MSU RETURNING MATERIALS: PIace in book drop to LJBRARJES remove this checkout from .—;-—. your record. FINES will be charged if book is returned after the date stamped beIow. “F5 t 5 199! ANALYSIS OF RISK AND RETURN ASSOCIATED WITH ALTERNATIVE CASH MARKETING STRATEGIES 0N MICHIGAN CORN. "HEAT, AND SOYBEANS By Gregory Scott Franklin A THESIS Submitted,to Michigan State University in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE Department of Agricultural Economics 1984 ABSTRACT ANALYSIS OF RISK AND RETURN ASSOCIATED WITH ALTERNATIVE CASH MARKETING STRATEGIES 0N MICHIGAN CORN. WHEAT, AND SOYBEANS By Gregory Scott Franklin ‘ The research presented in this study is designed to assist agricul- tural producers in the decision process of when to sell grain. Net re- turns to storing corn, wheat, and soybeans on-farm and commercially ih Michigan are examined in real 1983 dollars. Average net return and risk associated with alternative cash marketing strategies are developed and tested through the use of basic portfolio theory and linear programming. Optimal solutions are then obtained and analyzed in an (E, V) risk/re- . turn context. It is hypothesized the analysis will suggest that through careful selection of cash marketing strategies, average net returns to storage can be increased for an expected level of risk. ACKNOWLEDGMENTS I wish to thank members of my thesis committee: Drs. John Ferris, Jim Hilker, and Richard Simonds. Special thanks goes to Drs. Jim Hilker and Roy Black for their guidance in direction of the thesis. Most im- portant, sincere appreciation is extended to my parents for their assis- tance in providing the opportunity for a college education. ii TABLE OF CONTENTS Page LIST OF TABLES .......................... v LIST OF FIGURES ............... ' .......... vii CHAPTER I. INTRODUCTION ........... . . . . . ........ l 1.1 Objective ....................... 2 1.2 Methodology ...................... 3 1.3 Related Research ................... 4 1.4 Contribution ..................... 5 II. THEORETICAL CONSIDERATIONS ................. 7 2.1 Seasonal Price Movement ................ 7 2.2 Examination of Seasonal Prices . . . . . . ...... 11 2.2.1 Corn ...................... 11 2.2.2 Wheat ..................... 15 2.2.3 SoybeanS' .................... 16 2.3 Financial-Theoretical Concepts and Portfolio Theory ........................ 17 2.3.1 Economic Theory and Choice ........... 19 2.3.2 Theoretical Considerations of the Market. . . . 21 1 2.3.3 .Risk and Uncertainty ......... , ..... 25 , 2.4 Decision Analysis Considerations . . . .~ ....... 27 2.4.1 Bayesian Theory ........ . ........ 29 2.5 Planning Under Risk .................. 33 2.5.1 Linear Programming ............... 34 2.5.2 Some Assumptions of Linear Programming ..... 36 2.6 Quadratic Programming ................. 37 III. DEVELOPING THE MODEL ................. . . . 39 3.1 Storing the Commodity: Further Assumptions and Considerations .................... 39 3.1.1 Commercial Storage ............... 40 3.1.2 On-Farm Storage .......... . ..... 41 3.1.2.1 The Drying Process . . ..... 42 3.1.2.2 Management of Continuous- Flow Dryers ................ 44 3.1.2.3 Aeration ............... 44 . 3.1.2.4 Management of Aeration ........ 46 3.2 ‘Costs Incurred for On-Farm Storage .......... 47 3.2.1 Drying Costs .................. 47 3.2.2 Aeration Costs ................. 49 3.2.3 Other Variable Costs .............. 50 m 3.3 Change in Technology. . . .............. 51 3.4 Calculation of Net Return to Storage ......... 51 3.4.1 Opportunity Cost ................ 52 3.5 Developing the Linear Program Model (MOTAD) ...... 54 3.5.1 The Efficient Boundary ...... . . . . . . . 55 3. 5. 2 The Model - Using MOTAD ....... . . . . 57 3.6 Assumptions and Further Considerations of the MOTAD Model ...... . ........ . ...... 59 IV. ANALYSIS OF NET RETURNS T0 STORAGE. . . . . . . . ..... 61 4.1 Net Return to Storage ................ 61 4.1.1 Corn - Net Commercial Storage Margins ..... 62 4.1.2. Corn - Net On-Farm Storage Margins ....... 65 4.1.3 Wheat - Net Commercial Storage Margins ..... 67 4.1.4 Wheat - Net On-Farm Storage Margins . . . . . . 70 4.1.5 Soybeans - Net Commercial Storage Margins . . . 70 4.1.6 Soybeans - Net On-Farm Storage Margins ..... 73 4.2 Summary of Net Return Tables ............. 73 4.3 Net Return to Post-Harvest Marketing Strategies. . . . 75 4.3.1 Net Return to Storing Corn - Commercial (1958- 82). ................. 77 4.3.2 Net Return to Storing Corn - On- Farm (1958- 82) .............. . . . . 80 4.3.3 Net Return to Storing Corn - Commercial (1973- 82) ................... 84 4.3.4 Net Return to Storing Corn - On- Farm , (1973- 82) ................... 88 4.4 On- Farm Storage at 14 Percent MOisture (1958- 82). . . 95 4.5 On- Farm Storage at 14 Percent Moisture (Short Crop) (1973- 82) ....................... 96 4.6 -On- Farm Storage Basis Rule at 14 Percent Moisture (1973- 82) ........ . ......... . . . . . 100 4.7 Other Strategies ................... 103 4.8 Commercial Storage - Wheat (1958-82) ..... . . . . 103 4.9 On-Farm Storage - Wheat (1958-82) ........... 107 4.10 Commercial Storage - Soybeans (1958-82) ........ 111 4.11 On-Farm Storage - Soybeans (1958-82) ......... 111 4.12 Distribution of Net Returns. . . . . . . . . . . . . . 115 4.12.1 Cumulative Distribution Function ........ 115 4.12.2 Probability Distribution Function ....... 120 4.13 Sumary of Net Returns Associated with Alternative Sell Strategies .......... . .......... 123 V. CONCLUSIONS ........................ 126 5.1 Limitations and Need for Further Research ....... 130 APPENDICES ............................ 133 BIBLIOGRAPHY ........................... 147 iv Table mumm-fi 10 11 12 13 14 15 16 17 LIST OF TABLES Seasonalit of Michigan Farm Prices of Corn (1958-1983 .............. , ........ Seasonality of Michigan Farm Prices of Wheat (1958—1983) ...................... Seasonalit of Michigan Farm Prices of Soybeans (1958-1983 ................... . . . . Payoff Table ..................... Activity Table . . . . . . .......... . . . . Initial and Recommended Moisture Content for Storage . . Opportunity Cost ................... Net Commercial Storage Margins for Corn (1983 Dollars) ....................... Net On-Farm Storage Margins for Corn (1983 Dollars). . . Net On-Farm Storage Margins on Michigan Corn (1983 Dollars) 14 Percent Moisture Content ..... . . Net Commercial Storage Margins on Michigan Wheat (1983 Dollars) ..... . . ............. Net On-Farm Storage Margins on Michigan Wheat (1983 Dollars) .................... Net Commercial Storage Margins on Michigan Soybeans (1983 Dollars) .................... Net On- Farm Storage Margins on Michigan Soybeans (1983 Dollars) ..................... 0 Commercial Storage: Corn (1958-82) ..... . . . . . . Commercial Storage: Corn (Short-Crop)(1958-82). On-Farm Storage: Corn (1958-82) ......... . . . Page 12 13 14 32 35 43 54 63 66 68 69 71 72 74 82 83 86 Table 18 19 20 '21 22 23 24 25' 26 27 28' 29 30 31 32 33 0n-Farm Storage: Corn (ShortaCrop)(l958-82) ...... Commercial Storage:' Corn (1973-82) .......... Commercial Storage: Corn (Short-Crop)(l973-82) . . . . On-Farm Storage: Corn (1973-82) ............ On-Farm Storage: Corn (Short-Crop)(1973-82) ...... On-Farm Storage: Corn (1973-82) 14 Percent Moisture (1973-82) ....................... On-Farm Storage: Corn (Short-Crop) 14 Percent Moisture (1973-82) ................... Basis Decision Rule Table ............... On-Farm Storage: Corn - Basis Rule (1973-82) 14 Percent (1973-82) ............... . . . . . On-Farm Storage: Corn, Comparison of Alternative Strategies (1973- 82) .................. Commercial Storage: Wheat (1958-82) .......... On-Farm Storage: Wheat (1958-82) ........... Commercial Storage: Soybeans (1958-82) ........ Commercial Storage: Soybeans (Short-Crop)(1958-82) . . . On-Farm Storage: Soybeans (1958-82) .......... On-Farm Storage: Soybeans (Short-Crop. 1958-82). vi Page 87 9O 91 93 94 98 99 102 105 106 109 110 113 114 117 118 Figure 10 11 12 13 14 15 LIST OF FIGURES GraphicaI Representation of Relationship Between Cash and Futures Price. and Basis. ._ ......... Graphical Representation Depicting Seasonal Price Movement .................... Graphical Representation of the Risk/Return Trade-Off Among Alternative Investments. . . . . . . . . Graphical Representation Illustrating an Investor's Preference for Risk and Return . ........... Graphical Representation of Aeration Periods In a Normal Year ................ . . . . . Graphical Representation of Risk/Return Trade-Off; Depicting the Set of A11 Feasible Strategies ..... Average Net Return to Storage, Commercial Corn (1958-82) ....................... Average Net Return to Storage. Commercial Corn (Short-Crop. 1958-82) ............ ,. . . . - Average Net Return to Storage. 0n-Fami Corn (1958-82) ....... . .......... . . . . . Average Net Return to Storage, On-Farm Corn (Short-GNP, 1958-82) ooooooooooo a ooooo Average Net Return to Storage, Commercial Corn (1973-82) ............. . .......... Average Net Return to Storage. Commercial Corn (Short-Crop, 1973-82) .............. . . . Average Net Return to Storage, On-Farm Corn (1973-82). . ' ..................... Average Net Return to Storage, On-Farm Corn (Short- -Crop, 1973- 82) ................. Average Net Return to Storage, On-Farm Corn (1973-82) 14 Percent Moisture ............. vii Page 18 20 45 55 81' 81 85 85 89 89 92 92 97 Figure Page 16 Average Net Return to Storage. On-Farm Corn (Short-Crop, 1973-82) 14 Percent ............ 97 17 Average Net Return to Storage, On-Farm Corn (Basis Rule, 1973-82) 14 Percent ............ 104 18 Average Net Return to Storage, Alternative Strategies - (1973-82) 14 Percent ............ 104 19 Average Net Return to Storage, Commercial Wheat (1958-82) ........................ 108 20 Average Net Return to Storage, On-Farm Wheat (1958-82) ........................ 108 21 Average Net Return to Storage, Commercial Soybeans (1958-82) ........................ 112 22 Average Net Return to Storage. Commercial Soybeans (Short- Crop, 1958-82) .................. 112 23 Average Net Return to Storage, On-Farm Soybeans (1958-82) ........................ 116 24 Average Net Return to Storage, On-Farm Soybeans (Short- -Crop, 1958- 82) .................. 116 25 Cumulative Probability Distribution. On- Farm Corn (June, 1958- 82) ..... . ................ 119 26 Probability Distribution, On-Farm Corn (June, 1958-82) ........................ 121 27 Graphical Representation Approximating the Probability Distribution for Selling in June vs. the Normal Distribution ................. 122 viii CHAPTER 1 INTRODUCTION Each year agricultural producers choose from various marketing alter- natives. Each of the available alternatives accdmmodate strategy and decision making choices. Inherent to the overall decision process is the financial risk associated with the respective choices a producer makes. The cha11enging part in this decision framework is selecting a strategy or a combination of strategies which best accommodates farm management goals, objectives, and risk preference. A major decision producers face at harvest is whether to store and how long to store their grain to realize a positive net return to storage. This paper examines the net return to storage associated with alternative post-harvest market- ing strategies fbr corn, wheat, and soybeans in the State of Michigan. Simply defined, "returns to storage“ are the financial returns of storing a commodity from harvest to some future date after the costs of storing are taken into account. A producer who decides to store a com- modity at harvest is interested in whether the anticipated price increase during the post-harvest period will be sufficient to cover all the costs. This has become increasingly important to agricultural producers as price volatility (especially since the early 1970's) continues to play a major role in the marketing process. Seasonal price patterns for these grains also display varying amounts of deviation.1 Further, dramatic intra and inter-year price fluctuations have created instability in cash flow prac- tices for farming operations which in turn disrupts long range management plans and financial commitments. The volatility in prices complicates sales decisions. The producer must decide when to sell, how much to sell, and at what price. 1.1 Objective Each marketing plan or sales decision bears a certain amount of risk. Generally, efforts to attain a greater expected return entail a greater degree of risk. So it is expected that the sales decisions, when to sell, how much to sell, and so on, have a significant influence on average price received. The procedure to test this expectation is to measure how well various marketing strategies would have performed given histori- cal price and cost data from 1958 through 1983. The specific purpose of this paper is to present an evaluation of _cash marketing strategies designed to maximize net farm returns subject to a specified level of risk for corn, wheat, and soybean cash sales in the State of Michigan. Historical price and cost data provide the basis for calculating net returns associated with various selling strategies. Thus, the model used develops an efficiency frontier showing trade-offs between expected income and associated risk. It is hypothesized the analysis will suggest that, through careful selection of a marketing strategy compatible with farm management goals, 1John N. Ferris, "An Analysis of the Seasonal Cash Price Pattern on Michigan Corn, Wheat, and Soybeans," Agricultural Economics Staff Paper #79-6 (Michigan State University, East Lansing, Michigan, 1979). pp. 12-14. average net returns to storage can be increased for a given level of expected risk. More specifically, this research intends to provide more useful information in assisting the potential storer with the decision of when to sell grain. The study's scope is limited to evaluating cash marketing strategies. Further, the analysis considers storage of a commodity fer no more than one year. 1.2 Methodology» Data for this research was obtained from USDA publications, the Farm Credit Banks of St, Paul, Minnesota, grain and bean storage elevators in Michigan, and various other sdurces. Price and cost data is deflated to 1983 levels by the Consumer's Price Index. The CPI is an appropriate deflator as it measures the cost of a fixed bundle of goods that does not vary over time except for periodic revisions. Interest rates used in the calculation of returns to storage are defined as the cost of money to farmers through the Production Credit Association fbr production loans. storage loans, and or other operation expenses.‘ Real effective interest rates are used in calculating returns to storage. The effective rate takes into account loan fees and stock, which represents the effective cost to the farmer. Using real values in the analysis provides for a more accurate comparison with other current costs and prices and adjusts for the difference in purchasing power over time. A fortran computer program (presented in Appendix E) is developed to obtain net return to storage results. The analysis is further facili- tated by the use of a linear program. The linear program utilizes the MOTAD (minimization of total absolute deviations) approach developed by Hazell (1971) to measure the return associated with alternative marketing strategies. The methodology considers both on and off-farm storage and is based upon "producer sell decisions“ for determining the net return associated with alternative post-harvest marketing strategies for corn, wheat, and soybeans in the State of Michigan. To perform the analysis, basic port- folio theory and statistical methods provide the necessary framework. Lastly, the precept for which this research is largely based upon is that "we learn from history."1 1.3 Related Research . The proposed research is related to prior work. Most recent is the study by Rister, gt gl,, where a methodology based upon decision analysis is developed for determining economic returns to alternative post-harvest marketing strategies for grain sorghum in the Texas coastal bend region.2 The study uses stochastic dominance techniques to assess the impact of producerS' risk preference on "optimal" marketing strategies and assess the usefulness of price outlook information to producers. The study's results prove interesting and analysis of the evaluation of market out- look information is a major contribution. 1T. A. Hieronymous, "When to Sell Corn, Soybeans, Oats, Wheat," (University of Illinois, College of Agriculture, Cooperative Extension Service, Oct. 1966), p. 13. 2“Edward M. Rister, Jerry R. Skees, and J. Roy Black, "Evaluating Post-Harvest Marketing Strategies for Grain Sorghum and Assessing the Value of Outlook Information Using Stochastic Dominance," Joint Project, (Texas Agricultural Experiment Station TA 18098, Kentucky Agricultural Egpgyiment Station Paper No. 82-1-129, and Michigan State University. 8 . Cornelius examines alternative post-harvest marketing strategies for Pacific Northwest white wheat'producers.1 The study provides a simple-to-understand marketing plan from which agricultural producers can follow. Ferris evaluates seasonal behavior in prices and returns to storage for corn, wheat, and soybeans in the State ofMichigan.2 The study is useful in analyzing seasonal price variation and net returns to storage over time. Probability margins are designed and offer some indication of the risk and return associated with various marketing strategies. ' Each of the previously discussed studies provide interesting re- sults. The study proposed here is not intended to move beyond those by Rister, gt_gl,, and Cornelius but instead combine all pertinent informa- tion to formulate an "incorporated" approach for arriving at the returns associated with alternative post-harvest cash marketing strategies. This approach entails, for example, deflating all cost and price data to con- stant 1983 dollars and evaluating both on and off-farm marketing strate- gies.‘ A substantial time period of the historical data is developed for the analysis for which conclusions are then based. 1.4 Contribution This research is designed to contribute to the marketing material presented at the agricultural marketing workshops by Michigan State 1James C. Cornelius, "Marketing Management: Guidelines for Farm Level Wheat Sales Decisions," Working Draft (Oregon State University, Corvallis, Oregon, Department of Agricultural and Resource Economics, May 1982 . 2Ferris, p. 1. University faculty throughout Michigan. The results are expected to be a helpful guide in making storage and marketing decisions. CHAPTER II THEORETICAL CONSIDERATIONS 2.1 Seasonal Price Movement The returns associated with storing a commodity should not be ana- lyzed without first identifying the basic theoretical concepts underlying the reasons for storing. Relationships between differences in temporal prices and the movement of prices through time in relation to storage are explored in this section. The theory developed is particularly rele- vant to those commodities produced once a year but stored and consumed throughout the year and thereafter; (e.g.), corn, wheat, and soybeans. Following, seasonality of prices, the futures market, cash prices, and basis are used to explain the theoretical concepts of storing these com- modities. Most agricultural products are seasonal in nature with regard to production and marketing patterns. The price behavior of a seasonal crop is a repeating pattern, completed once every twelve months. Season- ality for grains arises from climatic factors and the biological growth process of plants. The usual price pattern for a seasonal crop is for the price to rise through the year as a function of the cost of storing the commodity. Thus, the commodity is allocated through the year by the relationship of current prices and expected prices to storage costs. Normally, prices fbr grains'of storable commodities are lowest at harvest time and then peak prior to the next harvest. To conceptualize the rise in prices throughout a "normal" crop year (to cover the cost of storage), the following example is given. Assume a "perfect market" in which all supply and demand factors as well as other information are known by all buyers and sellers. In such a case, cash and futures prices would f01low the hypothetical smooth pattern in Figure 1, representing "perfect knowledge” in the market. Price l l J J k “oath Figure 1 Grahpical Representation of Relationship Between Cash and Futures Price, and Basis The "basis" shown in Figure 1 represents the difference between a futures price and a cash price at a given point in time, which theoreti- cally accounts for the cost of storage plus delivery. As the delivery month approaches. the basis narrows. Depending on current inventories relative to expected supplies, a positive or negative basis may exist.1 Assuming a positive basis exists, the narrowing is a reflection of the decreasing cost of storage as the delivery month approaches. Simply. a producer stores a commodity if he/she expects the benefits from storage to at least equal the cost of storage. The perfect market concept discussed earlier may be viewed in equilibrium as FP - CP = CS: where: FP CP CS Price Price expected future price current cash price cost of storage between the two time periods 9, .................. E }Cosl of storage 1. to t; P ..... homo" end of season p------- \ b0---- % 12 months time Figure 2 Graphical Representation Depicting Seasonal Price Movement 1For a complete description of the subject matter see William G. Tomek and Kenneth L. Robinson, Agricultural Product Prices, (Cornell University, Ithaca, New York, 1972), p. 263. 10 In this context, the price of the commodity will rise from a low point at harvest by just enough to cover storage costs from the time of harvest to subsequent points in the year. As the next crop year approaches, price declines rather sudden to the next seasonal low. Figure 2 illus- trates these concepts.1 For a number of reasons, however, a "normal" seasonal price pattern does not often prevail within any given year. In essence this leads to imperfect knowledge and hence. producers may act on imperfect infbrmation; storing excess stocks, selling too much too soon, and so on. As a result, price may not increase enough to cover storage costs in a particular year. On the average, however, seasonal price increases should cover storage costs, otherwise, in the long run there would be no storage.2 Price changes within the year usually deviate from the smooth patterns depicted in Figures 1 and 2. The diagrams, however, emphasize the theoretical logic behind the seasonality component of prices. Since the real world is more complex and uncertain than in the theo- retical concepts just described, it seems agricultural producers would find it beneficial, over time, to implement strategies for improving upon their post-harvest marketing decisions. These decisions may include, for example, when and how much to store and sell and what marketing tools. to use. The basis fbr this study rests, in part, upon the assumption that producers do want to make better post-harvest decisions in order to more fully fulfill their marketing objectives. As such, further analysis 1 2 Ibid., p. 172. Ibid., p. 173. 11 in the following section provides a more detailed investigation of the seasonality component in prices. 2.2 Examination of Seasonal Prices As has been diScussed, seasonality in prices plays an important role in agriculture. To further understand the theoretical concept of seasonality in grain and soybean prices, a statistical analysis of the price data is examined. Tables' 1, 2, and 3 exhibit a "seasonal" analy- sis of the price data (1958-1983) for corn, wheat, and soybeans, respec- tively. An index, standard deviation, and trend value is given for each month in a year. To obtain the index value, a ratio is calculated for each month relative to a 12-month moving average. The ratio is then converted to a base of 100 and averaged fOr the entire period. 2.2.1 Corn Examining the month of November for corn indicates an index value of 91.8. This means that the average price of corn in November was 91.8 1 percent of the annual average for the 1958-83 period. Comparison of the monthly index values shows November averaging considerably lower than- all the other months and June through August ranging the highest among the indices. Prices generally average lowest at harvest (November) and increase (with exception of February and March) through the crop year up until August, and then decrease just prior to the following harvest (in September and October), when supply increases substantially. To measure the amount of variation in the indices. standard devia- tion is used. 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August shows a relatively high index of 104.0, however, it has the highest standard deviation, 8.9. 50, 68.3 percent of the time (approximately 2 out of 3 years), it would be expected that prices range between 95.1 and 112.9 percent of the annual average in August. The values labeled in the "trend" row indicate to what extent the seasonal price pattern for corn has been shifting over time. These trend values show the annual rate of change in the respective index. For example, nominal prices in March through October (with the exception of June through August), have trended slightly downward relative to the annual averages at approximately -.1 percent per year. 2.2.2 Wheat Index values fbr wheat indicate a seasonal price pattern different from that fbr corn. Prices at harvest (June-July) average lowest, how- ever, only increase until the following January instead of just prior to the next harvest. One could imagine that wheat prices would increase from the designated harvest month (July) until around May. This suggests Michigan wheat producers should carefully consider the decision to store past January. Negative trend values in the February to June period further indicate that careful consideration should be given to not storing past January. This period also contains some of the highest standard deviations of the 12 months. May and June are the highest with a standard 1Diana R. Harrington, "Modern Portfolio Theory and the Capital Asset Pricing Model - A User's Guide," (Englewood Cliffs, New Jersey: Prentice- Hall, Inc., 1983), p. 6. 16 deviation of 10.2 and 8.8, respectively. As-expected, standard devia- tions for wheat run consistently higher than those for corn. 2.2.3 Soybeans Seasonality in prices for soybeans is similar to that for corn. The index lows are in the harvest period (October and November) while there is a steady increase from November to June. In percentage terms. the increase over this period is 8 percent, which is lower than for corn. In absolute terms, however, the increase in cents per bushel is greater than that for corn simply because of the relative value of both commodi- ties. In other words, on a per unit basis, soybeans are worth more than corn. The variability in soybean prices (in absolute terms) over the 25 year period is greater than for corn and wheat, which is expected given the relatively higher prices. The January through March period displays the lowest variability in price, but, also shows a strong "trend" in downward price movement relative to other months. The theoretical concepts just described hopefully have offered a basic understanding of price movement for the grains considered. With this knowledge, one may and often does base store or sell decisions on expected and past price movement alone. As will be understood in follow- ing sections, however, many other factors are important in deciding whether or not to store or sell a commodity. Risk and the cost of storing, for example, are the most important factors fbr consideration. The following thus offers a general discussion on risk by examining some basic financial-theoretical concepts of relevance to this research. 17 2.3 Financial-Theoretical Concepts and Portfolio Theory So far no mention of the financial-theoretical concepts with respect to storing a commodity has been made. Just like a stock investor, the agricultural producer allocates resources (e.g., time and money) respec- tively among alternative risky prospects to increase his or her wealth. Each must choose a mixture from some available set of possibilities. This section first brings to light some of the basic theoretical concepts underlying portfolio choice under conditions of risk. Following, the subject matter is discussed as it applies to storing agricultural commo- dities. The subject area is extremely broad and will be discussed only in a general context. Perhaps a starting point for this topic is a discussion interpreting and explaining the term "portfolio theory." Portfolio theory (or Markowitz theory) delineates the decisions that will be made by a popula- tion of normal investors - each exercising his or her personal perfer- ences."1 Here and henceforth the term investor may be thought of as that defined in Webster's Dictionary: one who commits "(money) in order to earn a financial return 2: to make use of for future benefits or advan- tages." Thus, it is easy to see that an agricultural producer who stores grain in the hopes of higher financial return complies with this defini- tion, since the grain could have been sold for a certain amount of money. More specifically, portfolio theory holds that all investors are risk averse; other things being equal, all rational investors will avoid risk. One of the first models to deal explicitly with risk in a portfolio 1John L. Maginn and Donald L. Tuttle, Managing Investment Portfolios 1 A Dynamic Process (Boston: Warren, Gorham and Lamont, Inc., 1983), p. 92. 18 sense was devised by Harry Markowitz (1952). In general, the model states that the investor chooses among all possible investments on the basis of their risk (portfolio variance) and return (portfolio return). These two characteristics are plotted graphically fer a group of invest- ments in Figure 3. Each x represents a possible investment. It is possible for some x's to represent a single asset, whereas others may represent various combinations of assets. .Hence, the portfolios (x’s) constitute all possible combinations of the individual investment's alter- natives. Average Return (E,V) Frontier Risk (variance) Figure 3 Graphical Representation of the Risk/Return Trade-Off Among Alternative Investments In choosing among the possible portfolio alternatives the rational investor will choose investments that provide the highest return for expected level of risk or, those that offer the lowest amount of risk for a given return. As one can surmize from the graphical representation, the "best return" portfblios theoretically lie on the line. This line 19 represents the "efficient" (E, V) frontier in that no portfolio with this much average return has a lower variance. Those portfolios lying below the line are termed "inefficient" because, at any given x below the curve it is possible to obtain greater certainty of return with no less average return. Furthermore, it has been shown that a mixture of risky prospects (in an (E. V) context) provides for diversification of any given port- folio. In other words, specific amounts of diversification reduce vari- . ability in return. Hence, the risk averse investor will essentially be characterized by possessing a diversified portfolio, since diversity generally represents aversion toward risk. 2.3.1 Economic Theory and Choice All that has been stated thus far relates to an investor’s trade- off between two important dimenSions - risk and average return. The investor, however, has not been given any direction as to choose a par- ticular portfolio. .This is where the theory of choice intervenes. The theory proposes to solve this problem by first specifying those alterna- tives or options available to the investor and second, showing how to choose among those alternatives. Depending on the investor only some of the x's (portfolios) displayed in Figure 3 may be deemed "available" alternatives. Assuming the inves- tor has recognized these alternatives, the next step is to choose a port- folio among the available opportunities. The investor's preference fOr risk can be graphically represented by plotting the trade-offs between risk and average return. The line connecting the preferred risk-return trade-offs are called "utility" curves. Figure 4 illustrates this with 20 Average Return Utility Curves fficient Frontier Risk (variance) Figure 4 Graphical Representation Illustrating an Investor's Preference for Risk and Return the efficient frontier (line B-C) and a set of utility curves reflect- ing the investor's risk-return trade-off. Each curve represents a com- bination of risk and return equally satisfactory to the investor. As can be seen, as risk increases, the return required to induce the inves- tor to take the risk must also increase. Also, a point 0 exists, repre- senting the point at which any further amounts of risk acquired will re- sult in a decrease in average return. The final step in this second process is the matching of the avail- able investment alternatives with the investor's most desired alterna- tive. This selection of the optimal combination of risk and return from ,the efficient set of many alternatives is represented by point A in 21 Figure 4. The investor chooses point A because: (1) there are no.invest- ments on a higher utility curve; and(2) anything below point A would not yield as much utility (satisfaction) as an investment on the efficient frontier. Obviously, different investors with different attitudes toward risk will have different sets of utility (indifference) curves. The popula- tion of investors may agree on the efficient set of alternatives (line A-B), however, this does not mean that all investors will choose the same portfolio. Since each investor has his or her own set of indiffer- ence curves, the selection of investments will be wide-ranging depending on the amount of risk the investor is willing and able to assume. This risk comes in many forms as will be mentioned, however, it is first necessary to understand some of the basic underlying concepts of the "market" with respect to portfolio theory. 2.3.2 Theoretical Considerations of the Market When an investor selects an investment for purchase or sale he or she may proceed with the transaction without any prior information as to price, volume traded, etc. In fact, some have argued that the typical investor would do just as well and possibly better if investment selec- tions were made by "throwing darts," (known more frequently as the "ran- ”dom walk" theory). This is an equivocal reflection on the market in that information is not likely to be very helpful in making profitable deci- sions. Actually, this view is a derivative of the efficient market hypo- thesis (EMH).1 1Ibid., Maginn and Tuttle, p. 396. 22 In a general sense, the efficient market hypothesis states that it would be impossible consistently to outperform the market. The "efficient market" being one in which all infOrmation impacting an investment's 1 In this type of market the average return is reflected in its price. investor should expect to earn a fair return, and not a superior or in- ferior return. . As we saw earlier, the investor looks for the highest possible re- turn given the level of risk he or she is willing and able to assume. One can assume then that the investor's objective is to maximize the utility of wealth, where utility describes the differences in individual preferences. .It is assumed that while an investor may have preference for a given investment, he or she also has what is commonly termed a "diminishing positive marginal utility." Simply, this says that more wealth is preferred to less but, each incremental amount of wealth is enjoyed less than the last because each increment is less important in satisfying the basic needs and desires of the investor. Other less com- ” monly developed utility functions might include an investor with a pre- ference for risk. In this sense. the investor (risk taker) prefers more to less but, each increase in wealth makes the individual more acquisitive. As stated earlier investors make choices on the basis of risk'(vari- ance) and average return. The variance of any given portfolio is the only factor determining investors perceptions of risk. Average return is the only other influence on an investor's choice. Each of these factors 1Andrew Rudd and Henry K. Clasing, Jr.. Modern Portfolio Theory_- The Principles of Investment Management, (Homewood, Ill.: Dow Jones Irwin, 1982), p. 164. ' 23 are essential elements in portfolio theory and are described in the following text. The average of a past series may also be called the mean. The mean value .. a probability distribution is called the expected value of a random variable or uncertain event. Hence, the mean expected rate or value is a weighted average of possible outcomes with probabilities or frequencies used as the weights. From this point on, the term used to describe risk will be "standard deviation,“ which is the square root of the variance. Markowitzs' earlier work determined the standard deviation of a portfolio by: l) the standard deviation of each investment; 2) the correlation between each pair of investments; and 3) the amount invested in each investment. After a, b, and c are known, the standard deviation of the portfolio can be computed. Markowitzs' work showed that the higher the correlation among investment returns, the greater is the standard deviation of the 1' Although this earlier wdrk has proved useful. this research portfolio. utilizes this concept of standard deviation only in part, as will be seen in subsequent sections. Controversy over using standard deviation as a measure of risk has been an issue for many years. The problem in using standard deviation is that it is an accurate description of only normal distributions. Hence, it is possible for two given portfolios to have the same mean and standard deviation and offer quite different returns. This so called 1Harry M. Markowitz, Portfolio Selection - "Efficient Diversifica- tion of Investments," (New York: John Wiley & Sons, Inc., 1959), p. 19. 24 "skewness of returns" distribution is_ignored by simple portfolio theory's use of standard deviation as the sole measure of risk. Skewness. however, can and is an important factor in investment decisions. Solutions to correct skewness problems do exist.1 In reality many continue to use standard deviation as an appropriate measure of risk. The main reason is because it allows one to use the mean and standard deviation to describe an investment's relative attrac- tiveness. Further, it can now be concluded from previous discussion on portfolio theory's assumptions of investor's choice, that investors choose those portfolios with the highest rate of return for their pre-. ferred level of risk. However, is a certain or given level of risk viewed the same by all investors? Portfblio theory assumes that all investors' estimates of risk and return are similar. Hence, the theory creates a Single efficient frontier (the (E, V) frontier - as seen in Figure 3) in which all investors have a "consensus" on the estimated mean and standard deviation and thus of the relative value of each investment. Assuming investors have homogeneous expectations is not necessarily reality in the marketplace. Obviously, one can see that investors have different expectations about the future. The point, however, is whether this diversity affects prices. According to the efficient market hypo- thesis, the price of an asset is the best estimate of the future pros- pects for that asset. In summary, the assumptions of the efficient market hypothesis are not realistic. This is a widely known fact. The reason for its 1Harrington, p. 25, footnote no. 4. 25 continued use, however, is that if some explanation or forecast can be derived from the-model, it can be used to make better decisions. This research neither subscribes to nor hypothesizes any form of the efficient market hypothesis but, presents a discussion to provide for clarification of some characteristics in the market. 2.3.3 Risk and Uncertainty Throughout the paper the term risk has continually appeared but only been described in the context of standard deviation of a portfolio. This risk (standard deviation), being the possibility that the actual return from an investment will differ from the expected return. The previous overview has in fact been accomplished without describing where risk arises or explaining its economic origins. The following attempts to resolve these deficiencies. ‘Before going any further, however, a distinction should be made between risk and uncertainty. In this sense risk may best be described .in that the probabilities of various outcomes are known. Uncertainty. however, implies no knowledge of the probability distribution of the possible outcomes. Stated another way, there exists no reliable means ‘ of estimating the likelihood of an event occurring. An uncertainty associated with commodity prices, for example, may relate to unforseen political events (i.e., Soviet Grain Embargo). Althbugh many agricul- tural producers do not make the distinction between risk and uncertainty it is useful to do so in the sense that there are different types of risk. Two types of risk associated with the marketplace are "nonsystematic" and "systematic" risk. The former is described as risk that is non-market- related. It is defined as so because it is caused by changes that are 26 specified to the decision-maker. Erratic changes in management style. for example, may affect the net returns to storing. This type of risk. is thus considered unexpected or unpredictable with respect to manage- ment decisions. However, unsystematic risk can be diversified away (e.g., improvement in management decision behavior) and so it is assumed not to be important to the storers' forecast of future returns.1 Systematic risk may be defined as market related, or risk that is caused by economic and or political events that affect the returns of all assets. An example of this is the Soviet Grain Embargo of 1980. All those with grain in storage at that time were affected to some ex- tent. It should be clear now that it is this type of risk that cannot be diversified away and that which the storer of grain requires compen- sation for. I The previously discussed types of risk may come in various forms. These forms include interest rate risk, liquidity risk, purchasing power risk (the "inflation affect"), business risk (the risk of remain- ing solvent), and investment riSk - (i.e.)will it pay to invest in on- farm storage facilities. Each of these forms of risk could easily accom- modate lengthy explanations, however, it is not necessary that the reader understand in detail the various forms and thus only a general elicita- tion is presented. In retrospect, we can assume that the basic principles of portfolio theory previously discussed apply to the potential storer of grain. Re- member, according to the definition of investor, the potential storer is, in a sense, an investor. He or she commits an asset (grain) to storage 1Harrington, p. 14. 27 in hopes of attaining a return higher than what would.have been received if the grain was sold at harvest. Since we now consider the potential storer to be, in a general sense, an investor, we can assume the charac- teristics of the storer in the market not to be any different from that of a stock investor. In other words, we may expect the two investors to be identical with respect to the financial-theoretical concepts discussed. With the previously stated assumptions it is now possible to move ahead into other facets of importance regarding the theoretical concepts of storing grain. The following concepts and procedures to be discussed are relevant to the theoretical framework described thus far, however, they more specifically relate to basic concepts of decision analysis. We will be mainly concerned with risky choice (choosing between avail- able alternatives) in a managerial context. Hence, the following . utilizes only some of the basic concepts of "decision analysis" and probability theory as it pertains to the analysis set forth in this research. 2.4 Decision Analysis Considerations In general, decision analysis pertains to the systematic rationali- zation of risky choices among alternatives. It is a logical procedure for making risky choices. The approach indicates which alternatives the decision maker ought to take. Further, decision analysis (as it applies to this research) involves: (l) defining relevant acts and states and their outcomes; and (2) selecting the optimal strategy on the basis of maximizing expected utility.1 Much of the processes involved in decision 1Jock R. Anderson, John L. Dillon, and Brian Hardaker, A ricultural Decision Analysis, (Ames, Iowa: The Iowa State University Press, 1977), p. 12' 28 analysis is attempted in an intuitive manner. The more formal processes. however. enable a decision maker (manager) to better ensure that his risky choices suit his preferences and personal beliefs and that the outcome will be as close as possible to the expected outcome. Hence. the fbllowing is intended to show how decision analysis can be used to make better decisions in order to obtain personal goals. The discussion begins by identifying only those basic components of decision analysis utilized in this research. The acts or actions available to the decision maker, between which he must choose, may be referred to as an "act," denoted by a], a2...., aij' It is necessary these actions be defined as mutually exclusive and exhaus- tive, In other words, two or more outcomes cannot occur simultaneously and the sum of the outcomes' separate probabilities must be equal to one. For our purposes, however, the later will not be of any significant impor- tance as will be understood later in the paper. States of nature or possible events are termed "states." States are also defined as mutually exclusive and exhaustive and are denoted by S1, 52, S3,..., Si‘ State of nature variables may be continuous by nature (e.g., rainfall), however, a discrete representation is adequate for this analysis. The "outcome" depends on which state occurs and which act was chosen. Outcome may be measured in terms of utility, which can be represented from the ith state and jth activity; U(ajlsi). By measuring the outcome in terms of utility, all aspects of the decision maker's preferences are captured and correctly balanced.1 1Ibid.. p. 5. 29 Further. "predictions" are denoted as p1. p2..... pk. In decision analysis predictions are used as additional information about the states of nature. Hence. it is common to convert this information into "like- lihood" via conditional probability. The likelihood probability pertains to a specific experiment or empiricism. In general, it is the probability of observing prediction pk given that a particular state, the ith inter- val, prevails. This explanation may be denoted as P(pk|si). For pur- poses of this discussion the likelihood probabilities are based on per- sonal or subjective judgement but do not necessarily have to be. ' With the above information it is now possible to define a strategy. In this context, the term strategy may be thought of as the action that h strategy St, for th is taken in advance of some expected outcome. The tt example, could be defined as taking the jth act in the i state. or taking the it“ act when the kth prediction is observed. The previous discussion represents only a very few of the basic concepts used in decision analysis. The described components. however, are the basiCs in a multitude of theoretical approaches used to explain decision theory. They eventually are used to some extent in the analy- sis of this research but befbre it is seen how they are utilized, a widely used decision theory is discussed. Hence, the following provides a general understanding of Bayesian Theory. 2.4.1 Bayesian Theory More recently there has been a shift of emphasis from classical, or traditional, statistical inference, to the problem of decision making under conditions of uncertainty. The modern formulation has come to be known as statistical decision theory or "Bayesian decision theory." 30 This theory is based on the assumption that regardless of the type of decision. certain common characteristics of the decision problem can be discerned. In general. the Bayesian method utilizes the previously dis- cussed components of decision analysis in the following way.1 P(Si|pk) g :(si)P(kaS.) a P(:i. pk) (Pk) 1:]P(s1-)P(pk151-) The decision maker is aware of several possible states of nature si but does not know precisely which one of them truly prevails. Based on the present state of knowledge about the situation or perhaps some sort of empirical evidence. the decision maker may make some assessment of the probabilities P(Si) that reflects his or her personal beliefs as to how likely the various alternative states of nature are to prevail. These probabilities are called prior probabilities for the states of nature. The decision maker proceeds with observing various predictions or forecasts which are denoted by pk. He or she may then determine the probability of the kth prediction given it has been observed under a specific state of nature Si. These are the conditional probabilities P(pk,|si), i=1, 2,..., k. Bayes' Theorem then allows the decision maker to calculate the conditional probabilities, which are nothing more than an expression of the decision maker's revised belief concerning the different states of nature after observing the kth predictions. These revised probabilities are called posterior probabilities in that they form the basis for whatever inferences the decision maker wishes to make about the unknown state of nature. 1Ibid, p. 50. 31 Although the Bayesian approach to decision making has been criticized by some because prior probabilities can be affected by the decision maker's own biased viewpoint, it is a widely respected technique and used in many fields. Due to the limited scope of this research further discussion of Bayesian inference as it applies to decision analysis is not warranted. however. it is discussed in this section for reasons that will be apparent in the final text of this paper. Given the basics, a model can now begin to be developed. So that we may understand exactly what we are to begin to build, a review of the essential elements is presented as it applies to this research. (1) The decision-maker. In this case it is the potential storer. The decision-maker may be a single individual, a corporation, etc. (2) Alternative courses of action. The potential storer must choose among alternative actions or "acts." It is assumed he or she will choose the best alternative act or combinatibn of acts with respect to their preferred level of risk and return. Acts in the foregoing analy- sis represent selling in different months, any 12 of them. For example, a strategy may be "sell 1/2 of the harvested crop at harvest and 1/2 in January." (3) States of nature. These states are viewed as lying outside the control of the decision maker. States in the fbregoing analysis will represent years, 25 in all, crop years 1958-59 to 1982-83. (4) Outcome. A measure of net benefit to be received by the deci- sion maker under particular circumstances. The outcomes are summarized in a payoff table. which displays the consequences of each action selected and each state of nature that occurs. Outcomes in this research are de- noted in terms of net return per bushel as will be shown later. 32 (5) Objective function. This function is primarily concerned with the criterion of maximizing expected utility. The objective function in this context is total net return to storage. The payoff table. expressed symbolically in general terms, is given in Table 4. It is assumed there are 12 months in which the storer can sell grain. These months (acts) are denoted as a], a2,..., aij‘ These different possible courses of action are listed as column headings in the table. There exists also 5 possible states of nature, denoted s1. 52...., $25. Each state represents one year. The outcome resulting from each combination of an act and state of nature is designated by the symbol 0 with appropriate subscripts. Hence, 0 represents the net bene- fit or outcome of the selection of an act and the occurrence of a state of nature and further, can be treated most generally in terms of the utility of this consequence to the decision maker. In summary, the util- ity of selecting act a and having state 5 occur is denoted 012, for exam- ple. Note that the first subscript in these utilities indicates the state that prevails and the second subscript denOtes the act chasen. TABLE 4. Payoff Table ’ Acts State a1 a2 a3 . . aij 51 011 012 - 0112 S2 021 022 0212 O 9 0 S25 ‘ 0251 0252 02525 33 Notice in the previous discussion no mention of prior probability is made. In other words, no probabilities were subjectively assigned to the states of nature. One of the distinguishing characteristics of Bayesian decision theory, as we have seen, is the assignment of personal or subjective probabilities. Hence. this is important in that the analy- Sis proposed in this research does not attempt to use Bayesian statisti- cal procedures as a method for determining which strategy will be chosen. The approach to be used, however, does utilize all the components dis- cussed thus far. except the use of prior (or subjective) probabilities to states of nature or to the actions. With these concepts behind us, the following presents the method used by which the decision maker can reach his or her expected utility given preference for risk and expected return. 2.5 Planning Under Risk This section utilizes what has been diScussed-thus far (in Chapter 2) and systematically interrelates these concepts into a more workable form. Given the basic theoretical concepts underlying portfblio choice, we know that the intent of the decision maker is to maximize utility. The decision maker's objective is to find an optimal portfblio lying on the (E, V) frontier that accommodates his or her personal preference for risk and expected return. The foregoing application focuses on the allo- cation of resources among alternative risky prospects in order to maxi- mize utility. I To operationalize the methods of maximizing utility, several mathe- inatical programming techniques have been used, Among the most popular and successful models have been that of quadratic and linear programming. 34 Each of these techniques takes into account risk through mathematical programming formulations and sustains common variables with which an optimal solution is derived. 2.5.1 Linear Programming In linear programming. "linear" implies that all the mathematical functions in the model are required to be linear functions. "Programming" simply means arranging or planning the problem at hand. Thus, linear programming essentially deals with the problem of allocating limited resources among competing activities to obtain an "optimal" result. This result being one that best reaches the decision maker's preferred or de- sired level of risk and return among all given feasible alternatives. The following provides a basic understanding to the form of the model and defines a feasible solution as each apply to this research. Linear programming finds the optimal values of the variables x1, x2...., xj,..., X"; where xj represents the jth storage activity. "Activities“ represent all the possible alternatives that can be con- ducted by the decision maker and all possible ways of undertaking these alternatives. One can now imagine that there are any number (m) con- straints (of any kind) to which any number (n) activity levels are re- stricted. This may be viewed as: ' n (F2.l) .2 a 3:1 Xj'i=1}bi ”lit-mm ii In this formulation (F2.l) only one of the signs can hold for any given constraint. The other terms are represented as follows: bi denotes the ayailable amount of the ith resource, and a. is the technical input- 1i output coefficient which specifies the amount of the ith resource 35 required fer a unit of product from the jth activity. Further, a restric- tion on x1 is that it be nonnegative since, fbr example. negative amounts of grain sales do not exist. Hence, the constraints reflect competition between activities and their interrelationships for the limited alloca- tion of farm resources. Since xj is the level of activity j (a decision variable) for j = l, 2,..., n. allow Z to be the overall measure of effectiveness. The cj will then be the increase in Z that would result for a unit increase in x:j (j = l, 2,..., n), the decision variable. Given this, let the bi represent the constraints (amount of resource to be allocated), (l = l, 2,..., m). The above formulation and terms are summarized in Table 5. TABLE 5. Activity Table Resource Activities (aij) Constraints 1 2 3 ... n ‘ a‘11 a12 a1n b1 2 a21 a22 a2n m aml ‘ amn bm AZ/Unit C1 C2 ... Cn Level X1 X2 ... Xn 36 Given the technical constraints as seen in formulation (F2.l), we can now complete the model. This last part entails creating an "objec- tive function." This function maximizes net returns to storing subject to the constraints in (F2.l) and may be written as: n (F2.2) Z =ji]cjxj - F where Z is net profit and c. is the per unit net reveune generated from J the jth activity. F denotes fixed costs but, can be omitted since these costs do not vary with the levels of the activities. This omission can take place without affecting an optimal linear programming solution. As mentioned earlier, the xj variable represents the decision variable. Given the above information, a solution can be obtained. Simply, a feasible solution is one in which all the constraints are satisfied. It is possible, however, that this solution is not the desired solution of the decision maker since many feasible solutions may exist. Hence, an optimal solution is one that represents the most favorable value of the objective function and is most desirEd by the decision maker. 2.5.2 Some Assumptions of Linear Programming Various assumptions in linear programming are implicit in the pre- viously formulated model. However, to more easily evaluate how linear programming can be used in this research, it is helpful to highlight some of these assumptions. In linear programming the certainty assumption states that all the parameters of the model (the aij’ bi’ cJ values) are known constants. In reality, however, this assumption is seldom satisfied. In general, linear programming models are formulated to select some course of action 37 in the future. If the parameters used are based on a forecast of future conditions, this inevitably introduces some degree of uncertainty. Hence, one would expect these parameters to change over time. _ One other assumption is that of divisibility. Often. the solution obtained by linear programming is not in integer form. This being the case, activity units can be divided into any fractional levels. This is important so that non-integer values fbr the decision variables are per- missable. Other assumptions exist, however. it is sufficient for our purposes to mention only those previously discussed. _ One last assumption that is needed to carry out the later analysis is that the states of nature in the linear programming model are con- sidered "independent." In other words, information on the occurrence of the first state of nature 51 yields no information about the occurrence or nonoccurrence of the state of nature 52. Described another way, what happens in 51 has no influence on what is expected to happen in 52. Hence, the states of nature considered in this research are assumed to be inde- pendent of one another. 2.6 Quadratic Programming Quadratic programming is similar to linear programming in many re- spects. It is considered by many to offer more desireable properties in terms of solving problems. Perhaps the main difference between OP and LP is the method by which a solution is reached. OP utilizes the sum of the squared deviations about the mean to reach an optimal solution while LP (MOTAD) simply uses the absolute deviations from the mean. Further, OP requires a priori the variance and covariance relationships for each activity. 38 Using the MOTAD model as a substitute for QP may result in some loss in the reliability of the results but nonetheless has proved suffi- 1 cient and even superior to some instances. For the purposes and objec- tives of this research. the LP programming model is quite adequate. 1P. B. R. Hazell, "A Linear Alternative to Quadratic and Semi- variance Programming for Farm Planning Under Uncertainty," American‘ Journal of Agricultural EconOmics, Vol. 53, No. (1), pp. 53-62, 1971. CHAPTER III DEVELOPING THE MODEL 3.1 Storing the Commodity: Further Assumptions and Considerations As we have seen, many theoretical concepts interrelate with the storing of a commodity. Much of the discussion thus far has related to the basic theoretical concepts underlying the principles for storing grain and, further, has provided a general understanding of the impor- tance of these concepts. With this knowledge, attention now will focus more specifically on the assumptions directly relating to the analysis to be perfbrmed. It is now clearly evident. the reason for storing grain is the anticipation of price increases relative to the costs incurred. To ob- tain net commercial and on-farm storage margins, the casts of storing grain must be taken into account. This section discusses the processes involved in storing grain and soybeans, the costs incurred, and other pertinent information with respect to commercial and on-farm storage. In deciding whether or not to store grain, the decision maker should first determine the level of storage costs. The alternative methods of storage from which the decision maker has to choose are: (1) commercial storage; (2) on-farm storage in bins; or (3) on-farm storage in cribs. For purposes of this research the potential storer stores grain: (1) commercially at a local elevator; or (2) on-farm in bins, in which all equipment necessary to properly store grain is assumed to be owned by 39 40 the potential storer. Thus, depending on the method used for storing, costs will vary dramatically. 3.1.1 Commercial Storage The analysis to be performed assumes commercial storage is repre- sented by a local elevator. If a producer stores commercially, drying and other maintenance associated with storage'are the responsibility of the elevator. Further, a contract between the two parties is created whereby the amount of grain and other significant factors in storing grain commercially are agreed upon at the time of the transaction. Once the transaction is completed, all risks associated with the physical storing of the grain (e.g., damage, spoilage. etc.) are incurred by the commercial storer. The cost of the services perfbrmed by the commercial storer (drying, storing, etc.) are. of course, paid by the owner of the grain. Costs for these services are discussed in later sections. The costs for commercial storage varies slightly at any given time among commercial elevators in Michigan. Commercial storage cost data used to perform the analysis were obtained from various commercial stor- age elevators in Michigan. Thus, the data obtained from the various ' locations is considered representative of the cost of commercially stor- ing grain and soybeans fer the period analyzed. The commercial storage elevators where data was obtained included, for example, the Pigeon Coop in Pigeon, MiChigan and Mason Elevator, located in Mason, Michigan. Commercial storage costs for the commodities analyzed are presented in Appendix A. 41 3.1.2 On-Farm Storage The assumptions for on-farm storage are as follows. It is assumed the storer has the necessary equipment to properly store grain and that no other use is made of the facilities. Further. no charge is made fbr the fixed costs. Hence, the decision of using the facilities in a par- ticular year is not affected by the fixed costs. The fixed costs of storing on-farm include interest. depreciation, repair and maintenance, property tax, and insurance on the investment. Variable costs of storing on-farm are included in the on-farm analy- sis. These costs represent the additional costs incurred while the grain is in storage. They represent interest on the money tied up in grain (opportunity cost), insurance on the commodity, shrinkage and deteriora- tion (insect damage), and the cost of aerating. An assumption of the on- farm analysis, however, is that returns are considered to be the returns to any storage profit that might be realized. Costs fbr labor and manage- ment are not accounted for. In other words, the on-farm analysis assumes no costs for labor and management. One further assumption pertaining to the commercial and on-farm analysis is that no account is taken for drying costs (except for a sen- sitivity analysis as will be seen). Stated another way, in obtaining net return to storage values, the calculations begin with a standard bushel of grain and thus do not consider the cost of drying to the 15.5 percent level. For example, the "standard bushel" for corn is as follows: (1) weights 56 pounds per bushel (2; has a moisture content of 15:5 percent has less than 3 percent foreign mater1al (4) has less than 5 percent damaged kernels (5) has a test weight of 54 lbs. 42 The reason for assuming a standard bushel is that the grain has to be converted to a "standard" level of moisture whether it is sold at harvest ot during post-harveSt. ’If the grain is not converted to a standard level. discounting the value will adjust for the process. As will be seen, however. a sensitivity analysis will take into account drying costs to a certain extent. For this reason a discussion of the fellowing is necessary. 3.1.2.1 The Drying Process ' Drying to below "standard" levels generally takes place for grain that is to be stored for extended periods of time; fbr example, longer than 3-4 months. The process of drying grain fior storage is handled by machinery that either: (1) dries grain in batches; or (2) dries grain as it flows continuously through the equipment. Each of these grain drying systems includes an air—moving device, a means of introducing the air into the grain mass, and a chamber to hold grain. A heating system may or may not be a part of the drying facility. For simplicity, the analysis considers only a high temperature con- tinuous-flow column drying system in obtaining cost estimates for on-farm 1 This system is chosen for its general acceptability and wide storage. use in Michigan. The continuous-flow drying system requires equipment for an input of wet grain and removal of dry grain at a rate conSistent with the drying capacity of the unit. Dried grain may then be further conditioned, stored, or marketed. 1Roger Brook, Agricultural Engineer Extension Agent, Michigan State Engineering Department, personal interview, November, 1983. 43 The drying process is undertaken fbr several reasons. To store .grain in a safe environment free of mold infestation. the moisture con- tent of the product must be controlled. The major objective in drying grains is to reduce the moisture content so that spoilage will not occur before they are utilized. Table 6 provides infbrmation regarding initial and recommended moisture content for grain and soybean storage in a "normal" season in Michigan. TABLE 6. Initial and Recommended Moisture Content for Storage Usual When Through Winter Through Harvested (till April) Summer Shelled Corn 24 - 26% 15% 14% Wheat 13 - 16% 14% 13% Soybeans - 13 - 16% ' 14% 12% .The length of time that crops can be stored varies with the moisture content and the crop. To store a crop an entire year and especially through the summer months, however, its moisture content shduld be approx- imately l-2 percent below the moisture level that is considered safe for 3-4 months storage. Obviously some years are not "normal" with re- spect to temperature. humidity, etc. To simplify the analysis, however, each year in the historical time period is considered "normal." Lastly, it should be stressed that the process discussed above is essential to a successful operation. In other words, the decisions inanagement makes with respect to various levels of drying can have a 44 substantial impact on realized net returns. Since spoilage and damage can result in large losses. management decisions on drying levels are critical. 3.1.2.2 Management of Continuous-Flow Dryers Continuous-flow dryers are usually operated 16 hours a day or more and thus require careful management. The higher-than-normal air tempera- ture required to dry grain demands that careful attention be given to safety devices. Management includes proper maintenance of all mechanical equipment in addition to the fOllowing factors: (1) final grain moisture should be checked daily; (2) exit air temperature should be checked periodically at several places to assure that airflow is well distributed over the entire grain bed; (3) trash and fines should be cleaned out of the heated air plenum on a regular basis; and (4) metering devices should be checked regularly to assure that grain flow is not blocked by husks of foreign material. 3.1.2.3 Aeration Aeration is the practice of ventilating stored grain with low air- flow rates to maintain grain quality. Aeration: (l) prevents moisture migration by maintaining a uniform temperature throughout the grain mass; (2) cools the grain to reduce mold growth and insect activity; (3) removes storage odors; and (4) distributes fumigants in the grain 111655 .1 1Donald B. Brooker, Fred W. Baker, and Carl W. Hall, Drying,Cereal Grains, (Westport, Connecticut: The AVI Publishing Company, Inc., 1974), p. 179. 45 During a normal year in Michigan aeration may be used any number of times, however. it is common that the process take place 2-3 times; once near the harvest period and again when seasonal temperature changes be- gin to take place.‘I Figure 5 illustrates the general time periods in which aeration would take place in a normal year for the commodities analyzed. Temp. (‘F) 55' Figure 5 Graphical Representation of Aeration Periods In a Normal Year Aeration may also occur during unexpected or abnormal temperature changes. Preventing moisture migration and cooling the grain are the main purposes for aerating. 1Roger Brook, general discussions subsequent to previous interview. 46 3.1.2.4 Management of Aeration Aeration is usually started soon after the grain (corn and soybeans) is placed in storage. Aeration for wheat usually takes place near the ’ same time as that for other grains. This process is desirable to help equalize any moisture or temperature variations in the grain mass. The final grain temperature sought in the aeration process partially depends on the length of the storage period. Grain that will be removed from storage in the spring should register a final temperature of approxi- mately 50 degrees Fahrenheit. If conditions are hot and humid during this period while the grain is moved, there Will be little or no moisture condensation on the grain surface. If grain is to be stored through the summer months, it should be cooled to approximately 35 degrees. Fans need not be operated. however, when the air temperature is below 30 degrees. ' Grain at the surface of the storage facility may pick up moisture from warm, humid air with the advent of spring and summer. If this happens, damp grain can develop in localized areas and result in mold and insect growth. In this case. the operator can allow some severe local spoilage to develop or warm the entire bin of grain and fumigate. If he decides to warm the grain, it is possible that it will result in considerable weight loss through drying. Also. once the warming process is started, it must not stop until the entire grain mass is warmed. Condensation in the colder grain ahead of the warming zone forms a wetted layer adjacent to the warm grain. If this layer is allowed to remain in one place very long, the grain will spoil. During a normal year the aeration process for corn, wheat, and soy- beans generally takes place at the same time for the three commodities; 47 once soon after harvest and again around April when temperatures begin to climb. Aeration (for purposes of this analysis) is carried out by running the fan for 200 consecutive hours fbr each period. This process is derived by assuming the on-farm storage capacity is a 15.000-20.000 bushel bin filled to capacity with a 5 horsepower motor to drive the fan. Naturally. necessary fan time will vary according to the amount of grain that is placed in the bin, the bin size, and horsepower of the motor. Other maintenance or management time requires that the bin of grain be checked periodically to detect development of "hot spots" or mold pockets. Either of these factors has the potential of creating costly damage if not prevented. It is suggested that the bin be checked at least once every two days fbr proper care of the grain. 3.2 Costs Incurred for On-Farm Storage Each of the previously discussed practices incdr various costs in- herent to on-farm storage. As mentioned earlier, the cost of the on- farm storage facility and peripheral equipment needed to store grain properly is not included as a cost to the producer in the analysis of net returns to on-farm storage. The only variable costs included in the on-farm analysis are aeration costs and opportunity cost. Other costs (are not included for reasons previously mentioned. 3.2.1 Drying Costs Drying costs of grain to desired levels (as seen in Table 6) may be considered a major component of the total operating costs for on-farm storage. In a "normal" year in Michigan, corn is harvested having a moisture content of 24-26 percent. The desired moisture content for storage (at least through the winter months) is about 15-15.5 percent. 48 Prices fbr corn are reported in terms of a standard bushel (56 lbs., 15.5 percent moisture). This being the case, no drying cost is included in the on-farm net returns analysis. In other words, the on-farm analy- sis begins with a standard (dry) bushel of corn. If the producer knows at harvest, however, he is going to store grain at least until April, a drying cost is incurred at harvest in drying the grain from 15.5 per- cent to 14 percent. which results in some weight loss and hence is con- sidered an extra cost. Since the analysis is perfbrmed assuming a dry bushel, the extra drying represents an added cost to the storer. This enables the producer to hold grain through the summer months at a safe level if he so desires. ‘ The analysis in this study assumes the storage facility to be a continuous flow column drying system, demanding the use of electricity and propane to run it. Previous work by Brook and Bakker-Arkema indicates the drying cost ratio in cents per pound of water removed is 1.7 for this system.1 Based on historical energy prices for propane and elec- tricity, a conSistent time series was developed to estimate the cost of drying corn from the 15-15.5 percent range down to 14 percent. These costs are presented in Appendix B. If at harvest a producer anticipates storing corn past April, he may choose to dry the corn to 14 percent at that time in order to save on drying costs. His alternative is to dry to approximately 15 percent at harvest and dry further in April if grain for future sale remains in 1R. C. Brook and E. W. Bakker-Arkema, "Energy Utilization in Grain Drying/Alternate Drying Systems," (Cooperative Extension Service, Agri- cultural Engineering Information Service, Michigan State University. aeis no. 446, file no. 18.151, March 1981), p. 3. 49 the storage facility. The later is not a desired practice since depend- ing on the storage facility. grain may have to be moved again. Further- more. this later process is considered more expensive. Shrinkage of the grain is a main factor in the dryation process. As a result of drying to 14 percent from 15-15.5 range. shrinkage of the physical product takes place. The shrink factor of 2.24 percent is derived by the use of the Table presented in Appendix C. Thus, shrinkage is considered a cost to the producers since the total value of his corn decreases, reflecting the decrease in total weight of the product. ' Wheat and soybeans require little or no drying directly after har- vest in a "normal" season in Michigan. If storage of these commodities is to run an extended period of time and or drastic changes in tempera- ture arise. maintaining quality can usually be handled with aeration. For-this reason, on-farm storage of these commodities incur no drying costs in this study. One other concern for wheat is that of rodent and insect infesta- tion. (Generally, this problem is much more severe for wheat compared to other grains. For reasons of simplicity, however, no account is taken for the possible loss in grain mass. 3.2.2 Aeration Costs The cost of aerating stored grain may vary depending on the size of the storage facility, the amount of grain stored, management practices, and so on. The following formula exhibits how aeration cost is derived. Aeration at 1/5 cfm/bu: [(1/5 cfm/bu) (sp) 3000] * .75 * 200hr = kwhr/bu. 50 where: cfm - is a measure of airflow supplied by the fan; cubic feet per minute; bushel -'is a measure of grain volume; lbu = 1.25 ft3; cfm/bu - is the airflow per bushel of grain affected; total cfm divided by total bushels in the drying or storage bins; SP - is static pressure; assumed here to be 2 water column inches; kw - is a kilowat; kwhr - is kilowat hours; .75 - is a constant, kw = .75 * horsepower; 200hr - is the number of hours of fan time operation, consecutive hours; 3000 - is a constant pertaining to the efficient operating zone of air delivery by the fan.1 Kilowat hours per bushel are converted to cents per bushel in the final stage of the equation. An historical time series is thus created and includes a cost in the net return analysis. Aeration cost is the same for each of the commodities analyzed. 3.2.3 Other Variable Costs of Storing Other costs incurred for storage of these commodities include the purchase of fumigants and management and labor cost. Since the practice of fumigating varies widely from one operation to another, it is not in- cluded in the net return analysis for reasons of simplicity. As mentioned earlier. the costs of management and labor are also not included in the analysis of net returns to storage. 1Brooker, Bakker-Arkema, and Hall, p. 107. 51 3.3 Change in Technology A factor to consider in this study is the change in technology, over time. One would expect newer machinery involved in the storage process to become more efficient over the historical period. This would result in lower operation costs. Costs of energy, however, have in- creased over this same period and thus, have had an opposite effect on costs. Due to the limited scope of the study and for all practical pur- poses, it is assumed that these factors have remained constant. 3.4 Calculation of Net Return to Storage With the previous information it is now possible to construct the net return equations. The following equations represent net returns to post-harvest sales fbr commercial and on-farm storage. NRC1j= P ' ("*WC) ' 0C ' PHI. J = * .. .. - NRFij (Pij SHF) DCF 0C PHij m" 2? .rw «1921) OC - iil [Pij-l(1 + q )- - Pij-l] i=1 where: NRCi. - commercial net returns associated with a post-harvest sales J strategy in year i and month j ($/bu); NRFij - on-farm net returns associated with a post-harvest sales strategy in year i and month j ($/bu); Pij - post-harvest sales price in year i and month j ($/bu); M - number of months stored past harvest for which monthly cash storage costs are assessed; MSC - monthly storage costs, ($/bu); 0C opportunity cost associated with the money tied up in grain ($/bu); '52 PHij harvest-time sales price (price at harvest), (Slbu); SHF - shrink factor incurred when drying corn to 14.0 percent from 15.5 percent; DCF - drying cost associated with drying corn from 15. 5 percent to 14.0 percent ($lbu); r1d - effective real rate of interest. (taxes not included); q - equals 4, represents a quarterly compounding factor; TM - total number of months stored from harvest-time to the post-harvest sales month. The return to storage equations (NRC and NRF) are calculated in real, 1983 dollars. As mentioned earlier, calculating and evaluating net re- turns in real values (1983 dollars) provides for a more accurate compari- son with other current costs and prices and adjusts for the difference in purchasing power over time. More specifically, the standardization of the calculation permits the evaluation of net returns of selling in different months (e.g., selling all in February vs. selling all in June). Further, composite post-harvest sales alternatives (e.g., selling 25 percent in January and 75 percent in April vs. selling all in April) may be compared. 3.4.1 Opportunity Cost Simply, opportunity cost may be defined as the value foregone by not using a resource in the most profitable alternative way. In the storer's case the resource is revenue foregone by holding or storing grain. By storing at harvest a producer is foregoing the opportunity to pay off existing debt and or invest the sales revenue. As previously seen in equation form, the opportunity cost is (in simple form) the summation of price in month j-l multiplied by the effective rate of 53 1 Other computations interest (rij) for that same month. respectively. in the opportunity cost equation include compounding the effective rate of interest - quarterly, and subtracting back out Pij-l to obtain the opportunity cost. The reason for compounding quarterly is to represent. for example. a compounding interest in a savings account. Actually, the compounding factor is of no major significance since no matter how the interest is compounded. it will offer approximately the same results. The monthly interest rates used in obtaining net return margins are actual monthly average Michigan Production Credit Association short term loan rates. The rate takes into account loan fees and stock (which represent approximately one percent of the loan rate) reflecting the "effective" cost to the borrower. As mentioned earlier, the "real” ef- fective monthly rate is then obtained by subtracting out the correspond- ing monthly inflation rate. Lastly, it should be noted that the PCA rate is used in the net return calculations primarily because it is con- sidered to be the rate at which producers pay off existing short term loan debt. 1 To understand the concept of opportunity cost more clearly as it applies to this research. the following example is given. Assume a corn producer stores corn from November to March. The analysis is de- signed so that if the producer sells in March it is assumed the sale may take place between the lst of the month through the 15th. Given hypothe- tical data, Table 7 illustrates the concept of opportunity cost. 1Some ambiguity exists concerning the calculation of opportunity cost Iaeing consistent with (E, V) analysis, due to (E, V)'s use of pairwise comparisons. An alternative is to calculate opportunity cost as the price at hagvest times the prevailing interest rate times the number of months store . 54 TABLE 7. Opportunity Cost Nov. Dec.. Jan. . Feb. Mar.. 1 Price ($/bu) - 2.00 2.50 3.00 3.50 3.00 2 Interest Rate - .12 .12 .12 .12 .12 Opp. Cost (1*2) - .02 .025 .03 .035 = $.11 If the storer sells in March. the opportunity cost is $.ll/bu. (.02 + .025 + .03 + .035) in nominal terms. The $.1l/bu. is obtained by multiplying the price by the interest rate for each month respectively and then summing. For illustrative purposes this example is expressed in nominal terms, however, the calculations in the net return equations are in real 1983 dollars. To obtain net return to storage margins, a fortran computer programr was developed. The results and discussion of the net return margins are presented in the fbllowing chapter. These values thus are needed as input into a linear program to obtain the outcome of implementing various sell strategies. The following discusses the model used in this research to obtain results from implementing alternative post-harvest marketing strategies.‘ 3.5 Developing the Linear Program Model This analysis utilizes both those concepts discussed in Chapter Two and the input components previously discussed to evaluate efficient farm marketing strategies under risk. More specifically, the analysis uses MOTAD (minimization of total absolute deviation) to simulate alternative sell strategies for corn, wheat. and soybeans. Before developing the 55 model to be used in this analysis, however, concepts of the quadratic programing model are reviewed to some extent. and desired features of its expected income-variance criterion are considered. 3.5.1 The Efficient Boundary As we saw earlier, a potential storer holds perferences among alter- native risky choices on the basis of their expected income E and variance V. Thus. the potential storer has what is called an E-V utility function. Indifference curves convex to the origin are also a part of the optimal E-V farm plan. These precepts to portfblio analysis and quadratic pro- gramming are reviewed in Figure 6. E Increasing Utili ’,' ,' ‘ I I Efficient E-V Boundry :v’ "’ Set of all feasible strategies V Figure 6 Graphical Representation of Risk/Return Trade-Off; Depicting the Set of All Feasible Strategies 56 Given the assumptions in Figure 6. the potential storer rationally will choose those strategies for which the associated expected level of income is maximized for the given level of variance. Quadratic pro- gramming is thus used to develop a set of feasible strategies. What is created is called the efficient (E-Y) pairs which define an efficient boundary over the set of all feasible strategies. This is represented as the line segment AC in Figure 6. As previously discussed. the decision maker will choose a particular efficient strategy from the set of available alternatives depending on his or her preference fOr risk and expected income as described by his or her (E-V) utility function. The point at which the decision maker reaches highest utility is represented by point B in Figure 6. The utility function shown in Figure 6 is. however, in reality not easy to obtain. Thus, it makes more sense in the short run (to avoid complexity) to allow the decision maker to choose from the entire set of available alternative strategies. This allows for a certain amount of flexibility in addition to allowing the decision maker to make a choice among alter- natives in relation to a multiplicity of goals. The previous discussion represents only in a very broad context some of the basic concepts underlying quadratic programming. Other im- portant aspects not mentioned include data requirements, the specifica- tion of the model, its main advantages, and limitations. The discussion is thus presented in a nonmathematical general form: (1) for the purpose of understanding the very basis of operating within an (E-V) framework; and (2) to provide a general understanding of quadratic programming to propose the alternative model (MOTAD) used in this analysis. 57 3.5.2. The Model - Using MOTAD MOTAD was developed by Hazell as an approach which minimizes tatal absolute deviation rather than variance.1 The approach may be solved using a linear programming algorithm as opposed to a more complex design in quadratic programming. Thus, the MOTAD model used to develop risk efficient farm plans is a variation of linear programming and solution results closely parallel those of quadratic programming. Given the net return values computed from the fortran program, the MOTAD model is designed as follows. The mean absolute deviation of ex- pected farm profit is formulated as: S n (BJ)M=% sink .- Ei)x r=1 i=1 ‘3 J j l where s is the sample size. cij represents the net revenue observation h activity in the ith year, and the ED denotes the sample mean th 2 for the jt net revenue per unit of the j activity. It should be noted that M is an unbiased estimator of expected net return. In other words, the expectation, or mean, of M is equal to the parameter for which it is an estimator. That is, if M is an estimator of M, then M is unbiased if E(M) = M. Since M is used as a measure of uncertainty, it is reasonable to consider E and M as "the" parameters in the selection of a strategy. Thus, E-M strategies may be defined as having expected maximum net return 1Hazell, pp. 53-62. 2Anderson, Dillon, and Hardaker, p. 207. 58 subject to a given level of mean absolute deviation. Further, this may be termed the E(Z),M efficient set of farm plans.1 The above may also be approached in a slightly different manner. As described by Hazell, it is perhaps considered more adequate to design the model whereby the mean absolute value of "negative" deviations about the mean are calculated as: S 11 (F3.2) N = M/2 = 1' 2 | min[ 2 H 31(613' ' Ei’fi’ 03' Hence, the negative deviations can be measured by the following equation: n _ (F3.3) y1 = 321(c15 - cj)xj where yi denotes the summation of the total negative deviations about the mean. Given the above, expected profit can now be maximized with a parametric constraint on the sum of the negative deviations. Thus, the following represents the model used in this research to obtain the effi- cient farm plan. It is designed as follows: maximum net return is: 03.4) E(Z) = J3.3-iii subject to: n - . (F3.5) jE-lahjxj {1: :} bh h = 1,000, 111 n on. - (F3.6) J.210”. - c)xj + y; o i = 1,.... s Ibid. 59 s (F3.7) z y 1-1 1 5.1 = sM/2 A = 0 I'Ama X with x1 0 for j = l,..., n and y1 O for i = l...., s.1 In the above model formulation 3.4 represents maximizing net profit. The technical constraints are represented by formulation 3.5. In equa- tion 3.6. y1 measures the negative deviation of the total net revenue for each state i,.... 5 while the summation term computes the total de- viation. Thus, if (3.6) yields a positive value. the corresponding yi variable will be zero. This is so because: (1) the restriction on the yi variables is that they be nonnegative values; and (2) the total value of the objective function is limited through the parametric constraint on the sum of the y1 variables in (3.7). Further, only if the net reve- nue for any state in 3.6 is negative will the y1 be forced to an equiva- lent positive value. Hence. in 3.7 1amda (x) measures the sum of the total negative deviations over 5 states.2 Finally, the efficient fron- tier may be traced out by parameterizing 1amda (A), (the risk variable), from zero to its maximum value. 3.6 Assumptions and Further Considerations of the MOTAD Model As seen earlier, M represents the mean absolute deviation of net profit. Thus, it can be considered to examine the statistical proper- ties of the mean absolute income deviation as a substitute for the var- iance in deriving (E-V) farm plans. Hazell has shown that when the total 1Ibid.. p. 208. 2Ibid. 60 gross margin distributions are normal or approximately normal. the 1 It is implicitly assumed MOTAD model generates efficient strategies. then that the net margins obtained and used in the foregoing analysis are considered approximately normal. This assumption of approximate normality for activity net revenues (x5) implies that total net revenue 2 will also be approximately normally distributed. Hence. utility of the decision maker can be assessed in terms of the mean and total abso- lute deviation of Z. In retrospect, this can be regarded as a type of portfblio analysis. where the decision maker chooses (out of the utility- maximizing set of xj values), the optimal portfolio or strategy. This strategy being one which maximizes utility subject to the constraints of expressions 3.5 through 3.7. Lastly, the outcome of a strategy via MOTAD is stated in terms of mean net return and mean absolute deviation. For convenience, the (E. M) efficient set of strategies created by MOTAD is converted to an (E, V) locus. In other words, results are evaluated in terms of the mean-return and standard deviation assdciated to a given strategy. It is important to remember, however, that the (E, V) outcomes presented are only representations of the (E, M) efficient set created by the MOTAD model. (IHazell. PP. 53-62. CHAPTER IV ANALYSIS OF NET RETURNS TO STORAGE 4.1 Net Return to Storage We now know the objective of the storer is to earn a net return on the grain (asset) stored higher than what would have been received by selling at harvest. In deciding whether or not to store and for how long, the potential storer must take into account the costs that will be incurred, whether it be on-farm or commercial storage. Obviously, costs and prices will change over time, which makes each year unique with respect to the storage decision. To conceptualize the incorpora- tion of the costs and prices overthe last two and a half decades, the’ previous formulations of net return to storage equations are calculated for each of the grains and presented in the form of pay-off tables. The tables present the net return to storing a commodity versus selling at harvest and further show the mean and standard deviation of net return associated with each of the post-harvest sale months. The period analy- zed is the crop year of 1958-59 through 1982-83, for corn, wheat, and soybeans, and 1973-74 through 1982-83 for corn only. These results thus incorporate the previously discussed costs associated with commercial and on-farm storage, respectively, and are presented in real 1983 dollars. Further, the following net return to storage pay-off tables display a number of relevant statistical measures. As discussed earlier, the mean and standard deviation for each month over the period are presented. 61 62 The tables also present the median, kurtosis, skewness. and the highest and lowest values for the individual months. The mdeian represents the numerical value of the middle case or the case lying exactly on the 50th percentile, once all the values in the given month have been rank ordered from highest to lowest. More simply, the median is that point in a .distribution above and below which half the values fall. Kurtosis is a measure of the relative peakedness or flatness of the curve defined by the distribution of values for each month. A normal distribution will have a kurtosis of zero. If the kurtosis is positive. then the distri- bution is more peaked (narrow) than would be true for a normal distribu- tion, while a negative value means that it is flatter. Positive and negative skewness values represent skewness to the left and right of a normal distribution, respectively. Lastly, the high-low values repre- sent the highest and lowest net return values over the period for the month in question. The statistical measures median, kurtosis, and skewness are present- ed in the following pay-off tables, however, evaluation is delayed until the last section of the chapter. As will be seen, these measures com- bine to help explain the distribution of net returns to storage data. 4.1.1 Corn - Net Commercial Storage Margins Net commercial storage margins for corn are presented in Table 8. November is designated as the harvest month while December through Octo- ber is considered the storage period for which sales may take place. As can be seen. there may be substantial variation in net returns from year to year. Some years resulted in relatively high net margins while others showed negative returns. Although variation in net returns is prevalent, 63 ee... ee... e.m.. em.. .8. 3e. mee. e... .8. .m... .3. :3: .mm.~- .N...- em...- mee.~- nee..- e...~- .me..- .....- ~.~..- .e.. - .mm. to. em.. mme. .ee. eee..- eme..- eme.~- e.e.~- ~.e..- eme..- mmm. e... - mmoexesm eee. ..e. eem. - .ee.. 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Interestingly, the average standard deviation increases fairly consistently fromDecember through August. The average net mar- gins associated with each of the standard deviations, however, show a quite different pattern from the standard deviations. Average returns are favorable for the December through February period but decline sub- stantially from March on. Obviously, the negative average values repre- sent negative returns to storage which suggests that on the average. over time, it has not been a profitable decision to commercially store corn past March. As can be seen in the table, negative values are most fre- quent toward the second half of the crop year. Also in Table 8 are the average net return and standard deviation data for the 1973-74 through 1982-83 cropyear period. Net returnsfor this period are not as favorable compared to the overall period, 1958-83. December and January, however, still display the highest average net re- turn with relatively low standard deviations. The standard deviations for this later period are considerably higher than those in the 1958-83 period but display approximately the same relative magnitude between months. As one might expect, the 1973-4 through 1983-3 period show a higher standard deviation than the overall period. With regard to the analysis of the 1973-82 period, a certain amount of caution is warranted. Scepticism arises from the fact that the cal- culated means and standard deviations represent averages from a 65 relatively short period of history. Further, this period is noted as having the most volatile market behavior in history. The 10 year analy- sis nonetheless allows us to gain a better perspective on the changes in net returns to storage that have occurred over time. 4.1.2 Corn - Net On-Farm Storage Margins Table 9 presents net on-farm margins for'storing corn. As one might expect. on-farm net returns are substantially higher than those for com- mercial storage. This is primarily due to the fact that net on-farm storage margins only take into account the aeration and opportunity cost associated with storing on-farm. Remember, fixed costs are not included in the calculation of net returns to on-farm storage. In the on-farm results. considerably less negative net margin values appear compared to the commercial storage results. Thus, most of the negative values still frequent toward the end of the crop year. With exception of February, March, and April, average returns and stan- dard deviations display somewhat of a pattern, increasing throughout the crop year. August shows the highest average net return but is associated with the next to highest average standard deviation. As will be dis- cussed, the potential storer will evaluate the risk-return trade-off when deciding on a storage strategy. Again, average net return and standard deviation results are pre- sented for the 1973-74 through 1982-83 crop year period. As expected, average net returns are lower and standard deviations higher for this period. The relative risk-return trade-off between months, however, has remained about the same. Lastly, July and August average net returns are the highest values in this period, which again display a similar pattern to the overall period. 66 -..~ vo... omoé v3; 2.... 2o. 3... 8a.. 36.. .nué 3v. :3: ova..- mop..- mww..- «on..- saw... was..- moo..- Nvm..- .o...- nue. . mom. zed mom. a... wow. oo...- .00... oo..~i ..o.~i son..- mam. . moo. moo. mmoexuxm vvm. mum. . mo.. . «.0.. nvv.n mno.m ..v.m oo..v oom.~ vom.m .mm. m.moes:x ovo. i a... ..0. msn. m... mow. ..N. no.. mv.. map. mmo. ce.uoz vv... .mm. voo. uno. om... .vm. mum. who. .mm. ..v. omm. .o.m oso. 1 amp. ovu. vvw. om.. «mp. ouo. mv.. amp. mop. oo.. can: «minke. -..~ 3o.— voo... o3.— vB.. So. 3.. 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In the previous case of on-farm storage for corn, it was assumed the storer started with a standard, dry bushel of grain at harvest for which no drying costs were incurred. Table 10 however. represents drying down to the T4 percent level of moisture content at harvest. In this case, costs are incurred by the storer for drying from a l5.5 percent to a 14 percent moisture level. This scenario reflects the fact that the storer is interested in storing into the summer months, in which case "extra” drying at harvest is re- commended. If the extra drying is not done at harvest and grain is stored until mid summer, the lower moisture content may be achieved with aeration processes to prevent any major spoilage or damage. Thus, Table 10 reflects the net return associated with the extra drying at harvest by taking into account extra drying cost. As expected, average returns are somewhat lower, reflecting the added drying cost incurred at harvest. Standard deviations, however. are quite similar. The difference between the two periods is again as one might expect: lower average net returns and higher standard devia- tions in the later period as opposed to the overall period. 4.1.3 Wheat - Net Commercial Storage Margins In Table ll net commercial storage margins for wheat are presented.‘ July is the designated harvest month. As one would expect, average net returns and standard deviations are higher than those for commercial storage of corn. 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The rule says to sell at harvest if the cash price is above the loan rate and the July basis is narrow, and store otherwise. A representa- tion of the basis rule may be presented as follows: Narrow Wide Basis . Basis. Below Loan Store Store Above Loan‘ Sell Store The mechanics and design of the rule are as fellows. If at harvest prospects for an increase in corn price (to more than cover storage costs) are favorable and or expected, then a producer may decide to store until as late as July. The "normal" July basis is assumed to be approximately 35 cents per bushel, The "break-even" basis represents the amount of the normal basis plus storage costs incurred. Thus, if the July basis in any given year is approximately 15 cents greater than the break-even basis, a "wide" basis exists and this suggests storing. The 15 cents represents the cost of storing corn on-farm until July. If a "narrow" basis (less than 15 cent difference between the actual July basis and the break-even basis) exists, the decision-rule says to sell at harvest since it is likely that storing in anticipation of higher prices may not compensate for storage costs. 101 The other part of the basis rule is as follows. If the cash price is below the loan rate then store and if not, then sell at harvest. The methodology is that the producer is guaranteed the government loan rate price for his or her corn, provided he or she complies with the Government programs. The reason for holding grain at harvest if price is below the loan rate is that this gives the storer the opportunity to further evaluate the market. Once the full opportunity cost of hold- ing the corn exceeds the difference between the cash price and the loan rate (provided the cash price is under the loan rate), then the storer would sell the grain. Prices used fer the basis rule are Saginaw nominal cash prices. The loan rate is the USDA Government program rate, announced prior to planting for each year a rate exists. Further, the cash prices used in the basis rule analysis are an average of the third week in October to the second week in November. Using Chicago futures prices, basis averages correspond accordingly to the cash averages. Hence, the de- cision rule is based on nominal prices while the scenario is simulated in the model using real Michigan monthly average prices. The basis rule suggested to sell at harvest 3 years out of the 10 year analysis and store the remainder of the time. The outcome was that the wrong decision resulted 3 out of the ten years. Table 25 summarizes the basis rule for the 1973-82 period. The factor that decides if the outcome is right or wrong is simply the observed net return to storing for the year in question. If the return to storing was positive for most months, for example. then it paid to store that year. 102 TABLE 25. Basis Decision Rule Table Corn Saginaw July Loan Cash Basis Decision Year Rate Price Average Rule Outcome 1973-74 1.05 2.04 .478 sell wrong 1974-75 1.10 3.35 .57 sell right 1975-76 1.10 2.21 .726 store right 1976-77 1.50 2.03 .70 sell right 1977-78 2.00 1.53 .785 store right 1978-79 2.00 1.90 .61 store right 1979-80 2.10 2.17 .872 store right 1980-81 2.25 2.98 .984 store wrong 1981-82 2.40 2.22 1.02 store wrong 1982-83 2.55 1.98 .52 store right Figure 17 depicts alternative cash strategies and the efficient frontier for the basis scenario. Obviously, this decision rule increased average returns considerably while it also reduced the average risk fac- tor. Selling the entire crop in July, for example, averaged 36.8 cents per bushel with a s.d. of 42.6 cents compared to 24.4 and 83.2 cents respectively for implementing no decision rule over the same period. The basis rule, however, averaged considerably lower returns than did the short-crop rule. Lastly, it should be noted that the basis analysis did not prescribe how long the grain should be stored to receive the optimal average re- turn for a specified level of risk. Thus, one can see in Figure 17 that strategies on the efficient frontier and other strategies depict the average risk-return trade-off as they relate to the amount stored and sold. 103 4.7 Other Strategies To further examine the net returns associated with on-farm storage, the following analysis evaluates two strategies as they would have per- formed on the average in the short-crop and basis scenarios. These strategies are: (l) sell half the crop in January and half in June; and (2) sell half in January and one quarter in June and July, respec- tively. As with the previous analysis, the l973-82 period is examined including the extra cost of drying at harvest. The results of these two strategies are presented in Figure 18 and Table 27. The "Normal" represents implementing no decision rule while the "Basis" depicts strategy 1 and 2 under the Basis decision rule sce- nario. As can be seen, a considerable improvement has been made, how- ever, the "S-Crop" (short-crop) scenario averages considerably better. Combining the short-crop and basis scenarios with the two strategies offered an average closely between the two scenarios with respect to average net return. . . . , By selling a certain amount in January, the strategies didn't per- - form as well as if none were sold in that month. Thus, the amount of risk incurred decreased slightly by not selling the entire crop in the later months. June and July. Lastly, it is obvious to sell that the short-crop and basis decision rules considerably improved average re- turns to storage. 4.8 Commercial Storage - Nheat (1958-82) Figure 19 depicts net return to storage results for commercial storage. As one might expect, the risk and return associated with stor- ing wheat is considerably higher than that for commercially stored corn. NET RETURN (s/eu) NET RETURN (3/30) 0.35 J 0.25 J 0.15J 0.05 - 104 0.3 - 0.2 -1 ‘lJul +Jun + Moy 1’ AU9 4 Apr T 5.9 + Mor + Feb +Jon 4 Dec I I I r I I T 0.2 0.4 0.6 STANDARD DEVIATION (S/BU) D Eff. Frontier + Month Figure 17 Average Net Return to Storage On-Farm Corn (Basis Rule, 73-82) 14% 0.8 0.36 0.34 - 0.32 — 0.3 .. 0.28 ~ 0.26 - 0.24 - 0.22 - 0.2 - 0.18 J 0.16 - 0.14 -4 0.12 -' 0.1 -i 0.08 -‘ 0.06 .- 20 S—Crop lo S-Crop 2:- Combo 1A Combo 2+ Bosis' 1+ Boeis 20 Normal 10 Normal 0.04 I I I I I I I 0.2 0.4 0.6 STANDARD DEVIATION (S/BU) Figure 18 Average Net Return to Storage Alternative Strategies - (l973-82) 14% 0.8 105 TABLE 26. On-Farm Storage: Corn - Basis Rule 14 Percent Moisture (1973-82) Efficient Frontier (Harvest) Strategies 39!. ... MAY_ ... JUL_ MEAN S.D. l .7549 .1424 .1027 .076 .096 2 .2645 .4273 .3082 .228 .289 3 .2902 .7098 .339 .390 4 1.0 .368 .426 Entire Crop Sold In: MEAN S.D. DEC .015 .111 JAN .053 .126 FEB .067 .218 ' MAR .142 .263 APR .171 .357 MAY .269 .384 JUN .318 .469 JUL .368 .426 AUG .270 .584 SEP .176 .679 OCT -.098 .683 M 106 TABLE 27. On-Farm Storage: Corn,Comparison of Alternative Strategies (1973-82) QAN_ ... JUN_ JUL_ MEAN S.D. Normal: 1 .50 .50 .093 .615 2 .50 .25 .25 .109 .632 Basis Rule: 1 .50 .50 ' .186 .298 2 .50 .25 .25 .198 .287 Short-Crop Rule: 1 .50 .50 .304 .370 2 .50 .25 .25 .315 .411 Basis + Short-Crop: l .50 .50 .250 .258 - 2 .50 .25 ° .25 .259 .164 107 The average s.d. associated with approximately a 14 cent per bushel re- turn for wheat is about 75 cents compared 31 cents for corn, The higher potential return for wheat obviously is derived from the higher per unit value of the commodity and thus accommodates a relatively higher risk factor per unit compared to corn. Selling the entire crop in October, November, or December averaged relatively well over the 1958-82 period. This can be concluded by ob- serving how close these strategies are to the efficient frontier. Other strategies of selling the entire crop in a given month resulted in lass than desirable average returns. Selling in February, fer example, averaged only 1.3 cents over the period with a s.d. of $1.17. Table 28 depicts the strategies located on the efficient frontier. The efficient frontier strategies are simply comprised of selling amounts of wheat in October and December. Interestingly, no other months are considered optimal sale periods, which further indicates that these two months provide the highest average return for the stated level of risk. 4.9 On-Farm Storage (1958-82) Table 29 and Figure 20 present alternative sell strategies for on- farm storage of wheat. As expected, on-farm average returns were again higher than those for commercial storage. The efficient frontier has moved upward while accommodating approximately the same level of risk per unit of return. Selling in January not only is now located on the efficient frontier but also averaged the highest net return over the period. Selling in February may also now be considered a relatively profitable sale period. .OVerall, the on-farm analysis for wheat averaged substantially higher returns than those for the commercial analysis. NET RETURN (t/aU) NET RETURN (S/BU) 108 Average Net Return to Storage On-Farm Wheat (1958-82) ' 0.3 0.28 1 0.26 - 1OIDec: 0.2‘ q «2 *JOI‘I 15 0.22 - °62I 0.2 -‘ 113+ Nov 0.18 - ‘ 0.10 - 21?“. 0.14. 90 on Is” 023 0.1 2 7D +Aug 0.1 d 0.00 J 5" 0.08 -( 3:, 0.04 o b 0'02 ‘ +Feb o I r I W— I I I I I j fi‘I f I I I I T 0 0.2 0.4 0.6 0.8 . 1 1.2 1.4 1.6 1.8 2 , STANDARD DEVIATION (S/BU) Eff. Frontier + Month 0 Other Strategies Figure 19 Average Net Return to Storage Commercial Wheat (1958-82) 0.6 21. Jon 24°09 ’ ' 0.5 - n-bDec $2 15:) Q3 + Feb 11 a 0.3 _, +Oct ’0 ozs +Sep +Mor 7o 0.2 -< R! +Aug 0.1 .. 3° 10 +Apr O I fifi' I I I I I I I I I I I I I I I 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 STANDARD DEVIATION (S/BU) Eff. Frontier «- Month 0 Other Strategies Figure 20 109 TABLE 28. Commercial Storage: Wheat (1958-82) Efficient Frontier (Harvest) Strategies JUL_ Q§I_ Q§§_ MEAN S.D. l .8730 .05 .077 .028- .148 2 .8186 .05 .1314 .042 .222 3 .7642 .05 .1858 .057 .296 4 .7098 .05 .2402 .071 .370 5 .6549 .0513 .2938 .085 .454 6 .5973 .0599 .3428 .099 .530 7 .5399 .0684 .3917 .113 .592 8 .4823 .0770 .4407 .127 .681 9 .4247 .0856 .4897 .141 .740 10 .3673 .0941 .5396 .156 .832 11 .3097 .1027 .5876 .170 .908 12 .2522 1112 .6366 .184 .983 13 .1947 1198 .6855 .198 1.059 14 .1372 1283 .7345 .212 1.135 15 .0796 1369 .7835 .226 1.210 16 .0221 .1454 .8325 .240 1.286 17 .1031 .8969 .254 1.330 18 1.0 .260 1.363 Entire Crop Sold In:' MEAN S.D. AUG .114 .909 SEP .148 1.023 OCT .165 .906 NOV .201 1.090 DEC .260 1.363 JAN .246 1.626 FEB .013 1.171 MAR -.140 1.219 APR -.418 .957 _MAY -.621 1.006 JUN -.800 1.142 Other Strategies: . 19 - Sell 1/4 from Aug. through Nov. .157 .907 20 - Sell 1/2 in two lowest risk months .140- .871 21 - Sell 1/4 in four highest return months .218 1.254 22 - Sell 1/3 in three highest return months .236 1.376 110 TABLE 29. 0n-Farm Storage: Wheat (1958-82) Efficient Frontier (Harvest) Strategies QQL_... ygy_ ‘Q§§_ JAN, MEAN 1 .8802 .05 .0698 .055 2 .8252 .05 .1248 .083 3 ‘.7705 .05 .1795 .111 4 .7157 .05 .2343 .139 5 .6610 .05 .2890 , .166 6 .6060 .0516 .3424 .194 7 .5497 .0590 .3913 .222 8 .4934 .0664 .4402 .250 9 .4371 .0737 .4892 .277 10 .3808 .0811 .5381 .305 11 .3245 .0885 .5870 .333 12 .2682 .0959 .6359 .361 13 .2120 .1032 .6848 .388 14 .1631 .1106 .7337 .416 15 .0994 .1180 .7826 .444 16 .0431 .1253 .8316 .472 17 .1793 .9207 ..498 18 .8014 .1986 .514 19 .4983 .5017 .526 20 .2069 .7931. .536 21 . »- v 1.0 .544 Entire Crop Sold In: MEAN S.D. AUG _ .164 .907 ' ..164 ’ SEP ’ ‘ .247 1.021 . ' .247 OCT . .315 .903 .315 NOV .398 1.084 .507 DEC .507 1.356 .544 JAN .544 1.619 .361 FEB .361 1.165 .258 MAR .258 1.215 .029 APR .029 .949 -.125 MAY -.125 .990 -.255 JUN -.255 1.126 Other Strategies: MEAN 22 - Se11 1/3 in three highest return months .483 23 - Sell 1/4 in four highest return months .453 24 - Se11 1/2 in two highest return months .526 25 - Sell 1/10 from Aug. through May .270 (A 01 00 .619 .907 .903 .356 .619 .165 .215 .949 1.126 dddd 111 4.10 Commercia1 Storage - Soybeans (1958-82) Figure 21 represents average net returns to storing soybeans for the 1958-82 period. Selling in April and May averaged relatively well over the period while sales in other-months relative1y poor. Strategies on the efficient frontier averaged less for the same level of risk compared to those fOr commercial storage of wheat. This may be due in part to the higher commercial storage cost fOr soybeans. The short-crop decision rule is represented in Figure 22 and Table 31. This scenario represents those years where a 20 percent reduction in supply from the previous year occurred. The designated short-crop years for soybeans occurred in 1974-75 and 1980-81. Again the short-crop decision rule improved average returns to storage considerably. Sales in January, February, March, April, and , May further i11ustrate the improvement in average returns. The added number of selling periods on the efficient frontier create the opportun- ity for the decision-maker to further spread risk. Tab1e 31 further illustrates this concept. Overall, the short-crop scenario again in- creased average return while decreasing the per unit standard deviation. 4.11 0n-Farm Storage - (1958-82) Figure 23 displays average net returns for storing soybeans on-farm. Selling periods December, April, May and June again are located on the efficient frontier. The sell all in June strategy averaged the highest net return of 69.1 cents with a s.d. of $3.391. Both the risk and re- turn associated with this strategy averaged substantially higher than the highest average margin for commercial storage of soybeans. Overall, the NET RETURN ($/BU) NET RETURN (3/30) 112 0.12 0.11 - 0.1 .. 0.09 4 0.08 - 0.07 - 0.06 4 0.05 -l 0.04 .. 0.0:! ~ 0.02 . 0.01 d .Moy 70 5° +Apr +Jun 1:: 0.6 T T I j I T 1 2 3 STANDARD DEVlATION (S/BU) D Eff. Frontier + Month Figure 21 Average Net Return to Storage Commercial Soybeans (1958-82) 0.5 - 0.4 -‘ 0.3 - 0.2 "' 4 I May +Jun +Apr + Feb 4- Mar 3o 20 +Jon *AUQ + Doc *' JU‘ I I I l’ I 1 I 1 2 3 STANDARD DEVIATION (8/ BU) + M 0 Eff. F rontior onth Figure 22 1 Average Net Return to Storage Commercial Soybeans (Short-Crop, 58-82) 113 TABLE 30. Commercial Storage: Soybeans (1958-82) Efficient Frontier (Harvest) Strategies QQI, MAX. MEAN 1 .8553 .1447 .015 2 .7589 .2411 .025 3 .6142 .3858 .041 4 .4695 .5305 .056 5 .3248 .6752 .071 6 .1802 .8198 .086 7 .0355 .9645 .101 8 1.0 .105 Entire Crop Sold In: MEAN S.D. ' NOV -.086 .487 DEC -.042 .805 JAN -.093 1.158 FEB -.060 1.720 MAR - .017 2.031 APR .069 2.310 MAY .105 3.002 JUN .030 3.387 JUL -.277 2.239 AUG -.230 2.890 SEP -.914 2.242 TABLE 31. Commercial Storage: 114 soybeans (Short-Crop)(l958-82) Efficient Frontier (Harvest) Strategies Q§I_. 1 .6880 2 ' .3759 3 .0639 4 Entire Crop Sold In: MEAN NOV -.020 DEC .083 JAN .134 FEB .248 ' MAR .317 APR .400 MAY .498 JUN .471 JUL .088 AUG .143 SEP -.537 .. MAR ... .2226 .4453 .6679 S.D. .352 .623 .731 1.258 1.154 1.974 2.664 3.096 1.804 2.651 1.807 1.0 .0894 .1788 .2682 MEAN S.D. ($lbu.) .113 .608 .225 .1.216 .338 1.773 .498 2.664 115 efficient frontier for on-farm storage illustrates the relative attrac- tiveness of the average net margins compared to commercial storage. Figure 24 shows average net returns associated with implementing the short-crop decision rule for on-farm storage. Average net returns were improved considerably while the risk factor per unit return also slightly decreased. Selling the entire crop in June (except for short- crop years) averaged $1.07 with a $3.087 s.d. compared to 69.1 cents and 3.391 cents respective1y for commercial storage. Selling periods January, March, and July also improved substantially by implementing the short-crop decision rule. Again, overall average return and s.d. improved considerably. 4.12 Distribution of Net Returns The previous analyses provide useful information in deciding how long to store grain. Mean and standard deViation measures utilize con- cepts of portfolio theory discussed earlier to explain in part the risk- return trade-off among alternative post-harvest cash marketing strage- gies. In spite of the appreciation one may have developed for the re- sults obtained, further analysis is suggested. Specifically, cumula- tive and probability distribution functions may enable the decision- maker to further his or her knowledge about the risk and return associ- ated with alternative cash strategies. (4.12.1 Cumu1ative Distribution Function The cumulative distribution function may be defined as P(x 5_X*) or P(x 3_X*), Where X* is some particular value of the uncertain quan- tity x. This function, P(x 5_X*), says the probability of x is less than or equal to a particular value X*. This can be represented NET RETURN (R/BU) NET RETURN (8/8U) 116 0.8 0.7 - ... May fidun 7D 0.6 -( 50 #Aug +Apr 0.5 d 50 +Jul 0.4 - D +Mor 0.3 ~ 3° + Feb 0.2 - 3° +Jon +Doc 0.1 - to o T I I I T I l O 1 2 3 STANDARD DEVIATION (S/Bu) D Eff. Frontier + Month Figure 23 Average Net Return to Storage On-Farm Soybeans (1958-82) 1.2 1.1 -1 + May “Jun SD 1 7 ~ +Aug 0.9 J mpr +Ju| an 0.7 .. 30+ Mar 0.6 " 4, FQb 0.5 - Zn 0.4 "‘ +Jon . +Sop 0.3 - + Doc 0.2 - 1 D 0.1 - +Nov o I . I I I I I r 0 ‘1 2 STANDARD DEVIATION (8/30) a Eff. Frontier + Month Figure 24 Average Net Return to Storage On-Farm Soybeans (Short-Crop, 58-82) 117 cmN.N mcc.: mum Nam.N «mm. c=< oeN.N mcc. Nan pom.m pom. 22a moo.m ewe. >oz .o.m zcmzv savages; pcmpovoom ANm-mmm.V mcamnzom “maaaoum Eaaa-=o .Nm mumgmzv mum w3< 4:6 2:0 >oz “cH upom coco mgwucm FNMQ‘MQ mowmoamcpm gmwucogm ucwvupymu ANm-mmmpv Aaoau-oao;mv mcaonxom "amateum saaa-=o .mm mummN amen. on um.go m. =.ego cog: mmex:.gsm mam