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A STATISTICAL MODEL FOR CHARACTERIZING
PRICE VARIABILITY WITH APPLICATION TO

DAIRY INVESTMENT ANALYSIS
presented by

TERRY ROS S SMITH

has been accepted towards fulfillment
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Ph.D. degree in Dairy Science

 

 

 

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A STATISTICAL MODEL FOR CHARACTERIZING
PRICE VARIABILITY WITH APPLICATION
TO DAIRY INVESTMENT ANALYSIS

BY

Terry Ross Smith

A DISSERTATION

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

DOCTOR OF PHILOSOPHY

Department of Dairy Science

1980

ABSTRACT

A STATISTICAL MODEL FOR CHARACTERIZING
PRICE VARIABILITY WITH APPLICATION TO
DAIRY INVESTMENT ANALYSIS

BY

Terry Ross Smith

A statistical linear model was developed to estimate variability
for price time series data. The quarterly or yearly estimates result-
ing from the statistical model were used to generate simulated proba-
bility distributions for each of the variables considered. A set of
five interactive Fortran computer programs was developed to perform
the statistical and simulation procedures.

A linear one-way classification model with fixed effects was
used to compute quarterly and yearly solutions. The model (Yij = A1
+ Eij) is full ranked and unique solutions were computed under the
constraint that the overall mean (u) was zero. The estimable function
was u + Ai, which becomes A1 with u set equal to zero. Although small
sample sizes (one observation per month) did not permit a statistical
test for nonhomogeneity of variance between quarters or years, the
variances across time periods for the price series were assumed to be
different. Covariances between time periods were considered to be
non-zero, since the objective was to develop "best estimates" for time
series data, error terms were assumed to be correlated across observations.

As a result, autocovariances were used to develop the variance-covariance

Terry R. Smith

matrix used in the linear model to calculate quarterly estimates.

A simulation procedure was used to generate probability distributions
based on the statistical estimates. A triangular matrix derived from
the historical variance-covariance matrix and a random normal deviate
generator were used to simulate the price series. The simulation pro-
cedure generated a random series of normally distributed and appropriately
correlated probability distributions for each variable considered. The
procedure assumed that each variable reacts to the changes in the other
variable in a way that could be described by the variance-covariance
matrix.

The input time series were tested for normally distributed residuals.
The results of the Shapiro-Wilk test for normality indicated that for the
eleven time series selected, the normality assumption was not always met.
The price series were deflated by U.S.D.A. price indices in attempting to
normalize these data. While there was some improvement in the normality
of the time series when deflated, selection of the appropriate deflator
and the interpretation of deflated series were considered major problems.

Statistical tests were also performed to validate the output from
the statistical and simulation computer programs. Based on comparisons
made between original and simulated sample variances and means the
technique appears to generate reasonable probability distributions
based on the input time series.

By incorporating probability distributions, such as those generated
using the above described procedures for price, income or cost variables
into a capital budgeting model, the information generated by the analysis

represents a substantial improvement over conventional methods of

Terry R. Smith

incorporating risk into investment decision models. The technique was
illustrated with an example. The statistical estimates and simulated
probability distributions for two price series were applied to the
capital budgeting model.

The example compared the profitability of leasing 100 bred dairy
heifers with purchasing the same over a four year period. Michigan cull
cow and calf price probability distributions were used in the model.
The analysis demonstrated the difference between the net present value
that resulted using the expected values for each variable compared to
the distribution of net present values using probability analysis.
Probaility statements were made based on these results.

A users guide to the computer programs and source program listings

were included.

ACKNOWLEDGMENTS

I wish to express my gratitude to those who provided advice and
assistance throughout this study and over the course of my graduate
education. Special thanks are extended to my major professor, Dr.

John A. Speicher, for his continuous encouragement, support and sharing
of his knowledge, throughout my graduate experience. I would also like
to thank Dr. Ivan L. Mao for his valuable assistance and suggestions
during the preparation of this dissertation.

I would like to extend my thanks to Drs. John R. Brake and William
G. Bickert for serving on my guidance committee and for their time
spent on my behalf, in and out of the classroom.

An appreciative thank you is offered to Dr. Harold D. Hafs for his
encouragement and the financial support provided by the Department of
Dairy Science during my graduate study.

My deepest appreciation goes to my wife, Cynthia, for her constant
support and understanding throughout the course of my graduate education.
An a special thanks must go to our son, Ryan, for the added happiness

and joy he has brought into our family.

ii

TABLE OF CONTENTS

INTRODUCTION .

LITERATURE REVIEW.

MATERIALS AND METHODS
General Linear Model .
Development of the Variance— Covariance Matrix .

Simulation Model
Test for Normality

Application of the Statistical and Simulation Procedures

RESULTS AND DISCUSSION .

Evaluating the Normality Assumption . .

Statistical Comparison between Original and Simulated
Data .

Application of Simulation Technique to Capital
Budgeting . .

Leasing versus Purchasing Dairy Cattle - an Example .

A Guide to the Use of the Statistical and Simulation
Progran .

SUMMARY AND CONCLUSIONS
APPENDIX A .

REFERENCES .

21
22
24
27
29
30
32
32
35

39
4O

48

53

59

81

LIST OF TABLES

A Comparison between Original and Deflated Time
Series with respect to Normality.

A Comparison between Input Sample Variance and
Simulated Sample Variance for Selected Deflated
and Undeflated Time Series

Results of Wilcoxon's Rank Sum Test to Compare Means.

Animal Numbers from 100 Bred Dairy Heifers Over Four
Years . . . . . . . . . . . . . . . . . . . . . . . .

A Comparison of Approaches to Net Present Value (NPV)
Analysis of Costs and Returns for Leasing versus
Purchasing 100 Bred Dairy Heifers .

33

37

38

43

44

LIST OF FIGURES

l. Generalized Variance-Covariance Matrix (lower triangle). . . 26

Table

APPENDIX A: LIST OF TABLES

IMSL Routines .
Computer program variable list.

Source listings of interactive
Program One

Source listings of interactive
Program Two

Source listings of interactive
Program Three

Source listings of interactive
Program Four .

Source listings of interactive
Program Five .

Sample Output from Interactive

computer programs

computer programs

computer programs

computer programs
computer programs

Computer Programs.

59

60

63

67

71

72

75

77

INTRODUCTION

Price and income instability has been characteristic of American
agriculutre. Since the 1930's, price and income variability in agri-
culture have indirectly been met by various government price, income,
trade, resource and inventory programs. However, the price-supporting
features of many of these programs has focused more strongly on in-
creasing the level of farm income rather than reducing its variability.
As a result, these programs have often stimulated growth in production
capacity, thereby compounding problems of supply management and income
instability (Barry and Fraser, 1976). The combined effects of invest-
ment in larger, more efficient technologies and the expanded production
capacity requires high rates of financial growth in order to preserve
the economic viability of the farm business.

Producers have little capacity for influencing resource and
product prices. The prices paid, prices received and level of pro-
ductive capacity of a farmer is affected by internal as well as external
forces. Comodity price support programs, environmental regulations,
trade restrictions, wars, changes in demand and changes in government
monetary and fiscal policies, are externalities which have an impact
on the farm firm. Adverse weather, pests and disease outbreaks can be
disastrous to crop and livestock production. Family health and manage-
ment ability and continuity are examples of internal sources of uncertainty.

Since farm growth objectives and investment alternatives imply long-
term planning horizons for proper economic analysis of all flows, costs,

returns and cash, a manager's inability or reluctance to plan over

periods of sufficient length can lead to inadequate economic decisions.
Current methods designed to aid in the decision making process can be
improved upon and new ones need to be developed. The principal objective
of this study was to develop a statistically-sound procedure for incor-
porating yield, price and income variability into a capital budgeting
model.

Most dainrfarmers had quite favorable returns during the five-
year period, 1968 to 1972. Increasing feed costs caused a profit squeeze
in 1973, although milk prices were also rising (Knoblauch, 1976; U.S.D.A.,
1978). Many dairy farmers experienced negative returns to their labor
and management during the period, 1974 to 1977 (Kelsey and Johnson, 1979;
U.S.D.A., 1978). The Food and Agriculture Act of 1977 required a milk
price support level of not less than 80 percent of parity through March
1979 (then the minimum support level reverted to not less than 75 percent
of parity). The act also required that the support level be adjusted
semi-annually through March 31, 1981 to reflect changes in the parity
index. If Congress does not act again before September 30, 1981, the
support level will revert to 75 percent of parity. The difference
between 75 and 80 percent parity represents a difference of approximately
75 cents per hundredweight (cwt) of milk. Partially as a result of the
higher support prices, the financial conditions of dairy farmers have
improved and returns to labor and management were sharply higher in
1978 and 1979 (Kelsey and Johnson, 1979; U.S.D.A., 1978).

While empirical variability estimates are not necessarily identical
with the traditional concept of risk or uncertainty, they are objective
measures of past variability in income, prices and yields (Carter and

Dean, 1960). Knoublauch (1976) showed that the economic environment in

which Michigan dairy producers operate has become more variable in terms
of product prices and input costs. As yield and price were combined
into gross income per acre, more variation existed for all crops during
the 1970 through 1974 period than for the previous 10 year period. Base
milk prices increased slightly in variability during the 1970 to 1974
period but were the most stable of all farm product prices. Farm input
costs also increased in variability with the exception of farm wage rate
which remained about the same, while 6-24-24 mixed fertilizer cost
variability was less in the 1970 to 1974 period.

Barry and Brake (1) noted that multiperiod planning horizons intro-
duce risk not found, or at least well beyond that feund' in single period
production models. Future coefficient values may be specified as point
estimates of expectations or as probability distributions of future values
with mean and variance estimates or probability functions which describe
the kind of relationships which are not deterministic but more obscure.
In the latter case, variables may vary jointly but not in an exact
manner, resulting in relationships which are necessarily probabilistic
and subject to equation errors.

Revenue and cost infbrmation relevant to evaluating capital invest-
ment proposals should be expressed in terms of cash flows into and out
of the business during its expected economic lifetime. Cash flow
analysis aids in evaluating the impact of both alternative investment
and financing strategies used in acquiring resources, on liquidity
position, income expectations and the present worth of the firm (Barry
and Brake, 1971).

To collect and assemble realistic estimates for the key factors

which might be expected to impact an investment proposal, means to find<mrt

a great deal about them. Hence, the kind of uncertainty that is involved
in each estimate can be evaluated ahead of time. Using this knowledge

of uncertainty, decision makers can maximize the value of the available
information. The value of computer programs in developing clear portrayals
of the uncertainty associated with alternative investments has been shown
(Hertz, 1964, 1968; Bennett gt_al,, 1970).

To have calculations of the odds on all possible outcomes lends
some assurance to the decision-makers that the available information has .
been used with maximum efficiency. The probabilistic approach has the
inherent advantage of simplicity in that it requires only an extension
of the input estimates (to the best of our ability) in terms of probabi-
lities. Using this approach, managers can take the various levels of
cash flows, present value and other results of a proposed outlay and get
estimates of the odds for each potential outcome.

If one assumes that the future variability associated with a
particular enterprise or set of enterprises is closely related to past
variability, empirical variability estimates should provide a more
reasonable basis fer making both short and long run management decisions.
Instead of quantifying only single point estimates fer input and output
parameters (as is most commonly done), the uncertainty in these parameters
is quantified by computing best estimates of each parameter and then
using the associated variance-covariance matrix and a correlation tech-
nique similar to that described by Clements g£_31, (1971) to simulate
the outcomes of the correlated events. The resulting simulated distri-
butions provide a measure of risk or uncertainty which could be incorpor-
ated into a capital budgeting model and used to generate probability

distributions of projected net cash flows. The financial outcome of the

combined variables could be calculated each time the set of variables
are sampled and the resulting output would simulate the range and
distribution of possible outcomes from a proposed investment in terms
of the particular "profitability" critereon to be used or tested.

This research effort was designed to develop a method for charac-
terizing the price, yield and income variability facing dairy producers
in Michigan. The principal objective was to design a statistical model
to describe the variability associated with price variables as well as
the covariances between variables over time.

The statistical estimates, based on the input time series are used
as input to a simulation program. The simulation procedure generates
a probability density function for each variable which has the variance
and covariance characteristics of the original input time series. The
output from these procedures is examined statistically in attempting to
examine the significance of improvements made over existing methodologies.

The generated density functions are then incorporated into a
capital budgeting example in which leasing of dairy cattle is compared
with purchasing. This approach provides the decision-maker with an
estimate of the effect of price variability on the profitability of
alternative investments. This approach to investment analysis offers
the decision-maker a wealth of information that represents a substantial
improvement over more conventional attempts at incorporating risk into

investment analysis.

LITERATURE REVIEW

Knight (1931) defined risk as a condition where the probabilities
associated with outcomes are known (measurable). This contrasts with
uncertainty where probabilities associated with outcomes are not known
(not measurable). Johnson (1957) stated that risk exists when pre-
scriptive knowledge is sufficient to make a decision (decision mode).
Uncertainty, on the other hand, exists when specifications for pre-
scriptive knowledge are not met (learning mode). Johnson described a
decision-maker with sufficient knowledge to derive a frequency proba-
bility distribution as being in a risk-knowledge situation. With only
partial knowledge he is in a subjective risk-knowledge situation.
While recognizing the above distinctions made between risk and uncer-
tainty, the two terms will be used interchangeably in this thesis to
describe actions with more than one possible outcome where the like-
lihood of all possible outcomes is described by a probability distri-
bution or probability density function.

Heady ££_al, (1954) approached economic instability and crop
production choices involving risk using a mathematical approach which
involved measuring variances and correlation coefficients of income
from various enterprises. The mathematical formulae derived stated
that the variance for the whole farm is the sum of the variances of the
enterprises plus the covariances of the enterprises. This formula was
then expanded to account for the proportion of the enterprises making
up the total farm. They found that when the number of enterprises

combined was three or greater, the formula became complicated making

variability calculations difficult.

Carter and Dean (1960) attempted to provide a more objective
measurement of the uncertainty or variability associated with various
crops and cropping systems in California. Three types of crop varia-
bility were considered in their study; price variability, yield varia-
bility and income variability which arises from the interaction of
product yield per acre and product prices relative to vcosts. Bartlett's
test was used to test the homogeneity of the variance over time. Where
variance was not homogeneous the variance based on the most recent (5
year) period was taken as the best estimate of future variance. Little
year-to-year correlation between price and yield was evident for
California field crops, and yield variability for field crops was
relatively low. Therefore the most important factor contributing to
gross income variability of field crops was felt to be the variability
of prices. Correlation between the "random elements" of pairs of net
income time series were estimated using the variate difference method,
and were typically lower than those of the original series. The actual
net incomes of crops tend to be highly correlated because the major
economic influences (inflation, price cycles, wars, level of technology,
etc.) affect most enterprises similarly. Their data appeared to support
the hypothesis that considerable individual farm yield variability is
"averaged out" when state or even county series are used.

Greve 33 El. (1960) analyzed the variabiltiy of production, price
and gross returns of wheat, grain sorghum, steer and cow-calf enterprises
in Northwestern Oklahoma. A summary of the estimated coefficients of
variation for the specified enterprises indicated that grain sorghum

was more variable than wheat in production, price, gross income and

returns. The cow-calf system, ignoring inventory changes, is the more
stable of the two livestock enterprises. With inventory changes in-
cluded, the cow-calf enterprise shows the greatest relative degree of
variation in gross income and returns using actual and deflated prices.
The authors pointed out that the pattern, or sequence of favorable and
unfavorable years, may be at least as critical as the degree of variation
over years. They used the Wallis and Moore non-parametric test to check
for the randomnesss of a time series. Each of the three tests for
bunchiness (runs of specified duration, 4-year moving averages and non-
parametric statistical test) suggest the presence of cycles or "bunches"
in each of the series of data tested. Cow-calf enterprise data tended
to bunch near the mean with a low (4 percent) coefficient of variation
indicating the relative stability of cow-calf production. The small
amount of correlation between wheat returns and the returns from each
of the other chief farm enterprises implied a stabilizing effect if
enterpriSes= were combined.

Day (1965) studied the possible asymmetry or skewness of field
crop yield probability distributions. The contrast among the estimated
distributions suggests that decisions for maximizing profit and minimiz-
ing risk must be based not only on zexpected yields and variances but
upon skewness as well. In the case of cotton and corn, nitrogen
application not only increased average yields but concurrently reduced
positive skewness. However, in the case of oats, negative skewness
was increased by nitrogen application meaning that at higher nitrogen
levels above average yields can be expected more than half of the time.
These results suggest that a model of a single firm or a small relatively

homogenous region that uses average yields at low nutrient levels will

overpredict yields more often than under predict them and vice versa at
high levels. The author suggests that the mode or median may be pre-
ferred for prediction or forecasting purposes. If the objective were
to maximize the probability of coming close to the observed values the
mode should be used. If the objective were to have nearly equal chance
of over or underpredicting, the median should be used.

Luttrell and Gilbert (1976) analyzed average yields for a number
of major crops in the nation and the leading producing states. These
data provided little evidence that yields were either cyclical or bunchy
as a result of weather. However, there was some evidence of positive
autocorrelation in yields since 1933. Such results appeared to reflect
the uneven rate of application of high yield producing inputs rather
than a non-random influence of weather. No attempt was made to anlyze
yield patterns in areas smaller than states. The Durbin-Watson sta-
tistic was used to test for buchiness of regression analysis in terms
of first-order positive autocorrelation. The Wallis-Moore test (non-
parametric test of randomness) was used to test the randomness of
yields since it is not limited by the assumption of normally distributed
observations.

The empirical findings of Black and Thompson (1977) were consistent
with the existence of drought cycles for corn, soybeans and wheat yields.
However, not every year within a drought period exhibited below average
yields and vice versa. There was no evidence for a two-year cycle.

Barry and Fraser (1976) evaluated the feasibility and structural
implications of relevant risk responses available to producing firms
that differ by size and type. A table of annual coefficients of varia—

tion for monthly prices of selected commodities over 1959 to 1974 period

10

was presented. The coefficients reflected irregular influences on
prices, since the data were modified to account fer estimates of seasonal
price patterns in the respective commodities. The increasing effects of
irregular influences on intrayear price variation are clearly evident

for crops and to a lesser degree for livestock. Coefficients of varia-
tion for monthly prices of wheat, corn and soybeans and sorghum increased
greatly in 1973 and 1974.

Markowitz (1959) argued that if investors were faced with two
investment alternatives with the same levels of expected return but
with different variance, they would prefer the investment with the
smaller variance. Thus, the choice set would solve for investment
alternatives which minimized variance at each level of expected wealth.
Markowitz referred to this resulting choice set as the expected value-
variance (EV) efficient set and used quadratic programming to derive
the EV set, which for a given level of expected wealth minimizes the
variance.

Webster and Kennedy (1975) estimated sets of indifference curves
for five farmers who were willing to forego expected income for increases
in a probabilistically defined minimum income. Three of the five farmers
showed an increasing marginal utility for less variability in incomes.
The other two farmers showed increasing followed by decreasing marginal
utility fer variability of income. The information obtained was to be
used for predictive rather than normative purposes.

Aanderund (1966) estimated the income variability of enterprise
combinations for Northwest Oklahoma farm and ranch situations. The
study was designed to determine the probably effect on capital accumu-

lation and survival of farm operators using alternative farm plans.

11

The results were presented in an income opportunity framework allowing
the returns and variability .estimates of alternative farm organizations
to be examined with the farmer deciding on the level and variability of
income which meets his preferences.

The objective of a study by Bostwick (1963) was to determine prior
wheat yield variability and construct yield probability functions designed
to be of assistance to managers in making planning decisions. Probability
functions were derived from wheat yield observations (n=5000) over a
thirty-five year period using an extreme value statistical distribution
approach.

Johnson 33.31, (1967) incorporated a distribution of crop yields in
a deterministic linear programming model to demonstrate the impact of
random variations in yield on firm growth. Solutions to the stochastic
linear programming problem were approximated for three assumed farm
situations and compared with nonstochastic solutions in which means of
the probability distributions for crop yields were used. In each of the
three farm situations the nonstochastic solution was higher than the
estimated mean for the stochastic solution. It was pointed out that
this was due in part to interactions among household consumption witha
drawals, the investment policy and yield variability.

Boehlje and White (1969) applied a multi-period linear programming
model to a hypothetical farm firm over a ten-year planning horizon.

One of the two objective functions maximized the present value of the
annual disposable income, and the other maximized the net worth of the
firm at the end of the planning horizon. Technical coefficients and

input and product prices were assumed to be constant throughout the 10

years since their principal objectives were to analyze the impact of

12

resource availability and different optimizing criterea on farm firm
growth. The empirical results for the corn-hog farm situation indicated
that in all cases the farm specializes in the production of the most
profitable enterprise, which with the assumed technical and price co-
efficients was the production of hogs. The major limitation of the
multiperiod linear program was felt to be the difficulty of incorporating
elements of risk and uncertainty in the model.

How and Hazell (1968) used quadratic programming to incorporate
income variance into a linear programming model. While the combination
of farm enterprises obtained by the solution of a linear programming
model may provide the maximum expected income under given constraints,
the year to year degree of uncertainty from one year to the next might
not be acceptable to a farmer. The quadratic programming procedure
results in a set of solutions for each of which the variances of expected
income is at a minimum for the given expected mean income level, with
income level varying over the whole feasible range. This is accomplished
by setting the linear programming objective function as a constraint to
be varied parametrically and substituting the variance-covariance matrix
as the objective function to be minimized. The resulting EV efficient
pairs trace out a pattern of increasing expected mean income associated
with increasing income variance.

Scott and Baker (1972) used a quadratic programming model to generate
income variabilities. Their model incorporated income variance and co-
variance of possible enterprise combinations. The quadratic programming
model is the same as a linear programming model except that the quadratic
programming model contained a risk aversion coefficient which could be

varied. By varying the risk aversion coefficient, points on the efficient

13

frontier were generated. An efficient frontier allows the decision
maker to choose the level and variability of income which meets his
preferences. Since there has been little if any success in quantifying
the correspondence between a risk aversion coefficient and a decision
maker's utility function, risk aversion coefficients have had little
empirical use.

Hazell (1971) developed an approach which minimizes total absolute
deviations rather than variance, which may be solved using a linear pro-
gramming algorithm and which gives results remarkably similar to those
of quadratic programming. The model was solved to determine the set of
production activities that minimizes variance of net returns subject to
receiving a specified level of income or to develop the efficiency
frontier showing trade-offs between expected income and variance.

Thompson and Hazell (1972) later showed that a linear programming
approximation using the mean absolute deviation (MAD) could be used to
derive efficient EV farm plans when a suitable quadratic programming
routine was not available. A Monte Carlo simulation study demonstrated
that the MAD estimator of variance was only marginally less efficient
than that of the sample variance in ranking equal-income plans by their
variance. The percentage of additional loss in utility incurred by
employing the MAD exceeded 10 percent only in cases of high correlation
and/or large sample sizes.

Lin g£_al, (1974) compared utility maximization with profit max—
imization as predictors of farmer behavior. The EV (expectation, variance)
frontier for each farm shows a set of alternative production plans, each
providing minimum net income variance for specified levels of expected

net income. Empirically, the EV frontier from each farm was efficiently

14

derived using a quadratic programming model. The decision maker's
subjective probability distributions of price and yields were incorpor-
ated, insofar as possible, in the estimation of the expected net returns
and the variance-covariance matrix of net returns. While the means and
variances of net income for individual crops on each farm were estimated
subjectively, it proved impossible to obtain subjective estimates of
covariances. They also feund that combining historical covariances

with subjective variances let to inconsistencies revealed by a variance-
covariance matrix that was not positive semi—definite. Thus, to generate
estimates of covariances, time series of net incomes for each crop were
reconstructed by expressing the historical trend-corrected net incomes
for each crop in terms of standard normal deviates about the mean. Then,
substituting the standard deviations derived from the subjective net
income distributions, calculation of the variance-covariance matrix from
this reconstructed set of time series data preserved the subjective net
income variances, incorporated the historical relationships among crops
and guaranteed a positive semi-definite matrix.

Barry and Willman (1976) used a multiperiod, quadratic linear
program to model a firm's growth environment and to evaluate the effects
of external credit rationing on optimal levels of foward contracting.

The model's basic structure resembled that of other growth models (Barry
and Baker, 1971; Hazell and How, 1968) in terms of multiperiod linear
programming with the addition of risk information expressed as variances
and covariances on selected activities in each period. The program
derives an efficient EV solution, called a "growth plan", that minimizes
variance for a given net present value of income. The model was designed

to provide the decision maker with a set of a priori growth plans that

15

organize the business to accept alternative combinations of risk and
returns valued over the planning period. The decision maker then chooses
the plan that best satisfies his preferences toward risk and returns.

Robison and Brake (1979) define portfolio theory as "an efficiency
critereon that indentifies a set of investment plans that minimize
variance (maximize expected returns) for given levels of expected wealth
(variance) from which well—defined classes of decision makers can find
their expected utility-maximizing solution". This set of investment
plans, often referred to as the expected value-variance (EV) set, is
efficient because it restricts the search for preferred solutions to
those EV efficient plans. As a financial model, the assumptions and
restrictions of porthlio theory seem acceptable: production is linear,
. asset choices are mostly divisible and the variance is on the price
side. But as a farm planning tool, portfolio models seem less useful
because production is not linear, asset choices are seldom completely
divisible and variance on the output side is at least as important as
variance of the price side. Still as an empirical tool, it represents
an improvement over previously papular linear programming models (Brake
and Robison, 1979).

Robison and Barry (1977) used EV analysis to model farm plans by
limiting resources with a set of linear constraints and estimated the
variances and covariances of returns between production alternatives.
This application of portfolio analysis (referred to as risk programming)
has been useful in applied decision making problems when the choice set
was not predetermined. It has been shown that only if the probability
distribution of outcomes for each alternative is normally distributed

or if the investors possess quadratic utility functions can we be sure

16

that the EV set will include the expected utility maximizing choice
(Barry and Fraser, 1976). However, unless the probability distributions
of outcomes are highly skewed, the expected utility maximizing choice
will be very close to at least one alternative included in the EV set.

Halter and Dean (1965) used an empirical model to simulate both
the uncertain environment facing management and management's decision
made in response to that uncertain environment. The model simulated
range conditions and price relationships for a 40 year period. Simu-
lation was then used to evaluate the desirability of one specific change
in management policy: namely, consideration of alternative price expec-
tations models to be used in making the critical May-June decisions on
buying cattle directly for the feedlot. The authors concluded from
the results obtained from this study that it was difficult to make
marked improvements in either level of income or in reduction of income
variability by adjusting the buying decisions since range and price
conditions are essentially exogenous to the farm. They further state
that management may be forced to accept wide variability as unavoid-
able and must therefore investigate ways to improve the technical
efficiency of the firm (i.e. better feed conversion, faster gains and"
lower operating costs).

’Zusman and Amaid (1965) evaluated the performance characteristics
of various decision rules by simulating a beef cattle enterprise over
a period of 16 years with weather events constituting the main stochastic
input. Optimal decision rules were defined in terms of the net present
value and the~ coefficient of variations of the income flows. Plotting
the coefficient of variation against the present value of the income

stream showed that it was impossible to increase the present value of

17

annual income without simultaneously raising the coefficient of variation
and vice versa.

Patrick and Eisguber (1968) developed a simulation model of farm
firm behavior with managerial ability of the operator and farm capital
structure as the controlled variables. Managerial ability of the farm
operator was expressed in terms of the technical transformation rates
(i.e. yield per acre, etc.). Capital structure was divided into three
parts: interest rate, long-term loan limits and intermediate-term loan
limit. Managerial ability of the farm operator was the major factor,
among those considered, determining the rate of growth of the farm firm.
Improvement of the technical rates of transformation by 10 percent in-
creased the farmer's net worth about $2,000 per year or about 25 percent
at the end of the 20-year period.

Blackie and Dent (1976) incorporated risk and uncertainty into
simulation model predictions only to the extent that individual managers
defined their price and cost expectations. Although it is possible to
select prices and costs stochastically from an appropriate distribution
the authors believed that these investigations would be of greater general
interest if these effects were not included.

Sadan (1970) describes the opportunities facing the investment
decision maker in a farm firm under risk in terms of "efficiency frontiers",
the dimensions of which are the expected present value and the variance of
the farm's future net returns. A simulation model was developed to trace
possible efficiency frontiers for an actual Israel kibbutz. The outcome
of each "simulation experiment" included the unit's net worth, its con-
sumption allowance and the corresponding variances or coefficients of

variation. The point of operation for the actual farm was located on

18

the simulated efficiency frontier and the observed safety margin (defined
as the ratio of the internal rate of return on farm investments, divided
by the respective external rate, imputed as the sum of the rate of interest
and the depreciation rate) was interpreted as a measure of risk aversion.
An arc estimate of the slope of the frontier at the point of operation
indicated a marginal rate of substitution of 55 cents per unit variance

at that point. It was suggested that this and similar models be used to
stimulate the possible effects of alternative policies of the lending
agencies upon the expected value and variance of the firm's net returns.

Hertz (1968) described a computer simulation model used to assess
the variability associated with various investment policies and alterna-
tives. The technique requires three basic steps: 1) estimating the
range of values for each of the factors and, within that range the like-
lihood of occurence of each value, 2) randomly selecting from a distri-
bution of values for each factor one particular value to be combined
with one for each other factor to compute the net present value or rate
of return from that combination, and 3) repeating step 2) over and over
again to define and evaluate the odds of the occurence of each possible
outcome through simulation.

Clements g£_gl, (1971) developed a procedure for correlating events
in farm firm simulation models. Simulation models frequently incorporate
Monte Carlo procedures to represent uncertainty. The Monte Carlo
applications typically assume that the correlation between any two
events is either non-existent (zero) or perfect (one). This assumption
does not realistically represent the covariance between related events
and may even introduce artificial and unrealistic variability into an

analysis. The procedure developed by Clements (1971) can be incorporated

19

in Monte Carlo simulation models to correlate events at any desired level
from minus one to plus one. In general, the procedure involves defining
an upper triangular matrix of coefficients, calculating the numerical
values of these coefficients using the variance-covariance matrix and
combining the estimated coefficients with a series of random normal
deviates to generate the correlated outcomes.

Hertz (1964) points out that the controversy and furor associated
with the development of ways to improve our ability to discriminate
among investment alternatives has largely been resolved in favor of the
discounted cash flow method. As these techniques have progressed, the
mathematics involved has become more and more precise so that we can
now calculate discounted cash flows to the penny or rates of return to
a fraction of a percent. However, behind these precise calculations
are data which are not that precise. There is something more the
decision maker ought to know in addition to the expected net present
value or expected rate of return.

Hertz (1968) found in general that risk-based policies consistently
gave better results than those using single-point, deterministic decision
rules. The program also allowed management to ascertain the sensitivity
of the results to each or all of the input factors. Simple by running
the program with changes in the distribution of an input factor, it is
possible to determine the effect of added or changed information (or the
lack of information).

Bennet g£_al, (1970) developed methods for the financial evaluation
of mineral deposits. The objective was to select the best alternative
mining and processing methods and production rates. This was done by

performing a financial evaluation through computer simulation using

20

probablistic risk analysis and sensitivity analysis. The probabilistic
analysis approach, which was based on the premise that the expression

of a range and/or distributional characteristics of variables in estab-
lishing an estimate for a parameter, is more realistic than the choice

of a single point estimate. Numerous simulations were made using randomly
selected values of the input parameters resulting in a frequency distri-
bution of the rates of return and their relative probability of occurence.
The sensitivity analysis method was combined with the probabilistic
analysis to interrelate mineral reserves, capital investments, operating
costs and production rates. The evaluation of a mineral deposit was
performed using three alternative mining and processing methods at three
different quantities of resources and various production rates and

prices. The grade, recovery, operating costs and annual production

rates were entered as probability density functions, the capital
investments as normal distributions and working capital and ore as

point estimates. The results were reported for different ore prices

for the recovered product. Results were reported as the range and most

frequent rate of return (percent).

MATERIALS AND METHODS

A set of five interactive computer programs was developed to perform
the statistical analyses and the simulation procedures developed in this
study. The first four programs are used to obtain statistical estimates
based on the input data. These quarterly or yearly estimates are used
in the simulation program to generate a series of normal random vectors
with the prescribed correlation between variables. The statistical
estimates represent a major improvement over earlier work (Clements
g£_al,, 1972) in terms of generating input data for the simulation
procedure. The simulation procedure, adapted from Spence (1976), assumes
that the variation and covariation of each of the variables considered
will react to changes in the others in the way that was observed and
described by the variance-covariance matrix, during the time period
from which the data were taken. The generated vectors describe pro-
bability density functions for each variable that can be used as input
to a capital budgeting model. This approach to investment analysis
represents a substantial improvement over the more common "point estimate"
approach. The procedures developed in this study for incorporating
variability into investment analysis go beyond those which are based
on the decision-maker deriving optimistic, most likely and pessimistic
forecasts for the series of events considered. This technique recognizes
that the component cash flows are most often related to the movements of
other price series.

Data from U.S.D.A. Agricultural Prices, Michigan Agricultural

21

22

Reporting Service and Michigan State University Enterprise Budgets were
used to illustrate the technique and program output. The raw data and
deflated data (deflated by USDA price indices) and output data were

tested for the assumption of normally distributed residuals.

General Linear Model

 

The general linear fixed model Y = Xb + e has expectations E(e) = O
and E(y) = KB, and variance-covariance matrix Var (y) = Var(e) = V.

According to Searle (1971) least square estimation involves minimizing

(Y - Xb)’ V" (Y - Xb)
with respect to b where, ' refers to the transpose and " the inverse.

This leads to the least square solutions.
b = (X! V" X)" XI V" Y

For a full rank model, the unique inverse of X'X, (X'X)", exists.
Therefore, solutions for the elements in b are unique solutions or
estimates (b). For a non-full rank model, those involving fixed effects,
(X' V" X)" does not exist, and as a consequence the solutions are not.
unique. However, unique solutions can be obtained when certain con-
straints are imposed in the process of solving the normal equation.

The approach taken to compute quarterly estimates from monthly
observations on a variable, assumed that the residual random error
(Eij) was distributed with mean zero and variance-covariance matrix V.
In order to obtain the quarterly estimates, u was set equal to zero.
Since, u + A1 is an estimable function, with Ai set to zero, u + A1 is

equal to Ai' The consequence of this constraint is that the model is

23

full ranked and unique solutions exist. In this particular one-way
classification model situation, u + A1 is then equivalent to A1, since
A1 is really u + Ai’ which is an estimable function. Thus, the following

linear model was used,

where, Yij was the response of the jth observation in the ith quarter
(three monthly observations per quarter), and A1 the fixed effect for
quarter j.

From the model, the following normal equation is constructed,
X' V" X‘b = X' V" Y

where, Y is an observation vector of length n (number of months), X is

a known n x q (number of quarters) matrix derived from the data (con-
taining ones and zeros according to the presence or absence of Y's in
quarterly classes), b is a q x 1 vector of unknown quarterly constraints,
X' is the transpose of X, and V" is the inverse of the variance-covariance
matrix.

The normal equation is then solved for
b = (XI V" X)" X! V" Y

Although the small sample sizes (one observation per month) do
not permit a statistical test for non-homogeneity of variance between
quarters or years, the intuitive assumptions made for this analysis
were that variances over time are not alike and that the covariances

between time periods are non-zero.

24

Development of the Variance-Covariance Matrix

 

If observations from a population are random samples drawn in a
completely random and uncorrelated manner (drawn with replacements or
from an extremely large population), then all covariances would be zero.
Further, if all observations are random samples from the same population
with a constant variance or from different populations with homogenous
variances, the variances would be equal to a constant variance. In
practice the sampling scheme is designed to make certain that V s 021,
or the assumption is made that this is the case when writing the linear
model.

However, if for example, one assumes that the price of a commodity
this month is related to the price of the same commodity_ last month
and the month prior to that, and so forth, then the error terms can be
expected to be correlated across observations. This phenomenon is
termed autocorrelation and V does not equal 021. Autocorrelation is
most often a characteristic of time series data, when data are collected
from the same observational unit at successive points in time. An
appropriate procedure to use when faced with situations where the error
terms are correlated or have unequal variances is refered to as general-
ized least squares. This method uses the information about the variance
and covariances of the error terms to increase the accuracy of the
estimators. In the case of observations with error terms of unequal
variance, the procedure effectively gives greater weight to those obser-
vations whose error terms have smaller variances. In the autocorrelated
case, this procedure transforms the variables in such a way that the
error terms implicit in the transformed variables are uncorrelated.

With autocorrelation it is difficult to state a priori the magnitude of

25

the correlation among succesive error terms. Given the values X1, X2,...

..,Xn; the (n-l) pairs (X1,X2), (X2,X3),......, (X X“) constitute a

n-l’
set of bivariate values which have a correlation coefficient associated
with them, as do the (n-Z) pairs and so on. The coefficient (k-l)
terms apart, i.e. of Xt and Xi + 1’ is called the serial or autocorre-
lation of order k.

The autocorrelation principle was used to develop the covariances
of the variance-covariance matrix (V) needed to calculate the quarterly
solutions. In this way autocovariances were used as estimates of
covariances across months within quarters, between quarters within
years and between quarters across years. For the input time series

(Wi), i = 1,......,n, autocorrelations (ACj), j = l,....n-l were

computed using maximum likelihood estimates.

 

1 “'j - —
—_-.- 2: (X.-X) (x. .-X)
AC. = n 3 i=1 1 1+3 , j = l,......k
J 1 “‘1 — 2
E .2 0‘1"”
1:1

Where, X is the mean and n is the number of observations in the input
time series.

A variance-covariance matrix (lower triangle) of generalized form
is depicted below to identify the individual elements (Figure 1). Each
row or column represents a month of the year. Therefore, a variance-
covariance matrix for monthly observations over a five year period

would be dimensioned 60 x 60.

26

Figure l. Generalized Variance-Covariance Matrix

v1

C1,1

1
1,1 v1

c2,1 v2

2,3 c2,2 C2,1 v2

C C C

1,2
2,3 2,2

2,4

OOOO<

0000

2.5 2,4 2,3 2,2 2,1 V2\

27

where, V ...... Vn is the estimated variances (diagonal) for quarter

1,
one through the nth quarter, respectively; Cij is the covariance for
the ith quarter for i = l, ...... ,n and the jth lag for j = l, ...... n-l.
The variance within a quarter was the basic unit used to generate the
covariances (off-diagonals) for the variance-covariance matrix.

The second Fortran program (PRGZ) calculates the estimated variances
and covariances for an input time series for each variable (up to 10
variables) for up to five years of monthly observations. The variance-
covariance matrix, for each variable is written on a tape from which
the third Fortran program (PRGS) calculates (V") the inverse for each
variable. The inverted matrices then are entered into the normal

equation in the fourth program (PRG4), and used in calculating the

quarterly solutions (b) for each variable.
b = (X! V" X)" X' V" Y

The IMSL Library (1979), an extensive collection of mathematical
and statistical subroutines written in Fortran, was used to perform
most of the matrix manipulations and computations. Appendix A, Table
1 contains a complete list and brief description of the subroutines
used. The Fortran programs (Appendix A) are well documented with
comment cards interspersed throughout to allow one with a basic knowledge

of Fortran to follow them.

Simulation Model
A correlation technique adapted from that described by Clements
23.2}: (1971) was used to simulate events based on the quarterly

solutions computed by Program Four (PRG4). Variability estimates were

28

based on the quarterly estimates, a triangular matrix derived from the
historical variance-covariance matrix and the assumption that the input
data were normally distributed. The simulation procedure generates for
each variable a random series which is normally distributed and appropri-
ately correlated with the series generated for the other variables con-
sidered. This procedure assumes that each variable will react to changes
in the other variables in the way that was observed during theperiod
(up to 5 years) which provided data for the variance-covariance matrix
(calculated in PRGl) for all variables over the entire time period.
According to Anderson (1958), if Z is a normal vector with mean u
and variance V, there exists a unique upper triangular matrix C such

that,
X = C Z + u

In this case (X-u) has a variance-covariance matrix V = CC'.
In order to obtain C from V the so-called "square root method"
can be used which provides a set of recursive formulaes for the com-

putation of the elements of C.

 

_ 11
C11 - 8 , l :_1 §.m
a.
11
1'1 1
i' = (oii' Z C..2)1, 1 < i :_m
J k=l 13
j-l

C.. ..- 2 C. C. C.., 1 < ‘ < ‘ < m
1] (OIJ k=1 1k Jk)/ JJ 3 1-

29

Since C is an upper triangular matrix, (Cij = 0 for all j < i) after
obtaining the elements of C, all components of X can be determined
from Z as weighted sums: Xi = sum (Cij Z1 + Hi).

The generation of a random vector X with mean u and variance-
covariance matrix V can be programmed in the following steps.

1) Obtain the triangular matrix C from V.

2) Generate m independent random normal variates.

3) Perform the matrix—vector multiplication and vector addition.

The result is a normal vector from the multivariate distribution
defined by u. V and a probability density function of X. In this way
two or more (up to 10) correlated random normal variates can be generated

from independent normal variates by the transformation process of X =

CZ + u.

Test for Normality

 

According to Gill (1978) the assumption that the errors are normally
distributed is not essential to the partition of variance and develop-
ment of point estimators but is critical in probability statements about
the reliability of estimates (confidence intervals) and decisions based
on tests of hypotheses. Two statistical facts lending support to the
appropriateness of the assumption of normality for a large proportion
of practical cases are; l) the central limit theorem which establishes
approximate normality of means from all but very small samples; and 2)
the f test of the hypothesis of treatment effects is known to be robust,
i.e. the probabilities of errors of Type I and Type II are little affected
by moderate departures from normality.

The analysis of variance test for normality described by Shapiro

30

and Wilk (1965) is designed to provide an index or test statistic to
evaluate the assumed normality of even small samples (n < 20). For

n < 50) this test compared to goodness of fit utilizing chi-square
distributions or Kolmogorov-Smirnov non-parametric test, is considered

a somewhat better approximation for small samples and is quite sensitive
to a wide range of departures from normality such as skewness and kurtosis
of the distribution (Gill, 1978).

The W statistic used to test for normality is scale and origin
invariant and hence supplies a test of the composite null hypothesis
of normality. Non-normality is indicated when the calculated "W"
statistic is smaller than the appropriate critical value. By studying
the distributional characteristics of a body of data one may be encour-
aged to consider, for example, normalizing transformations, the use of
distribution-free techniques, as well as detection of gross peculiarities
such as outliers or data errors.

By the central limit theorem, the actual distribution of a variable
can sometimes deviate considerably from the normal distribution without
significantly affecting the final results. The major consequence of
non-normality is that, without knowledge of the distribution of the
variables, precise probability statements cannot be made. The Shapiro-
Wilk test was used to test the normality assumption for the input data
(raw and deflated) as well as the quarterly solutions and generated

output .

An application of the statistical procedures and simulation technique

 

The statistical estimates and simulated probability distributions

were used to analyze a practical example, involving a comparison between

31

leasing and purchasing dairy cattle. A computerized dairy herd growth
model (TELPLAN 52) was used to project cull cow and bull calf numbers
over a four year period for 100 bred heifers. Nott and Sargent (1975)
developed the recursive, deterministic growth model capable of calculat-
ing dairy livestock inventories for individual farm situations. The
model was verified with farm accounting and DHIA summary statistics.

The statistical estimates for cull cow and calf prices in Michigan

for the period 1975-1979 were used in the example. A discussion of

the results of the capital budgeting analysis is included in the

Results and Discussion section, to follow.

RESULTS AND DISCUSSION

Evaluating the Normality Assumption

 

The Shapiro-Wilk method was used to test the assumption of normally
distributed residuals, since the simulation procedure assumes a normal
distribution. The 11 sample time series (Table l) were analyzed over
four, three year time periods, (1967-1969, 1970-1972, 1973-1975, 1976-
1978) and for one year (1979) individually. The results generally
indicated that the price series were non-normally distributed. An
attempt was made to rid the series of the effects of changes in the
general price level by deflating the original data. It was felt that
by removing the effects of movements in the general price level, the
deflated price series would be a better representation of price
variability related to factors other than the movements in the general
price level. It was also felt that the deflated series would more
closely approximate a normal distribution, in which case the assump-
tions of the statistical and simulation procedures would be met.

Two USDA monthly indices were used to deflate the selected price
series. The index of prices received by farmers for feed grains and
hay (IPRFGH) and the index of prices received by farmers for dairy
products (IPRDAIRY) were the two indices used. Table 1 shows a com-
parison with respect to the test for normality between the deflated
and undeflated price series for 11 monthly price series for the years

1967 through 1979.

32

33

TABLE 1. A Comparison Between Original and Deflated Time Series with
Respect to Normality

 

Price Series Years
I T r 1 fi' I f I I
67- 69 70- 72 73- 75 76- 78 79
Milk D D D D D
*
Milk cows D A D D C)
Cull cows D D D D D
Calves F D D F D
*
Soybean oil meal D F C) N F
16% grain mix D N F F F
Alfalfa hay F F F F D
Corn N D F N D
* 1'
Wheat N C) N F C)
Oats N F F F D
Soybeans N N N N D

 

1Letter designation in each cell represents form of data most closely
approximating a normal distribution.
N = not deflated
D : deflated by index of prices received by farmers for dairy products.
F a deflated by index of prices received by farmers for feed grains and
hay.

*

Indicates a probability (P<.Ol) of rejecting the non-normality hypothesis.

The results presented in Table 1 would appear to indicate that
deflating generally resulted in a more normally distributed price series
for those examined. The IPRDAIRY index seemed to be the appropriate
deflator to use for milk, milk cows, cull cows and calf prices across
time periods. The choice of a deflator for the remaining price series
would not be as obvious. Undeflated soybean price appeared to be more
normally distributed than when the series was deflated by IPRFGH. The

IPRFGH index was most effective in terms of normalizing the soybean oil

34

meal, 16 percent dairy concentrate, alfalfa hay and oat price series,
although was not consistent across time periods.

Tomek and Robinson (1972) point out that if the index selected
contains the price series being deflated, then the coefficients relat-
ing the deflated variables to other variables are likely to be biased.
The amount of bias depends upon the relative weight which the price
series being deflated has in the index. In other words, since the
index changes in part as a function of the change in the particular
commodities price, deflating by the index tends to cancel the influence
of the price change. An alternative to deflating would be to include
the price indices as separate independent variables. However, since the
price indicies often have strong trends, they are often highly inter-
correlated with other explanatory variables.

Based on the results of this portion of the study, deflating the
price series appears to have some merit in terms of normalizing the
series. It should be noted that the results of the statistical analysis
using the Shapiro-Wilk test of normality indicate that the undeflated
price series (of those analyzed) do not approximate the characteristics
of a normal distribution as well as the deflated series do. This becomes
an important consideration with respect to the simulation procedure,
which is based on the normality assumption in terms of generating the
simulated data. Assuming the appropriate deflator were chosen, the
question of how the output from the deflated model is to be interpreted
remains an important one. If one were to develop a model to predict
future indices then: there may be some value in deflating the price
series and using the index prediction model to predict future prices

based on the model of changes in the general price level. This predictive

35

capability could certainly add a great deal to the techniques developed

in this study.

A statistical comparison between the original and simulated data

The means and variances computed for the simulated data are con-
sidered "best estimates" of the population parameters, u and 02,
respectively. Statistical tests were performed to compare the means
and variances of the simulated population with those of the original
input time series.

In testing the difference between the input data sample variance
and the generated sample variance, the ratio of the two sample variances

was used to calculate the f statistic. The function (812/012) /

(522/022) is the F distribution with r and r degrees of freedom

1-1 2-1
for independent samples of size r1 and r2 for two normally distributed
variables. Therefore, the hypothesis that 012 = 022 is tested against

the hypothesis that the population variances are not equal. The value
of the test statistic is the ratio of the sample population variances,
f = Slz/SZ2 and is compared with respective critical values, £1-0/2,
rl-l, rZ-l found in an F distribution table. Thus, if the test
statistic, f = S12/822 is larger that the upper-tail critical value,
one may reject the hypothesis that the population variances are equal
and conclude, (for the selected significance level, l-a) that the S1
sample data shows less variability than the 82 sample.

The results of comparing the sample variance (512) associated with
the original input time series (n = 60, i.e. five years) with the sample

variance ($22) for the simulated data (n = 1000, e.e. five years) were

as might be expected. The test statistic was in general consistently

36

less than the upper-tail critical value (£0.025,999,59 = 1.48) for cases
2 2 2 2

in which 52 > S1 , and £0.025,59,99 g 1.39 for cases in which S1 > 52 .
The values for the sample population standard deviations are shown in
Table 2, along with the results of the statistical test used to compare
sample variances. Since this test should be applied to normally dis-
tributed variables.the results of the non-deflated data were compared
with the deflated data, since the non-deflated data were shown to be
more normally distributed. While deflating the data obviously removed
a large portion of the variation associated with movements in the general
price level, the simulated sample variances were statistically different
for the input sample in six of the 54 cases tested, compared with two
incidences of dissimilar variances for the non-deflated data. The simu-
lated sample variance appeared to be consistently larger, though not
necessarily statistically different than the input sample variance for
most cases tested. No explanation was made for this trend in the data.

The simulated and input data were also compared with respect to
their respective sample means. The test of difference between the two
means was made using the Wilcoxon's rank sum test. This test is used
fer comparing the means of two populations having the same but unspecified
distributions. The IMSL Library (1979) subroutine NRWRST was used to
perform the test. The results showed that the hypothesis of equal means
(simulated vs actual) would be rejected only in those few cases indicated
in Table 3, for the selected time series.

These analyses were performed as an attempt at verifying the validity
of the statistical and simulations procedures carried out by the computer

programs. Based on these analyses one should have some confidence in

concluding that the sample variances and sample means for the original

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38

Table 3. Results of Wilcoxon's Rank Sum Test Comparing Simulated Mean
with Actual Input Data Mean

 

Years

 

Price series 1967 - 71 1972 - 76 1975 - 79

 

Corn *
Wheat *

Oats

Soybeans * *

Hay *
Milk

Milk cow *

Cull cow

Calf

 

*
Indicates hypothesis of equal means was rejected (a > .10)

data are quite similar to those generated by the simulation procedure.

The major shortcoming of the computer programs developed to perform
the statistical analysis are with respect to the design of their data
handling capabilities. Each data point is in effect handled only as a
monthly observation. In other words, variables which generally call
fer annual observations would be treated as if the observations were
monthly. As a result, three annual crop yields would be treated as a
"quarter" (i.e. three months). The output data must be used and inter-
preted assuming the input data represents monthly observations. The
statistical programs and simulation procedure are designed to handle
the computations for from two to ten variables. However, each variable
must necessarily have the same number of observations as all other

variables considered during one pass through the program sequence.

39

Thus, for example, annual crop yield and monthly crop price data are not
compatible in the same model. It would however, be possible to run the

program with the monthly data initially and then use the yearly data.

Application of Simulation Technique to Capital Budgeting

The statistical and simulation procedures can be used to incorporate
the variability of a wide variety of events into an investment analysis.
In addition to the leasing versus purchasing example described below,
ones imagination is the only limit to the application potential of this
technique. Events such as seasonal and year-to-year variations in milk
production levels, and yearly crop yields would be expected to have a
major impact on various managerial decisions. The variability in the
level of milk production and milk price, for example, over the course
of expanding the size of a dairy enterprise should certainly be incor-
porated into a complete farm budget analysis. Analysis of variations
in estimated gross income per acre or per animal unit are examples of
variables which include the variability owing to both physical produc-
tion and prices.

A relatively simple example was developed to illustrate the use ”
of the statistical and simulation procedures within a capital budgeting
framework. The example used compares the relative profitability of
leasing 100 bred dairy heifers versus purchasing the same. The example is
designed to demonstrate the kinds of infermation the capital budgeting output

provides the decision maker with when the simulation technique is incorporated

40

into the analysis. The results of the "point estimate" approach are
compared with those generated when the statistical solutions and sim-

ulation technique is used.

Leasing versus_purchasing dairy cattle - an example

 

In general, a lease is a contract by which the lessee acquires
sole use of an asset in return for lease payments. The lease payments
reflect the value of the leased asset and the lessor's (owner of the
asset) carrying costs and his profits. Thus, a lease is similar to a
conventionally financed installment loan. A dairyman entering into a
dairy cattle leasing agreement generally agrees to maintain a specified
number of cows for the duration of the agreement (usually for 3 to 7
years).

Leasing agreements differ greatly with respect to culling leased
cows, replacing culled cows, disposition of calves born to leased cows
and disposition of leased cows at the time the lease expires. The
dairyman typically receives the milk income from the leased cows.
Therefore, when comparing leasing with purchasing one may assume that
milk receipts will be the same, if the dairyman is comparing purchasing
a group of cows with leasing the same group of cows. Thus, a comparative
analysis need only include differences in costs and returns due to
leasing or purchasing.

Determining whether a change in the farm business will be profitable
or not does not require a complete budget. A partial budget is a plan
that lists only the receipts and expenses which are expected to change
with the proposed change. In order to evaluate and compare the costs

and returns over time, for any method of acquiring control of an asset,

41

one must recognize the time value of money. Net present value (NPV)
analysis or capital budgeting is a technique used to evaluate the pro-
jected cash flows for an investment or to compare investment alternatives.
This technique dismants the projected cash flows (outflows or costs and
inflows or returns) to their present values (their equivalent worth
today).

There are seven basic types of information required for an NPV
analysis. Namely, l) the initial investment of equity (owned) capital;
2) the annual cash flows (cost and returns) attributed to the investment;
3) the length of time over which the analysis is being made; 4) the
salvage value (if any) of the investment; 5) the interest or discount
rate or required rate of return; 6) the applicable marginal income tax
rate for generating results on an after-tax basis; and 7) the depreciation
method used for old or new assets.

In the example chosen, the only costs associated with leasing are
the monthly lease fees since only those costs and returns that differ
with respect to leasing versus purchasing need to be considered. In
this particular example the leasing company would receive all cull
leased cows and pay the dairyman a fixed rate ($30/head) for all bull“
calves born from leased cows. The cost side of the comparison for the
purchasing example includes only the purchase price. The returns
associated with the purchase alternative include, the market value of
culled cows and the actual market value for the sale of bull calves.

In addition to these costs and returns there are important tax considera-
tions that need to be included in the analysis.

In the case of the leased animals, the lease fees are the only

deductible expenses associated with leasing. The example assumes that

42

the lessor chooses to pass the investment tax credit to the lessee,
although it does not enter the analysis since it is assumed that the
same tax credit would be available under the purchase alternative.

In the case of the purchased animals, depreciation, and the interest
paid on borrowed capital are included because of the effect they have
as tax shields on cash flows.

The acceptability and ranking of investments based on discounted
cash flows depends on the sign (positive or negative) and the relative
magnitude of the resulting NPV figures. In this example, since it was
assumed that actual cash receipts from milk sales would be the same
whether the animals were purchased or leased, the resulting NPV figures
would be expected to be negative as the costs (negative NPV) outweigh
the tax credits and income from the sale of calves (leasing and purchas-
ing) and the sale of cull cows (purchasing). Therefore, the smallest
negative NPV would be the favored investment.

The dairy herd growth model (TELPLAN 52) was run for 100 purchased
bred heifers to project the number of culls and bull calves to be
expected over a four year planning horizon. Average death losses and
cull rates were used to generate the results. Death losses for purchased
bred heifers were assumed to be 2, 4,3 and 2 percent for years one
through four, respectively. The average assumed cull rates for purchased
bred heifers were assumed to be 35, 33, 28 and 26 percent for years one
through four, respectively. Death losses for bull calves from birth to
sale as newborn calves was assumed to be five percent. The average age
at which heifers freshened was set at 24 months and an average calving
interval of 13 months was used. The heifer raising strategy used to

generate the desired results was one in which no heifers were raised as

43

replacements. The results from the animal inventory analysis are

presented in Table 4.

Table 4. Animal Numbers from 100 Bred Heifer Over Four Years

 

Year
1 2 3 4
Cull cows 34 21 10 1
Bull calves 47 31 21 15

 

Table 5 presents a net present value analysis of the costs and
returns associated with leasing versus purchasing 100 bred heifers
using the conventional "point estimate" approach and the simulation
technique developed in this thesis.

Since —$29,702 (NPV purchase) is less negative than -$3l,l89
(NPV lease) based on expected values, one would conclude that purchas-
ing is the preferred alternative. However, the range and distribution
of NPV's around the expected value resulting from the simulation pro-
cedure indicate there are times when the leasing option should be the
preferred option in this example.

According to Gill (1978), for any normally distributed random
variable Y, the transformation, 2 = (Y - uy/o, always leads to a normal
distribution with mean 0 and variance 1, where "y is the population
mean and o is its' standard deviation. The probability density function

for the standard normal form is:

44

Table 5. A Comparison of Approaches to Net Present Value (NPV)
Analysis of Costs and Returns for Leasing versus
Purchasing 100 Bred Heifers1

 

LEASE Present Value
Costs:
Lease fees ($10.50/head/mo) $41,400
Total costs $41,400
Returns: 2
Lease fees (tax credit) $ 7,988
Sale of bull calves ($30/head/ml) $ 2,223
Total returns: $10,211
Net Present Value = (-$41,400 + $10,211) = $~31,189
PURCHASE
Costs:
Purchase price ($600/head) $60,000
Total costs: $60,000
Returns: 2 3
Interest (tax credit) ’ 2 4 $ 1,744
Depreciatgon (tax credit) ’ $ 4,064
Cull cows : Expected P.V. $20,115
Range $16,513 - $24,083
Standard deviation $ 1,708
Bull calves: Expected value $ 4,375
Range $ 3,686 - $ 4,963
Standard deviation
Total Returns: Expected value $30,298
Range $26,817 - $34,024
Standard deviation $ 1,742
Net Present Value: Expected value -$29,702
Range -$33,183 - $-25,976

Standard deviation $ 1,742

 

110% discount rate over 48 months

22095 marginal tax rate

31095 annual rate of interest
4straight-line depreciation: salvage value of $200/head

Sassumes average of 12.5 cwt/head

45

FZ(z) = (1//§F) exp (22/2)
and the cumulative density function is:

z.
p(z < z) = F2 (2) = _£ 1 f(z) dz

Initially, one must establish with some degree of confidence that
the distribution approximates a normal distribution, prior to making
standard normal transformations and probability statements.

The third and fourth moments for the 1000 simulated net present
values were 0.080 and -0.643, respectively. The moments of a sample
can indicate a great deal about the shape of the parent population's
distribution. The third moment is used to determine whether a distri-
bution is symmetric or skewed about the mean. Skewness is estimated by
calculating the ratio of the third central moment to the standard
deviation cubed. The third moment is equal to zero when the distribution
is symmetric about the mean. A distribution is said to be skewed to the
right or positively skewed when the third moment is positive and left
or negatively skewed when the third moment is negative. It is generally
accepted that a distribution is symmetric about the mean when the value
of the third moment lies between -0.5 and 0.5. The fourth moment
(kurtosis) is used to interpret the flatness or peakedness of a distri-
bution. Kurtosis is estimated by dividing the fourth central sample
moment by the standard deviation raised to the fourth power and then
subtracting three. The fourth central moment is also zero for a random
variable distributed exactly normal (Gill, 1978). This would indicate

that this sample population has a distribution which is skewed slightly

46

to the right and is somewhat peaked. However, the sample appears to
approach normality. A normal distribution would be expected since the
simulation procedure assumed normality in that random normal deviates
were used to generate the simulated values. The Shapiro-Wilk test for
normality was not uSed because the sample sizes were greater than 50
which is theelimit for this procedure.

Once the values of the random variable have been transformed to
units of 2 (standard deviations of Y) probabilities can be evaluated.
In order for the leasing option to be equivalent or preferred over the
purchase option, the NPV of the purchase option must be -$3l,l89 or less.
The NPV for the lease (-$31,189) can be transformed to standard normal
form in order to evaluate the probability of leasing being the more
profitable alternative. The transformation procedure for this example

is shown below.

A = (Y - u)/o
21 = ((-31,189) - (-29,702))/1742
z. = -0.8536
1

Probability (Y < ~31,189)

Probability (Z < -0,8536)

Probability (z < -0.8536) = 0.1967

This result indicates that although the "point estimate" approach
using expected values would result in a decision maker accepting the
purchase option as the most favorable option the results of the simula-
tion procedure indicates there would be times in which the leasing option

would be preferred. For this specific example, the calculated probability

47

would indicate that leasing would have been the more profitable alter—
native 20 percent of the time.

However, Gill (1978) warns that probabilities cannot be evaluated
without knowing the values of ”y and o for the defined population of
interest. In cases where large amounts of empirical evidence about the
values of my and a have been accumulated, one may assume the values of
the parameters and proceed directly with questions of probability. It
can be shown that with information from 1000 subjects from the relevant
p0pu1ation, one may be approximately 95 percent confident that the
population variance lies in the interval 0.95 Sy to 1.05 Sy i.e., with
1000 observations the sample variance still is subject to an error of
5 percent (or more). However, somewhat fewer observations are required
for equivalent reliability of estimates of the oppulation mean, unless
the coefficient of variation is relatively large.

Therefore, even with 1000 observations (simulations) on Y, in the
example, in which Sy = 1742., one can be 95 percent confident that
the population variance lies within 0.95 and 1.05 times the sample
variance. Thus Sy would lie (with 95 percent confidence) between 1698

and 1785. Therefore, the probability that leasing would have been the

preferred option would lie between 19.1 and 20.2 percent, as shown below.

Zj = ((-3l,189) - (-29,702)/1968 = -0.8757
Probability (Z < -O.87S7) = .1906
Zk = ((-3l,189) - (-29,702))/1785 = -O.833l

Probability (Z < -0.8331) = 0.2024

48

Agguide to the use of the statistical and simulation programs

The techniques developed in this study were designed to provide
an objective measurement of the uncertainty or variability associated
with prices, cost and income, based on historical price series. If the
desire for income stability is strong, farm managers may consider com-
bining enterprises that could be expected (based on historical data) to
reduce the variability of annual incomes even at the cost of some re-
duction in average income over a period of time. The resulting variance
of income from a combination of more than one enterprise is dependent
upon the variability (variance) of the individual enterprises to be
considered, and the degree of association or correlation (covariance)
of the returns of these enterprises. If for example, farm resources
were divided equally among two or more enterprises, total variance for
the farm business would be reduced provided the variances (measure of
variability) of the individual enterprises were approximately equal and
there was less than a perfect correlation (price movements in the same
direction for inputs and outputs) between enterprises.

Decisions made with respect to capital expenditures are among the
most difficult managerial problems. Most investments occur over a
considerable time in the future, and therefore considerable effort is
needed to predict probable costs and returns of each alternative.
Secondly, often times most, if not all, of the capital must be laid out
immediately, while benefits or returns occur over time. Thus, a
decision-maker must balance added returns that will occur in future
years against an expenditure that will be made immediately. Under-
standing that the value of money is influenced by time is important in

evaluating the profitability of investment opportunities. A dollar

49

received or spent some time in the future is not worth a dollar today.
Discounting is the process used to find the present value of a given
amount of money to be received or paid in the future. Net present
value is simply the difference between the present value of benefits
and the present value of costs.

Because there are few cases in which prices, costs and income
levels are known precisely beforehand, an analysis of alternative
outcomes is essential to making a good decision based on the infor-
mation available. One common way in which risk or uncertainty is
incorporated into an investment analysis, is to estimate different
levels of costs and benefits and compute the net present value for each
combination. These estimates are typically considered: 1) the best or
most reasonable; 2) pessimistic; 3) optimistic. Such estimates provide
a basis for taking into account the consequences of unexpected or
unforseen situations associated with the investment. Calculating the
net present value for each estimate represents only a few points on a
continuous distribution of possible combinations of future events.
Since every factor that enters into the evaluation of a specific
decision is subject to some uncertainty, the decision maker needs a
portrayal of the effects that the uncertainty surrounding each of the
significant factors has on the returns he is likely to achieve. The
overall objective of this study was to develop a procedure which combines
the variabilities inherent in the factors considered. Historical price
data is used to compute quarterly or yearly estimates for each variable
entered into the procedure. These estimates are then used as input to
the simulation model. The purpose of the simulation model is to generate

a distribution (1000 generated values) for each input price series. This

50

goes beyond representing the distribution of prices with only a few
points. The simulation procedure also recognizes that prices fer
different variables are interrelated or correlated. The simulation
technique generates distributions for each variable, that are correlated
with the other variables as dictated by the historical input price
series. A probability distribution represents the odds of achieving a
particular value or range of values based on the input time series.

The following discussion describes, for the user's benefit, the
set of five interactive computer programs developed in this study. The
first four programs perform the statistical procedures described in
detail in the materials and methods section of this thesis. Monthly
data can either be read from a computer disk file or input directly by
the user typing it into the computer terminal. The programs will accept
monthly observations for from two to ten variables over a time period
of up to five years for each variable. The statistical programs (PRGl
thru PRG4) compute both quarterly and yearly best, linear, unbiased
estimates (B.L.U.E.) from the input series. A variance-covariance
matrix (which is a representation of the variability associated with
each variable over time, the "co-variability" between variables and
across time periods) is used to generate another matrix (unique trian-
gular matrix) which is used in the fifth program to generate the simu-
lated data. If the user is interested, he can elect to have the
variance-covariance matrix, correlation matrix and unique triangular
matrix printed out.

After the statistical computations are completed by the first
four programs the simulation program (PRGS) uses the statistical

solutions to generate the simulated data. These simulated data are

51

normally distributed about the statistical estimates and appropriately
correlated across variables and across time periods. These simulated
data represent a "best estimate" of a sample probability distribution
for each variable, thereby providing an estimate of the distribution
of values for each variable.

By incorporating these distributions into a capital budgeting
model, a computer program can be used to carry out the discounted
cash flow calculations a large number of times to generate an output
to which probabilities can be attached. In other words, instead of
outputting a single value as a measure of an investment's worth (as is
most commonly done), a decision maker would be presented with a distri-
bution of net present values or rates of return. Sensitivity analysis
could then be used to determine how sensitive the results are to a
change in the tax rate or discount rate, by rerunning the program for
each change. This thesis contains an example in which the generated
distributions for dairy calf and cull cow prices were incorporated
into a capital budgeting model to illustrate this technique.

Listings of the source programs (PRGl thru PRGS) are included
in Appendix A, Tables 3 through 7. A sample output is also presented-

in Appendix A, Table 8.

Procedure for running_the prpgram
1) Sign onto the computer (See CDC Interactive Terminal User's Manual).
2) Type: ATTACH,EX,TRSEXEC. (Hit RETURN after each input line.)

3) Type: EX.

4) The program will ask the user if he would like a brief description

of the programs. ALL yes/no questions should be answered with:
l=YES; 0=NO.

5)

6)

7)

8)

9)

52

The programs now begin executing in sequence.

Each program will ask the user several questions relative to the
data input during execution of program 1 (PRGl). When the correct
response has been entered the program will continue to execute.

Program 1 will ask whether the data is written on a disk file or
whether it will be entered by the user. If the data resides on a
disk file (data for one variable followed by data for the next)
the program will not execute properly until the data file has been
attached as TAPEl (i.e., ATTACH,TAPE1, user's data file).

The user can receive a copy of the full matrix output should he
desire it, by indicating this in response to a question asked in
the beginning of program 1. (Certain statistical results will
always be printed during execution).

After having completed the series of five programs, TAPEll will
contain the simulated data, with one variable followed by another
in the order in which they were input. This file may be saved and
cataloged for future use by typing: CATALOG,TAPEll,your data file
name.

If at any time the user wants to stop execution of the programs

he can do so by pressing the ESC key. This will immediately terminate

the program sequence. To rerun the program follow the instructions

outlined previously.

For those interested in altering the programs in any way, the

following steps should be followed:

ATTACH,P*,TRSPRG*. (where, * is the program number)
SYSTEM,FORTRAN.
OLD,PE,FR,1,BY,1.
If you want a program listing, type: LIST.
SAVE,XX,NS.
RETURN,P*.
CATALOG,P*,your file name.

The user's copy of the program can now be altered without affecting

the existing program.

SUMMARY AND CONCLUSIONS

Conclusions

 

The statistical procedure used in this study to compute unique
solutions for price time series data represents a major improvement
with respect to the input to a simulation model designed to generate
probability functions for each variable considered. These quarterly
or yearly solutions (best linear unbiased estimates) were incorporated
into a simulation procedure designed to generate correlated time series.
Although, the simulated sample variances appeared consistently larger
than the original series, a statistical test used to compare the
variances indicated they were not significantly different. It was
demonstrated that the deflated time series were more normally distri-
buted than the non-deflated series. It was concluded that there is a
need for further study in the area of characterizing the distributional
characteristics of time series data. An example was provided illustrat-
ing how the simulated distributions could be incorporated into an
investment analysis. The output from the capital budgeting analysis
provides a decision-maker with both the expected return based on the
probabilities of all possible returns and more importantly, the expected
variability in returns. Decision-makers can take the various levels
of possible cash flows, and get estimates of the odds for each potential
outcome from a net present value analysis. Such a procedure could also
be used to produce valuable information about the sensitivity of possible

outcomes to the variability of input factors and to the likelihood of

53

54

achieving various net present values or rates of return. To have
calculations of the odds on all possible outcomes (based on statistical
estimates) lends some assurance to the decision-maker that the available
information has been used with maximum efficiency.

The model, Yij = A1 + E.., is a one-way classification model with

11
fixed effects. Unique solutions are obtained from what would typically

be a non-full ranked model by constraining the overall mean (u 0).
Only in this way is the model a full-ranked model with a unique inverse
(X' V" X)", and therefore unique solutions. The interactive computer
programs compute the quarterly estimates for monthly time series data
for up to ten variables over a time period of up to five years.

Recognizing that the error terms for time series data are typically
correlated across observations, the autocorrelations were used in the
development of the variance-covariance matrix. This procedure gives
greater weight to those error terms with smaller variances. Autoco-
variances were used as estimates of covariances across months within
quarters, between quarters within years and between quarters across
years. In this way, the variance-covariance matrix V used in the normal
equation to compute unique solutions incorporated trend into the linear
model.

If one assumes that the future variability associated with a
particular set of conditions is closely related to past variability,
empirical estimates do provide a reasonable basis for making short
and long run decisions. Therefore, the problem becomes one of selecting
a length and time period most representative of expected variability in

the future. Five years were selected as the maximum period over which

the statistical analysis can be performed. This decision was based

55

principally on computer memory limitations and computing costs. The
approximate cost of inverting a variance—covariance matrix for each
input time series represents the major cost associated with executing
these programs. The approximate cost of inverting a variance-covariance
matrix representing five years of monthly observations (60 x 60) is
$1.00 per variable. As the size of the matrix increase the rate of
increase in the cost of computing the inverse also increases. Thus,

it was felt that a five year time period was a reasonable period of
time in terms of characterizing price variability and at the same time
holding down the compuuurmemory and computational requirements.

The simulation model used to generate the probability distributions
is based on the assumption that the input variables are normally dis-
tributed. Statistical tests were conducted on 11 price series (milk,
milk cow, cull cow, calf, corn, wheat, oats, soybeans, hay, soybean
oil meal, 16 percent concentrate) for three year time periods from 1967
through 1979. The results from the Shapiro-Wilk test for normality
indicated that removing the effects of changes in the general price
level by deflating, improved the distributional characteristics of the
input time series. The index of prices received by farmers fer dairy"
products seemed to be an appropriate deflator for milk, milk cow, cull
cow and calf prices across those time periods used. The index of
prices received for feed grains and hay was effective as a deflator in
terms of normalization, for soybean oil meal, 16 percent dairy concen-
trate, alfalfa hay and oat price series. Although these results were
not consistent across time periods. The basic problem associated with
deflating time series is the bias introduced as a result of the series

itself being a part of the index. The choice of an appropriate index

56

and the interpretation of the output from a deflated model are difficult.
Further research needs to be done in the area of describing the distri-
butional characteristics of price series. All too often, researchers
assume normal distributions without making any attempt at veryifying
their assumption.

The tests used to compare the original input price series with the
simulated series proved to be encouraging. The variancesand means did
not appear to be greatly different for those time series tested, thus,
validating the performance of the statistical and simulation procedures.
Although the simulated sample variance appeared consistently larger,
this difference was not shown to be significant.

The value of computer programs in developing clear portrayals of
the risk or uncertainty associated with alternative investments has
been demonstrated. An investment analysis using discounted cash flows
(capital budgeting) should provide the decision maker with more infor-
mation than the expected net present value or internal rate of return.
By incorporating probability distributions for price, income or cost
variables into a capital budgeting model, the infbrmation generated by
the analysis represents a substantial improvement over conventional
methods of incorporating uncertainty into an investment decision model.

An illustration of the risk analysis technique using capital
budgeting was presented using the correlated price series for two
variables. Michigan cull cow and bull calf price data for the four-year
period 1976-1979 were used to compare leasing dairy heifers with
purchasing the same. The procedure used the statistical estimates and
resultant simulated probability distributions for the input price series,

to generate a random normal series of net present values for the

S7

purchasing option. Values of the random series were transformed to
standard normal form and probabilities evaluated. The numerical results
using expected values (point estimates) would have caused one to select
the purchase option. However, the results differed when the probability
distributions for the two correlated price series were incorporated into
the model. These results would offer the decision maker a probability
distribution of net present values to which the probability that a value
will occur can be computed. In the example, the leasing option should
have been the preferred option approximately 20 percent of the time.
However, it should be recognized that the generated results will nec-
essarily be indicative of conditions during the historical price series
for the four-year period selected. If one assumes that the variability
of the recent past is a relatively good indicator or predictor of future
variability, then this approach to investment analysis offers some
improvement over other attempts at incorporating measures of risk.

The major advantage of a probability analysis of investment
alternatives is that it results in a distribution of values to base
decisions on. Whether the results are in terms of net present value,
internal rate of return or other criteria, probability statements can“
be made.

There are limitations to the techniques and procedures developed
and examined in this study and a great deal of room for improvement.

The key to improving decision making aids or models is using the avail-
able data and information, be it historical series or predictions, to
the fullest extent possible. A good decision is one in which the
resource allocations are most likely, in a probabilistic sense, to

produce favorable outcomes. Good decisions only improve the chance of

58

favorable outcomes, they do not guarantee them!

APPENDIX A TABLES

Appendix Table 1.

BECORI-

BECOVM-

BEMMI-

FTAUTO-
GGMML-
LINVZF-
USMMMX-
USWFM-
VCVTFS-
VCVTFS
VMULFF-
VMULFM¥

VMULFP-

59

International Mathematical and Statistical Laboratory
(IMSL)

Estimates of means, standard deviations, correlation
coefficients

Estimates means and variance-covariance matrix

Estimates of means, standard deviations, third and fourth

moments

Estimates variance, autocorrelation, autocovariance

Normal

Matrix

or Gaussian random deviate generator

inversion, full storage mode, high accuracy version

Determines minimum and maximum values in a vector

Prints
Matrix
Matrix
Matrix
Matrix

Matrix

a matrix stored in full storage mode
storage mode conversion-full to symmetric
storage mode conversion-symmetric to full
multiplication-full by full
multiplication-transpose of A by B

multiplication-A by transpose of B

Appendix Table 2.

A

AMAT

AMINPUT
AMQRTS
AP

C

CV

D

DENOM

EINV
F

FVCVU

IDGT

IER
INCD
IVNBR

NOBSSM

NQOBS

NQRTS

60

Computer program variable list

MATRIX (NMONTHS X NMONTHS) FULL STORED VBLOWUP (TAPES)
GOES INTO INVERSION PROGRAM

MATRIX (NVARS X NVARS) UNIQUE TRIANGULAR FROM TRSPRGl (AMAT)

MATRIX (NVARS X NVARS) UNIQUE TRIANGULAR MATRIX (UPPER TRIANGLE)
OF A- GOES INTO TRSPRG4

MATRIX (NQOBS,NVARS) OF INPUT VARIABLES

MATRIX (3 X NQRTS) OF MONTHLY OBSERVATIONS WITHIN QUARTERS
MATRIX (NVARS X NVARS) UNIQUE TRIANGULAR MATRIX (LOWER TRIANGLE)
MATRIX (NQRTS x NMONTHS) X-PRIME*V-INVERSE MATRIX PRODUCT S*A
COEFFICIENT OF VARIATION

MATRIX (NQRTS,1) X-PRIME*V-INVERSE*Y

DENOMINATOR FOR WEIGHTED MEAN CALCULATION SUBROUTINE: WEIGHT
MATRIX (NQRTS,NQRTS) X-PRIME*V-INVERSE*X

MATRIX (NQRTS,NQRTS) INVERSE (X-PRIME*V-INVERSE*X)

MATRIX (NQRTS,1) SOLUTIONS

FULL STORAGE OF VAR-COV MATRIX (ALL VARIABLE-UNADJUSTED FOR
TIME

NUMBER OF SIGNIFICANT DIGITS TO BE USED IN ACCURACY TEST
SUBROUTINE: LINVlF (IMSL)

ERROR PARAMETER FROM IMSL SUBROUTINES
VECTOR (NVARS*(NVARS+l)/2 WORKSPACE SUBROUTINE: BEMMI
VECTOR (6) INPUT TO VAR-GOV SUBROUTINE

NUMBER OF OBSERVATIONS PER VARIABLE IN SUBMATRIX ENTERED
INTO VAR-COV SUBROUTINE

NUMBER OF OBSERVATIONS PER QUARTER (NQOBS 3)

NUMBER OF QUARTERS (N YEARS*4)

Table 2.

NSUBM

NTOBS

NVARS

NVCVU

NYEARS

PROD

SCORR

V

VAL

VBLOWUP

VCOVT

VCVT

VCVU

VMEANS

VTEMP

WKAREA

61

(continued)

NUMBER OF SUBMATRICES ENTERED INTO VAR-COV SUBROUTINE

TOTAL NUMBER OF OBSERVATIONS (NYEARS*12+1) INCLUDING ONE FROM
PREVIOUS YEAR FOR SERIAL CORRELATION

NUMBER OF VARIABLES

NUMBER OF VALUES IN SYMMETRICALLY STORED VAR-COV MATRIX
(NVARS*(NVARS+1)/2

NUMBER OF YEARS CONSIDERED FOR EACH VARIABLE

MATRIX (1,NQRTS) MATRIX MULTIPLICATION PRODUCT SUBROUTINE:
WEIGHT

MATRIX CORRELATION COEFFICIENTS SUBROUTINE: SECOR
VECTOR (NVARS*IN) RAMDOM DEVIATES

VECTOR CONTAINING STD. DEV. FOR VARIABLES SUBROUTINE: SECOR
VECTOR CORRELATION COEFFICIENTS SUBROUTINE: SECOR

MATRIX (3,NVARS) STD. DEV., 3RD, 4TH MOMENTS

MATRIX LAGGED DATA BY QUARTERS SUBROUTINE: SECOR

MATRIX (NMONTHS,NMONTHS) FULL STORED EXPANDED
VAR-COV MATRIX WITH DERIVED COVARIANCES INSERTED

COVARIANCES DERIVED FROM VARIANCES AND SERIAL CORRELATIONS
SUBROUTINE SECOR

VECTOR (NQRTS*(NQRTS+1)/2 SYMMETRIC STORAGE OF VAR-COV MATRIX
FOR NQRTS PERIODS

VECTOR (NVARS*(NVARS+1)/2 SYMMETRIC STORAGE OF VAR-COV MATRIX
UNADJUSTED FOR TIME

VECTOR (NVARS) OF MEANS FOR EACH VARIABLE
VECTOR (NVARS) FOR WORKING STORAGE

VECTOR (NQRTS**2+3*NQRTS) WORKSPACE SUBROUTINE: LINV2F
(IMSL)

VECTOR RAW DATA- TRSFRGZ

62

Table 2. (continued)

XM VECTOR CONTAINS MEAN FOR VARIABLES- SUBROUTINE: SECOR
XMAX VECTOR (NVARS) MAXIMUM VALUE FOR EACH VARIABLE

XMEAN VECTOR (NVARS) MEAN FOR EACH VARIABLE

XMIN VECTOR (NVARS) MINIMUM VALUE FOR EACH VARIABLE

XPRIME MATRIX (NQRTS X NMONTHS) X-PRIME FOR NORMAL EQUATION
Y VECTOR RAW DATA LAGGED ONE PERIOD- TRSPRG2

Y MATRIX (NMONTHS,1) Y MATRIX FOR NORMAL EQUATION

ZNUM NUMERATOR FOR WEIGHTED MEAN CALCULATIONS SUBROUTINE: WEIGHT

63

Appendix Table 3. Source listing - Program 1 (PRGl)

IOOQIQOD.

PROGRAP TRSPRGT(INPUTQCUTPUTOTAPEToTAPEZcTAPE3)

THIS PRCGRA! READS INPUT DATA (TAPET) IN FREE FORMAT AND CALCULATES
A VARIANCE°COVARIANC£ MATRIX AND UNIQUE TRIANGULAR MATRIX

FOR UP TO TEN VARIABLES.

TAPET ' INPUT DATA READ IN FREE‘FORHAT ONE VARIABLE AFTER ANOTHER

TAPEZ 3 OUTPUT UNIOUE TRIANGULAR MATRIX (AMAT)

DIPENSICN APINPUTIOZ.5C),VTEPPISO),VPEANS(SO),VCVU(TZ7$),

*IVNBR(6),PVCVU(SO,SO),APAT(53,30),INCD(1275),
OXFOPENTI3OSG)

REWIND1

REBINOZ

REHIND!

IBUCIC

PRINT*,' '

PRINT',' ""CXECUTINC TRSPRC1tte'"

PRINT‘," '

IFLAC’C

PRINTF,” ENTER NUMBER CF VARIABLES TO BE READ IN“

RCAD',NVARS

PRINT0,' ENTER NUMBER OF YEARS"

READ-QNYEARS

Nn-NVARs-WVEARS

PRINT',' ENTER NUPBER CF OBSERVATIONS PER YEAR”

READ',NYROBS

IftNVARS.LE.1)PRINT~o' CANNOT CALCULATE VAR-COV MATRIX-
PRINTF,“ ARE THE OBSERVATIONS FOR EACH VARIABLE WRITTEN "
PRINT'," TC A TAPE? (181C5:SINO)'

REAthflY

PRINT',” ENTER NUMBER OF OBSERVATIONS TO BE READ IN FOR EACH'
PRINTe,' VARIABLE INCLUDING THO OBSERVATION FRO” LAST TWO PERIODS"
PRINTO,” OF PREVIOUS YEAR.”

READ'ONTUIS

PRINTE,” DO YOU WANT FULL OUTPUT OF MATRICES PRINTED ON TERSINAL?“
R£A60,VY

IT(YY.ED.1)IFLAC'1

IFIWV.EG.1160 To 313

- DATA READING LCOP

22

00 3c I81,NVAPS

DO 25 121,Wroas

READO.APINPUT(J¢I)

CONTINUE

PRINT'," HERE ALL VALUES ENTERED CO'RECTLY?”
READ',NY

IFINY.EO.C)GO re 31
URITE(1,')(APINPUT(NJ,I),NJ81,NTOES)

GO T0 35

Table 3.

64

(continued)

' DATA CORRECTION LOOP

31

35

310
33

S
6

13

PRINT',” COUNT DOWN FROM FIRST VALUE ENTERED TO INCORRECT VALUE?“
PRINT-s“ TYPE IN POSITION OF INCORRECT VALUE AND CORRECT VALUE'
PRINT‘p” CA SAN! LINE SEPARATED BY A CONRA"

READ*,INCOR,OKCOR

PRINT 359INCOROOKCOR

PORrArto OBSERVATION t,I3,- HAS BEEN CHANGED TO 'oFTO.S)
PRI‘T'," Is THIS TNE CORRECT CHANGE?“

READ',NV

IF(NV.EO.C)GO TD 31

PRINTO," ARE TNERC ANY OTHER CHANCES?“

REAO',NY

IFtWV.£O.1)GO TO 31

URIIECTO'AIANINPUTINJOI)QNJITQNTOBS)

CONTINUE

GO TO 32

READ(1,*)((AHINPUTIJ,I),JI1,NTOBS),I!1,NVARS)

PRINT 33oNVARSeNTOBS

FORPAT('ODATA HAS BEEN READ IN FOR ‘,IZ,A VARIABLESo',I3,

‘9 OBSERVATIONS PER VARIABLE.)

J31

DO 10 I'1eNVAR5

PRINT 5,I,APINPUT(J,I)

EORFATIROFIRSI OBSERVATION VARIABLE 'pIZo' 8P,F19.Z)
PRINT 6;I,APINPUT(NTOBS'I)

FONFRIIP LAST OBSERVATION VARIABLE *OIZO' 8'.F1D.2)
JET ‘

CONTINUE

' ESTIMATES OF PEAN, STD. DEV., 3RD, 6TH ”OPEVTS 8 CORRELATIONS

dru

16
17

19

REHINDT

unuroes-z

DO 1 I81,NVARS

READ(1,9)(VTENP(II),IIs1,2)

DU 2 J'ION

REAO(1,-)AIINPUT(J,I)

CONTINUE

CONTINUE

CALL BeretIANINPUT.N.WVARS.62.VIEFP.anFENT.VCVU.INCO.IERI
Pirate," '

CALL USUSFI18NCORRELATION EATRIX,TE,VCVU,NVARS,Z)
IF(IBUC.EO.1)FRINT',IER,' SUBROUTINE: BENPI'
DU 15 I'IOHVARS

PRINT 16.I.VTEFP(I)

PORFAT('OVARIABLE',IZ,' ARITN'ETIC ‘EAN 8*,T3S,F8.3)
PRINT 17,annEn1(1,I)

FORPAII- c.12x.oSTAWOAeo DEV. av.735.rto.$1
cvaxnonenrt1,r)IvtENPtIJA100.

PRINT 15,cv

FOReArtc t,1zx,oc05rr. VAR. (PCT) ..,r3s,;3.3)
PRINT 19.XFOWENT(2.I1

FORPAI(A ',12x,'THIRD POPENT 8*,T35,F10.S)

Table 3.

2?
15

65

(continued)

PRINT ZO,XRUHENT(3,I)

FURFATC* ',TZX,'PUURTN HOHENT 3',T3$,FTC.5)
CONTINUE
HPITEC3O')(CAPINPUT(J9I)OJ'19NIQI'1QNVARS)

REHINO!

REAC(3,')(CAPINPUTCJ,I),J|1,NYQOES),I31,”N)

CALL BEPNI(APINPUT,NYROBSINN,6Z,VTEHP,XNOPE”T,VCVU,INCD,IER)
IFCIBUE.EO.1)PRINT'oIEPo' SUBROUTINE: BEPNI'
PRINT',” ”

IFCIFLAG.EO.1)CALL USUSNCTEHCORPELATION PATRIX,18,VCVU,NN,Z)
UVCVU‘(NNP(NN*1)/Z)

IVNBRCTI3NN

Ivnaa(2)s~vnoas

IVNER(3)-NY!OES

IVNBRCA):T

IVNBRC6)'°

CALL BECOVPCANINPUT,52.IVNER,VTEFP,V'EANS,VCVU,IER)
IF<IBUG.£O.1)PRINT*,IER;" SUBROUTINE: BECOVF-T"
CALL VCVTSF<VCVU,NN,FVCVU,5CJ

IFCIBUG.EO.T)PRINT'9' SUBROUTINE: VCVTSF-T'
PRINT',” '

O PRINT CUT VAR‘COV PATRIX (UNAOJUSTED)

PRINTP," ”
IFCIFLAGoECoTICALL USUPHCTOHVAR C CUV .‘TRIXQ16QFVCVUQSOONNQ~NQZ)

* CALCULATE UNIQUE TRIANGULAR NATRIX "APAT' FRO! VAR‘COV MATRIX

CALL APATRIXCFVCVU,NN;APAT)
PRINT'," '

o PRINT OUT UNIQUE TRIANGULAR 'ANAT' MATRIX

10
O

IF(IFLA6.EO.1)CALL USUFN(11HANAT PATRIX.11,ANAT,$O,NN,NN,Z)
HRITECZ,')((APATCI,J),JIT,NN),I'1,NN)

PRINT',” "

PRINT-o” BE SURE TO CATALOG TAPEZ - UNIQUE TRIANGULAR FATRIX'
PRINT*,' TO BE USED LATER IN THE SIPULATION”

PRINTO," “

STOP

END

SUBROUTINE ANATRIXCFVCVU,NVARS.A)

OINCNSION FVCVU(SO,SC),A(SO.SO)

ICOLsNVARS

IROHINVARS

ISTCP-NVARs-T

CNEKTTIO.

DO TO I81,NVARS

00 19 J‘TONVARS

AII.J)-c.

CONTINUE

0 CALCULATE TNE NVARS TN CCLURN

'

A(VVARS,NVAPS)'SORT(FVCVU(NVARS,NVARS)J
DO SO I81,ISTOP

66

Table 3. (continued)

50 A(I,NVARS)IFVCVU(I,NVARS)/ACNVARS,NVARS)
Q

0 NEXT DIAGONAL ELENENT

t

90

ICCL’ICOL‘1
IR098ICOL
IROURTsIROU¢1
SUPSO.
OO TOO K-IROUP1,NVARS

1CD SUH'SUIOACIROHQKIO'Z
ITTSUN.CT.A(IROU.ICOL))sunsA(IROU.ICOL)
A(IROH,ICCL)'SORT(PVCVU(IROUoICOL3'SUF)
IFCICOL.EO.1)GO TO 220

O

. COMPLETE THE COLUNN
'
ICOLPT'ICOL*1
ISTOPSISTOP-T
DO ZOO JI1,ISTOP
IRON‘IROH-T
IT(IROU.EO.C)GO TO 210
SU'*O.
DO 15¢ KIICOLP1,NVARS
TSO SUHSSUP+ACIROU,K)*A(ICCL,K)
A(IROH.ICOL)I(FVCVU(IROU,ICOL)'SUN)IACICOL,ICOL)
zoo CONTINUE
210 GO TO 92
220 CONTINUE
RETURN
END

67

Appendix Table 4. Source listing - Program 2 (PRGZ)

OQOQNIOQQ

O. ‘.

PROGRAM TRSPRGZ(INPUTQOUTPUToTAPEToTAPEboTAPESQTAPEZO)
DIMENSION 1(133),VMEANS(3),R(3),SCORRCL‘),S(AL),

NAPORTSC3,£6),VTENP(AA),VCVTC1B30),VCOVT(66),VALC3,Z),IVNBR(E),
OXMCZ),VELOHUP(6¢,60),VCVTFCZO,ZO),U(S),AC(6D),ACV(6O),PACV(GO),
*HKAREACTZ),VARS(60)

TAPET ' INPUT DATA READ IN FREE FORMAT ONE VARIABLE AFTER ANOTPER

TAPEA ' OUTPUT- FULL STORED VAR-COV MATRIX' TIME ADJUSTED

TAPES ' EXPANDED VAR‘COV MATRIX‘ TO GO INTO INVERSION PROGRAM

TAPEZD 8 AUTO°CORRELATIONS TO BE USED TO CALCULATE AUTO-COVARIANCES

REUINDT
REHINDA
REHINDS
REUINOZO

IOUSIO

PRINT-g" '

PRINT',” 9"*EXECUTIN6 TRSPREZ***"

PRINT',‘ ”

PRINT'," ENTER NUMBER CF VARIABLES TO BE READ IN"

REAO'ONVARS

PRINTO," ENTER NUMBER CF YEARS DATA REPRESENTS”

REAC*,NYEARS

PRINT',” ENTER NUMBER CF OBSERVATIONS TO BE READ IV FOR EACN'
PRINT-9' VARIABLE INCLUDING THO OBSERVATIONS FROM LAST THO PERIODS

.N

PNINTt,” OF PREVIOUS YEAR."
REAO'QNTOPS
NNCNTHSsNTEAOSnTZ

NOOES'3

NORTSnNYEARS*A

DO 19 IIT,NVARS

PRINT'," ”

PRI‘T'," VARIABLE: ",E
READCTo')(X(L)9L'ToNTOES)

NVCVT8(NDRTS*(NORTS’1))IZ

CALCULATE AUTO-CORRELATIONS AND AUTO‘COVARIANCES

CALL SECOR(X,NTOBS,NDRTS,NDOBS,VMEANS.VCVT¢VCOVT9NVCVT)

INSERT DERIVED CCVARIANCES INTO VBLOWUP

CALL INSERTCVCVT,VCOVT,NORTS,NMONTHS,VBLOUUP)

HRITE EXPANDED VAR'COV MATRIX TO GO INTO INVERSION PROGRAM

Table 4.

O

- PUT

SS
99

68

(continued)

HRITE(5,')((VBLOUUPCIX,JX),JX81,NFONTHS),IX'T,NPONTNS)
CONTINUE

PRINT',“ '

PRINT’," CATALOG TAPES TO GO INTO TNE NATRIX INVERSION PROGRAP”
STOP

END

SUBROUTINE SECORCX,NTOES.NORTS,NOOBS,VMEANS,VCVT;VCOVT,NVCVT)
DIMENSION 2(133),VMEANS(3),R(3),SC‘A),HCS),AC(6O),ACV(6D),P

¢ACVCDETQ
‘UKAREAC12),VARS(60),VBLOHUP(63,60),

*APDRTS(3,A£),VTEMP(€6),VCVTC1B3O),VCOVT(AL),VAL(3,Z),IV"BR(6),X'(Z
1')

N31
NMONTNSINORTS*3

CALCULATE AUTO COVARIANCES

LAI1

DO 99 ISIT,NORTS

DO 88 Jc1,5

V(J)!X(N)

NINOT

CONTINUE

LUIS

KsLIZ

ISN'T

CALL FTAUTO(U,LH,X,L,ISU,AMEAN,VAR,ACV,AC,PACV,HKAREA)
DO 55 IA81,2

ONE AND THO LAC AUTO-CORRELATIONS INTO VCOVT
VCOVTCLA)'ACCIA)

LAILA+1

CONTINUE

NON-2

CONTINUE
NVCVT!CNORTS'(NORTS*1)IZ)
IVNBRC1)ONDRTS
IVNORCZ):NOOES
IVNBRT3)INOOBS

IVNDRCA)81

IVNBRC6)-C

' CALCULATE VAR‘COV MATRIX FCR THE I‘TN VARIABLE BY QUARTERS
o URITE DATA INTO THREE OBSERVATIONS PER QUARTER

N82

DO 90 IS81,NORTS
DO 80 J81,3

N8N¢1
ANDRTS(J,IS)8X(N)
CONTINUE
CONTINUE

Table 4.

69

(continued)

CALL BECOVMCAMORTS NOOBS, IVNBR ,VTEMP,VMEANS, VCVT, IER)
PRINT'Q' 0K.
Natl-NIT

' VCOVT: COVARIANCES DERIVED FROM VARIANCES AND SERIAL CORRELATIONS

21

DO 21 K81,NORTS
VCOVTCNTSVCOVT(P10VCVT(I!)
Nin+1
VCOVT(P)!VCOVT<P)NVCVT(IK)
"ono1

N8N01

IKSIK¢N

CONTINUE

' CONVERT SYMPETRIC VCV MATRIX TO FULL STORAGE ”OE

CALL VCVTSF(VCVT¢NORTSoVOLONUPobO)
HRITE(6,*)((VBLONUPCIV,JV),JV.1,NORTS),IV'1,NORTS)
JX‘T

' EXPAND DIAEONAL OF VCV (3!): TO GO INTO VCV-EXPANDED ”ATRIX

600
SOD

'EULL

150
ZTD
130

OD SOC ITIToNORTS

DO 60C JT-1,NOOBS
VARSCJX)¢VELOUUP(IT,IT)

JX8JX*1

CONTINUE

CONTINUE

[x31

NAC'NTOBS‘3

CALL PTAUTOCXQNTOBSUN‘COODS9AHEAN9VARgACVQACQPACVQHKAREA)
HRITECZO,*)(ACCLLJ.LL-1,NAC)

MATRIX

DO TOO IO'1.NFONTNS

DO 200 J081.NHONTNS

LA68IABS(JO- IO)

IFCLA6.ED.:)GO TO 150
VBLOWUP(IO,JO)-AC(LA6)*VARS(IO)

GO TO 230

VBLOHUPCIO.JO)8VARSCIO)

CONTINUE

CONTINUE

CALL VCVTFSCVDLONUPoNNONTNSoéOoVCVT)
RETURN

END

SUBROUTINE INSERTCVCVT,VCOVT,NORTS,NPONTNS,VELONUP)
DIMENSION VCVT(1830)oVCOVT(AO).VELOHUP(60o631
NVCVT'N'ONTNS'CNMONTNS’T)IZ

K89

JIZ

I82

N81

VCVTCIJIVCOVTCN)

I8I¢J

NEXT8N01

VCVTCI)8VCCVT(NEXT)

Table 4.

99

70

(continued)

III'1
VCVTtI)8VCOVT(N)
IF(I.ED.NVCVT-1)GO TO 99
III¢C

J8J*3

{8(06

N3N+2

GO TO 1

CONTINUE

CALL VCVTSF(VCVT,NHONTHS,VBLOHUP,60)
RETURN

END

71

Appendix Table 5. Source listing - Program 3 (PRGS)

n t .0 0

PROGRAM TRSPRG3(INPUT,OUTPUT,TAPES,TAPEé)
OIPENSION A(60,60),AINV(60,DO),HKAREA(378C)

TAPES l EXPANDED VAR-COV FATRIX- TO GO INTO INVERSION PROGDAP

TAPEé 8 INVERSE DP Exflauosc VAR-CDV MATRIX

REHINOS

REHINOD

IBUE3O

PRINT’,“ ‘

PRIVT'," "*'EXECUTING TRSPR63"'*"

PRI‘T'Q' '

PRINT',“ ENTER ”UNBER CF VIRIAELES”

READ',NVIRS

PRINT'," ENTER "HUBER CF TEARS OITI REPRESEnTS"
REAO'QNYEARS

PRIOT‘,' ENTER "UNBER CF OBSERVATIONS PER TEIR“
QE‘C',NOBS

NHCNTHS'NTEARS'NOBS

IOGT'O

DO 77 I'T,NVARS
REAC‘5,*)((‘(IN,JN),JN31,NFONTHS’er31,NFONTHS)

* INVERSION OF "I” USING HIGH ACCURACY INVERSION ROUTIHE
* INVERSE OF “A“ 8 'AINV'

77

IE(IBU‘.E0.0)CALL UERSET(T,5)

CALL LI”VZF(A,NNONTHS,CO,AINV,IDGT,HKAREA,IER)
IFIIBUGoEOoT)PRINTOQIERQ' SUBROUTINE: LINVZ' VARIABLE: ',I
IFIIER.E0.0)PRINT'," INVERSE COMPUTEO‘ VARIABLE ”,3
IF(IEP.EO.TE9)PRINT*," PITRIX IS lLGORITNHICALLT SIVGULAR”
I'(IER.EO.T31)PRINT'," PATRIX T00 ILL‘CCNOITIONED FOR ITERATIVE I‘
*PROVEPENT'

IEIIER.EO.35)PRINTC,' INVERSE COFPUTEO‘ ACCURACY TEST FAILED"
HRITEID,’)((‘INV(12,JZ),JZ'1,N"ONTHS),IZ'1,HNONTFS)

CONTINUE

PRINT'," “

FRINT',' CITILOO TIPED: HHICH CONTLIVs-INVERSES EUR ”,VVARS,' VIRI
*‘OLEIS” ”
PRINT'Q' '

END

72

Appendix Table 6. Source listing - Program four (PGR4)

ODI'UOUQ...IO.G.D

PROGRAP TRSPRG6(INPUT,CUTPUT,TAPE1,TAPEb,TlPE6,TAPE7,TAFEE,TAPE?,

0TAPE109TAPE11)

DIRENSION XPRINE(20,60),C(23,6£),Y(6C,1),D(ZO,1),ECZO,ZO),

‘EIVV(ZO,ZC),F(20,1),HKADEA(660),AIVV(60,DO)

IOUG'O

TAPET I TRSSYIATRIX37-RATRIX

TARES 8 FULL STCRAGE OF Vll‘COV NATHICES’E‘CR VARIABLE
TAPEE ' INVERSE OF BLOWN-UP VAR¢COV HATRIX: v-Inv
TAPE? ' HEIGHTEO REAR TO GO INTO SIHULATION

TAPES 3 RIGHT¢HAND SIDE: (X-PRIRE V-INV Y)

TAPE9 8 (X’PRIFE V'IVV X)

TAPETO ' SOLUTIONS: (X'PRIHE V-INV X)INV 'Y

TAPETT ' SIHULATEO DITA VECTORS

REHINDT

REHINO‘

REBINOE

REHINOE

REHIND°

REHINOTO

L31

PRINT'.’ '

PRINT'," "’*EXECUTINO TRSPRG“"‘"

PRINT*,' ”

PRINT'," ENTER RUPEER OE VARIABLES”

RE‘O'QNVRRS ’
PRINT',” ENTER NUMBER OF TEARS DITA REPRESENT5”
RE‘O'pNTE‘RS

PRINT‘," ENTER NUMBER CE OBSERVATIONS PER TEIR"
READ'QNOOS

”PONTHE'NTERRSRNOOS

H3~HO~THSII

HORTStfl

IVEIRSUNHONTHS/TE

0 M AND IA (NPONTPS) ARE EOU‘L TO SIZE OF BLOWN-UP VAR-COV MATRIX

VUN’ONTVS
IA8NNONTHS
IDGTSC

. READ IN THE INVERSE OF THE BLOHN-UP VARIANCE COVARIANCE MATRIX

DO ECO INa1,NVAR$
PRINT*,' V‘RIABLE: "pIN
READ‘OQ”‘(RINV(IDJ,QJ'1ON’DIaTQN)

73

Table 6. (continued)

9 GENERATE THE X-PRIME MATRIX
CALL XPATRIX(M,NMONTHS.XPRIME)

'PREMULTIPLT V INVERSE BY X PRIME

371 CALL VPULFF(XPRIME,AINV,R,H,N,ZO,DO,C,ZO,IER)
IF(IOUG.EO.1)PRINthIERq' SUBROUTINE: VMULfF-T'
IF(IER.EO.129)PRINT 129

129 FORMAT(* *,'TERPINAL ERROR (129) INDICATES MATRICES BEIVG'I,
" ‘ULTIPLIED HERE DIMENSIONED I!CORRECTLY*)

'READ T MATRIX AND PREMULTIPLT 87 X PRIME V INVERSE
J81

' DELETE FIRST THC VALUES FROM RAH DATA (USED FOR SERIAL CORRELATION)
DO 33 II'TQZ
READ(1,')DELETE

33 CONTINUE

O READ IN T-MATRIX
READ(19')(T1I91)9I'T¢N)
CALL VMULFF(C.Y,M,N,L.ZO,60,D.ZC,IER)
IF(IBUG.ED.1)PRINT',IER.' SUBROUTINE: VMULPF-Z“
IFIIER.EO.129)PRINT 129

t MULTIPLICATION OF X PRIME V IMVERSE ET TRANSPOSE OP X PRIME IE. X
CALL VPULFP(C,XPRIME,M,N,M,2C,ZO,E,2C.IER)
IF(IBUG.ED.1)PRINT*,IER,' SUBROUTINE: VMULFP'
IF(IER.EO.129)PRINT 129

- CALCULATE INVERSE; X PRIME V INVERSE X
IF(IBUG.EO.CJCALL UERSET(1,L)
CALL LIMVZF (EA!,ZO,EINVpIDGT,UKAREA,IERD
IF(I3UG.EO.1)PRINT',IER,” SUBROUTINE: LINVZF”
IF(IERoEO.129)PRINT'v' MATRIX ALGORITHMICALLY SINGULAR'

. NULTIPLT RNS BY X PRIME V INVERSE X INVERSE TO GET ESTIMATES
CALL VMULFP(EIVV,O,M,M,L,ZO.20,F,ZO.IER)
IFTISUG.EO.1)PRINTtoIERo' SUBROUTINE: VPULFF‘E'
IF(IER.EO.129)PRIMT 129
J81
HRITEITO,*)(F(I,L},I31,F
IL81
PRINT*," VARIABLE ".IN.'GDARTERLY SOLUTIONS:"
DO 1 It1,NYEARS
PRIMT',“ YEAR: ”,I
DO 2 IJ'TQ‘
PRINT',” QUARTER: ",F(IL.L)
IL'IL’1

2 CONTINUE

1 CONTINUE

EDD CONTINUE
STOB
END
SUBROUTINE XRATRIX(ROUQCOLU‘NoXPRIME)
DIMENSION XPRIME(23,DD)
INTEGER ROH,COLUMN
L81
VaCCLUNN/RDD

Table 6.

(DON

1O

74

(continued)

NNON

DD 10 I81,ROH
DO 3 J81,COLUMN
IF(J.EO.L.AND.J.LE.N)GC TD 7
XPRIMECI.J)=O
GO TO 6
XPQINETI,J)31
IF(J.ED.L)L=L+1
CONTINUE

NsNoNN
LaN-(NN-T)
CONTINUE

RETURN

END

75

Appendix Table 7. Source listing - Program 5 (PRGS)

D
PROGRAM TRSPRGS(INPUT,CUTPUT,TAPE2,TAPE1D,TAPE11)
DIMENSION R(1600),A(SD,50),XNEAN<SD),XNINTSC),XMAX(SD),
*V(3,50),HTMEANS(SO),INCDISS),SMEAN(5),UA(1300,1C),C(55)
IBUC-O

Q

. TAPEZ ' UNIDUE TRIANGULAR 'A“ AMATRIX (AMAT) FROM 9961

. TAPETD 8 QUARTERLY SOLUTIONS FROM PRG‘. ’

- TAPE11 I SIMULATED DATA

I
RENINDZ
RERIND13
REHIND11

* NVARS 3 NUMBER OF VARIABLES ° DIMENSION OF VAR-COV MATRIX

A

IN 8 NUMBER OF DEVIATES TO BE GENERATED
' NVEARS 8 NUMBER OF YEARS
PRINT'o' '
PRINTA," ARRREXECUTINC TRSPRCS'RRR'
PRINTA,‘ '
PRINTE," ENTER NUMBER OF VARIABLES"
READ'gNVARS
PRINTt," ENTER NUMBER OF YEARS”
REAO',NYEARS
NNONVARS'NYEARS
IN-1DDD/NTEARS
READ(2.*)((A(I.J),JI1,NN),I81,NN)
'
. CALCULATE YEARLY MEANS FRO! DUARTERLT SOLUTIONS
ISUM=D.D
DD 10 I81,HN
DO 11 J81,‘
READ(1D,')SPEAN(J)
ZSUM'ZSUMoSPEANTJ)
11 CONTINUE
HTPEANscIltzSUM/L.
ZSUM'D.D
10 CONTINUE
D

DSEEDI1C.DC

* HARM-UP RANDOM NUMBER GENERATOR
NITCO
CALL GGNML€DSEED,N,R)
CONTINUE

Q

0 RANDOM OR GAUSSIAN RANDOM DEVIATE GENERATOR
CALL GGNML(DSEED,IN,R)
ICUM82.C
(:1
DO 5 IITQNN
DO A M81,IN
DC 3 J21,NN

76

Table 7. (continued)

Z!A(I,J)'P(X)
ZCUM'ZCUPTZ
(8&01
IE(‘ 0‘! QIN)K.T

3 CONTINUE
USZCUMAUTMEANSCI)
HRITE(119')U
ICUM'Z'C.G

A CONTINUE
CALL GENMLtDSEED,IN,R)

S CONTINUE

I

t FIND MINIMUM AND MAXIMUM VALUES GENERATED FOR EACH VARIABLE
NININTEARS'IN
REHINDTT

DO 12 I81,NVARS
REAO(11,O)(R(J),J81,NIN)
CALL USMNMX(R,NIN,1,LMIN,ZMAX)
XMIN(I)IZMIN
XMAX(I)'ZMAX

12 CONTINUE

'

c ESTIHATES OE SIFULATED ‘EANS; STD. DEV..3RD, 5TH NOMENTSpCORRELATIONS
REHINDTT
READI11,*)((UA(J,I),J'1,NIN),I81,NVARS)
CALL BEMMI(UR,NIN,NVARS,TDOO,XMEAN,V,C,INCO,IER)
I'CIOUOoEOoT)RRINT'QIERQ' SUBROUTIHE: OEFFI'
PRINT'," ”
CALL USHSPITENCORRELATION MATRIX,18,C,NVARS,Z)
DO 90 I81,NVARS
PRINT 919IOXHEAN1I)

91 FORMATIAOVARIABLE ',IZ,' SIMULATED MEAN 8*,T35,FE.3)
PRINT 92,V(1,I)

92 FORMATI' ',12X,*STANDARD DEV. 8',T35,F1O.5)
CV'V119ITIXMEANIIIO1OD. , -
PRINT 97,CV

97 FORMAT(* ',12X,OCOEFF. VAR. (PCT) ",T35,F8.3)
PRINT 93,V(2,I)

93 FORMAT10 OQTZXQOTNIRD MOMENT '09T359F1Oo5)
PRINT 9A,V(3.I)

9L FORMATIM ',12X,*FOURTH MOMENT ",T35,FTO.S)
PRINT 9S,XFIN(I)

95 FORMATI' 'QTZXv'FINIMUP VALUE M'vT359FBo3)
PRINT 96,XMAX(I)

96 FORMAT(0 ',12X,'MAXIPU' VALUE '*,TSS,FS.3)

9O CONTINUE

FRINT'Q' '

PRINT',” '9‘. END OF SIHULATIO" """
RRINT'," "

STOP

END

77

Appendix Table 8. Sample output from interactive computer programs

09“EXECUTIHE TPSPREIOOO.

ENTER NUMBER OF VRRIRBLES TO BE PERD IN
ENTER NUMBER OF (ERRS

:EHTER NUMBER OF OBSERVRTIOHS PER YEPR

HPE THE OBSERVHTIOMS FOR EACH HHIHBLE MRITTEM
TO H TRPE? vIaYEszoana)

ENTER NUMBER OF OBSERVRTIOHS TO BE PERD IN POP ERCH
'vRRIRBLE INCLUDING THO OBSERVRTIOH FROM LRST THO PERIODS
OF PREVIOUS .EHR.

“a

DO 'fOU MHHT PULL OUTPUT OF HRTRICEE PRINTED ON TERHIHRL7

.0)

BATH HRS BEEN RERD IN FOR 4 VRRIRBLES: 62 OB? ERVRTIOHS PER 'RRRIRBLE

FIRST OBSEPVRTIOH VRRIRBLE I i 3.60
LRET OBSERVRTIOH VRRIRBLE ' 6.30

FIRST OBSERVRTIOH VRRIRBLE C 370.00
LRST OBSERVRTIOH VfiRIRBLE 395.00

1
3
3
FIRST OBSERVRTIOH HRRIRBLE 3 O 16.30
3
a
4

LRST OBSERVRTIO" VRRIRBLE ' 30.60

FIRST OBEERVRTIOH VERIRBLE
LRET OBSERVRTION VRRIRBLE

CORRELRTIOH HRTRIX

- 26.50
. 42.00

(0
w

4
1 1.00000
3 .T6613 1.00000
3 .37733 .31711 1.00000
4 .7313: .03752 .3331?
1. 00000
VHRIPBLE 1 fiRITHMETIC MEAN t 3.311
3THMDHRD DEV. a .LTL48
COEFF van. (PCTE - 4.939
THIRD MOMENT - -.01369
FOURTH MOMENT - -.334E3
VRRIRBLE a HPITHMETIC MEfiM - 335.000
STHMDHRD DEV. a 39.00022
COEFE. VHR. (PCT) - 11.60?
THIRD MOMENT - -.05??E
FOURTH MOMEMT - -1.3?355
VRPIRBLE 3 RPITHMETIC MEHM a 19.013
2TRMDM RD DEV. - 3.03054
:OEFF. 009. (PCT) - 10.030
THIPD MOMEMT . -.03663
FOURTH MOMEMT - -1.E0391
VRPIHBLE 4 RPITHHETIC MEHM - 33.133
STRMDRPD DEV. - 3.17354
COEFF. HEP. IPCTb - 15.337
THIRD MOMENT I .34?.-
FOURTH MOMENT - -1.S013

BE ZUHE TO I-HTHLCE THPEa - UflIQUE TPIHHGliLFP murprg
TO BE USED LHTEP IN THE -IMuLHTIOM

78

Table 8. (continued)

n‘- -.. -_-.
. a-

ooooexscurxne rpsppeaoooo

.Enrsp nunsee as waazeaLss ru es REED In

ENTER nunssp OF wean: DfiTfi pspnsssnr:
.J

aura? NUMBER OF OBSEPVHTIOMS TO 3: Pain IN sop earn
weeteats INCLUDING run as: aver ? 9' ‘r ' a
as P9501002 7259. a ION. = on LH.T run Psarun-
oea

vnpzaeLe: 1

on

URRIRBLES 3
OK

LL.

WHQIHBLE:
OK
VRPIHBLES 4
OK

careuogprness TO so INTO THE nernrx rmvsasron 990599"
' .740 up seccnns EXECUT' n r ‘
ox-errspa.rnzpneztso.uet.ea. .n Ins
HTTRCHEMG-TRSPREELGO

HPL 3.77

535: sesun.13.ns.oa.

ooooexecurrns rnsppssoooo
.Enrsn nunaea as VHPIEBLES
.EMTER nunsea up weeps nave pepnssenr:
.agrsn nunsea or ozsaavnrrons pep veep

rnvspss compursn- VRPIHBLE 1

INVERSE.CONPUTED- RECUPPCY TEST PRILED'
.IHVEREE,COBPUTED- RQQUEEQY‘TEET FRILED

Table 8. (continued)

79

°°¢°EXECUTIMG TRSPRE4ooo¢
§NTER NUMBER OF TRDIHBLES

ENTER NUMBER OF YERPS DPTH REPREEEHTS
ENTER NUMBER OF OBEERVRTIONS-

013
THPIRBLE 1
1 EH“ 8
nuHFTEP:
QUHRTEP:
QURQTER:
QUEPTER:
YERR: &
QUHRTER:
QuHRTEP:
QURRTER:
OUHPTEP:
TERR: 3
@UHRTER:
QURRTER:
OUHRTER:
QUHRTER:
'I’EFTP 3 4
OUHPTEP:
QURRTER:
OURPTEP:
OUHPTER:
TERR: S
QURRTER:
@UHPTEP:
QUHRTEP:
QUARTER:
VHRIRBLE a
TEHH: 1
QURRTEP:
QUERTEP:
OURRTEP:
@URRTEP:
'I’ ERR: &
QURPTER:
QUARTER:
OUHPTER:
QURRTER:
TEfiRz 3
QufiPTpr
QURRTEP:
QUHRTER:
QUHPTER:
TEPR: a
QUPRTEP:
QURPTEP:
OURPTEP:
OUHPTEP:
:ERP: .
OURPTEP:
@UHPTEP:
QUERTEP:
QURRTER:
”PPIFBLE 3
'5992 1
qupsrgp:

PER '
OUHRTEPLT. .OLUTIOMS:
1 3.:4ussxesssea
a 3.419T1934436!
3 3.4?6363749492
4 3.60093036303?
1 3.73 3993394612
3 3.?43949929329
3 3.631439239433
4 3.763313973663
1 3.323643003983
E S.970??643634E
3 3.?SE9TSS164&
4 5.943924 4730206
1 3.995694409969
E 5.9EESSTJTI9SI
3 3.364?37635903
4 6.163929319313
1 9.319ESEST4303
a 6.10:9:944205493
3 0.01390a~’5576
4 5.139345: 3044
JUHRTERLT SOLUTIONS:
1 ET3.303911T 339
E 334.1910909?06
3 397.637ET3333
4 237.0133431973
I 396.1126400849
E 301.?399980333
3 311.4aa35ra112
4 310.9100TBS176
I 304.324TTIEI47
a 331.3920950TS
3 343.49?9873322
d 353.71490?6164
1 939.2207936999
a 348.3339330737
3 342.939937349
4 963.603329971?
1 334.7969331991
a 399.4006365199
9 391.359T949309
4 395.711933439
QUFPTERLT tOLUTIOMS:
1 16.17370073793

80

Table 8. (continued)

00°~EHECUTINE TRSPRESOOOO
ENTER NUMBER OF TRRIRBLES
0;:

§NTER NUMBER OF TERRS

O

CUPRELMTIUH MHTPIX
I

1 1.00000
3 .r039a 1.00000
3 .03123 .09??9 1.00000
THRIRBLE 1 SIMULATED MERN - 5.349
ETHNDHRD DEV. a .39334
CCEFE. VRP. flPCTF 8 4.930
THIRD MOMENT a -.0T99a
FOURTH MOMENT . -.13494
MINIMUM TfiLUE I 3.013
MAXIMUM TRLUE 8 6.653
T9RIRBLE a tIMULfiTED MERN . 330.41?
STRNDRRD DEV. 8 34.69460
COEFF. TAR. (PCT? a 10.49?
THIRD MOMENT a .3533?
FOURTH MOMENT - -.9096?
MINIMUM TRLUE - 333.727
MRXIMUM TRLUE - +£2.39?
TRRIRBLE 3 SIMULRTED MERN - 13.124
ETRNDRRD DEV. - 10.95301
COEFF. 09R. uPCTfi a 50.444
THIRD MOMENT - -.3&T91
FOURTH MOMENT a 1.31360
MINIMUM TRLUe - 937.033
MRXIMUM VRLUE - . 64.665
0000 END OF SIMULATION 0009
:TOP

UK - 1.?2? CP SECUHDS EXECUTION TIME

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