At average and low levels of yield basis risk, the welfare-equivalent IYCI
premiums show relatively small increases when price and yield are negatively correlated,
suggesting a decrease in the relative performance of AYCI when futures and options are
added. Likewise, with no price-yield correlation and average yield basis, the welfare-
equivalent IYCI premium increases. However, when the price-yield correlation is zero
and there is little yield basis risk, the welfare-equivalent IYCI premium actually

decreases slightly.

198

The results suggest the addition of futures and options to the portfolio can impact
the relative performance of AYCI to IYCI, but the impact depends ambiguously on the
level of yield basis risk. However, at the average level of yield basis risk IYCI still
outperforms AYCI under current trigger yield restrictions when futures and options are

added to the portfolio.

7 .4 Government Program

The current government deficiency payment program provides a mechanism to
manage price risk. The deficiency payment program has many of the same
characteristics as a put option in terms of reducing price risk. The program also
provides an implicit subsidy to farmers who participate in the program. Including the
government program in the portfolio with crop insurance may impact the use of the
insurance instrument as well as the welfare of farmers. In addition, the relative
performance of IYCI and AYCI may change as a result of adding the deficiency payment

program to the portfolio.

W

The selected trigger yields and willingness-to-pay measures under alternative
trigger yield restrictions are reported in the top half of Table 7 .5 for portfolios that
include the government deficiency payment program. The insured acreage level is set

at the base acreage and the IYCI premium loading is 35%.

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With negative price-yield correlation, the optimal trigger yield increases slightly
when the government program is included in the portfolio unless the maximum trigger
yield is a constraint. The willingness-to-pay measures, which include the implicit
program subsidy, are increased significantly when the government program is included
in the portfolio.

When price and yield are not correlated, the unrestricted trigger yield decreases
slightly for IYCI and remains unchanged for AYCI at the maximum trigger yield
constraint when the government program is added to the portfolio. The willingness-to-
pay measures increase less than in the negative correlation case. The increase in
willingness-to—pay measures are larger for AYCI than for IYCI which indicates adding
the government program to the portfolio improves the relative performance of AYCI.

With the government program included in the portfolio, the willingness-to-pay for
the AYCI portfolio remains below that for the IYCI portfolio under current trigger yield
restrictions. However, the difference in welfare from the two yield indices is
significantly reduced when the government program is added to the pOrtfolio. When
price and yield are negatively correlated, the difference in willingness-to-pay at current
trigger yield restriction levels is $2.22 per acre with the government program in the
portfolio as opposed to $3.28 per acre in the crop insurance only portfolio and $3.86 in
the futures, options, and crop insurance portfolio.

The optimal insured acreage and trigger yield levels, and willingness-to-pay for
alternative trigger yield restriction levels are reported in the bottom half of Table 7 .5,

when the insured acreage is allowed to be selected. Once again, the optimal insured

201
acreage levels are significantly above the base acreage level when the trigger yield

restrictions are set at low levels. However, the opportunity costs of the restriction are
relatively small, ranging from $0.00 to $1.59 per acre. The opportunity costs are highest
at low trigger yield levels, and approach zero as the trigger yield restrictions are
removed. Once again, there appears to be little gain in relaxing the current restrictions

on insured acreage.

W
The risk management performance of AYCI under alternative yield basis risk
levels relative to the performance of IYCI is studied in this section when the government

program is included in the portfolio.

No Digger Yield Restriction

The welfare-equivalent IYCI premiums for alternative levels of AYCI yield basis
risk and willingness-to-pay measures are shown in the top half of Table 7 .6 when there
are no trigger yield restrictions and the government deficiency payment program is
included in the portfolio with crop insurance. The insured position is fixed at the base
acreage level.

The willingness-to-pay is significantly higher when the government program is
included in the portfolio instead of futures and options. The major reason for the

increase is the government program provides an implicit subsidy of about $33 per acre.

202

Table 7.6 Welfare-Equivalent IYCI Premiums for Alternative AYCI Basis Levels
with Government Program

 

Negatixecemlation ZerQ Correlation
No Trigger Yield Restrictions

 

 

 

2.149 $44.33 0.63 1.998 $49.80 0.63
1.196 56.89 0.83 1.153 64.33 0. 83
1.054 61.44 0.94 1.016 68.79 0.94

Under Current Restriction

2.449 $40.74 0.63 2.657 $45.80 0.63
1.528 49.23 0.83 1.388 57.11 0.83
1.118 54.40 0.94 1.013 61.34 0.94

 

Note: Government program is included in the portfolio with crop insuranCe. Yield basis
risk is expressed by the farm- and county-level yield correlation; WTP is the willingness-

to-pay.

The welfare—equivalent IYCI premium is reduced when the government program
is included in the portfolio, below the levels when futures and options were included in
the portfolio. This suggests that including the deficiency payment program in the
portfolio improves the performance of AYCI relative to IYCI. At the average level of
yield basis risk, the farmer will prefer AYCI to IYCI at the 35 % IYCI premium loading.
The main reason is that, because mean income is increased by the subsidy implicit in the
government program, the farmer cares less about the risk due to the CRRA preference.
The basis risk associated with AYCI is relatively less important than IYCI actuarially

unfair premium loading.

203

Current Restrictions

The tradeoff between IYCI and AYCI is studied again under the current trigger
yield restrictions. The welfare-equivalent IYCI premium and willingness-to—pay
measures are shown in the bottom half of Table 7 .6.

Adding the government program to the portfolio decreases the welfare-equivalent
premiums for farmers with average or high levels of yield-basis risk relative to those in
the crop insurance only portfolio and the crop insurance, futures and options portfolio.
At low levels of yield—basis risk, adding the government program to the portfolio
increases (decreases) the welfare-equivalent premiums slightly relative to the other
portfolios when the yield-basis risk is negative (zero).

The most significant decreases are realized by farmers with high levels of yield-
basis risk; again, because the mean income increase causes the farmer to care less about
income risk and the high yield basis risk doesn’t harm AYCI as much as before.
However, IYCI still strongly outperforms AYCI when basis risk is high. While adding
the government program to the portfolio improved the relative performance of AYCI,
IYCI is still preferred by farmer’s with the average level of yield-basis risk at the 35 %
IYCI premium loading. Once again, farmers with low levels of yield-basis risk favor

AYCI.

7.5 Other Contract Design Parameters
In the previous sections, we studied participation and welfare effects for a number

of alternative design parameters including the yield index, maximum trigger yield

204

restriction, and insured acreage restriction. There are other contract design parameters
that may affect the risk management performance of crop insurance, such as the premium
rate and the price index used to value yield shortfalls.

The IYCI premium needs to be set above the actuarially fair level for the typical
farmer to help the program become financially feasible. The cost of the premium loading
is that the higher premium will cause farmers to buy less insurance (select lower trigger
yield), exposing them to higher yield risk and a lower farmer welfare level. The
premium for AYCI could be slightly above the actuarially fair level because there might
be some transaction costs to implement the scheme. However, without moral hazard,
adverse selection, and high operating cost, the transaction costs associated with AYCI
are generally low which keeps the premium at levels close to actuarially fair.

The price index used to calculate the indemnity payout is another parameter that
must be specified for both IYCI and AYCI. The specification of the price index may
have implications for the risk management performance for each contract. Numeric
analysis was undertaken for a number of alternative price indices: the expected cash price
at harvest, the expected futures price at harvest, the realized cash price at harvest, and
the realized futures price at harvest. The results suggest that using stochastic prices,
such as cash price or futures price at harvest, is generally dominated by using
deterministic prices, such as expected cash price or futures price, although by a very
small margin.

Interestingly, allowing the representative farmer to select a deterministic price

index level has an identical effect to allowing them to select an insured acreage level

205

because each of the levels is a multiplying factor to the indemnity received less premium
cost in our model. Therefore, the results regarding the insured acreage level apply here.
Under current trigger yield restrictions, a higher deterministic price is preferred for both

IYCI and AYCI, but the welfare gain is small.

7.6 Summary and Conclusion

Historically, crop insurance programs have been designed around individual farm
yield indices (IYCI) to determine indemnification payouts. Transaction costs associated
with moral hazard, adverse selection, spatial correlation, implementation, and
measurement have combined to hinder the financial soundness of these programs. Efforts
to reduce these transaction costs have resulted in restrictions on insurance contract
design. These restrictions have still failed to result in a financially sound program and
have reduced the attractiveness of the crop insurance instruments to farmers. Premiums
above the actuarially fair level for most farmers have been used to help finance the
programs.

Recently, experimental crop insurance programs have been developed which use
area yield indices (AYCI) to determine indemnification payouts in an effort to reduce the
problems associated with IYCI. If the use of an area yield index can reduce these
problems sufficiently, AYCI can be offered at premium levels that are close to actuarially
fair levels.

The primary disadvantage of AYCI is that farmer yield is generally not perfectly

correlated with the area yield index, which reduces the ability of the instrument to

206
manage individual farm yield. However, the current AYCI design features also include

a number of design restrictions which are similar to IYCI. Because AYCI does not face
the same transaction difficulties as IYCI, it may be feasible to eliminate some restrictions
and improve the performance of AYCI. Two of the restrictions examined are the
maximum trigger yield level and amount of insured acreage.

The results suggest that IYCI is preferred to AYCI under the current levels of
trigger yield restrictions by farmers with average yield basis risk. However, removing
the AYCI trigger yield restrictions makes AYCI preferred to IYCI with a 35% premium
loading. The use of the area yield index essentially eliminates the adverse selection and
moral hazard problems associated with the individual yield index, allowing relaxation of
trigger yield restriction in AYCI.

Current restrictions on insured acreage levels were found to have relatively small
impacts on farmer welfare and the relative performance of IYCI and AYCI. The
opportunity costs of the current 100% insured acreage restrictions are highest when the
trigger yields are restricted to low levels, but essentially disappear as the trigger yield
restrictions are removed. Thus, there appears to be little gain from removing the
restrictions on insured acreage, particularly if the trigger yield restrictions are relaxed
for AYCI.

Adding futures and options or a government deficiency payment program into the
portfolio increases farmer welfare and improves the relative performance of AYCI for
high levels of yield-basis risk. The government program improves the relative

performance of AYCI more than futures and options at high levels of yield basis risk.

207

However, IYCI is still preferred to AYCI under current trigger yield restrictions by
farmers with average yield basis risk, even when the government program is included in
the portfolio.

The major policy implication is that AYCI can be an attractive substitute for IYCI
if the trigger yield restrictions are removed for AYCI. There appears to be no clear
reason to institute the current restrictions on trigger yield levels when an area yield index
is used to determine indemnity payouts.

The results depend significantly on the level of yield basis risk with AYCI
becoming more desirable as the yield basis risk decreases. Consequently, the more
localized the area yield index, the more desirable AYCI becomes in general. This
suggests an area index which is defined over a broad geographic region will decrease the

competitiveness of the AYCI instrument relative to an index defined over a smaller

geographical region.

Chapter VIII
ECONOMIC EVALUATION OF THE GOVERNMENT DEFICIENCY

PAYNIENT PROGRAM

Recent political debates have focused on the reduction and/or elimination of the
government deficiency payment and target price programs provided by USDA. The issue
facing farmers in this discussion is not increased exposure to price risk, because futures
and options can be used to manage price risk. The real loss to farmers from elimination
or reduction of the government program is the implicit subsidy, which for the base case
was estimated to be about $33 per acre in 1994.

Current federal budget difficulties have increased discussions about the need for
and level of subsidization provided to farmers. The government subsidy level will
inevitably be reduced in the future. In our model, based on the farm programs in place
during 1994, the level of subsidy can be reduced by increasing set aside acreage and
flexible acreage, and/or reducing the level of the target price. These design changes not
only impact the level of subsidy but also the risk reduction effect from participation in
the program. As a result, changes in the program design may have significant impacts
on participation in the program and other instruments, as well as on farmer welfare.

The design parameters of the government deficiency payment program are often

adjusted each year. The previous analysis used 1994 program specifications. The

208

209
impacts of the changes in the 1994 program design on participation and farmer welfare

are studied in this chapter. The target price, p,, acreage reduction amount, r, and

flexible acreage, q, values which are considered in this chapter are listed in Table 8.1.

Table 8.1 Alternative Government Program Design Parameters

 

 

1994 Changes
Parameter Notation Value Evaluated
Target price p, $2.75/bu -$0. 10 -$0.25
Flexible acreage q 15% +15% -15%
Acreage reduction program r 0 +5 % +10%

 

Each design parameter is changed individually while holding all the remaining
parameters at their values in the base model. The target price, 1),, is decreased by $0. 10
and by $0.25 per bushel. The flexible acreage is decreased to 0% and increased to 30%
of base acreage, to evaluate a range of possible levels. The acreage reduction program
(ARP) is increased to both 5% and 10% of base acreage which encompass requirements
in recent years.

Changes in the design of government program only affect cases where the
program is included in the portfolio, so only these cases are analyzed and reported.
There are six portfolios studied in this section: 1) government program only; 2)
government program, futures, and options; 3) government program and IYCI; 4)
government program and AYCI; 5) government program, futures, options, and IYCI; and

6) government program, futures, options, and AYCI, respectively.

210
8.1 Change in Target Price

The change in target price level directly affects the expected subsidy to farmers,
the cost to the government, and the farmer’s income risk. In 1994, the target price was
set at $2.75/bu for corn, higher than the expected cash price of $2.48/bu. Participation
in the program essentially eliminates all price variability below the target price level, and
also increases the expected price the farmer receives.

The impacts of decreasing the target price on market positions and farmer welfare
are studied in this section. Market participation levels and the resulting willingness-to-
pay levels associated with alternative target prices are reported in Tables 8.2 and 8.3
respectively. For convenience, the results from the 1994 base model are also reported
in each table.

Two general results are immediately apparent from the tables. First, the
willingness-to—pay levels indicate that participating in the government deficiency payment
program is always preferred to not participating, when the target price changes within
the allowed range. And second, the trigger yield is always selected at the maximum
level, 75% for IYCI and 90% for AYCI, whenever crop insurance is available.

A ceteris paribus decrease in the target price leads to a decrease in futures and
option positions. As the target price decreases, the farmer tends to sell more put options
and buy less (or sell more) futures. The reason for the change with no price-yield
correlation is that a decrease in the target price increases price variability so that the

farmer tends to sell more futures to manage the price risk. While in the negative

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213

correlation case, a decrease in the target price increases price variability and less futures
are bought to cross hedge the yield risk.

Willingness-to-pay generally decreases as the target price decreases, primarily
because the value of implicit subsidy is reduced. In the negative price-yield correlation
case, when the government program is the only instrument in the portfolio, the decrease
in willingness-to—pay is only $0.22 per acre when the target price is reduced $0.25 per
bushel, because the gain in risk reduction nearly offsets the loss in subsidy value due to
the negative price-yield correlation. In the portfolios that contain other risk instruments,
the decrease in willingness-to-pay ranges from $12.93 to $16.12 per acre when the target
price drops $0.25 per bushel.

In the zero price—yield correlation case, when the government program is the only
instrument in the portfolio, the willingness-to—pay drops $18.47 per acre when the target
price is reduced $0.25 per bushel. This is much more than in the negative correlation
case because the decrease in the target price now decreases both the risk management
value of the program and the subsidy. When other risk management instruments are
included in the portfolio, the willingness-to-pay drops by levels similar to in the negative
price-yield correlation case, ranging from $13.59 to $15.45 per acre as the target price
decreases $0.25 per bushel.

The results show that farmer welfare change in response to a change in the target
price is generally robust to the price-yield correlation when other instruments are
included in the portfolio. This is because the welfare change comes mainly from the

implicit subsidy which is independent of the correlation, while the income risk is

214

managed using other instruments. However, the welfare change becomes sensitive to the
price-yield correlation when the government program is the only instrument in the
portfolio, because the correlation affects the risk distribution and the government

program is inflexible and ineffective in managing income risk.

8.2 Change in Flexible Acreage

Like the target price, a change in flexible acreage can impact the subsidy level
and risk management capability of the government program. However, unlike the target
price, a change in flexible acreage does not impact the level of subsidy per bushel but,
instead, affects the number of acres eligible to receive the deficiency payment. As a
result, increasing flexible acres impacts both the level of subsidy per base acre to the
farmer as well as the proportion of production that receives the minimum price from the
deficiency payments.

The impacts of changing flexible acreage on market positions and farmer welfare
are studied in this section. Market participation levels and the resulting willingness-to-
pay levels associated with alternative flexible acreage levels are reported in Tables 8.4
and 8.5 respectively. For convenience, the results from the base model under the 1994
program design are again reported in the tables. Once again, the crop insurance trigger
yield is always selected at the maximum level, and the farmer always participates in the
government program.

A ceteris paribus change in flexible acreage leads to a change in futures in the

opposite direction and a change in options in the same direction. That is, when flexible

215

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217
acreage is increased (decreased), the farmer tends to sell less (more) options and buy less

(more) or sell more (less) futures. As discussed earlier, the government program is a
substitute for options and futures. When the flexible acreage increases (decreases), less
(more) put options will be sold, and less (more) futures will be bought in the negative
price-yield correlation case, and more (less) futures will be sold in the zero price-yield
correlation case. However, the change in the positions are all small.

The willingness-to-pay for the portfolio decreases as the flexible acreage
percentage increases, because there is less acreage that can claim the deficiency payment
which reduces the implicit subsidy inherent in the government program.

In the negative price-yield correlation case, when the government program is the
only instrument in the portfolio, then the willingness-to-pay change is only $0.07, as a
result of the subsidy/risk tradeoff discussed earlier. With other risk instruments in the
portfolio, the willingness-to-pay increases between $4.37 and $5 .00 per acre when the
flexible acreage percentage decreases 15 % and declines between $4.43 and $4.98 per
acre when the flexible percentage increases 15%. In this case, futures and options can
substitute for the government program in managing risk, so the change in subsidy is the
main cause of the willingness-to-pay change.

In the zero price-yield correlation case, when the government program is the only
instrument in the portfolio, the change in willingness-to-pay is much higher than the
negative correlation case, about $5.00 per acre, when the flexible acreage changes by

15% of the base acreage. When at least one other risk management instrument is

218
included in the portfolio, the change in the willingness-to-pay ranges from $4.75 to $5.28

per acre, slightly higher than the negative price-yield correlation case. .

For both negative and zero price-yield correlation cases, the changes in futures
and options positions and willingness-to-pay for a 15 % increase in flexible acreage are
smaller than the corresponding changes caused by a $0.25 per bushel decrease in the
target price. These results show that decreasing the target price to the level of expected
cash price reduces farmer welfare more than increasing flexible acreage to 30% , because

the government subsidy is reduced more.

8.3 Change in Acreage Reductions

The ARP is the pr0portion of land that must be left idle in order for the farmer
to participate in the program. If the ARP is zero, as in the 1994 base caSe, the program
provides subsidy and price risk protection to the farmer for free. As with flexible
acreage, change in ARP does not affect the subsidy per bushel directly, but the subsidy
per base acre. Furthermore, it affects the output levels because it requires a portion of
land to be removed from production.

The impacts of increasing ARP on market positions and farmer welfare are
studied in this section. Market participation levels and the resulting willingness-to—pay
levels associated with alternative ARP levels are reported in Tables 8.6 and 8.7
respectively. For convenience, the results for the 1994 base model are again reported

in these tables.

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Once again, the trigger yield is always selected at the maximum level when crop
insurance is available. However, unlike the previous cases, when only the government
program is included in the portfolio, the ARP is set at 10% of the base acreage, and
price and yield are negatively correlated, the farmer becomes indifferent about whether
participating in the program or not because the program now brings zero welfare gain
to him/her. When other instruments are included in the portfolio, the farmer continues
to participate in the government program.

A 10% acreage increase in ARP induces changes in futures and options positions
and the direction of the changes is sensitive to both the price-yield correlation and the
portfolio. The changes can be partly explained by the substitution relationship between
the government program and futures and options in managing price risk, just as in the
flexible acreage change case. Furthermore, the reduced output from the land removed
under ARP introduces additional opportunity cost to the farmer, which contributes to the
differences in market position changes relative to the flexible acreage case. However,
the changes are all small compared with the changes caused by a $0.25 per bushel
decrease in the target price or a 15 % increase in flexible acreage.

The willingness-to-pay decreases as the ARP increases, and the decrease is more
rapid than for a comparable increase in flexible acreage because the increase in ARP not
only reduces the subsidy effect of the government program, but also increases the
opportunity cost of the land removed from the program.

In the case of negative price-yield correlation, when the government program is

the only instrument in the portfolio, the decrease in willingness-to-pay is only $0.35 per

222
acre when the ARP increases by 10%. Although the decrease is small in this case, the

willingness-to—pay for the government program drops to zero. However, with other risk
instruments in the portfolio, the decrease in willingness-to—pay ranges between $18.24
and $23.14 per acre.

With no price-yield correlation, the willingness—to-pay decreases as the ARP
increases because the subsidy value decreases, risk management value decreases, and the
opportunity cost of setting land aside increases. The magnitude of the change is similar
to the negative correlation case when at least one other instrument is included in the
portfolio, ranging from $18.04 to $23.15 per acre as the ARP increases to 10% of base
acreage. This is because other pricing instruments help manage the change in risk
exposure eaused by the change in the government program, so that the welfare change
is basically caused by the subsidy change which is independent of the price-yield
correlation. However, the change in willingness-to-pay is much larger than the negative
correlation case when the government program is the only instrument in the portfolio,
$17.33 per acre. This is because the farmer’s risk exposure is reduced as APR increases
in the negative price-yield correlation case which help offsets the negative welfare effect
of subsidy reduction.

The results suggest farmer welfare is more sensitive to a change in the ARP than
to a change in flexible acreage because of the decreased production from the set aside
acreage. Unlike the target price and flexible acreage, ARP imposes costs to the farmer.

As ARP increases beyond a particular point, e. g. 10% in the negative price-yield

223

correlation case, the set-aside cost begins to dominate the benefit from subsidy and risk

management, and the farmer may choose not to participate in the program.

8.4 Summary

Under the 1994 design, the government deficiency payment program provides a
significant subsidy to the farmer, which usually makes it more valuable than the other
pricing instruments that offer little, if any, profit opportunities. However, the farmer is
only allowed to choose between participating or not in the program, so that the
government program provides little flexibility to deal with the farmer’s risk management
needs. As a result, the program provides significantly more welfare gain when at least
one other instrument is included in the portfolio to help manage risk.

Farmer welfare declines as the target price decreases, the flexible acreage
increases, and/or the ARP increases. Welfare changes significantly with the change in
target price, and is relatively more sensitive to ARP changes than to flexible acreage
changes. The change in futures, options and crop insurance participation levels when the
government program parameters change is primarily caused by the substitution effect
between the program and futures and options, and the change is generally small.

A special result from increasing ARP is that production is reduced. Because it
is believed that the low market price is caused by over-supply, increasing ARP is
expected to help reduce supply and increase the market price. The increased market
price will then reduce the subsidy per bushel the farmer receives and the government

pays, and also cause a positive extemality to farmers not participating in the program.

224

However, as the domestic market gets more open to the world market promoted
by international agreements, such as NAFTA and GATI‘, it gets harder to raise the
market price by unilateral production reduction. In this case, a percentage increase in
ARP won’t increase market price much because the reduced supply is easily offset by
imports, so that little cost to the government is saved above that obtained from an
increase in flexible acreage, while the loss to the farmer is much higher. Therefore,
decreasing the target price is the more effective and safe way to reduce the government
subsidy while still providing the program as an alternative risk management instrument
to farmers.

Farmer welfare changes are generally robust to the level of price-yield correlation
as the government program policy changes in the portfolio setting. In the portfolio
setting, other instruments in the portfolio help manage risk changes that result from a
change in the program, and the welfare change is basically the result of the expected
subsidy change, which is independent of the price-yield correlation. However, when the
government program is used in isolation the farmer welfare change is sensitive to the
price-yield correlation. In this case the welfare change is a result of changes in both
subsidy and risk reduction/increasing effect, and the latter is greatly affected by the

price-yield correlation.

Chapter IX

CONCLUSION

Risk is one of the most important factors that affect production and marketing
decisions in the agricultural sector. Price and yield risk are two major sources of income
risk. Because of the existence of market instruments and government programs, such as,
futures, options, crop insurance and deficiency payment programs, farmers can manage
much of this income risk.

This dissertation studies a representative farmer’s decision to use futures, options,
crop insurance, and a government deficiency payment program when he faces both price
and yield risks. The results show the optimal use of the instruments depends on the
instruments included in the portfolio. This suggests models which consider incomplete
portfolios of risk instruments may provide misleading results.

Even though FCIC realizes financial loss in providing crop insurance and a
premium loading is imposed on typical farmers, the fact is that FCIC has been subsidized
by the government, and the government deficiency payment program also provides a
subsidy to farmers. These programs impose large costs to the federal government. In
the current political and budget environment, it appears the subsidy inherent in the design

of the current programs will be reduced or eliminated at some time in the future. Thus,

225

226

the impacts of design changes in the crop insurance and deficiency payment program on
farmer behavior and welfare are explored in this study.

Expected utility is chosen to characterize farmer decision making under risk in
this research. Numerical methods are used to solve the farmer’s expected utility problem
in order to study the farmer’s incentive to use the risk management instruments, measure
farmer welfare, and evaluate implications of policy changes.

Cash price, futures price, farm-level yield and county-level yield are all stochastic
when the farmer makes his marketing decisions. Previous research has shown that these
prices and yields follow complex joint distributions that make modelling the farmer’s
decision problem analytically difficult. A bivariate ARCH model with seasonality is used
to describe prices while a deterministic trend model with nonnormal errors is used to
characterize yields. A joint distribution for the prices and yields is generated numerically
using Taylor’s method.

The data used to parameterize the model are representative of a corn farm in
southwest Iowa in the 1994 crop year. The simulated 1994 joint distribution of price
yield has a negative price—yield correlation, higher yield risk than price risk in terms of
the coefficient of variation, and a yield basis risk represented by a farm-county yield
correlation of 0.84. These three attributes have very important impacts on the role, the
participation decisions, and the welfare value of each risk management instrument.

The optimal portfolio for the representative farm in the base model includes
futures, options, government program and IYCI. Under the base model conditions, the

government deficiency payment program is valuable because it provides an implicit

227
subsidy to farmers; IYCI outperforms AYCI; and both types of crop insurance are more

valuable than futures and options.

There is a substitution relationship between futures and options, and between
options and the government deficiency payment program. The optimal positions of these
instruments depend on the design specification of other instruments in the portfolio. For
example, the options position is very small in the base solution, however, as the
government program design changes so that the target price is lowered or ARP or
flexible acreage increases, the options position increases significantly.

The correlation between price and yield is another important parameter in the
model. In areas like southwest Iowa where corn is planted densely and yields are high,
the output level in the area may influence market price causing the correlation to be
negative; while in areas that com is planted less densely and/or yields are lower, the
correlation may be close to zero. Both negative and zero correlation cases were
considered in this research. The major impact of going from negative to zero price-yield
correlation is that optimal futures positions switch from long to short and farmer welfare
is increased when crop insurance is included in the portfolio. Also, when the
government program is used in isolation, its value is almost zero in the negative price-
yield correlation case because participating in the program actually increases the income
variability which nearly offsets the value of the subsidy. However, the program value
is over $33 per acre in the zero correlation case, because it reduces income variability

and provides an implicit subsidy at the same time.

228

The farmer’s income risk structure and risk preference also impact the use and
the value of the instruments. Generally, the more risk averse the farmer is, the higher
he values the risk reducing effect of each instrument, but the less he cares about the
income mean change brought about by the instrument. The change in farm yield
variability changes the relative importance of reducing price risk and yield risk, and
consequently changes the relative value of price and yield instruments. In addition, the
change in yield basis risk changes the risk management effectiveness of AYCI. For
example, a farmer whose absolute relative risk aversion is low, farm yield variability is
low, and/or yield basis is low, prefers AYCI is to IYCI.

The current form of IYCI, Multiple Peril Crop Insurance, is facing financial
problems, because of the existence of moral hazard, adverse selection, high operating
cost, and geographical yield correlation. The high cost of providing IYCI has led policy
makers to implement a form of AYCI which reduces moral hazard, adverse selection and
operating cost. Unfortunately, yield basis risk may prevent AYCI from providing
effective yield risk reduction. A comparison between IYCI and AYCI under alternative
crop insurance contract designs was also conducted in this research.

The results show that under the current contract specifications AYCI is generally
inferior to IYCI because of the yield basis risk. However, AYCI has the potential to
outperform IYCI when: 1) basis risk level is decreased, 2) IYCI premium loading is
increased, and/or 3) the AYCI trigger yield restriction is removed. The farmer’s yield
basis risk level can be reduced when the insurance area is more localized, because the

effect of the stochastic factors, such as weather and soil type, is more homogeneous.

229

Increases in the IYCI premium may occur because FCIC faces financial loss currently
even with a government subsidy. It seems reasonable to remove or relax the AYCI
trigger yield restriction, because the restriction is not necessary with little moral hazard
or adverse selection problems. The results also show that allowing farmers to select
insured acreage or the price index used to value yield shortfalls may slightly increase the
welfare of IYCI and AYCI, but the improvement is generally small.

The impact of changes in the government program on farmer welfare is also
studied in this research. The value of the government program decreases significantly
as the target price decreases, the flexible acreage increases, or the ARP increases,
primarily because of the reduction in government subsidy. However, the same
percentage increase in flexible acreage causes less welfare loss than ARP because ARP
imposes extra cost to the farmer as a result of the set-aside requirement. Although the
risk management effect of the government program is also affected by the change in these
parameters, the effect is relatively less important than its subsidy reduction because the
value of the government program comes mainly from its subsidy, and the use of futures
and options can adjust to manage the risk affected by the change in the parameters.

This research computes optimal risk management decisions for a competitive farm
in a market structure with futures, options, deficiency payment program, and crop
insurance in a portfolio, when both price and yield are uncertain. The impact of the
farmer’s optimal decisions and welfare for alternative crop insurance and government

program designs are also studied. Sensitivity analysis is conducted on various parameters

230

that describe the characteristics of the representative farmer, so that the results can be
generalized to farmers in other regions and with different characteristics.

The results can be used as a reference by farmers in their risk management
decision making, by government policy makers as they revise the existing farm and crop
insurance programs, by economists in farm risk analysis, and by others such as, food
processors, traders, insurance agents, and environmentalists, who are interested in
changes in farm production and marketing strategies when relevant government policies
change.

This research considers only one crop in a one period static setting. Though
restrictive, it provides some basic results on farmer’s risk management behavior. Future
research should generalize the result to include multiple crops, and/or a multiple year
dynamic analysis, where the correlation among prices and yields for different crops, the
effect of crop rotations, discount rate and timing of market transactions will all affect the

decisions on risk management instruments.

APPENDICES

APPENDIX A

STATISTICAL HYPOTHESIS TESTING
El 11' 'E II . E T

There are three models in Table 4.1. The first one has no trend while the other
two have differently specified trends. If the data have unit roots, the as in those models
should equal one.

Perron suggests that equation (0) be run first, and if the Z(<I>3) is small, which
means the hypothesis that 3 =0 and 7 =1 can’t be rejected, Z(<I’2) should then be checked.
If Z(<I>,_) is small, the hypothesis that p=0, fl=0 and 'y=l can’t be rejected for this
model. However, the conclusion of a unit root still can’t be drawn yet, because model
(c) is not correctly specified.

Also, the small 203), large Z(tfi), and ambiguous Z(t,,) and 2(7) indicate the
hypotheses B=O can’t be rejected at 10%, [.4 = 0 can be rejected at 5%, and 7=1 can
be rejected at 10% but not at 5% for this model. Therefore, equation (b) is checked.

Z(<I>1) can be obtained through the regression of equation (b). If Z(¢I>,) is small,
the hypothesis that p' = 0 and 7' =1 cannot be rejected, then equation (a) should be
checked. Z(<I>,) is large enough to reject the hypothesis at 5 % level which suggests either

I" = 0, or 7' = 1, or both, is violated, but doesn’t tell which case is true. So, Z(7'),Z(t1.)
and Z(t’.) are checked. The large Z(tu‘) rejects the hypothesis p' = 0, but 2(7’) and
Z(t1.) are not large enough to reject the unit root hypothesis 7‘ = l at 5% level.

LR " ' r hP'

The Log-likelihood Ratio (LR) test states that the twice of the difference between
the log-likelihoods of the unrestricted model and the restricted model is X2 distributed
with the number of restrictions as the degree of frwdom.

Ho-‘¢1=ll’l=0 Ha-'¢1¢00rll’l¢0
LR = 2(Lu ' La) ~ X20)

where LU and L, are the log-likelihoods of the unrestricted model and the restricted
model respectively.

231

232

Starting from I = 6, equation (4.3) are repeatedly run for I = 5,4,3,2 and l, the
log-likelihoods and LR tests are listed in Table 3.1.

Table A.1 LR Tests for Degrees of Frequencies in Seasonality

 

 

I Log-likelihood LR Prob Ho
Futures Price .
6 -548.183 5.103 .078 4),, = 0 up, = 0
5 -550.734 0.266 .875 5 = 0 $5 = 0
4 -550.868 4.860 .088 4}, = 0 414 = 0
3 -553.298 5.340 .069 4), = 0 4': = 0
2 -555.968 4.184 .123 «b, = 0 4'2 = 0
1 -558.060
Cash Price
6 -562.683 4.724 .094 4),, = 0 dz, = 0
5 -565.045 5.416 .067 d), = 0 dz, = 0
4 -567.753 5.140 .077 4), = 0 w, = 0
3 -570.323 0.108 .948 «b, = 0 w, = 0
2 -570.376 4.846 .087 d, = 0 $2 = 0
1 -572.799

 

The null hypothesis of no seasonality versus the alternative of ‘ one degree of
seasonality can be rejected at 5% significant level. All other levels of seasonality can
be rejected in pairwise test against the preceding lower degrees of seasonality which
provides some evidence that the seasonal factor may be characterized by a first order
frequency of sine and cosine variables.

AW
It is necessary to introduce Ljung-Box Q test (Harvey, 1990, p212) before

investigating the model specifications, because it is an important statistic used to tests
whether the model is correctly Specified.

The Ljung-Box Q statistic is defined as:

233

2

D
Q = 717+»: r, (A.l)

7.1 T‘T

where T is the sample size, D is the level of autocorrelation being tested, and r, is the
1th sample autocorrelation in the residuals. The null hypothesis, H0, is that there is no
autocorrelation in the residuals at lag D. Under the null hypothesis, Q has a x2
distribution with D degrees of freedom asymptotically. A high value of Q leads to a
rejection of the null hypothesis.

 

When testing the autocorrelation in the standardized squared residuals for an
ARCH model, each residual is squared first, and then divided by its conditional variance.

The bivariate deterministic variance model with first degree seasonality is
estimated first and Ljung-Box Q statistics suggest there are autocorrelations in the
standardized squared residuals of futures prices at lag 1. As a result, an ARCH(l) model
with the seasonality component is estimated. All the three ARCH(l) coefficients at lag
1 in the variance-covariance equations are significant at the 10% level.

However, while the lower level ARCH effects are eliminated by ARCH(l)
specification, the statistics still show there exist higher level ARCH effects in the futures
price equation. The model is then specified with ARCH(9) for the futures price variance
equation while keeping the cash price variance and covariance equations ARCH(l). The
estimated ARCH coefficients at lag 2, 4, 5, 6, 7, and 8 are zeroes when restricted to be
non-negative, and all the diagnostic tests suggest there is no autocorrelations left
unexplained in the variance process.

APPENDIX B

SIMULATING BIVARIATE NORMAL DISTRIBUTIONS
The technique of adjoining two standard normal variables is illustrated below.
X is a vector of bivariate independent standard normal variable, and Y is a vector

of bivariate standard normal variable with correlation p. We want to find a transform
matrix M so that m a Y.

 

 

 

 

 

 

 

X 0 1 o
X = ' ~ N( , ) (3.1)
X2 0 0 1
Y o 1
Y=[1~N( ,[p]) (3.2)
I"2 O p l
M is a lower triangular transform matrix that has the form of (B3):
M = M" 0 , (3.3)
M21 M22
Let MK 5 Y,
Then
0 1 0
MX ~ N( , M M’ ). (BA)
0 0 1
We need
MM _ M121 Mllel _ [1 p] (3.5)
MIIMZI M221+M222 p 1 ,

and then, solve equation (8.5), we have

234

235

M11 = 1
M21 = p (B06)
M22 = l-p2
Consequently,
I 0
M = ,
9 Wm2
(3.7)
X1
Y = MX = .
le+V1'92X2

The technique of adjoining two non-standard normal variables with zero mean is
similar.

X1
X2

0% 0 ) (B‘s)

2
002

0

Xa-
0

~N(

 

 

 

 

where 03, is the variance of the ith variate, i = 1,2.

 

 

 

Y= Y‘ ~ N( O . of ”'02 ) (3'9)
Y2 O polo, 02
l 0
M = p.02 1_p2 9 h y
al _,
(3.10)

APPENDIX C

HYPERBOLIC SINE TRANSFORMATION
D E . .

Johnson (1949) found that the inverse hyperbolic sine transformation (IHST) can
transform a skewed and/or kurtotic distribution into a normal distribution.

The hyperbolic sine transformation (HST) is defined as:

 

_ . _ e”-l
y - srnh(x) - (C.1)
2e’

so that the inverse of this function, IHST, is:

x = sinh"(y) = 1n(t/2y2+1 + y) (C.2)
.1

Burbidgee, Magee, and Robb ( 1988) modified this transformation to allow
normality as a special case. They defined the modified IHST as below so that as 0
approaches zero, x approaches y.

x = M (C.3)

This modified version is used to estimate the yield model.

M
The HST can be shown to be an even function of 0, which means the function has
the same value for both 0 and -0, so that only the magnitude, and not the sign, of the

parameter 0 affects the shape of the transformed distribution. From equation (4.9) we
know:

236

237

= sinh (0 (v + 6))

 

 

 

e
i,,:,,,2,2.flll_n (C.4)
We want to prove that :
sinh(0(v+5)) = sinh(-0(v+6)) (C.5)
0 —6

which is equivalent to:

820(u+5)_1 - e-20(v+8)_1

29mm - -20e‘°‘”"’

(C45)

Multiplying both the top and bottom of the right hand side of equation (C.6) byemvm
to obtain:

e20(u+5) e-20(v+5)_l - 1_e29(v+5)

em") -20e ’“”"’ -20e"”"

(C.7)

After multiplying the top and bottom of the right side of equation (C.7) by -1, it
becomes exactly the left side of (Q6).

Second, we can show that the HST of a normal distributed variate becomes more
and more leptokurtic as the magnitude of 0 increases. Some numerical evidence is
provided in the paper of Burbidge, Magee and Robb (1988). In addition, a simulation
was conducted to obtain 5000 HS transformations from a standard normal variate when
restricting the skewness parameter 5 to be zero. The statistics of the nonnormal variates
corresponding to a series of 0 values ranging from -3 to 3 are provided in Table C. 1.

As expected from the above proof, these simulated data are symmetric around
their mean, 0, which is also the mean of the standard normal distribution. As 0 deviates
from 0, the mean, standard deviation, skewness and kurtosis are found to be nonzero.
The differences in the mean and skewness as 0 deviates from zero are trivial when
considering the wide dispersion of the simulated data in these cases, while the change of
kurtosis is drastic. So, 0 can be thought of as the kurtosis parameter in the yield model.

Introducing the skewness parameter, 5, doesn’t change the characteristics of HST
as an even function of 0, but it affects the skewness as well as the kurtosis. Table C.2
shows the statistics describing the simulated HST distributions when 6 ¢ 0. The data

238

Table C.1 Distribution Statistics of HST of a Standard Normal Variable (5=0)

 

 

0 min max mean standard deviation skewness kurtosis

-3.0 -55 87 13945 6.7933 291 .9 29. 860 1340.43
-2.0 ~259.9 478.2 0.3098 13.98 13.042 455.33
-1.0 -16.1 1 21.86 0.0302 1.777 0.8529 19.219
-0.5 -5.50 6.46 0.0174 1.147 0.0852 4.3934
-0.1 -3.54 3.87 0.0148 1.005 0.0127 3.0617
-0.03 -3.48 3.79 0.0147 1.001 0.0110 31260
-0.005 -3.47 3.78 0.0147 1.000 0.0108 3.0226
-0.0005 -3.47 3.78 0.0147 1.000 0.0108 3.0225
0 -3.47 3.78 0.0147 1.000 0.0108 3.1225
0.0005 -3.47 3.78 0.0147 1.000 0.0108 31225
0.005 -3.47 3.78 0.0147 1.000 0.0108 31226
0.03 -3.48 3.79 0.0147 1.001 0.0110 3.0260
0.1 -3.54 3.87 0.0148 1.005 0.0127 3.0617
0.5 -5 .50 6.46 0.0174 1.147 0.0852 4.3934
1.0 -16.11 21.86 0.0302 1.777 0.8529 19.219
2.0 -259.93 478.2 0.3098 13.98 13.042 455.33
3.0 -55 87 13945 6.7933 291 .9 29.860 1340.43

 

show that when 0 is close to zero, the distribution is close to normal. Although it skews
to the right as 6 getting larger, it is still very symmetric at the level of -3. As 0
increases, the distribution tends to skew to the left as 5 becomes negative and to the right
as 6 increases. When 0 is -0.5, the distribution is almost the same as that of 9 = 0.5,
again consistent with the properties of an even function, where the slight difference is
caused by simulation error. Generally, the simulation results show that distribution
skews more and more to the right as 6 getting larger.

Clearly, the collaborative effects of all the three parameters, (:2, 0 and 6, are
complicated, and the shape of yield distribution is jointly determined by all the
parameters. Care must be taken not to compare individual parameters without
conditioning on the remaining parameters in the transformation.

239

Table C.2 Distribution Statistics of the HST of a Standard Normal Variable

 

 

0 6 min max mean st.dev. skewness lalrtosis
.0005 —3.000 -6.474 0.7782 -2.9854 1.000 0.010577 3.0226
-1.000 -4.474 2.7783 -0.9853 1.000 0.010728 3.0226
-0.500 -3.974 3.2783 -0.4853 1.000 0.010766 3.0226
-0.050 -3.524 3.7284 -0.0353 1.000 0.010800 3.0226
0.050 -3.424 3.8284 0.0647 1.000 0.010808 3.0226
0.500 -2.973 4.2785 0.5147 1.000 0.010842 3.0226
1.000 -2.473 4.7787 1.0148 1.000 0.010880 3.0226
3.000 -0.473 6.7795 3.0149 1.000 0.011032 3.0227
.5 -3.000 -25.41 0.7980 -4.7853 2.778 -1.46827 6.5396
-l.000 -9.255 3.7620 -1.1615 1.293 -0.76958 4.7880
-0.500 -7. 154 4.9564 -0.5547 1.176 -0.37809 4.3745
-0.050 -5 . 650 6.295 1 -0.0393 1. 140 0.03736 4.3662
0.050 -5 .357 6.6334 0.0740 1.141 0.13312 4.4266
0.500 -4.196 8.3741 0.5905 1.312 0.55390 4.9672
1.000 -3.154 10.812 1.2007 1.312 0.95891 5.9137
3.000 -0.478 29.606 , 4.8669 2.853 1.70710 8.7932
-.5 -3.000 -27.78 0.7778 4.837 2.828 -l.529 7.326
-1.000 -10.13 3.722 -1.183 1.312 -0.8057 5.147
-0.500 -7.844 4.906 -0.5733 1.191 —0.4061 4.569
-0.050 -6.206 6.233 -0.05615 1.154 0.01468 4.426
0.050 -5 .888 6.568 0.05747 1.154 0.1113 4.461
0.500 -4.627 8.293 0.5747 1.197 0.5341 4.907
1.000 -3.499 10.71 1.185 1.323 0.9388 5.788
3.000 -0.6627 29.33 4.840 2.871 1.681 8.592

 

240
'on wi H T Yi 1

Starting from the HST yield model, which is equation (4.9), we want to maximize
the log likelihood density function of the nonnormal disturbance e, '

T
Max 2 ln[g(e,)] (C.8)

1-1

in which g(e,) is the ' f e,. Also, from equation (4.10) we
know if e, comes from the HST of the normal variate v, whose mean is zero and variance
is 6,, then the density function of v, is fill).

flv,)= 1 e7”

(C.9)
J2 wcz '

 

where v, is the IHST of e, . From equation (C.2) and (C.3) we have:

= —sinh“(0e,) - 6

1n[0le,+‘/(0e,)2+1] - 6.

(C.10)

eel—e

Now, let G(- ) and F(o ) are the cumulated density functions of e, and v,
respectively, then, M

M

(C.11)

 

 

We take derivative of equation (C. 10) with respect to e,:

241

0+ 202e,

h’(e)"1 “my”
I 0 0e,+‘/(0e‘)2+1 (C.12)

= 1

«(199)2 +1 .

When (C.9), (C.11) and (C.12) are substituted into (C.8), the MLE model
becomes

 

T 2
Max § -_;.[ln(21r)+lnc2+_:_'2.+ln((0e,)2+l)] , (C-13)

which is equivalent to

T 2

Max 2 '%0“¢2+%*1"((9e.)2+1)] _ (C.14)

1-1

APPENDIX D

CORRELATION MATRICES WITH ZERO PRICE-YIELD CORRELATION

Table D.1 Correlation Matrices of Simulated Prices and Yields with p,.,=0

 

Low Xl'eld Correlation Medium xield Correlation High rm mm
f p 1" YC f p Y’ 1‘ f p Y’ Y‘

Low Risk
f 1.00 .843 .026 .027 1.00 .843 .055 .027 1.00.843 .027 .027
p 1.00 .017 .010 1.00 .039 .010 1.00 .018 .010
1" 1.00 .623 1.00 .830 1.00 .941
Yc 1.00 1.00 1.00
Medium Risk
f 1.00 .843 .029 .027 1.00 .843 .051 .027 1.00.843 .026 .027
p 1.00 .018 .010 1.00 .035 .010 1.00 .016 .010
Y' 1.00 .627 1.00 .831 1.00 .941
1’c 1.00 1.00 1.00
High Risk
f 1.00 .843 .032 .027 1.00 .843 .048 .027 1.00.843 .024 .027
p 1.00 .018 .010 1.00 .031 .010 1.00 .014 .010
Y 1.000 .628 1.000 .830 1.00 .937
I" 1.000 1.000 1.00

 

242

BIBLIOGRAPHY

BIBLIOGRAPHY

Ahsan, S. M., A. A. G. Ali, and N. J. Kurian ”Toward a Theory of Agricultural
Insurance“, American Journal of Agricultural Economics, 65(1982), P520-529

Anderson, J. R. , J. L. Dillon, and J. B. Hardaker Agricultural Decision Analysis,
Ames, IA: The Iowa State University Press, 1977

Ardeni, Pier G. ”Does the Law of One Price Really Hold for Commodity Prices?”
American journal of Agricultural Economics 71(August 1989), P661-669

Arrow, K. J. Essays in the Theory of Risk Bearing, Chicago: Markham, 1971
Bachelier, L. J. B. A. Theorie de la Speculation, Paris: Gauthier-Villars, 1900

Baillie, R. T., and R. J. Myers ”Bivariate GARCH Estimation of the Optimal
Commodity Futures Hedge”, Journal of Applied Econometrics, 6( 1991), P109-124

Baquet, Alan E. , and J. Skees "Group Risk Plan Insurance: An Alternative Management
Tool for Farmers”, CHOICES, (first quarter 1994)

Bollerslev, T. "Generalized Autoregressive Conditional Heteroskedasticity”, Journal of
Econometrics, 31(1986), P307-327

Bollerslev, T. “Modelling the Coherence in Short-run Nominal Exchange Rates: A
Multivariate Generalized ARCH Model. " The Review of Economics and Statistics,
(1990), P498-505

Chambers, R. G. 'Insurability and Moral Hazard in Agricultural Insurance Markets”,
American Journal of Agricultural Economics, 71(1989), P604-616

Chavas, J. P. and M. T. Holt ”Acreage Decisions Under Risk: The Case of Corn and
Soybeans”, American Journal of Agricultural Economics, 72(1990), P529-538

Coble, K. H., T. 0. Knight, R. D. Pope, and J. R. Williams "An Empirical Test for

Moral Hazard and Adverse Selection in Multiple Peril Crop Insurance”, Selected Paper
at AAEA Annual Conference in Orlando, Florida, 1993

243

244

Danthine, J. P. “Information, Futures Prices, and Stabilizing Speculation”, Journal of
Economic Theory, 17(1978), P79-98

Day, R. H. "Probability Distribution of Field Crops”, Journal of Farm Economics,
47(1965), P713-741

Dillon, J. L. 'Bernoullian Decision Theory: Outline and Problems" , Risk, Uncertainty
and Agricultural Economics, edited by James Roumasset, Ames, IA: Iowa State
University Press, 1979

Engle, R. F. ”Autoregressive Conditional Heteroskedasticity with Estimates of the
Variance of UK. inflation”, Econometrica, 50(1982), P987-1008

Feder, G., R. E. Just, and A. Schmitz ”Futures Markets and the Theory of the Firm
under Price Uncertainty”, Quarterly Journal of Economics, 94(1980), P317-328

GAUSS 2.1, Aptech Systems, Inc., Kent, Washington, 1984-1991

Gordon, J. D. ”The Distribution of Daily Changes in Commodity Futures Prices.”
Technical Bulletin 1702 ERS, USDA, (July 1985)

Hadar, J ., and W. Russell ”Stochastic Dominance and Diversification.” Journal of
Economic Theory, 3(1971), P288-305

Halcrow, H. G. ”Actuarial Structures for Crop Insurance" Journal of Farm Economics,
31(1949), P418443

Hall, J. A., B. W. Brorsen, and S. H. Irwin ”The Distribution of Futures Prices: A Test
of the Stable Paretian and Mixture of Normals Hypotheses”, Journal of Financial
Quantitative Analysis, 24(1989), P105-116

Hansen, L. P. and KJ. Singleton "Stochastic Consumption, Risk Aversion, and the
Temporal Behavior of Asset Retums", Journal of Political Economics, 91(1983), P240—
265

Hanson, S. D ”Price Level Risk Management in the Presence of Commodity Options:
Income Distribution, Optimal market Positions, and Institutional value” , Unpublished
Ph.D Dissertation, Department of Economics, Iowa State University, Ames, Iowa, 1988

Hanson, S. D., and G. W. Ladd "Robustness of the Mean-Variance Model with
Truncated Probability Distribution”, American Journal of Agricultural Economics,
73(1991), P436-445

245

Holthausen, D. M. “Hedging and the Competitive Firm Under Price Uncertainty”, The
American Economic Review, 69(1979), P989-995

Hull, J. Options, Futures, and Other Derivative Securities, Englewood Cliffs,
NJ:Prentice hall, 1989

Kang, Taehoon, and B.W. Brorsen. ”GARCH Option Pricing with Asymmetry.”
Working paper, Department of Agricultural Economics, Oklahoma State University, 1993

Lapan, H., G. Moschini, and S. D. Hanson ”Production, Hedging, and Speculative
Decisions with Options and Futures Markets” , American Journal of Agricultural
Economics, 73(1991), P66-74

Love A. and S. T. Buccola ”Joint Risk Preference-Technology Estimation with a Primal
System”, American Journal of Agricultural Economics, 73(1991), P765-774

Markowitz, J. "Portfolio Selection", Journal of Finance, 7(1952), P77-91

Meyer, J. ”Two-Moment Decision Models and Expected Utility Maximization", The
American Economic Review, 77(1987), P421-430

Miranda, M. J. "Area-Yield Crop Insurance Reconsidered", American Journal of
Agricultural Economics, 73(1991), P233-242

Moss, C. B., and J. S. Shonkwiler "Estimating Yield Distributions with a Stochastic
Trend and Nonnormal Errors” , American Journal of Agricultural Economics, 75(1993),
P1056-1062 M

Myers, R. J. ”Time Series Econometrics and Commodity Price Analysis: A Review."
Review of Marketing and Agricultural Economics 62(August 1994):2, P167-182.

Myers, R. J. ”The Value of Ideal Contingency Markets in Agriculture", American
Journal of Agricultural Economics, 70(1988), P252-267

Myers, R. J ., and S. D. Hanson. "Pricing Commodity Options When the Underlying
Futures Price Exhibits Time-Varying Volatility. " American Journal of Agricultural
Economics, 75(February 1993), P121-130

Nelson, C. H. ”The Influence of Distributional Assumption on the Calculation of Crop
Insurance Premia", North Central Journal of Agricultural Economics, 12(1990), P71-78

Nelson, C. H., and E. T. Loehman ”Further Toward a Theory of Agricultural
Insurance”, American Journal of Agricultural Economics, 69(1987), P523-531

246

Nelson, C. H., and P. V. Preckel ”The Conditional Beta Distribution As a Stochastic
Production Function", American Journal of Agricultural Economics, 71(1989), P370-378

Perron, P. “Tends and Random Walks in Macroeconomic Time Series: Further Evidence
from a New Approach. " Journal of Economic dynamics and Control 12(1988), P277-301

Poitras, G. ”Hedging and Crop Insurance", The Journal of Futures Markets, 13(1993),
P373-388

Pope, R. D. and R. E. Just ”On Testing the Structure of Risk Preferences in Agricultural
Supply Analysis“ American Journal of Agricultural Economics, 73( 1991), P743—748

Pratt, J. ”Risk Aversion in the Small and in the large”, Econometrica, 32(1964), P122-
136

Robison, L. J ., and P. J. Barry The Competitive Firm ’s Response to Risk, New York:
Macmilan, 1987

Saha, A., C. R. Shumway, and H. Talpaz ”Joint Estimation of Risk Preference
Structure and Technology Using Expo-Power Utility", American Journal of Agricultural
Economics, 76(1994), P173-184

Sakong, Y., D. J. Hayes,a nd A. Hallam ”Hedging Production Risk with Options”,
American Journal of Agricultural Economics, 75(1993), P408-415

Samuelson, P. A. ”The Fundamental Approximation Theorem of Portfolio Analysis in
Terms of Means, Variances, and Higher Moments”, Review of Economic Studies,
37(1970), P537-542

Sandmo, A. "On the Theory of the Competitive Firm under Price Uncertainty",
American Economic Review, 61(1971), P65-73

Schweikhardt, D. , B. Gardner, J. Hilker, and G. Schwab ”Explanation and Worksheet
to Evaluate participation Decisions in the 1994 USDA Wheat and Feed Grain Programs” ,
Staff Paper 94-14, Department of Agricultural Economics, Michigan State University,
East lansing, Michigan, 1994

Skees, J. R. and M. R. Reed ”Rate-making and Farm-Level Crop Insurance: Implications
for Adverse Selection” , American Journal ongricultural Economics, 68(1986):P653-659

Swinton, S. M., and R. P. King "Evaluating Robust Regression Techniques for
Detrending Crop Yield Data with Nonnormal Errors”, American Journal of Agricultural
Economics, 73(1991):P446—451

247

Szpiro, G. G. " Measuring Risk Aversion: An Alternative Approach" Review of
Economics and Statistics, 68(1986) P156-159

Taylor, C. R. "Two Practical Procedures for Estimating Multivariate Nonnormal
Probability Density Functions. " American journal of Agricultural Economics
76(February 1994), P128—140

Tobin, J. "Liquidity Preference as Behavior Toward Risk" , Review of Economics Studies,
25( 1958), P65-86

Turvey, C. G., and T. G. Baker "A Farm-Level Financial Analysis of Farmers’ Use of
Futures and Options Under Alternative Farm Programs" , American Journal of
Agricultural Economics, 72(1990), P946-957

Tsiang, S. " The Rationale of the Mean-Standard Deviation Analysis, Skewness
Preference, and the Demand for Money." American Economic Review, 62(1972), P354-
371

Vercarnmen, J. "Hedging with Commodity Options when Price Distributions Are
Skewed." American Journal of Agricultural Economics, 77(1995), P935-945

Vereammen, J. and G. C. van Kooten "Moral Hazard Cycles in Individual-Coverage
Crop Insurance." American Journal of Agricultural Economics, 76(1994), P250—26l

Von Neumann, J. , and O. Morgenstem Theory of Games and Economic Behavior,
Princeton, NJ: Princeton University Press, 1944

Williams, J. R., G. L. Carriker, G. A. Barnaby, and J. K. Harper "Crop Insurance and
Disaster Assistance Designs for Wheat and Grain Sorghum", American Journal of
Agricultural Economics, 75(1993), P435-447

Yang, S., and B. W. Brorsen "Nonlinear Dynamics of Daily Cash Prices", American
Journal of Agricultural Economics, 74(1992), P706-715

US General Accounting Office. "Crop Insurance Additional Actions Could Further
Improve Programs’s Financial Condition" Report to the Ranking Minority Member,
Committee on Agriculture, Nutrition, and Forestry, US Senate Washington DC
GAO/RCED—95-269 September 1995

HICHIGRN STQTE UNIV.

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