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MAKING GRAIN PRICING DECISIONS BASED ON PROFIT MARGIN HEDGING AND REAL OPTION VALUES By HYUN SEOK KIM Bachelor of Science Hankuk University of Foreign Studies Seoul, Korea 2002 Master of Science Seoul National University Seoul, Korea 2004 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY July, 2008 ii MAKING GRAIN PRICING DECISIONS BASED ON PROFIT MARGIN HEDGING AND REAL OPTION VALUES Dissertation Approved: Dr. B. Wade Brorsen Dissertation Adviser Dr. Brian D. Adam Dr. Kim B. Anderson Dr. Timothy L. Krehbiel Dr. A. Gordon Emslie Dean of the Graduate College iii ACKNOWLEDGMENT I wish to express my sincere appreciation to my academic adviser Dr. Wade Brorsen for his intelligent supervision and invaluable guidance and encouragement throughout my Ph.D. program at Oklahoma State University. I also wish to thank all members of my academic committee, Dr. Brian Adam, Dr. Kim Anderson, and Dr. Tim Krehbiel for their helpful advice and invaluable comments during the preparation of this dissertation. My appreciation is extended to the Department of Agricultural Economics for providing a research opportunity and financial support for my Ph.D. program. Friendly faculty and staff in the department have contributed to my academic success. In particular, I would like to thank my parents, Ho Tak Kim and Yoon Jeung Cho, for their continuous support through prayer, hope and love which have helped me to stand firm throughout the whole process of completing this program. My wife Hae Jung and my two sons Eugene and Ian have been my continuous source of inspiration and love. Working on my Ph.D. was a lot easier with Hae Jung, Eugene and Ian on my side. Finally, I dedicate this work to Jesus Christ, my Lord and Savior. He has begun the good work in me and has proven Himself to be faithful in finishing it. To Him be the glory forever and ever. iv TABLE OF CONTENTS Chapter Page I. PROFIT MARGIN HEDGING·········································································································1 Introduction················································································································· 1 Theory························································································································· 2 Expected Target Utility···························································································· 2 Mean Reversion ······································································································ 6 Data ····························································································································· 8 Procedures················································································································· 10 Measure of Expected Utility···················································································11 Variance Ratio Test ······························································································· 12 Results······················································································································· 14 Summary and Conclusions ······················································································· 17 References················································································································· 20 Notes ························································································································· 23 II. CAN REAL OPTION VALUE EXPLAIN WHY PRODUCERS APPEAR TO STORE TOO LONG? ······················································································································· 34 Introduction··············································································································· 34 Theory······················································································································· 35 Data ··························································································································· 37 Procedures················································································································· 38 Estimation of Price Process Parameters ······························································ 38 A Universal Lattice Model ···················································································· 41 Simulation ············································································································· 45 Results······················································································································· 46 Summary and Conclusion························································································· 48 References················································································································· 50 iv LIST OF TABLES Table Page Table I−1. Average Prices (cents/bu) for Hedging Strategies at September 20th (1975 2005)················································································································ 27 Table I−2. Paired Differences tRatios of Average Prices, (19752005)··························· 28 Table I−3. Expected Utilities for Hedging Strategies at September 20th (19752005) ····· 29 Table I−4. Paired Differences tRatios of Expected Utilities, (19752005)······················ 30 Table I−5. Average Prices and Expected Utilities for Hedging Strategies for Multiple Crops Scenario (19752005)············································································ 31 Table I−6. Paired Differences tRatios of Average Prices and Expected Utilities for Multiple Crops Cases (19752005)·································································· 31 Table I−7. Variance Ratio Tests for Futures Prices (19752006) ······································ 32 Table I−8. Joint Variance Ratio Tests for Futures Prices (19752006) ····························· 32 Table I−9. Variance Ratio Tests for Futures Prices Using Jackknife Approach(19752006) ························································································································· 33 Table II−1. Parameter Estimateα of Seasonal Mean Reversion Price Process ··············· 53 Table II−2. Per Period Storage Costs (Percentage of Price) ············································· 54 Table II−3. Parameter Estimates of Seasonal Mean Reversion Price Process by Nonparametric Bootstrapping·········································································· 55 Table II−4. Sales Dates and Net Returns for Corn···························································· 56 Table II−5. Sales Dates and Net Returns for Soybean······················································ 57 Table II−6. Sales Dates and Net Returns for Wheat ························································ 58 Table II−7. Paired Differences tRatios of the Mean Net Returns between Seasonal Mean Reversion and Mean Reversion Model (19752006) ······································ 59 v LIST OF FIGURES Figure Page Figure I−1. Expected utility of a crop producer as futures price 0 f p changes when α equals to β ······································································································ 24 Figure I−2. Expected utility of a crop producer as futures price 0 f p changes when α is less than β ······································································································ 25 Figure I−3. Expected utility of a crop producer as futures price 0 f p changes when α is greater than β ································································································· 26 Figure II−1. An example trinomial lattice ········································································ 60 Figure II−2. Seasonality of change in corn price······························································ 61 Figure II−3. Seasonality of change in soybean price························································ 62 Figure II−4. Seasonality of change in wheat price ··························································· 63 Figure II−5. Cutoff price of mean reversion price process for corn using half storage and interest costs ···································································································· 64 Figure II−6. Cutoff price of seasonal mean reversion price process for corn using low storage and interest costs ················································································· 64 Figure II−7. Cutoff price of mean reversion price process for soybeans using low storage and interest costs······························································································ 65 Figure II−8. Cutoff price of seasonal mean reversion price process for soybeans using low storage and interest costs ·········································································· 65 Figure II−9. Cutoff price of mean reversion price process for wheat using low storage and interest costs ···································································································· 66 Figure II−10. Cutoff price of seasonal mean reversion price process for wheat using low storage and interest costs ················································································· 66 1 I. CHAPTER I PROFIT MARGIN HEDGING Introduction Some extension economists and others often recommend profit margin hedging, in which a producer sells a crop preharvest by short hedging whenever prices are above a target. However, this strategy recommendation is without a research base. The strategy is also included in undergraduate textbooks such as Purcell and Koontz (pp. 329330). With recent high prices, producers have forward contracted more of their crops, which provides evidence that some producers follow such a strategy. However, the theoretical assumptions that would justify such a strategy have never been developed. Some empirical studies (Leuthold and Mokler 1980; Kenyon and Clay 1987; Johnson et al. 1991) have found that a profit margin hedging strategy is profitable for producers or investors but did not include significance tests. Girma and Paulson (1998) studied the statistical behavior of crack spreads (the price difference between refined energy products and crude oil) and they conclude that historically simple buy and hold trading strategies are profitable and in many instances are significantly greater than zero. These previous studies, however, do not provide any theoretical justification for using profit margin hedging. 2 This research focuses on answering the question, “What assumptions for producer’s utility and price process can justify profit margin hedging?” The paper determines the producer’s utility function and price processes where profit margin hedging is optimal. A statistical test of mean reversion in agricultural futures prices is conducted. Simulations are conducted to compare the expected utility of a profit margin hedging strategy with the expected utility of other strategies such as always hedging and selling at harvest. Theory Expected Target Utility The goal of this section is to derive a theoretical model where profit margin hedging is optimal. The meanvariance (EV) model is the most commonly used to analyze choices under uncertainty. Optimal hedging strategies under EV, such as those of Johnson (1960), Stein (1960), and more general models such as Lence (1996) do not lead to profit margin hedging strategies being optimal. Simiarly, mean semivariance and mean targetsemivariance models such as those of Dejong et al. (1997), Lien and Tse (1998, 2000), Chen et al. (2001), and Turvey and Nayak (2003) do not lead to profit margin hedging rules being optimal. Some previous studies argue that EV analysis has several wellknown theoretical shortcomings (Fishburn 1997; Holthausen 1981). Fishburn (1977) proposed a meanrisk model which generalized the meantarget semivariance model (Markowitz 1959; Mao 1970; Hogan and Warren 1974; Porter 1974) to address the shortcomings of the EV model. The widely known shortcoming of the EV model is that if the outcome distributions are not of a locationscale form (such as 3 normal) or the utility function is not quadratic, the EV model is not consistent with expected utility. Fishburn’s model measured return as the mean of the outcomes, but defined risk as weighted deviations of outcomes below target and the model assumes risk neutrality above the target. Holthausen (1981) adapted Fishburn’s model by using the same measure of risk but defining return as weighted deviations above the target to avoid the risk neutrality restriction. To measure producer’s expected utility, this study adopts Holthausen’s model in which the utility function is: for all π ≥ t (1.1) − − − = α β π π π ( ) ( ) ( ) k t t U for all π ≤ t . where π indicates profit, t represents the target, k is a positive constant, and α and β reflect the risk preferences. If α <1 (α >1) , then the producer is risk seeking (averse) below the target. Then, the expected target utility can be written as (1.2) ∫ ∫ −∞ ∞ = − − − t t EU π π t β f π dπ k t π α f π dπ ( ) ( ) ( ) ( ) ( ) where f (π ) is the probability density function of π which is normally distributed with mean π and variance 2 π σ .1 If producers hedge preharvest without basis risk, then the profit is (1.3) p F p F f 0 π = (1− ) + where p is the price of crop at harvest, F is a hedge ratio, and 0 f p indicates the futures price at the time of the hedge. Then, equation (1.2) is rewritten as (1.4) ∫ ∫ −∞ ∞ = − + − − − − − A f A f EU( ) {p(1 F) p F t} f ( )dp k {t p(1 F) p F} f ( )dp π 0 β π 0 α π where 4 (t p F) F A f 0 1 1 − − = , f ( p) is the probability density function of p which is normally distributed with mean p and variance 2 p σ , and F is the choice variable. Equation (1.4) will be optimized when the first derivative with respect to F equals zero. The first order condition of equation (1.4) is (1.5) ∫ ∞ − − ′ = − − + − A f f EU ( ) {p(1 F) p F t} ( p p ) f ( )dp π β 0 β 1 0 π ∫ ∞ − ∂ ∂ − − + − A f pF dp f p F p F t 2 2 0 2(1 ) ( ) { (1 ) } σ σ π π β 2 0 0 (1 ) { (1 ) } ( ) F t p A F p F t f f f − − − − + − β π ∫ −∞ − − − − − − A f f k {t p(1 F) p F} ( p p ) f ( )dp α 0 α 1 0 π ∫ −∞ − ∂ ∂ + − − − A f pF dp f k t p F p F 2 2 0 2(1 ) ( ) { (1 ) } σ σ π π α 2 0 0 (1 ) { (1 ) } ( ) F t p k t A F p F f f f − − − − − − α π . The third and the last term in equation (1.5) become zero since A equals ( ) (1 ) 0 t p F F f − − . If α and β are equal, and k is one, then equation (1.5) can be rewritten as (1.6) ∫ ∞ − − ′ = − − + − A f f EU ( ) {p(1 F) p F t} ( p p ) f ( )dp π α 0 α 1 0 π ∫ ∞ − ∂ ∂ − − + − A f pF dp f p F p F t 2 2 0 2(1 ) ( ) { (1 ) } σ σ π π α 5 ∫−∞ − − − − − − A f f {t p(1 F) p F} ( p p ) f ( )dp α 0 α 1 0 π ∫ −∞ − ∂ ∂ + − − − A f pF dp f t p F p F 2 2 0 2(1 ) ( ) { (1 ) } σ σ π π α . Then, the first and the third terms in equation (1.6) cancel out. If price 0 f p is above the target and all the crop is hedged − that is, F equals one − then the second and the last terms are zero and equation (1.6) will be zero. If price 0 f p is below the target and producers do not hedge − that is, F equals zero − then there is no interior solution and the optimum is the lower bound of zero. The shape of expected utility when price 0 f p is below the target, in figure I−1, confirms that expected utility is highest when the hedge ratio is zero. Therefore, profit margin hedging where the producer hedges all when prices are above the target and none when prices are below the target is shown to be an optimal strategy under a highly restricted target utility function where a producer has the same level of risk preferences above and below the target, and k equals one. In the case of relaxing the assumptions that α and β are equal and k equals one, it cannot be solved analytically, so numerical methods must be used instead. Figures I−1 through I−3 show expected utilities as futures price 0 f p changes for alternate on values of α and β . If α equals β , as in figure I−1, then a producer hedges all of the crop when the futures price is greater than the target, but does not hedge when the futures price is less than the target. In this case, profit margin hedging is optimal, confirming our theory. In the case of α smaller than β , in figure I−2, the producer’s hedging behavior is equivalent to the case of α = β . In contrast to figure I−1, which is monotonic, figures I−2 and I−3 are nonmonotonic. The reason why figures I−2 and I−3 are not monotonic 6 is that the utility function is not concave or convex; because the utility function has a different form above and below target, risk seeking dominates producers’ risk preferences until some point of the hedge ratio but risk aversion dominates their risk preferences after that. Figure I−3 shows that if α is greater than β , producers hedge all of the crop when the futures price is greater than the target, but also hedges a portion of the crop when the futures price is less than the target. These numerical solutions show that profit margin hedging can be an optimal strategy when α does not equal β , but the optimal strategy is not always all or none. Some studies (eg. Lence 1996) showed that the optimal hedge ratio typically decreases in the presence of basis risk, yield risk, transaction costs or multiple crop outputs. Moschini and Lapan (1995), for example, showed that increasing basis risk results in a lower futures hedge ratio, and increasing yield risk also results in a lower hedge ratio. Bond and Thompson (1985) found that a rise in the transaction or storage cost leads to a decrease in the optimal hedging ratio. Fackler and McNew (1993) showed that, under a multiproduct approach, the fullyhedged position is not optimal and it is not optimal to hedge all commodities in the same proportion. Relaxing these assumptions is expected to also reduce optimal hedge ratios under profit margin hedging. Mean Reversion Profit margin hedging has been suggested as a profit increasing strategy. Zulauf and Irwin (1998) suggest that the success of selling before harvest depends on whether a price bias exists. That is, mean reversion is a needed attribute of price behavior for profit margin hedging to be a successful strategy. Therefore, this study provides a proof that 7 profit margin hedging is more profitable than the other strategies such as always hedging and selling at harvest if futures prices are mean reverting. The mean reverting futures price process can be written as (1.7) − = λ ( − ) +ε 0 0 f f p p p p where p is random cash price which equals the futures price at the terminal point of the hedge (no basis risk), 0 f p is futures prices at the time of the hedge, p is the longrun average price, and ε is an error term with mean zero and variance 2 ε σ . The estimated coefficient λ is the mean reversion speed by which 0 f p revert toward p . If a onetime period model is used, producers’ expected profit function can be obtained by taking the expected value of equation (1.3). (1.8) [ ] [(1 ) ] 0E E F p p F f π = − + If futures price follows a mean reversion process, equation (8) can be rewritten as (1.9) E[ ] E[ F {p p p } p F] f f f 0 0 0 π = (1− ) +λ ( − ) + = E[ F p F p p p F] f f f 0 0 0 (1− ) + λ (1− )( − ) + = F p F p p p F f f f 0 0 0 (1− ) + λ (1− )( − ) + = p p F F p p p F f f f f 0 0 0 0 − + λ (1− )( − ) + = (1 )( ) 0 0 f f p +λ − F p − p If 0 f p is greater than p , then F =1 and we can rewrite equation (1.9) as (1.10) E[ PMH p p] p E[ AH] p p p E[p] E[ SH] f f f f π , > = = π > +λ ( − ) = = π 0 0 0 0 where PMH indicates profit margin hedging, AH indicates always hedging, and SH is Selling at harvest. 8 If 0 f p is less than p , then F = 0 and we can rewrite equation (1.9) as (1.11) E[ PMH p p] p p p E[ SH] p E[ AH] f f f f π 0 < = 0 +λ − 0 = π > 0 = π , ( ) . Since λ is greater than zero and the futures price 0 f p is less than the longrun average price p , the expected profit conditioned on profit margin hedge is greater than 0 f p which is the expected profit conditioned on always hedging. Therefore, profit margin hedging is more profitable than other strategies such as always hedging and selling at harvest, if futures prices are mean reverting. It is important to note that the above derivation is based on a static oneperiod model. If the problem is made dynamic and producers are allowed to hedge at anytime, the profit margin hedging rule would still be profitable, but would no longer be optimal. The optimal rule could be derived similarly to what Fackler and Livingston (2002) derived for cash prices. In their model, if grain prices are below the mean, it is best to store since prices will revert to the mean. If grain prices are unusually high, it is best to sell immediately. If prices are near the mean, there can be a real option value from waiting since there is the opportunity to wait and select a time to sell when prices are unusually high. In that case, the expected profit maximizing target price would decrease as harvest approaches since there is less opportunity for price to increase above the mean. Data The chosen agricultural commodities are Oklahoma hard red winter wheat, Illinois soft winter wheat, soybeans, and corn. This study uses the July futures contract prices for wheat from the Kansas City Board of Trade (KCBT) and from the Chicago 9 Board of Trade (CBOT), November futures contract prices for soybeans and December futures contract prices for corn from the CBOT. Futures prices for KCBT wheat are obtained from KCBT and for CBOT wheat, soybean, and corn are obtained from Prophet Financial Systems, Inc. To test mean reversion, this study uses daily data. The sample period extends from August 1975 through May 2006 for KCBT and CBOT wheat, from December 1975 through October 2006 for soybeans and January 1975 through December 2006 for corn. July observations for KCBT and CBOT wheat, November observations for CBOT soybeans and corn are deleted since these observations are for the delivery period. Markets are thin during this time and can be quite volatile. No price changes across contract years are used. To conduct the simulation, Oklahoma June average wheat cash prices and prices for the July futures contract on September 20th from 1975 to 2005 are used. We use 31 years data for simulations whereas 32 years data for a mean reversion test since we do not have the 2006 economic cost of production. Five year moving averages of basis and yield of crops are used to make hedging decisions with basis risk and yield risk. For the case of multicrop producers, the study used Illinois June average cash price for wheat, and Illinois October average price for soybean and corn. For Illinois wheat, the July futures contract prices on September 20th from 1975 through 2005 are used. For Illinois soybean and corn, futures prices for the November contract and the December contract for soybean and corn, respectively, at May 10th from 1975 through 2005 are used. The Illinois monthly average cash prices are obtained from National Agricultural Statistics Service (NASS) of the United States Department of 10 Agriculture (USDA). This study used 70% of economic costs of production as targets for KBCT wheat and 80% for CBOT wheat and 100% for CBOT corn and soybean since the economic costs include many types of cost and it is too high to use as the target in KBCT and CBOT wheat. These choices depend on the number of hedges for the 32 year period. If the cost is set too high, a producer would seldom hedge. If cost is set too low, a producer would always hedge. For the costs assumed here, a hedge is placed 16 times of 31 years for KBCT wheat, 17 times for CBOT wheat, 16 times for CBOT corn, 20 times for CBOT soybean. The economic costs of production for the three crops from 1975 to 2005 are obtained from the Economic Research Service (ERS) of USDA (2008). The yield data of Oklahoma wheat at Garfield County, Oklahoma, and Illinois wheat, soybean, and corn in Livingston County, Illinois from 1975 to 2005 are obtained from NASS of USDA. Procedures This paper has two main procedures − simulation to compare the expected utility of profit margin hedging strategy with the always hedging and the selling at harvest strategies, and mean reversion testing for KCBT and CBOT wheat July futures prices, CBOT soybean November futures prices, and CBOT corn December futures prices. The expected utility is measured by taking the average utility across 31 years. To test mean reversion, the variance ratio test is employed. 11 Measure of Expected Utility Five scenarios are considered for each of three hedging strategies − hedging without risk, with basis risk, with yield risk, with yield and basis risk and with multiple crops. To measure expected utility without basis risk, a perfect foresight model is used which assumes actual harvest basis is known at the time of the decision. In this case, under a profit margin hedging strategy, if the sum of futures price at the time of the decision and foresighted basis is greater than the target return, then producers hedge all crops, otherwise they hedge none and sell the crops at harvest. If basis risk were considered, producers hedge all when the sum of futures price at the time of the decision and average basis is greater than the target return otherwise they do not hedge. With yield risk, producers hedge all crops if the average returns that is the sum of futures price at the time of the decision and the foresighted basis multiplied by average yield is greater than the target multiplied by average yield. In the case of multiple crops, the producer hedges all crops when the total returns − that is, the sum of futures price at the time of the decision and foresighted basis multiplied by quantity produced for each crop − is greater than the total target that is the sum of target multiplied by quantity produced for each crop. We assume producers are risk averse above the target and risk seeking below the target and pick 0.5 as the value of α andβ . We also assume a transaction cost of 1.2 cents per bushel but we do not consider margin calls. After calculating utility for 31 years from 1975 through 2005, expected utility is calculated as the average of utilities (14) Σ= = 31 1 ( ) 31 1 ( ) i i EU π U π . 12 Variance Ratio Test The idea behind the variance ratio test is that if the natural logarithm of a price series Pt is a random walk, then the variance of kperiod returns should equal k times the variance of oneperiod returns (Cochrane 1988; Kim et al. 1991; Lo and MacKinlay 1988; Poterba and Summers 1988). The general kperiod variance ratio, VR(k) is defined as (15) VR(k) = (1) ( ) 2 2 σ σ k k where ( ) 2 σ k is the variance of the k differences and (1) σ 2 is the variance of the first differences. The null hypothesis of interest is that VR(k) equals one. That is, VR(k) equal to one implies that futures price follows a random walk process, whereas a variance ratio of less than one implies a mean reversion process. Lo and MacKinlay (1988) show that the variance ratio estimator can be calculated as (16) Σ= − = − − nk t k t t k P P k m k ( ˆ ) , 1 ( ) σ 2 μ 2 where = − + − nk k m k(nk k 1) 1 and (17) Σ= − − − − = nk t t t P P nk 1 2 1 2 ( ˆ ) , ( 1) 1 σ (1) μ in which Σ= − = − = − nk t t t nk P P nk P P nk 1 1 0 ( ) 1 ( ) 1 μˆ , 13 where P0 and Pnk are the first and last observations of the price series. Since futures returns have been shown to exhibit conditional heteroscedasticity (Yang and Brorsen 1993), we computed the asymptotic variance of the variance ratio, φ (k) , under heteroscedasticity. The standard normal test statistic (Lo and MacKinlay 1988) is (18) (0, 1) [ ( )] ( ) 1 ( ) 1/ 2 N k VR k Z k a→ − = φ where Σ= − = k j j k k k 1 2 ˆ( ) 2( 1) φ ( ) δ and 2 1 2 1 1 2 1 2 1 ( ˆ ) ( ˆ ) ( ˆ ) ˆ( ) − − − − − − = Σ Σ = − = + − − − − nk t t t nk t j t t t j t j p p p p p p j μ μ μ δ . Chow and Denning (1993) derived the joint period test where the null hypothesis is ( ) i VR k equals one for i =1,K, l . The test statistic can be written as (19) max ( ) 1 i i l ZV Z k ≤ ≤ = which asymptotically follows the studentized maximum modulus distribution (Stoline and Ury 1979) under the martingale null hypothesis. This study also conducts a variance ratio test using a new jackknife method because of possible nonnormality. Specially, we use a jackknife approach where each year is treated as a unit, so we delete each year of observations from the data set for each sample. Then, the jackknife estimate of kperiod variance ratio of futures price ( ) ~ θ k is defined in the usual manner. Let θ (k) be the kperiod variance ratio of futures prices and 14 ( ) ~ k i θ be the kperiod variance ratio when the ith year observations are deleted from the data set. Since we use 32 years data set from 1975 through 2006 for a mean reversion test, the jackknife estimate of kperiod variance ratio is calculated as the average of ( ) ~ k i θ (21) Σ= = 32 1 ( ) ~ 32 1 ( ) ~ i i θ k θ k . The jackknife estimate of the standard error of ( ) ~ θ k is (22) { } 1/ 2 32 1 2 ~ ( ) ~ ( ) ~ 32 31 ~ ( ) − = Σ= i i σ k θ k θ k θ . Then, the test statistic should be (23) ( 31) ~ ~ ~ ( ) ( ) 1 ~ ( ) = − = df t k k t k θ σ θ . This jackknife approach is similar to that used with the Agricultural Resource Management Survey (ARMS) and described by Dubman (2000). Results Table I−1 shows that achieves the highest average prices in all scenarios for all crops, followed by always hedging and selling at harvest. Table I−2 presents the results of the paired difference tests of average prices for the profit margin hedging and other strategies. The average prices of paired profit margin hedging and always hedging are not significantly different from zero at a 5% significance level in all scenarios for all crops except in no risk and yield risk scenarios for CBOT soybeans. The average prices of profit margin hedging and selling at harvest strategies are significantly different from 15 zero at the 5% significance level in all scenarios for all crops except in basis risk scenario for KCBT wheat. Table I−3 shows the expected utilities for the hedging strategies. The expected utilities of profit margin hedging are higher than the other strategies which confirm the theoretical findings. Always hedging has a higher expected utility than selling at harvest in all scenarios. The results of the paired difference tests of expected utilities for the profit margin hedging and other strategies are presented in table I−4. The expected utilities of paired profit margin hedging and always hedging are not significantly different from zero at a 5% significance level in all scenarios for all crops except the cases of no risk and yield risk for CBOT soybean. The expected utilities of profit margin hedging and selling at harvest are significantly different from zero at the 5% significance level except in basis risk, and yield and basis risk scenarios for KCBT wheat and basis risk scenarios for CBOT soybean. Thus, the expected utility results reflect that preharvest prices were higher than harvest prices during this time period. Table I−5 shows average prices and the expected utilities in the multiple crops scenario. In the case of multiple crops, the average prices of the independent profit margin hedging are the highest, followed by profit margin hedging, always hedging and selling at harvest, respectively. The expected utility of the independent profit margin hedging strategy is the highest followed by profit margin hedging, always hedging, and selling at harvest strategy, respectively. Table I−6 presents the results of the paired difference tests of average prices and expected utilities for multiple crops case. We do not find any evidence that the average prices and expected utilities of profit margin hedging and always hedging strategies are 16 significantly different from zero at a 5% significance level. However, the average prices and expected utilities of profit margin hedging and selling at harvest are significantly different from zero at the 5% significance level. The table also shows that the average prices and expected utilities of profit margin hedging and independent profit margin hedging is significantly different from zero at a 5% significance level. These results show that adding both price risk and yield risk reduces the expected utility of the profit margin hedging rule. Also, if producers grow multiple crops, the profit margin hedging rule would not be optimal even with a target utility function. Thaler’s (1980) mental accounting where producers consider their price risk and yield risk separately or divide their profits for each crop into separate pockets of money is proposed as a possible explanation of the popularity of profit margin hedging. The result of the variance ratio tests in tables I−7 and I−8 show that both Zstatistic and ZVstatistic values are not significantly different from 1.0 at the 5% significance level for all crops which means there is little evidence of mean reversion in futures prices for all crops. Table I−9 shows the result of the variance ratio test using the jackknife approach. None of the tstatistics show significant differences from one at the 5% significance level for all crops which confirms the results that there is little evidence of mean reversion in futures prices for all crops. Many of the estimated variance ratios are even greater than one (although insignificant) which would indicate trend following rather than mean reversion. If prices were trend following then using a technical analysis rule would be optimal. As shown in the theory section, profit margin hedging can be profitable if futures prices are mean reverting. However, our mean reversion test results show no evidence of 17 mean reversion in future prices. Furthermore, simulation results show that although profit margin hedging is more profitable than selling at harvest in most cases, there is little evidence that profit margin hedging is more profitable than always hedging except in a few cases. Two possible explanations have been offered for the profitability of preharvest hedging over this time period. One is that more buyers than sellers are wanting to lock in prices and so the buyers are paying a risk premium. If this hypothesis were correct, then the recent introduction of index funds (Sanders et al. 2008) would serve to make preharvest hedging even more attractive. The other hypothesis is that the market priced a small probability catastrophic event, which never happened during this time period. If this hypothesis were correct, adding the 2008 crop year might remove the apparent profitability of preharvest hedging. Summary and Conclusions Some extension economists and others often recommend profit margin hedging in choosing the timing of crop sales. This paper determines producer’s utility function and price processes where profit margin hedging is optimal. Profit margin hedging is an optimal strategy under a highly restricted target utility function even in an efficient market. Profit margin hedging can be profitable if prices are mean reverting. Simulations are conducted to compare the expected utility of profit margin hedging strategies with the expected utility of other strategies such as always hedging and selling at harvest. A variance ratio test is conducted to test for the existence of mean reversion in agricultural futures prices. The simulation results show that the expected utility of profit margin hedging is higher than always hedging and selling at harvest 18 strategies except in a few scenarios such as yield and basis risk for CBOT wheat and yield risk and yield and basis risk for CBOT corn. Therefore, this result suggests that the profit margin hedging would give the highest expected utility to producers in most cases under the specified utility function. However, if a producer grows multiple crops or considers both price risk and yield risk, the expected utility of profit margin hedging strategy could be reduced and not be optimal even with a target utility function. Mental accounting where producers consider their price risk and yield risk separately or divide their profits for each crop into separate pockets of money is proposed as a possible explanation of the popularity of profit margin hedging. The paired differences tests of average prices and expected utilities for the profit margin hedging and the other two strategies shows that, in most cases, both average prices and expected utilities of profit margin hedging strategies are not significantly different from those of always hedging strategies, but are higher than those of selling at harvest strategies except in some yield and basis risk scenarios. This may be the result of the time period being a time of unusually stable prices or it could be due to buyers being more eager to lock in prices than seller. With the variance ratio test, there is little evidence that futures prices of all crops follow a mean reverting process. The results of variance ratio test using jackknife approach confirm the result that there is insufficient evidence to conclude that futures prices of all crops follow a mean reverting process. Since we do not find evidence of mean reversion in futures prices and profit margin hedging is not more profitable than always hedging except in a few cases, we rely primarily on the theoretical proof using the shape of utility functions in figures I−1 19 through I−3 as the primary justification to argue that profit margin hedging can be the preferred strategy. 20 References Bond, G.E., and S.R. Thompson. 1985. “Risk Aversion and the Recommended Hedging Ratio.” American Journal of Agricultural Economics 67:870–72. Chen, S., C. Lee, and K. Shrestah. 2001. “On a MeanGeneralized Semivariance Approach to Determining the Hedge Ratio.” Journal of Futures Markets 21:581−98. Chow, K.V., and K. Denning. 1993 “A Simple Multiple Variance Ratio Test.” Journal of Econometrics 58:385−401. 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Department of Agricultural and Consumer Economics, University of Illinois at UrbanaChampign. Stein, J.L. 1961. “Simultaneous Determination of Spot and Futures Prices” American Economic Review 51:1012−25 Stoline, M.R., and H.K. Ury. 1979. “Tables of the Studentized Maximum Modulus Distribution and an Application to Multiple Comparisons Among Means.” Technometrics 21:87−93 Thaler, R. 1980. “Toward a Positive Theory of Consumer Choice.” Journal of Economic Behavior and Organization 1:39−60. Turvey, C.G., and G. Nayak. 2003. “The SemivarianceMinimizing Hedge Ratio.” Journal of Agricultural and Resource Economics 28:100−15. U.S. Department of Agriculture (USDA), Economic Research Service (ERS). 2008. Available at http://www.ers.usda.gov/Data/CostsAndReturns/testpick.htm. Accessed on 12 February 2008. U.S. Department of Agriculture (USDA), National Agricultural Statistic Service (NASS). 2008. Available at http://www.nass.usda.gov/QuickStats/. Accessed on 12 February 2008 Yang, S.R., and B.W. Brorsen. 1993. “Nonlinear Dynamics of Daily Futures Prices: Conditional Heteroskedasticity or Chaos?” The Journal of Futures Markets 13:175−91. Yoon, BS., and B.W. Brorsen. 2005. “Can Multiyear Rollover Hedging Increase Mean Returns?” Journal of Agricultural and Applied Economics 37:65−78. Zulauf, C.R., and S.H. Irwin. 1998. “Market Efficiency and Marketing to Enhance Income of Crop Producers.” Review of Agricultural Economics 20:308−31. 23 Notes 1. The variance of profit, 2 π σ , equals 2 2 (1 ) p − F σ where 2 p σ is variance of p . 24 Figure I−1. Expected utility of a crop producer as futures price 0 f p changes when α equals to β Hedge Expected Utility Hedge Ratio α = 0.5 β = 0.5 t = 3.37 3.43 0 = f p 3.40 0 = f p 3.34 0 = f p 3.31 0 = f p 25 Figure I−2. Expected utility of a crop producer as futures price 0 f p changes when α is less than β Expected Utility Hedge Ratio α = 0.1 β = 0.5 t = 3.37 3.43 0 = f p 3.40 0 = f p 3.34 0 = f p 3.31 0 = f p 26 Figure I−3. Expected utility of a crop producer as futures price 0 f p changes when α is greater than β Expected Utility Hedge Ratio α = 0.9 β = 0.5 t = 3.37 3.43 0 = f p 3.40 0 = f p 3.34 0 = f p 3.31 0 = f p 27 Table I−1. Average Prices (cents/bu) for Hedging Strategies at September 20th (19752005) Strategies Commodity Scenario Profit Margin Hedging Always Hedging Selling at Harvest KCBT Wheat No risk 330.10 319.78 311.03 Basis risk 327.13 319.78 311.03 Yield risk 329.11 318.24 311.03 Yield and basis risk 328.09 318.24 311.03 CBOT Wheat No risk 320.73 312.58 297.29 Basis risk 319.93 312.58 297.29 Yield risk 319.81 308.83 297.29 Yield and basis risk 318.99 308.83 297.29 CBOT Corn No risk 241.26 238.94 224.81 Basis risk 239.97 238.94 224.81 Yield risk 238.65 230.26 224.81 Yield and basis risk 237.32 230.26 224.81 CBOT Soybean No risk 631.02 605.66 581.52 Basis risk 618.94 605.66 581.52 Yield risk 626.50 597.92 581.52 Yield and basis risk 613.21 597.92 581.52 28 Table I−2. Paired Differences tRatios of Average Prices, (19752005) Paired Differences Commodity Scenario Profit Margin Hedging vs. Always Hedging Profit Margin Hedging vs. Selling at Harvest KCBT Wheat No risk 1.47 2.39* Basis risk 1.13 1.78 Yield risk 1.30 2.55* Yield and basis risk 1.26 2.19* CBOT Wheat No risk 1.23 2.88* Basis risk 1.10 2.79* Yield risk 1.37 2.88* Yield and basis risk 1.25 2.78* CBOT Corn No risk 0.47 3.16* Basis risk 0.13 2.86* Yield risk 1.10 3.19* Yield and basis risk 0.82 3.26* CBOT Soybean No risk 2.22* 4.22* Basis risk 1.43 2.45* Yield risk 2.26* 4.29* Yield and basis risk 1.47 2.16* Note: tcritical value with 30 degrees of freedom at 5% significance level is 2.042. * indicates significance at 5% level. 29 Table I−3. Expected Utilities for Hedging Strategies at September 20th (19752005) Strategies Commodity Scenario Profit Margin Hedging Always Hedging Selling at Harvest KCBT Wheat No risk 2.15 0.93 0.57 Basis risk 1.79 0.93 0.57 Yield risk 1.82 0.85 0.57 Yield and basis risk 1.74 0.85 0.57 CBOT Wheat No risk 1.94 0.73 0.81 Basis risk 1.75 0.73 0.81 Yield risk 1.87 0.55 0.81 Yield and basis risk 1.67 0.55 0.81 CBOT Corn No risk 0.48 0.25 2.28 Basis risk 0.03 025 2.28 Yield risk 0.20 0.86 2.28 Yield and basis risk 0.22 0.86 2.28 CBOT Soybean No risk 6.44 3.81 2.55 Basis risk 5.03 3.81 2.55 Yield risk 6.25 3.44 2.55 Yield and basis risk 4.75 3.44 2.55 Note: We use α = β = 0.5 as levels of risk preference below and above target. 30 Table I−4. Paired Differences tRatios of Expected Utilities, (19752005) Paired Differences Commodity Scenario Profit Margin Hedging vs. Always Hedging Profit Margin Hedging vs. Selling at Harvest KCBT Wheat No risk 1.64 2.51* Basis risk 1.46 1.49 Yield risk 1.28 2.47* Yield and basis risk 1.40 1.81 CBOT Wheat No risk 1.71 3.07* Basis risk 1.36 2.92* Yield risk 1.78 3.13* Yield and basis risk 1.41 2.94* CBOT Corn No risk 1.19 3.64* Basis risk 0.39 3.17* Yield risk 1.51 3.75* Yield and basis risk 0.91 2.89* CBOT Soybean No risk 2.42* 4.43* Basis risk 1.52 1.84 Yield risk 2.44* 4.38* Yield and basis risk 1.54 1.63 Note: tcritical value with 30 degrees of freedom at 5% significance level is 2.042. * indicates significance at 5% level. 31 Table I−5. Average Prices and Expected Utilities for Hedging Strategies for Multiple Crops Scenario (19752005) Strategies Item Profit Margin Hedging Independent Profit Margin Hedging Always Hedging Selling at Harvest Average Prices ($/bu) 75.46 76.46 75.39 70.53 Expected Utilities 29.57 63.99 20.99 10.58 Note: The dates of decision making are September 20th for CBOT wheat and May 10th for CBOT soybean and corn. We use α = β = 0.5 as levels of risk preference below and above target. Table I−6. Paired Differences tRatios of Average Prices and Expected Utilities for Multiple Crops Cases (19752005) Paired Difference Item Profit Margin Hedging vs. Always Hedging Profit Margin Hedging vs. Selling at Harvest Profit Margin Hedging vs. Independent Profit Margin Hedging Average Prices 0.07 3.47* 2.42* Expected Utilities 0.98 3.66* 2.99* Note: tcritical value with 30 degrees of freedom at 5% significance level is 2.042. * indicates significance at 5% level. 32 Table I−7. Variance Ratio Tests for Futures Prices (19752006) Commodity Return Horizon (kdays) Variance Ratio Zstatistic KBCT Wheat 2 1.020 0.892 5 0.984 0.232 10 0.977 0.200 20 0.938 0.377 CBOT Wheat 2 1.007 0.357 5 0.952 0.738 10 0.922 0.842 20 0.848 1.056 CBOT Corn 2 1.033 1.857 5 1.046 0.547 10 1.048 0.510 20 1.094 0.643 CBOT Soybean 2 1.001 0.068 5 0.995 0.091 10 0.963 0.398 20 0.991 0.068 Note: July observations are deleted. Standard normal distribution Z at 5% significance level is 1.96. Table I−8. Joint Variance Ratio Tests for Futures Prices (19752006) Commodity VR(2) ZV KCBT Wheat 1.020 0.892 CBOT Wheat 1.006 1.056 CBOT Corn 1.033 1.857 CBOT Soybean 1.001 0.431 Note: July observations are deleted. Studentized maximum modulus distribution with 20 and infinity degree of freedom at 5% significance level is 3.643. 33 Table I−9. Variance Ratio Tests for Futures Prices Using Jackknife Approach(1975 2006) Commodity Return Horizon (kdays) Variance Ratio tstatistic KBCT Wheat 2 1.024 1.148 5 1.000 0.004 10 1.013 0.241 20 1.012 0.153 CBOT Wheat 2 1.010 0.571 5 0.969 1.130 10 0.944 1.528 20 0.898 1.650 CBOT Corn 2 1.036 1.723 5 1.056 1.986 10 1.069 1.532 20 1.147 1.996 CBOT Soybean 2 1.005 0.270 5 1.011 0.399 10 0.999 0.037 20 1.071 1.128 Note: July observations are deleted. tcritical value with 30 degrees of freedom at 5% significance level is 2.042. * indicates significance at 5% level. 34 II. CHAPTER II CAN REAL OPTION VALUE EXPLAIN WHY PRODUCERS APPEAR TO STORE TOO LONG? Introduction Some studies show that producers store longer than is profitable (Anderson and Brorsen 2005; Hagedorn et al. 2005). One possibility is that producers store crops longer than makes economic sense due to myopic loss aversion, which means that producers get more disutility from a loss than they get utility from receiving an equally sized gain. An alternative explanation results from producers’ decisions to sell grain being irreversible. Fackler and Livingston (2002) show that this irreversibility can create a real option value from waiting to sell grain. The key to generating a real option value is for prices to follow a mean reverting process. In the case of grain as considered by Fackler and Livingston (2002), if grain prices are low it makes sense to wait to sell because prices will revert to the mean. If prices are unusually high, it is best to sell. If prices are near the mean, there can be a real option value from waiting because there is the opportunity to wait and select a time to sell when prices are higher than currently. There are some recent studies that implement real options in agriculture. Purvis et al. (1995) examine the technology adoption of freestall dairy housing under 35 irreversibility and uncertainty and find that there can be a return to waiting to adopt in some cases. Ekboir (1997), WinterNelson and Amegbeto (1998), and Khanna et al. (2000) also used real options to analyze the investment decision of producers under uncertainty. This research focuses on answering the question, “Can real option values explain why producers appear to store too long?” To answer this question, this study first models and estimates the price process. The model attempts to capture two important features of agricultural commodity prices: mean reversion and seasonality. The price process which is modeled in this study differs from Fackler and Livingston (2002). The price process used here allows price to be a random walk within a season, but mean reverting across crop years. After estimating the price process, a universal lattice model is used to determine the cutoff price at which the producer is indifferent between selling and holding a crop. Simulations using cash prices of wheat, corn, and soybean are used to determine net returns under two different price processes, which is simple mean reversion and the new seasonal mean reversion price process. This empirical work shows that real option values cannot explain why producers appear to store too long. Theory A producer who holds stocks can be viewed as holding an American option since the producer has the option to sell at any time. The optimal storage problem is equivalent to the optimal stopping problem of an American call option which is exercised at the current price. If selling stock is irreversible the producer does not just hold stocks but holds stocks and a call option which can be exercised at the current price. An American 36 option is an optimal stopping problem of determining the optimal time to exercise an option. The decision to exercise the option is the same as with financial options. The option holder exercises the option, whenever its intrinsic value, which is the value of immediately exercising the option, is greater than its total value. Because of the early exercise possibility, American options are solved as a dynamic programming problem. Our derivation is based on risk neutral valuation rather than riskless arbitrage as in Black and Scholes (1973). The typical American call option under risk neutral evaluation can be expressed in terms of a value function, t V : (2.1) [max(0, ( ) )] rh t h h T t t V E p X e− + ∈ − = − where t h p + is the price of the underlying asset at time t + h , r is the riskfree interest rate, and X is the exercise price. The optimal storage problem differs from (2.1). First, holding stocks of a commodity incurs positive holding charges, whereas holding an option does not incur holding cost. Second, the exercise price of the optimal storage problem is current market price, which is not discounted as in (2.1). Finally, the storage problem has an initial price which is the current cash price whereas a usual American option does not have an initial value and so the option value is zero if the option is not exercised. Then, the value function of the optimal storage problem can be defined as (2.2) [max(0, )] [max( , )] ( ) (r s)h t t h h T t t r s h t h h T t t t V p E p e p E p p e − + + ∈ − − + + ∈ − = + − = where t p is cash price of a commodity at time t , T is the expiration date, and s is a per period storage cost which is a percentage of price. The producer sells stocks under the condition that 37 (2.3) [max( )] (r s)h t h h T t t p E p e − + + ∈ − ≥ That is the producer sells stocks whenever the expected return to store and sell at time t + h is less than or equal to the current market price. Data The chosen agricultural commodities are corn, soybeans and wheat. Thursday cash prices of South Central Illinois corn and soybean data from the National Agricultural Statistics Service (NASS) of the United States Department of Agriculture (USDA) are obtained from a computer database compiled by Farmdoc, University of Illinois at UrbanaChampaign (2008). Thursday cash prices of wheat at Medford, Oklahoma, are obtained from the Oklahoma Market Reports of USDA. The sample period extends from October 1975 through September 2007 for corn and soybeans, and from June 1975 through May 2007 for wheat. These primary data have some missing values for Thanksgiving and Christmas season. For these missing data, the most recently observed data are used. Annual state average prices from the National Agricultural Statistics Service (NASS) are obtained from the United States Department of Agriculture (USDA) website (2008). To estimate the price processes, 5year moving averages of annual average prices for each crop are used as mean prices. Corn and soybean storage costs from 1995 through 2004 are from Irwin et al. (2006). We calculate the previous 20 years of storage costs from 1975 to 1994 using producer price index from website of United States Department of Labor, and assume that storage costs of 2005 and 2006 equal the cost of 2004. Storage costs of wheat from 1975 38 to 2006 are obtained from Oklahoma Cooperative Extension Service at Oklahoma State University. The interest cost is calculated at the prime rate for that year plus 2%. The prime rate is the prime charged by banks in June for that year, quoted from the Kansas City Federal Reserve Bank (2008). Procedures Three main procedures are used: estimation of price process parameters, determining cutoff price, and simulation of the trading rules. A universal lattice model (Chen and Yang 1999) and discrete stochastic dynamic programming are used to determine cutoff prices. Estimation of Price Process Parameters The model of prices used here attempts to capture two important features of agricultural commodity prices, mean reversion and seasonality. A number of studies documented mean reversion in commodity cash prices (Brennan 1991; Lence et al. 1993; Dixit and Pindyck 1994; Bessembinder et al.1995; Wang and Tomek 2007). Also, some other studies have found that futures prices follow a near random walk within a contract month (Bessler and Covey 1991; Yoon and Brorsen 2005), but are mean reverting when prices across multiple contract months are used (Schroeder and Goodwin 1991). Seasonality in the mean level of price has been also well documented in commodity. For example, prices of seasonally produced goods tend to rise during the marketing season to cover the cost of storage. Price process in this research is not focused on seasonal volatility but on seasonal mean reversion. While seasonal volatility 39 is statistically significant due to the large sample size, it is relatively small and is not included here to simplify the model. A price process model which represents mean reversion can be described by (2.4) t t t t t t p − p = a + β p − p +ε ln ln − (ln ln ) 1 where t p indicates the cash price at time t, t p is the seasonal mean price, t a is a seasonal function, t represents number of weeks after harvest, β is a parameter to be estimated, and t ε is a normally distributed error term with zero mean and constant variance σ 2 . We allow prices to follow a random walk within a season, but to be mean reverting across crop years. Such a price process can be rewritten as if 0 t < t (2.5) + − + + − = − t t t t t t t t a p p a p p β ε ε (ln ln ) ln ln 1 if 0 t ≥ t where 0 t is a time within a season when the mean reverting process begins. Equation (2.5) imposes a random walk with drift in the early part of the storage season. This assumption is tested by estimating a more general model: if 0 t < t (2.6) + + − + + − + − = − t t t t t t t t t t a p p a p p p p α β ε α ε ( )(ln ln ) (ln ln ) ln ln 1 if 0 t ≥ t Restricting α to be zero gives equation (2.5). We find no evidence of differences among fourth, fifth, and sixth power polynomial functional forms. Visual inspection of a fifth power polynomial seasonality function suggests that it is more realistic than the other powers or a sinusoidal function. Therefore, we adopt a fifth power polynomial functional form for the seasonal function, t a , which is 40 (2.7) Σ= = 5 i 0 i t ia γ t where the γ s are the parameters to be estimated. If we impose a continuity restriction on the seasonal function a(t) then the change of seasonality at harvest in the current year is equivalent to the change of seasonality at harvest next year. Since this study uses weekly cash price data, we can impose a continuity condition, 0 52 a = a . Using (2.7) this can be rearranged as (2.8) 52 (52) 5 2 1 Σ= − = i i i γ γ and then, 1 γ can be obtained by other estimated parameters. Equation (2.6) is estimated using cash prices of three crops – wheat, corn, and soybean and the coefficient α is not significantly different from zero (table II−1). Therefore, α is restricted to be zero and then (2.6) can be rewritten as if 0 t < t (2.9) + − + + − = − t t t t t t t a p p a p p β ε ε (ln ln ) ln ln 1 if 0 t ≥ t , and then we can also define the simple mean reversion price process as (2.10) t t t t t p − p = a + β p − p +ε ln ln − (ln ln ) 1 . The specified value of 0 t can be determined by substituting numerical values from 0 to 51 for 0 t and selecting the one which gives the highest log likelihood value. Since this study uses a 5 year moving average as mean price p , the model is not stationary. If cost of production data had been available to use instead of the 5year moving average, the model would be stationary. The standard errors on the computer print out in this case are conditional 41 standard errors and are valid conditional on the true value of 0 t being selected and no standard errors are provided for 0 t . Therefore, a nonparametric bootstrap is used to obtain estimates of standard errors of 0 t . Ten thousand samples of size 1,738 for wheat, 1,639 for corn, and 1,637 for soybean are resampled and used to estimate the parameters. A Universal Lattice Model While the terms used in the option pricing literature are quite different than the dynamic programming terminology used by Fackler and Livingston (2002), the approaches are equivalent in that pricing an American option requires solving a stochastic dynamic program. There are many models for pricing options. Black and Scholes (1973) developed an option pricing model for European options. Cox et al. (1979) developed the binomial option pricing lattice which is widely used within finance to price American type options as it is easy to implement and handles American options relatively well. However, the binomial model assumes that the option price can just either go up or down over a time step. It does not assume that the price may remain unchanged. In 1996, Boyle introduced the trinomial option pricing model, which is similar to the binomial method in that it employs a lattice type method for pricing options. The trinomial method is more accurate than the binomial one and gives the same results as the binomial one with a fewer steps. In the trinomial lattice, the branches are up, flat, and down by an increment of change in underlying value 0p . That is, (2.11) p p p i t i t = + 0 3, , , 42 i t i t p p 2, , , = p p p i t i t = − 0 1, , , Figure 1 shows an example trinomial lattice. The branches are down, flat, and up with the risk neutral probabilities 1 R , 2 R , and 3 R , respectively, which satisfy the following three equations (2.12) i t i t i t i t i t i t i t i t R p R p R p p 1, , 1, , 2, , 2, , 3, , 3, , , , + + = +μ 2 , 2 , , 2 3, , 3, , 2 2, , 2, , 2 1, , 1, , ( ) ( ) ( ) ( ) i t i t i t i t i t i t i t i t i t R p + R p + R p − p +μ =σ 1 1, , 2, , 3, , + + = i t i t i t R R R where i t p , is the ith node of p at time t, n i t p , , is the nth lowest possible node at time t + 0t , and i,t μ and 2 ,t i σ are the expected change and the variance of i t p , during the next time interval 0t , respectively. However, in the trinomial lattice, if there is mean reversion in the process, the risk neutral probabilities of all nodes in the lattice could be negative. To solve this problem, Hull and White (1990) propose four alternative branching schemes. These alternatives include the branches of the lattice to go three ups, two ups, and one up; two ups, one up, and flat; flat, one down, and two downs; and one down, two downs, and three downs. Chen and Yang (1999) argue that in the alternative trinomial lattice there seems to be no consistent way to construct the lattice in which all probabilities are guaranteed to be positive. Thus, they extend Hull and White’s (1990) model and propose a general form of alternative branching schemes. This study uses Chen and Yang’s (1999) universal lattice model to determine real option value. With Chen and Yang’s lattice model, the three branches can be written as (2.13) p p j k p i t i t = + ( + )0 3, , , 43 p p j p i t i t = + ( )0 2, , , p p j k p i t i t = + ( − )0 1, , , where the variable j and k provide flexibility for the branches to yield nonnegative probabilities with any level of mean and variance, respectively. With this branching method, the risk neutral probabilities can be obtained solving (2.12) and then the results are (2.14) 2 2 2 , , , 1, , 2 ( )(( ) ) k p j p j k p R i t i t i t i t 0 0 − + 0 − + = μ μ σ 2 2 2 , 2 , 2, , ( ) 1 k p j p R i t i t i t 0 − 0 + = − μ σ i t i t i t R R R 3, , 1, , 2, , =1− − . To guarantee the convergence of the model, the constraints of 0 1 , , ≤ ≤ n i t P translate into the following two sets of sufficient conditions: (2.15) p k p i t i t 0 ≤ ≤ 0 , , σ 2σ and p k p j p k p i t i t i t i t 0 + 0 − ≤ ≤ 0 − 0 − 2 , 2 2 , 2 , 2 2 , μ σ μ σ and (2.16) p k i t 0 > , 2σ and p k p p k j p k p p i t i t i t i t 0 0 − − 0 + − ≤ ≤ 0 0 − − 0 2 , 2 2 , 2 , 2 2 , 2 μ σ μ σ or 44 p k p p k j p k p p k i t i t i t i t 0 0 − − 0 ≤ ≤ + 0 0 − + 0 + − 2 , 2 2 , 2 , 2 2 , 4 2 4 2 μ σ μ σ or p k p p j p k p p k i t i t i t i t 0 0 − + 0 ≤ ≤ 0 0 − + 0 + 2 , 2 2 , 2 , 2 2 , 4 2 μ σ μ σ . Since this study assumes constant volatility, which means k = 1, the risk neutral probabilities and the sets of sufficient conditions for constraints of 0 1 , , ≤ ≤ n i t P can be rewritten as (2.17) 2 2 , , 1, , 2 ( )(( 1) ) p j p j p R i t i t i t 0 0 − + 0 − + = μ μ σ 2 2 2 , 2, , ( ) 1 p j p R i t i t 0 − 0 + = − μ σ i t i t i t R R R 3, , 1, , 2, , =1− − and (2.18) p k p 0 ≤ ≤ 0 σ 2σ and p p j p p i t i t 0 + 0 − ≤ ≤ 0 − 0 − 2 2 , 2 2 , μ σ μ σ . A summary of this procedure is that 0p and 0t are chosen, and then the variable j is chosen using equation (2.18). After that, the risk neutral probabilities are obtained from equation (2.17). Finally, as mentioned in the theory section, since the optimal storage problem is equivalent to an American call option, using equation (2.2) and the value function of optimal storage problem can be determined as 45 (2.19) [max{( ) , 0}] ( ) 1, 1, 2, 2, 3, 3, t r s h t h t h t h t h t h t h h T t t V = E R p + R p + R p e− + − p + + + + + + ∈ − Then the cutoff price, which is the current market price where producer sells stocks since the expected return to store and sell at time t + h is less than or equal to the current market price, can be determined by (2.20) [ : {max( ) }] ( ) 1, 1, 2, 2, 3, 3, r s h t h t h t h t h t h t h h T t t t t C p p E R p R p R p e − + + + + + + + ∈ − = = + + We use a universal lattice model on a grid of values in p and t to determine a cutoff price. The value function is computed for each time period beginning with 52 to 1. To determine a cutoff price, we use the average cash price at harvest over the 32year period as an initial value for each crop ($3.23 for wheat, $2.38 for corn, and $5.99 for soybeans), and assume that price increments are 15 cents for corn, 38 cents for soybean, and 20 cents for wheat. Selling at harvest is the expected profit maximizing strategy when full storage and interest costs are used. Hagedorn et al. (2005) also show that selling at harvest is the best strategy when they use full storage and interest costs. However we also consider lower costs of half of the storage and interest costs as well as full storage and interest costs to determine cutoff price. Some producers are net lenders and have their own storage (so only marginal costs would affect the decision), so such lower costs are relevant for some producers. Simulation Simulations are conducted to determine net returns of the optimal strategy under two different price process: mean reversion and seasonal mean reversion. For the simulations, we design two different scenarios which depend on the level of storage and 46 interest costs. One scenario includes full storage and interest costs and another one includes half of storage and interest cost. Using equation (2.19), the simulations are conducted with weekly cash price data for corn, soybean, and wheat to find the first selling date of crops and value of selling the crop. Net returns for each crop year for each crop were computed using the first price that exceeds the specified cutoff price function. The net returns are computed as the value of sales less stockholding costs, discounted to the harvest time (2.21) ( ) r (T t ) T r T t T p e sTp e π = − − − − − where T is the sales date, t is the first date of the marketing season assumed to be the first weekday in June for wheat and the first weekday in October for corn and soybean, and s is a per period storage cost that is a percentage of price (table II−2). Results The estimated nonparametric bootstrap parameters of the seasonal mean reversion process are presented in table II−3. Mean reversion occurs late July with 3.6% weekly for corn, mid or late July with 4.2% weekly for soybean, and early or mid March with 2.3% weekly for wheat. That is, the total percentages of mean reversion for a marketing year are 28.5% for corn, 41.6% for soybean, and 32.2% for wheat. Figures II−2 through II−4 show the shapes of seasonality for corn, soybean, and wheat, respectively. After harvest, prices for corn and soybeans rapidly increase until the beginning of December and then slowly decrease. For wheat, prices also increase rapidly after harvest until early August and then slowly decrease. These seasonal price changes turn negative in early June for corn, early July for soybeans, and early March for wheat. 47 Thus, the seasonal function turns negative before mean reversion begins. This clearly indicates that producers will be selling before mean reversion begins (as most of them do), so real option values do not explain why producers appear to store too long. Optimal cutoff prices are illustrated in figures II−5 through II−10. The shapes of the graphs of the model which uses a mean reversion price process are very different from the model using a seasonal mean reversion price process. Only at extremely low prices is there ever an incentive to store to the point where seasonal mean reversion begins. Since producers would rationally sell before mean reversion begins, the real option value almost always disappears. This result contrasts with Fackler and Livingston’s (2002) model, which is that if grain prices are near the mean there can be a real option valuing from waiting to sell because there is the opportunity to wait and select a time to sell when prices are unusually high. They conclude that irreversibility confers an additional return in the form of an option to sell stocks in the future. This finding of a large real option value that can explain why producers appear to store too long is not supported. The results of simulations for corn, soybeans, and wheat are presented in tables II−4, II−5 and II−6 respectively. The difference of average net returns over the 32 years between the mean reversion model and the seasonal mean reversion model is small and the result of paired difference tests in table II−7 shows that all t values are not significant at the 5% level except a case that include full storage and interest cost for corn. Therefore, we can conclude that there is little evidence that, for most scenarios, the net returns over 32 years between the mean reversion model and the seasonal mean reversion model are different. As Brorsen and Irwin (1996) argue, statistical insignificance is a 48 typical result of marketing strategy simulation studies. The difference between marketing strategies is usually small, the variation is high, and with only one observation per year, the number of observations is small. Therefore, we rely primarily on the results in table II−3 in reaching the conclusion that the seasonal mean reversion model is preferred. Summary and Conclusion Previous studies suggest that producers tend to store crops longer than is profitable (Anderson and Brorsen 2005). Since decisions to sell are irreversible, there can be a real option value from waiting to sell grain. This research focuses on determining whether real option values can explain longer storage We estimate a new seasonal mean reversion price process using a nonparametric bootstrap rather than estimating a simple mean reversion price process. After estimating the price process, a cutoff price at which the producer is indifferent between selling and holding the crop is determined using a universal lattice model. Simulations are conducted to determine net returns under simple mean reversion and the new seasonal mean reversion price process. The estimated nonparametric bootstrap parameters of the seasonal mean reversion process show that mean reversion occurs mid or late July for corn, early July for soybean, and early March for wheat. The shapes of seasonality show that the seasonal function turns negative before mean reversion begins, which suggests that real option values are relatively unimportant in determining when producers sell their grain. The graphs of cutoff price when assuming a seasonal mean reversion price process show that producers sell before mean reversion begins except when prices are 49 extremely low. This result contrasts with Fackler and Livingston’s (2002) conclusion that irreversibility confers an additional return in the form of an option to sell stocks in the future. Therefore their finding of a large real option value that can explain why producers store too long is not supported. The simulation results represent that the difference of average net returns over the 32 years between the mean reversion model and the seasonal mean reversion model is very small and the result of paired difference tests conclude that there is little evidence that the net returns over 32 years between the mean reversion model and the seasonal mean reversion model are different. Based on the nonparametric bootstrap estimation of price process, we can conclude that the seasonal mean reversion model is preferred. 50 References Anderson, R.W. 1985. “Some Determinants of the Volatility of Futures Prices.” The Journal Futures Markets 5:331–48. Anderson, R.W., and J.P. Danthine. 1983. “The Time Pattern of Hedging and the Volatility of Futures Prices.” The Review of Economic Studies 50:249–66. Anderson, K.B., and B.W. 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Estimated unrestricted model is t < t0 , + + − + + − + − − = ( ) ( )(ln ln ( )) ( ) ( ) (ln ln ( )) ( ) ln ( ) ln ( 1) a t p p t t a t p p t t p t p t α β ε α ε 0 t ≥ t . Estimated restricted model is 0 t < t , + − + + − − = ( ) (ln ln ( )) ( ) ( ) ( ) ln ( ) ln ( 1) a t p p t t a t t p t p t β ε ε 0 t ≥ t . 54 Table II−2. Per Period Storage Costs (Percentage of Price) Commodities Year Corn and Soybean Wheat 1975 0.0021 0.0035 1976 0.0022 0.0035 1977 0.0024 0.0042 1978 0.0026 0.0049 1979 0.0028 0.0049 1980 0.0032 0.0053 1981 0.0036 0.0056 1982 0.0038 0.0056 1983 0.0039 0.0060 1984 0.0041 0.0060 1985 0.0042 0.0060 1986 0.0043 0.0060 1987 0.0044 0.0060 1988 0.0046 0.0060 1989 0.0048 0.0060 1990 0.0051 0.0060 1991 0.0053 0.0060 1992 0.0055 0.0060 1993 0.0057 0.0060 1994 0.0058 0.0060 1995 0.0060 0.0060 1996 0.0060 0.0060 1997 0.0060 0.0060 1998 0.0060 0.0060 1999 0.0060 0.0060 2000 0.0060 0.0060 2001 0.0060 0.0070 2002 0.0060 0.0070 2003 0.0060 0.0070 2004 0.0060 0.0070 2005 0.0060 0.0070 2006 0.0060 0.0070 55 Table II−3. Parameter Estimates of Seasonal Mean Reversion Price Process by Nonparametric Bootstrapping Corn Soybean Wheat Coefficient Standard Deviation Coefficient Standard Deviation Coefficient Standard Deviation α 0.0356 0.0117 0.0416 0.0194 0.0230 0.0115 0 γ 0.0060 0.0034 0.0037 0.0029 0.0078 0.0037 2 γ 5.7E04 1.3E04 2.7E04 1.4E04 1.4E04 1.5E04 3 γ 2.8E05 6.8E06 1.3E05 7.3E06 2.5E06 7.0E06 4 γ 5.8E07 1.5E07 2.7E07 1.6E07 2.9E09 1.5E07 5 γ 4.4E09 1.2E09 2.1E09 1.2E09 2.3E10 1.1E09 σ 2 0.0013 8.0E05 0.0011 5.9E05 0.0012 6.5E05 0 t 44 6.1469 42 7.2619 38 10.8677 Note: Estimated model is if 0 t < t , + − + + − = + + + 1 1 1 ( ) (ln ln ) ( ) ln ln t t t t t g t p p f t p p α ε ε if 0 t ≥ t . 56 Table II−4. Sales Dates and Net Returns for Corn Sale Dates (Weeks from Harvest) Per Bushel Net Returns ($/bu) Scenario 1 Scenario Year 2 Scenario 1 Scenario 2 Model 1 Model 2 Model 1 Model 2 Model 1 Model 2 Model 1 Model 2 1975 0 17 0 32 2.61 2.31 2.61 2.46 1976 15 17 20 32 2.27 2.17 2.24 2.05 1977 27 17 33 32 2.13 1.91 2.15 2.12 1978 25 16 32 19 2.03 1.98 2.19 2.03 1979 0 0 0 17 2.62 2.62 2.62 2.26 1980 0 0 0 16 3.05 3.05 3.05 3.08 1981 0 0 0 0 2.38 2.38 2.38 2.38 1982 0 0 15 0 2.00 2.00 2.14 2.00 1983 0 0 0 16 3.41 3.41 3.41 3.00 1984 0 0 17 15 2.68 2.68 2.46 2.46 1985 15 0 33 16 2.15 2.10 2.10 2.22 1986 34 0 37 0 1.41 1.42 1.50 1.42 1987 18 0 35 16 1.68 1.63 1.81 1.74 1988 0 0 0 16 2.68 2.68 2.68 2.36 1989 0 0 0 15 2.27 2.27 2.27 2.12 1990 0 0 0 15 2.19 2.19 2.19 2.19 1991 0 0 0 15 2.42 2.42 2.42 2.31 1992 0 0 25 16 2.06 2.06 1.96 1.92 1993 0 0 13 16 2.23 2.23 2.74 2.66 1994 0 0 17 15 1.95 1.95 2.07 2.08 1995 0 0 0 15 2.94 2.94 2.94 3.21 1996 0 0 0 15 2.90 2.90 2.90 2.49 1997 0 0 15 15 2.45 2.45 2.46 2.46 1998 14 0 34 15 1.85 1.81 1.68 1.90 1999 15 0 31 15 1.74 1.77 1.91 1.82 2000 12 0 35 14 1.88 1.63 1.45 1.95 2001 0 0 31 15 1.85 1.85 1.67 1.82 2002 0 0 0 16 2.46 2.46 2.46 2.12 2003 0 0 14 16 2.03 2.03 2.26 2.43 2004 0 0 33 16 1.76 1.76 1.68 1.72 2005 0 0 18 16 1.67 1.67 1.88 1.78 2006 0 0 0 15 2.41 2.41 2.41 3.30 32 year average 2.25 2.22 2.27 2.25 Note: Scenario1 includes storage and interest costs. Scenario2 includes half of storage and interest costs. Model1 assumes that price follows a mean reversion process. Model2 assumes that price follows a seasonal mean reversion process. 57 Table II−5. Sales Dates and Net Returns for Soybean Sale Dates (Weeks from Harvest) Per Bushel Net Returns ($/bu) Scenario 1 Scenario Year 2 Scenario 1 Scenario 2 Model 1 Model 2 Model 1 Model 2 Model 1 Model 2 Model 1 Model 2 1975 0 0 0 34 5.15 5.15 5.15 5.02 1976 0 0 0 29 5.95 5.95 5.95 9.37 1977 11 0 25 34 5.53 5.05 6.37 6.27 1978 0 0 0 0 6.17 6.17 6.17 6.17 1979 0 0 0 0 6.79 6.79 6.79 6.79 1980 0 0 0 0 7.51 7.51 7.51 7.51 1981 0 0 0 0 5.96 5.96 5.96 5.96 1982 0 0 15 0 5.00 5.00 5.22 5.00 1983 0 0 0 0 8.42 8.42 8.42 8.42 1984 0 0 0 0 5.77 5.77 5.77 5.77 1985 0 0 38 0 4.88 4.88 4.36 4.88 1986 0 0 36 0 4.80 4.80 4.77 4.80 1987 0 0 0 0 5.26 5.26 5.26 5.26 1988 0 0 0 0 7.89 7.89 7.89 7.89 1989 0 0 0 0 5.50 5.50 5.50 5.50 1990 0 0 0 0 6.01 6.01 6.01 6.01 1991 0 0 0 0 5.67 5.67 5.67 5.67 1992 0 0 15 0 5.17 5.17 5.31 5.17 1993 0 0 0 0 5.88 5.88 5.88 5.88 1994 0 0 0 0 5.22 5.22 5.22 5.22 1995 0 0 0 0 6.24 6.24 6.24 6.24 1996 0 0 0 0 7.25 7.25 7.25 7.25 1997 0 0 0 0 6.16 6.16 6.16 6.16 1998 0 0 40 0 4.92 4.92 3.18 4.92 1999 0 0 37 0 4.66 4.66 4.11 4.66 2000 0 0 37 0 4.73 4.73 3.80 4.73 2001 0 0 35 0 4.26 4.26 4.28 4.26 2002 0 0 0 0 5.18 5.18 5.18 5.18 2003 0 0 0 0 6.77 6.77 6.77 6.77 2004 0 0 0 0 4.98 4.98 4.98 4.98 2005 0 0 0 0 5.24 5.24 5.24 5.24 2006 0 0 0 0 5.27 5.27 5.27 5.27 32 year average 5.75 5.74 5.68 5.88 Note: Scenario1 includes storage and interest costs. Scenario2 includes half of storage and interest costs. Model1 assumes that price follows a mean reversion process. Model2 assumes that price follows a seasonal mean reversion process. 58 Table II−6. Sales Dates and Net Returns for Wheat Sale Dates (Weeks from Harvest) Per Bushel Net Returns ($/bu) Scenario 1 Scenario Year 2 Scenario 1 Scenario 2 Model 1 Model 2 Model 1 Model 2 Model 1 Model 2 Model 1 Model 2 1975 0 0 0 23 2.91 2.91 2.91 3.03 1976 0 0 0 24 3.36 3.36 3.36 2.16 1977 0 0 25 23 1.92 1.92 2.21 2.26 1978 0 0 0 0 2.90 2.90 2.90 2.90 1979 0 0 0 0 3.40 3.40 3.40 3.40 1980 0 0 0 0 3.40 3.40 3.40 3.40 1981 0 0 0 0 3.83 3.83 3.83 3.83 1982 0 0 0 0 3.44 3.44 3.44 3.44 1983 0 0 0 0 3.39 3.39 3.39 3.39 1984 0 0 0 0 3.34 3.34 3.34 3.34 1985 0 0 0 0 2.89 2.89 2.89 2.89 1986 0 0 25 0 2.23 2.23 1.85 2.23 1987 0 0 0 0 2.32 2.32 2.32 2.32 1988 0 0 0 0 3.05 3.05 3.05 3.05 1989 0 0 0 0 3.77 3.77 3.77 3.77 1990 0 0 0 0 2.94 2.94 2.94 2.94 1991 0 0 0 0 2.52 2.52 2.52 2.52 1992 0 0 0 0 3.48 3.48 3.48 3.48 1993 0 0 0 0 2.63 2.63 2.63 2.63 1994 0 0 0 0 3.10 3.10 3.10 3.10 1995 0 0 0 0 3.91 3.91 3.91 3.91 1996 0 0 0 0 5.37 5.37 5.37 5.37 1997 0 0 0 0 3.73 3.73 3.73 3.73 1998 0 0 0 0 2.70 2.70 2.70 2.70 1999 0 0 27 0 2.36 2.36 1.65 2.36 2000 0 0 0 0 2.40 2.40 2.40 2.40 2001 0 0 0 0 2.88 2.88 2.88 2.88 2002 0 0 0 0 2.85 2.85 2.85 2.85 2003 0 0 0 0 2.83 2.83 2.83 2.83 2004 0 0 0 0 3.50 3.50 3.50 3.50 2005 0 0 0 0 3.03 3.03 3.03 3.03 2006 0 0 0 0 4.54 4.54 4.54 4.54 32 year average 3.15 3.15 3.13 3.13 Note: Scenario1 includes storage and interest costs. Scenario2 includes half of storage and interest costs. Model1 assumes that price follows a mean reversion process. Model2 assumes that price follows a seasonal mean reversion process. 59 Table II−7. Paired Differences tRatios of the Mean Net Returns between Seasonal Mean Reversion and Mean Reversion Model (19752006) Commodity Include Storage and Interest Costs Include Half of Storage and Interest Rate Corn 2.30* 0.54 Soybean 1.00 1.67 Wheat N/Aa 0.04 Note: tcritical value with 30 degree of freedom at 5% significance level is 2.042. We calculate paired differences by subtracting the net return assuming a simple mean reversion model from a net return assuming seasonal mean reversion model. a A paired difference tratio for wheat of including storage and interest costs is not available since there is no difference of net returns between the two models, so variance of paired difference is zero. 60 Figure II−1. An example trinomial lattice x −2Δ x x −Δ x x −Δ x x x x x+Δ x x +Δ x x +2Δ x x +3Δ x x +2Δ x x +Δ x x x −Δ x x −2Δ x x −3Δ x 61 0.015 0.01 0.005 0 0.005 0.01 Oct N ov Dec Jan Feb Mar Apr May Jun Jul Aug Sep t % Figure II−2. Seasonality of change in corn price 62 0.01 0.008 0.006 0.004 0.002 0 0.002 0.004 0.006 Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep t % Figure II−3. Seasonality of change in soybean price 63 0.01 0.008 0.006 0.004 0.002 0 0.002 0.004 0.006 Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May t % Figure II−4. Seasonality of change in wheat price 64 0 0.5 1 1.5 2 2.5 3 Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep t $/bu Figure II−5. Cutoff price of mean reversion price process for corn using low storage and interest costs 0 0.5 1 1.5 2 2.5 3 3.5 Oct Nov Dec Jan Feb Mar Apr May Jun Jul A ug Sep t $/bu Figure II−6. Cutoff price of seasonal mean reversion price process for corn using low storage and interest costs store sell store sell store 65 5 5.5 6 6.5 7 Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep t $/bu Figure II−7. Cutoff price of mean reversion price process for soybeans using low storage and interest costs 0 1 2 3 4 5 6 7 Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep t $/bu Figure II−8. Cutoff price of seasonal mean reversion price process for soybeans using low storage and interest costs store store sell sell 66 2 2.5 3 3.5 4 4.5 5 Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May t $/bu Figure II−9. Cutoff price of mean reversion price process for wheat using low storage and interest costs 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May t $/bu Figure II−10. Cutoff price of seasonal mean reversion price process for wheat using low storage and interest costs store store sell sell VITA HYUN SEOK KIM Candidate for the Degree of Doctor of Philosophy Thesis: MAKING GRAIN PRICING DECISIONS BASED ON PROFIT MARGIN HEDGING AND REAL OPTION VALUES Major Field: Agricultural Economics Biographical Personal Data: Born in Seoul, Korea, on March 8, 1975, the son of Ho Tak Kim and Yoon Jeung Cho. Education: Graduated from YoungDong High School, Seoul, Korea, in February 1994; received a Bachelor of Science degree in Trade from Hankuk University of Foreign Studies, Seoul, Korea, in February 2002. Received a Master of Science degree in Agricultural Economics from Seoul National University, Seoul, Korea, in February 2004. Completed the requirements for the degree of Dotor of Philosophy in Agricultural Economics at Oklahoma State University in July 2008. Experience: Graduate Research Assistant, Department of Agricultural Economics and Rural Development, Seoul National University, March 2002 − February 2004. Assistant Researcher, Market Analysis Team, Korean Food Research Institute, February 2004 − August 2004. Graduate Research Assistant, Department of Agricultural Economics, Oklahoma State University, August 2004 − July 2008. Award: Spielman Scholarship for Excellence in Academic Performance, Department of Agricultural Economics at Oklahoma State University, March 2007. Professional Membership: American Agricultural Economics Association. Name: Hyun Seok Kim Date of Degree: July, 2008 Institution: Oklahoma State University Location: Stillwater, Oklahoma Title of Study: MAKING GRAIN PRICING DECISIONS BASED ON PROFIT MARGIN HEDGING AND REAL OPTION VALUE Pages in Study: Candidate for the Degree of Dotor of Philosophy Major Field: Agricultural Economics Scope and Method of Study: This study contains two essays. The first essay is preharvest pricing decision making and the second essay is postharvest decision making. The purpose of the first essay was to determine producer’s utility function and price processes where profit margin hedging is optimal. A statistical test of mean reversion in agricultural futures prices is conducted. The simulations were also conducted to compare the expected utility of profit margin hedging strategy with the expected utility of other strategies such as always hedging and selling at harvest. The purpose of the second essay was to determine whether real option values can explain why producers appear to store too long. To determine the real option value, we modeled and estimated a seasonal mean reversion price process which allowed price to be a random walk within a season, but mean reverting across crop years. After estimation of the price process, a universal lattice model was used to determine cutoff price. This study conducted simulations using cash prices of crops to determine differences of net returns of optimal strategy under two different price processes, which are a simple mean reversion price process and a new seasonal mean reversion price process. Findings and Conclusions: Theoretical results from the first essay showed that profit margin hedging is an optimal strategy under a highly restricted target utility function even in an efficient market. Profit margin hedging is profitable if prices are mean reverting. Simulation results showed that profit margin hedging gives the highest expected utility to producers under the highly restricted target utility function. With the variance ratio test, there is little evidence that futures prices of crops follows a mean reverting process. In the second essay, the estimated nonparametric bootstrap parameters of the seasonal mean reversion process show the seasonal function turns negative before mean reversion begins, which suggests that real option values are relatively unimportant in determining when producers sell their grain. The graphs of cutoff price when assuming a seasonal mean reversion price process show that producers sell before mean reversion begins except when prices are extremely low. Therefore, Fackler and Livingston’s (2002) finding of a large real option value that can explain why producers store too long is not supported. The simulation results show that there is little evidence that the net returns between the mean reversion model and the seasonal mean reversion model are different. ADVISER’S APPROVAL: B. Wade Brorsen
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Title  Making Grain Pricing Decisions Based on Profit Margin Hedging and Real Option Values 
Date  20080701 
Author  Kim, Hyun Seok 
Department  Agricultural Economics 
Document Type  
Full Text Type  Open Access 
Abstract  The purpose of the first essay was to determine producer's utility function and price processes where profit margin hedging is optimal. A statistical test of mean reversion in agricultural futures prices is conducted. The simulations were also conducted to compare the expected utility of profit margin hedging strategy with the expected utility of other strategies such as always hedging and selling at harvest. The purpose of the second essay was to determine whether real option values can explain why producers appear to store too long. To determine the real option value, we modeled and estimated a seasonal mean reversion price process which allowed price to be a random walk within a season, but mean reverting across crop years. A universal lattice model was used to determine cutoff price. Simulations are conducted to determine differences of net returns under two different price processes, which are a simple mean reversion and a new seasonal mean reversion price process. Theoretical results from the first essay showed that profit margin hedging is an optimal strategy under a highly restricted target utility function even in an efficient market. Profit margin hedging is profitable if prices are mean reverting. Simulation results showed that profit margin hedging gives the highest expected utility to producers under the highly restricted target utility function. With the variance ratio test, there is little evidence that futures price follows a mean reverting process. In the second essay, the estimated nonparametric bootstrap parameters of the seasonal mean reversion process show the seasonal function turns negative before mean reversion begins, which suggests that real option values are relatively unimportant in determining when producers sell their grain. The graphs of cutoff price when assuming a seasonal mean reversion price process show that producers sell before mean reversion begins except when prices are extremely low. Therefore, Fackler and Livingston's (2002) finding of a large real option value that can explain why producers store too long is not supported. The simulation results show that there is little evidence that the net returns between the mean reversion model and the seasonal mean reversion model are different. 
Note  Dissertation 
Rights  © Oklahoma Agricultural and Mechanical Board of Regents 
Transcript  MAKING GRAIN PRICING DECISIONS BASED ON PROFIT MARGIN HEDGING AND REAL OPTION VALUES By HYUN SEOK KIM Bachelor of Science Hankuk University of Foreign Studies Seoul, Korea 2002 Master of Science Seoul National University Seoul, Korea 2004 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY July, 2008 ii MAKING GRAIN PRICING DECISIONS BASED ON PROFIT MARGIN HEDGING AND REAL OPTION VALUES Dissertation Approved: Dr. B. Wade Brorsen Dissertation Adviser Dr. Brian D. Adam Dr. Kim B. Anderson Dr. Timothy L. Krehbiel Dr. A. Gordon Emslie Dean of the Graduate College iii ACKNOWLEDGMENT I wish to express my sincere appreciation to my academic adviser Dr. Wade Brorsen for his intelligent supervision and invaluable guidance and encouragement throughout my Ph.D. program at Oklahoma State University. I also wish to thank all members of my academic committee, Dr. Brian Adam, Dr. Kim Anderson, and Dr. Tim Krehbiel for their helpful advice and invaluable comments during the preparation of this dissertation. My appreciation is extended to the Department of Agricultural Economics for providing a research opportunity and financial support for my Ph.D. program. Friendly faculty and staff in the department have contributed to my academic success. In particular, I would like to thank my parents, Ho Tak Kim and Yoon Jeung Cho, for their continuous support through prayer, hope and love which have helped me to stand firm throughout the whole process of completing this program. My wife Hae Jung and my two sons Eugene and Ian have been my continuous source of inspiration and love. Working on my Ph.D. was a lot easier with Hae Jung, Eugene and Ian on my side. Finally, I dedicate this work to Jesus Christ, my Lord and Savior. He has begun the good work in me and has proven Himself to be faithful in finishing it. To Him be the glory forever and ever. iv TABLE OF CONTENTS Chapter Page I. PROFIT MARGIN HEDGING·········································································································1 Introduction················································································································· 1 Theory························································································································· 2 Expected Target Utility···························································································· 2 Mean Reversion ······································································································ 6 Data ····························································································································· 8 Procedures················································································································· 10 Measure of Expected Utility···················································································11 Variance Ratio Test ······························································································· 12 Results······················································································································· 14 Summary and Conclusions ······················································································· 17 References················································································································· 20 Notes ························································································································· 23 II. CAN REAL OPTION VALUE EXPLAIN WHY PRODUCERS APPEAR TO STORE TOO LONG? ······················································································································· 34 Introduction··············································································································· 34 Theory······················································································································· 35 Data ··························································································································· 37 Procedures················································································································· 38 Estimation of Price Process Parameters ······························································ 38 A Universal Lattice Model ···················································································· 41 Simulation ············································································································· 45 Results······················································································································· 46 Summary and Conclusion························································································· 48 References················································································································· 50 iv LIST OF TABLES Table Page Table I−1. Average Prices (cents/bu) for Hedging Strategies at September 20th (1975 2005)················································································································ 27 Table I−2. Paired Differences tRatios of Average Prices, (19752005)··························· 28 Table I−3. Expected Utilities for Hedging Strategies at September 20th (19752005) ····· 29 Table I−4. Paired Differences tRatios of Expected Utilities, (19752005)······················ 30 Table I−5. Average Prices and Expected Utilities for Hedging Strategies for Multiple Crops Scenario (19752005)············································································ 31 Table I−6. Paired Differences tRatios of Average Prices and Expected Utilities for Multiple Crops Cases (19752005)·································································· 31 Table I−7. Variance Ratio Tests for Futures Prices (19752006) ······································ 32 Table I−8. Joint Variance Ratio Tests for Futures Prices (19752006) ····························· 32 Table I−9. Variance Ratio Tests for Futures Prices Using Jackknife Approach(19752006) ························································································································· 33 Table II−1. Parameter Estimateα of Seasonal Mean Reversion Price Process ··············· 53 Table II−2. Per Period Storage Costs (Percentage of Price) ············································· 54 Table II−3. Parameter Estimates of Seasonal Mean Reversion Price Process by Nonparametric Bootstrapping·········································································· 55 Table II−4. Sales Dates and Net Returns for Corn···························································· 56 Table II−5. Sales Dates and Net Returns for Soybean······················································ 57 Table II−6. Sales Dates and Net Returns for Wheat ························································ 58 Table II−7. Paired Differences tRatios of the Mean Net Returns between Seasonal Mean Reversion and Mean Reversion Model (19752006) ······································ 59 v LIST OF FIGURES Figure Page Figure I−1. Expected utility of a crop producer as futures price 0 f p changes when α equals to β ······································································································ 24 Figure I−2. Expected utility of a crop producer as futures price 0 f p changes when α is less than β ······································································································ 25 Figure I−3. Expected utility of a crop producer as futures price 0 f p changes when α is greater than β ································································································· 26 Figure II−1. An example trinomial lattice ········································································ 60 Figure II−2. Seasonality of change in corn price······························································ 61 Figure II−3. Seasonality of change in soybean price························································ 62 Figure II−4. Seasonality of change in wheat price ··························································· 63 Figure II−5. Cutoff price of mean reversion price process for corn using half storage and interest costs ···································································································· 64 Figure II−6. Cutoff price of seasonal mean reversion price process for corn using low storage and interest costs ················································································· 64 Figure II−7. Cutoff price of mean reversion price process for soybeans using low storage and interest costs······························································································ 65 Figure II−8. Cutoff price of seasonal mean reversion price process for soybeans using low storage and interest costs ·········································································· 65 Figure II−9. Cutoff price of mean reversion price process for wheat using low storage and interest costs ···································································································· 66 Figure II−10. Cutoff price of seasonal mean reversion price process for wheat using low storage and interest costs ················································································· 66 1 I. CHAPTER I PROFIT MARGIN HEDGING Introduction Some extension economists and others often recommend profit margin hedging, in which a producer sells a crop preharvest by short hedging whenever prices are above a target. However, this strategy recommendation is without a research base. The strategy is also included in undergraduate textbooks such as Purcell and Koontz (pp. 329330). With recent high prices, producers have forward contracted more of their crops, which provides evidence that some producers follow such a strategy. However, the theoretical assumptions that would justify such a strategy have never been developed. Some empirical studies (Leuthold and Mokler 1980; Kenyon and Clay 1987; Johnson et al. 1991) have found that a profit margin hedging strategy is profitable for producers or investors but did not include significance tests. Girma and Paulson (1998) studied the statistical behavior of crack spreads (the price difference between refined energy products and crude oil) and they conclude that historically simple buy and hold trading strategies are profitable and in many instances are significantly greater than zero. These previous studies, however, do not provide any theoretical justification for using profit margin hedging. 2 This research focuses on answering the question, “What assumptions for producer’s utility and price process can justify profit margin hedging?” The paper determines the producer’s utility function and price processes where profit margin hedging is optimal. A statistical test of mean reversion in agricultural futures prices is conducted. Simulations are conducted to compare the expected utility of a profit margin hedging strategy with the expected utility of other strategies such as always hedging and selling at harvest. Theory Expected Target Utility The goal of this section is to derive a theoretical model where profit margin hedging is optimal. The meanvariance (EV) model is the most commonly used to analyze choices under uncertainty. Optimal hedging strategies under EV, such as those of Johnson (1960), Stein (1960), and more general models such as Lence (1996) do not lead to profit margin hedging strategies being optimal. Simiarly, mean semivariance and mean targetsemivariance models such as those of Dejong et al. (1997), Lien and Tse (1998, 2000), Chen et al. (2001), and Turvey and Nayak (2003) do not lead to profit margin hedging rules being optimal. Some previous studies argue that EV analysis has several wellknown theoretical shortcomings (Fishburn 1997; Holthausen 1981). Fishburn (1977) proposed a meanrisk model which generalized the meantarget semivariance model (Markowitz 1959; Mao 1970; Hogan and Warren 1974; Porter 1974) to address the shortcomings of the EV model. The widely known shortcoming of the EV model is that if the outcome distributions are not of a locationscale form (such as 3 normal) or the utility function is not quadratic, the EV model is not consistent with expected utility. Fishburn’s model measured return as the mean of the outcomes, but defined risk as weighted deviations of outcomes below target and the model assumes risk neutrality above the target. Holthausen (1981) adapted Fishburn’s model by using the same measure of risk but defining return as weighted deviations above the target to avoid the risk neutrality restriction. To measure producer’s expected utility, this study adopts Holthausen’s model in which the utility function is: for all π ≥ t (1.1) − − − = α β π π π ( ) ( ) ( ) k t t U for all π ≤ t . where π indicates profit, t represents the target, k is a positive constant, and α and β reflect the risk preferences. If α <1 (α >1) , then the producer is risk seeking (averse) below the target. Then, the expected target utility can be written as (1.2) ∫ ∫ −∞ ∞ = − − − t t EU π π t β f π dπ k t π α f π dπ ( ) ( ) ( ) ( ) ( ) where f (π ) is the probability density function of π which is normally distributed with mean π and variance 2 π σ .1 If producers hedge preharvest without basis risk, then the profit is (1.3) p F p F f 0 π = (1− ) + where p is the price of crop at harvest, F is a hedge ratio, and 0 f p indicates the futures price at the time of the hedge. Then, equation (1.2) is rewritten as (1.4) ∫ ∫ −∞ ∞ = − + − − − − − A f A f EU( ) {p(1 F) p F t} f ( )dp k {t p(1 F) p F} f ( )dp π 0 β π 0 α π where 4 (t p F) F A f 0 1 1 − − = , f ( p) is the probability density function of p which is normally distributed with mean p and variance 2 p σ , and F is the choice variable. Equation (1.4) will be optimized when the first derivative with respect to F equals zero. The first order condition of equation (1.4) is (1.5) ∫ ∞ − − ′ = − − + − A f f EU ( ) {p(1 F) p F t} ( p p ) f ( )dp π β 0 β 1 0 π ∫ ∞ − ∂ ∂ − − + − A f pF dp f p F p F t 2 2 0 2(1 ) ( ) { (1 ) } σ σ π π β 2 0 0 (1 ) { (1 ) } ( ) F t p A F p F t f f f − − − − + − β π ∫ −∞ − − − − − − A f f k {t p(1 F) p F} ( p p ) f ( )dp α 0 α 1 0 π ∫ −∞ − ∂ ∂ + − − − A f pF dp f k t p F p F 2 2 0 2(1 ) ( ) { (1 ) } σ σ π π α 2 0 0 (1 ) { (1 ) } ( ) F t p k t A F p F f f f − − − − − − α π . The third and the last term in equation (1.5) become zero since A equals ( ) (1 ) 0 t p F F f − − . If α and β are equal, and k is one, then equation (1.5) can be rewritten as (1.6) ∫ ∞ − − ′ = − − + − A f f EU ( ) {p(1 F) p F t} ( p p ) f ( )dp π α 0 α 1 0 π ∫ ∞ − ∂ ∂ − − + − A f pF dp f p F p F t 2 2 0 2(1 ) ( ) { (1 ) } σ σ π π α 5 ∫−∞ − − − − − − A f f {t p(1 F) p F} ( p p ) f ( )dp α 0 α 1 0 π ∫ −∞ − ∂ ∂ + − − − A f pF dp f t p F p F 2 2 0 2(1 ) ( ) { (1 ) } σ σ π π α . Then, the first and the third terms in equation (1.6) cancel out. If price 0 f p is above the target and all the crop is hedged − that is, F equals one − then the second and the last terms are zero and equation (1.6) will be zero. If price 0 f p is below the target and producers do not hedge − that is, F equals zero − then there is no interior solution and the optimum is the lower bound of zero. The shape of expected utility when price 0 f p is below the target, in figure I−1, confirms that expected utility is highest when the hedge ratio is zero. Therefore, profit margin hedging where the producer hedges all when prices are above the target and none when prices are below the target is shown to be an optimal strategy under a highly restricted target utility function where a producer has the same level of risk preferences above and below the target, and k equals one. In the case of relaxing the assumptions that α and β are equal and k equals one, it cannot be solved analytically, so numerical methods must be used instead. Figures I−1 through I−3 show expected utilities as futures price 0 f p changes for alternate on values of α and β . If α equals β , as in figure I−1, then a producer hedges all of the crop when the futures price is greater than the target, but does not hedge when the futures price is less than the target. In this case, profit margin hedging is optimal, confirming our theory. In the case of α smaller than β , in figure I−2, the producer’s hedging behavior is equivalent to the case of α = β . In contrast to figure I−1, which is monotonic, figures I−2 and I−3 are nonmonotonic. The reason why figures I−2 and I−3 are not monotonic 6 is that the utility function is not concave or convex; because the utility function has a different form above and below target, risk seeking dominates producers’ risk preferences until some point of the hedge ratio but risk aversion dominates their risk preferences after that. Figure I−3 shows that if α is greater than β , producers hedge all of the crop when the futures price is greater than the target, but also hedges a portion of the crop when the futures price is less than the target. These numerical solutions show that profit margin hedging can be an optimal strategy when α does not equal β , but the optimal strategy is not always all or none. Some studies (eg. Lence 1996) showed that the optimal hedge ratio typically decreases in the presence of basis risk, yield risk, transaction costs or multiple crop outputs. Moschini and Lapan (1995), for example, showed that increasing basis risk results in a lower futures hedge ratio, and increasing yield risk also results in a lower hedge ratio. Bond and Thompson (1985) found that a rise in the transaction or storage cost leads to a decrease in the optimal hedging ratio. Fackler and McNew (1993) showed that, under a multiproduct approach, the fullyhedged position is not optimal and it is not optimal to hedge all commodities in the same proportion. Relaxing these assumptions is expected to also reduce optimal hedge ratios under profit margin hedging. Mean Reversion Profit margin hedging has been suggested as a profit increasing strategy. Zulauf and Irwin (1998) suggest that the success of selling before harvest depends on whether a price bias exists. That is, mean reversion is a needed attribute of price behavior for profit margin hedging to be a successful strategy. Therefore, this study provides a proof that 7 profit margin hedging is more profitable than the other strategies such as always hedging and selling at harvest if futures prices are mean reverting. The mean reverting futures price process can be written as (1.7) − = λ ( − ) +ε 0 0 f f p p p p where p is random cash price which equals the futures price at the terminal point of the hedge (no basis risk), 0 f p is futures prices at the time of the hedge, p is the longrun average price, and ε is an error term with mean zero and variance 2 ε σ . The estimated coefficient λ is the mean reversion speed by which 0 f p revert toward p . If a onetime period model is used, producers’ expected profit function can be obtained by taking the expected value of equation (1.3). (1.8) [ ] [(1 ) ] 0E E F p p F f π = − + If futures price follows a mean reversion process, equation (8) can be rewritten as (1.9) E[ ] E[ F {p p p } p F] f f f 0 0 0 π = (1− ) +λ ( − ) + = E[ F p F p p p F] f f f 0 0 0 (1− ) + λ (1− )( − ) + = F p F p p p F f f f 0 0 0 (1− ) + λ (1− )( − ) + = p p F F p p p F f f f f 0 0 0 0 − + λ (1− )( − ) + = (1 )( ) 0 0 f f p +λ − F p − p If 0 f p is greater than p , then F =1 and we can rewrite equation (1.9) as (1.10) E[ PMH p p] p E[ AH] p p p E[p] E[ SH] f f f f π , > = = π > +λ ( − ) = = π 0 0 0 0 where PMH indicates profit margin hedging, AH indicates always hedging, and SH is Selling at harvest. 8 If 0 f p is less than p , then F = 0 and we can rewrite equation (1.9) as (1.11) E[ PMH p p] p p p E[ SH] p E[ AH] f f f f π 0 < = 0 +λ − 0 = π > 0 = π , ( ) . Since λ is greater than zero and the futures price 0 f p is less than the longrun average price p , the expected profit conditioned on profit margin hedge is greater than 0 f p which is the expected profit conditioned on always hedging. Therefore, profit margin hedging is more profitable than other strategies such as always hedging and selling at harvest, if futures prices are mean reverting. It is important to note that the above derivation is based on a static oneperiod model. If the problem is made dynamic and producers are allowed to hedge at anytime, the profit margin hedging rule would still be profitable, but would no longer be optimal. The optimal rule could be derived similarly to what Fackler and Livingston (2002) derived for cash prices. In their model, if grain prices are below the mean, it is best to store since prices will revert to the mean. If grain prices are unusually high, it is best to sell immediately. If prices are near the mean, there can be a real option value from waiting since there is the opportunity to wait and select a time to sell when prices are unusually high. In that case, the expected profit maximizing target price would decrease as harvest approaches since there is less opportunity for price to increase above the mean. Data The chosen agricultural commodities are Oklahoma hard red winter wheat, Illinois soft winter wheat, soybeans, and corn. This study uses the July futures contract prices for wheat from the Kansas City Board of Trade (KCBT) and from the Chicago 9 Board of Trade (CBOT), November futures contract prices for soybeans and December futures contract prices for corn from the CBOT. Futures prices for KCBT wheat are obtained from KCBT and for CBOT wheat, soybean, and corn are obtained from Prophet Financial Systems, Inc. To test mean reversion, this study uses daily data. The sample period extends from August 1975 through May 2006 for KCBT and CBOT wheat, from December 1975 through October 2006 for soybeans and January 1975 through December 2006 for corn. July observations for KCBT and CBOT wheat, November observations for CBOT soybeans and corn are deleted since these observations are for the delivery period. Markets are thin during this time and can be quite volatile. No price changes across contract years are used. To conduct the simulation, Oklahoma June average wheat cash prices and prices for the July futures contract on September 20th from 1975 to 2005 are used. We use 31 years data for simulations whereas 32 years data for a mean reversion test since we do not have the 2006 economic cost of production. Five year moving averages of basis and yield of crops are used to make hedging decisions with basis risk and yield risk. For the case of multicrop producers, the study used Illinois June average cash price for wheat, and Illinois October average price for soybean and corn. For Illinois wheat, the July futures contract prices on September 20th from 1975 through 2005 are used. For Illinois soybean and corn, futures prices for the November contract and the December contract for soybean and corn, respectively, at May 10th from 1975 through 2005 are used. The Illinois monthly average cash prices are obtained from National Agricultural Statistics Service (NASS) of the United States Department of 10 Agriculture (USDA). This study used 70% of economic costs of production as targets for KBCT wheat and 80% for CBOT wheat and 100% for CBOT corn and soybean since the economic costs include many types of cost and it is too high to use as the target in KBCT and CBOT wheat. These choices depend on the number of hedges for the 32 year period. If the cost is set too high, a producer would seldom hedge. If cost is set too low, a producer would always hedge. For the costs assumed here, a hedge is placed 16 times of 31 years for KBCT wheat, 17 times for CBOT wheat, 16 times for CBOT corn, 20 times for CBOT soybean. The economic costs of production for the three crops from 1975 to 2005 are obtained from the Economic Research Service (ERS) of USDA (2008). The yield data of Oklahoma wheat at Garfield County, Oklahoma, and Illinois wheat, soybean, and corn in Livingston County, Illinois from 1975 to 2005 are obtained from NASS of USDA. Procedures This paper has two main procedures − simulation to compare the expected utility of profit margin hedging strategy with the always hedging and the selling at harvest strategies, and mean reversion testing for KCBT and CBOT wheat July futures prices, CBOT soybean November futures prices, and CBOT corn December futures prices. The expected utility is measured by taking the average utility across 31 years. To test mean reversion, the variance ratio test is employed. 11 Measure of Expected Utility Five scenarios are considered for each of three hedging strategies − hedging without risk, with basis risk, with yield risk, with yield and basis risk and with multiple crops. To measure expected utility without basis risk, a perfect foresight model is used which assumes actual harvest basis is known at the time of the decision. In this case, under a profit margin hedging strategy, if the sum of futures price at the time of the decision and foresighted basis is greater than the target return, then producers hedge all crops, otherwise they hedge none and sell the crops at harvest. If basis risk were considered, producers hedge all when the sum of futures price at the time of the decision and average basis is greater than the target return otherwise they do not hedge. With yield risk, producers hedge all crops if the average returns that is the sum of futures price at the time of the decision and the foresighted basis multiplied by average yield is greater than the target multiplied by average yield. In the case of multiple crops, the producer hedges all crops when the total returns − that is, the sum of futures price at the time of the decision and foresighted basis multiplied by quantity produced for each crop − is greater than the total target that is the sum of target multiplied by quantity produced for each crop. We assume producers are risk averse above the target and risk seeking below the target and pick 0.5 as the value of α andβ . We also assume a transaction cost of 1.2 cents per bushel but we do not consider margin calls. After calculating utility for 31 years from 1975 through 2005, expected utility is calculated as the average of utilities (14) Σ= = 31 1 ( ) 31 1 ( ) i i EU π U π . 12 Variance Ratio Test The idea behind the variance ratio test is that if the natural logarithm of a price series Pt is a random walk, then the variance of kperiod returns should equal k times the variance of oneperiod returns (Cochrane 1988; Kim et al. 1991; Lo and MacKinlay 1988; Poterba and Summers 1988). The general kperiod variance ratio, VR(k) is defined as (15) VR(k) = (1) ( ) 2 2 σ σ k k where ( ) 2 σ k is the variance of the k differences and (1) σ 2 is the variance of the first differences. The null hypothesis of interest is that VR(k) equals one. That is, VR(k) equal to one implies that futures price follows a random walk process, whereas a variance ratio of less than one implies a mean reversion process. Lo and MacKinlay (1988) show that the variance ratio estimator can be calculated as (16) Σ= − = − − nk t k t t k P P k m k ( ˆ ) , 1 ( ) σ 2 μ 2 where = − + − nk k m k(nk k 1) 1 and (17) Σ= − − − − = nk t t t P P nk 1 2 1 2 ( ˆ ) , ( 1) 1 σ (1) μ in which Σ= − = − = − nk t t t nk P P nk P P nk 1 1 0 ( ) 1 ( ) 1 μˆ , 13 where P0 and Pnk are the first and last observations of the price series. Since futures returns have been shown to exhibit conditional heteroscedasticity (Yang and Brorsen 1993), we computed the asymptotic variance of the variance ratio, φ (k) , under heteroscedasticity. The standard normal test statistic (Lo and MacKinlay 1988) is (18) (0, 1) [ ( )] ( ) 1 ( ) 1/ 2 N k VR k Z k a→ − = φ where Σ= − = k j j k k k 1 2 ˆ( ) 2( 1) φ ( ) δ and 2 1 2 1 1 2 1 2 1 ( ˆ ) ( ˆ ) ( ˆ ) ˆ( ) − − − − − − = Σ Σ = − = + − − − − nk t t t nk t j t t t j t j p p p p p p j μ μ μ δ . Chow and Denning (1993) derived the joint period test where the null hypothesis is ( ) i VR k equals one for i =1,K, l . The test statistic can be written as (19) max ( ) 1 i i l ZV Z k ≤ ≤ = which asymptotically follows the studentized maximum modulus distribution (Stoline and Ury 1979) under the martingale null hypothesis. This study also conducts a variance ratio test using a new jackknife method because of possible nonnormality. Specially, we use a jackknife approach where each year is treated as a unit, so we delete each year of observations from the data set for each sample. Then, the jackknife estimate of kperiod variance ratio of futures price ( ) ~ θ k is defined in the usual manner. Let θ (k) be the kperiod variance ratio of futures prices and 14 ( ) ~ k i θ be the kperiod variance ratio when the ith year observations are deleted from the data set. Since we use 32 years data set from 1975 through 2006 for a mean reversion test, the jackknife estimate of kperiod variance ratio is calculated as the average of ( ) ~ k i θ (21) Σ= = 32 1 ( ) ~ 32 1 ( ) ~ i i θ k θ k . The jackknife estimate of the standard error of ( ) ~ θ k is (22) { } 1/ 2 32 1 2 ~ ( ) ~ ( ) ~ 32 31 ~ ( ) − = Σ= i i σ k θ k θ k θ . Then, the test statistic should be (23) ( 31) ~ ~ ~ ( ) ( ) 1 ~ ( ) = − = df t k k t k θ σ θ . This jackknife approach is similar to that used with the Agricultural Resource Management Survey (ARMS) and described by Dubman (2000). Results Table I−1 shows that achieves the highest average prices in all scenarios for all crops, followed by always hedging and selling at harvest. Table I−2 presents the results of the paired difference tests of average prices for the profit margin hedging and other strategies. The average prices of paired profit margin hedging and always hedging are not significantly different from zero at a 5% significance level in all scenarios for all crops except in no risk and yield risk scenarios for CBOT soybeans. The average prices of profit margin hedging and selling at harvest strategies are significantly different from 15 zero at the 5% significance level in all scenarios for all crops except in basis risk scenario for KCBT wheat. Table I−3 shows the expected utilities for the hedging strategies. The expected utilities of profit margin hedging are higher than the other strategies which confirm the theoretical findings. Always hedging has a higher expected utility than selling at harvest in all scenarios. The results of the paired difference tests of expected utilities for the profit margin hedging and other strategies are presented in table I−4. The expected utilities of paired profit margin hedging and always hedging are not significantly different from zero at a 5% significance level in all scenarios for all crops except the cases of no risk and yield risk for CBOT soybean. The expected utilities of profit margin hedging and selling at harvest are significantly different from zero at the 5% significance level except in basis risk, and yield and basis risk scenarios for KCBT wheat and basis risk scenarios for CBOT soybean. Thus, the expected utility results reflect that preharvest prices were higher than harvest prices during this time period. Table I−5 shows average prices and the expected utilities in the multiple crops scenario. In the case of multiple crops, the average prices of the independent profit margin hedging are the highest, followed by profit margin hedging, always hedging and selling at harvest, respectively. The expected utility of the independent profit margin hedging strategy is the highest followed by profit margin hedging, always hedging, and selling at harvest strategy, respectively. Table I−6 presents the results of the paired difference tests of average prices and expected utilities for multiple crops case. We do not find any evidence that the average prices and expected utilities of profit margin hedging and always hedging strategies are 16 significantly different from zero at a 5% significance level. However, the average prices and expected utilities of profit margin hedging and selling at harvest are significantly different from zero at the 5% significance level. The table also shows that the average prices and expected utilities of profit margin hedging and independent profit margin hedging is significantly different from zero at a 5% significance level. These results show that adding both price risk and yield risk reduces the expected utility of the profit margin hedging rule. Also, if producers grow multiple crops, the profit margin hedging rule would not be optimal even with a target utility function. Thaler’s (1980) mental accounting where producers consider their price risk and yield risk separately or divide their profits for each crop into separate pockets of money is proposed as a possible explanation of the popularity of profit margin hedging. The result of the variance ratio tests in tables I−7 and I−8 show that both Zstatistic and ZVstatistic values are not significantly different from 1.0 at the 5% significance level for all crops which means there is little evidence of mean reversion in futures prices for all crops. Table I−9 shows the result of the variance ratio test using the jackknife approach. None of the tstatistics show significant differences from one at the 5% significance level for all crops which confirms the results that there is little evidence of mean reversion in futures prices for all crops. Many of the estimated variance ratios are even greater than one (although insignificant) which would indicate trend following rather than mean reversion. If prices were trend following then using a technical analysis rule would be optimal. As shown in the theory section, profit margin hedging can be profitable if futures prices are mean reverting. However, our mean reversion test results show no evidence of 17 mean reversion in future prices. Furthermore, simulation results show that although profit margin hedging is more profitable than selling at harvest in most cases, there is little evidence that profit margin hedging is more profitable than always hedging except in a few cases. Two possible explanations have been offered for the profitability of preharvest hedging over this time period. One is that more buyers than sellers are wanting to lock in prices and so the buyers are paying a risk premium. If this hypothesis were correct, then the recent introduction of index funds (Sanders et al. 2008) would serve to make preharvest hedging even more attractive. The other hypothesis is that the market priced a small probability catastrophic event, which never happened during this time period. If this hypothesis were correct, adding the 2008 crop year might remove the apparent profitability of preharvest hedging. Summary and Conclusions Some extension economists and others often recommend profit margin hedging in choosing the timing of crop sales. This paper determines producer’s utility function and price processes where profit margin hedging is optimal. Profit margin hedging is an optimal strategy under a highly restricted target utility function even in an efficient market. Profit margin hedging can be profitable if prices are mean reverting. Simulations are conducted to compare the expected utility of profit margin hedging strategies with the expected utility of other strategies such as always hedging and selling at harvest. A variance ratio test is conducted to test for the existence of mean reversion in agricultural futures prices. The simulation results show that the expected utility of profit margin hedging is higher than always hedging and selling at harvest 18 strategies except in a few scenarios such as yield and basis risk for CBOT wheat and yield risk and yield and basis risk for CBOT corn. Therefore, this result suggests that the profit margin hedging would give the highest expected utility to producers in most cases under the specified utility function. However, if a producer grows multiple crops or considers both price risk and yield risk, the expected utility of profit margin hedging strategy could be reduced and not be optimal even with a target utility function. Mental accounting where producers consider their price risk and yield risk separately or divide their profits for each crop into separate pockets of money is proposed as a possible explanation of the popularity of profit margin hedging. The paired differences tests of average prices and expected utilities for the profit margin hedging and the other two strategies shows that, in most cases, both average prices and expected utilities of profit margin hedging strategies are not significantly different from those of always hedging strategies, but are higher than those of selling at harvest strategies except in some yield and basis risk scenarios. This may be the result of the time period being a time of unusually stable prices or it could be due to buyers being more eager to lock in prices than seller. With the variance ratio test, there is little evidence that futures prices of all crops follow a mean reverting process. The results of variance ratio test using jackknife approach confirm the result that there is insufficient evidence to conclude that futures prices of all crops follow a mean reverting process. Since we do not find evidence of mean reversion in futures prices and profit margin hedging is not more profitable than always hedging except in a few cases, we rely primarily on the theoretical proof using the shape of utility functions in figures I−1 19 through I−3 as the primary justification to argue that profit margin hedging can be the preferred strategy. 20 References Bond, G.E., and S.R. Thompson. 1985. “Risk Aversion and the Recommended Hedging Ratio.” American Journal of Agricultural Economics 67:870–72. Chen, S., C. Lee, and K. Shrestah. 2001. “On a MeanGeneralized Semivariance Approach to Determining the Hedge Ratio.” Journal of Futures Markets 21:581−98. Chow, K.V., and K. Denning. 1993 “A Simple Multiple Variance Ratio Test.” Journal of Econometrics 58:385−401. 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Accessed on 12 February 2008 Yang, S.R., and B.W. Brorsen. 1993. “Nonlinear Dynamics of Daily Futures Prices: Conditional Heteroskedasticity or Chaos?” The Journal of Futures Markets 13:175−91. Yoon, BS., and B.W. Brorsen. 2005. “Can Multiyear Rollover Hedging Increase Mean Returns?” Journal of Agricultural and Applied Economics 37:65−78. Zulauf, C.R., and S.H. Irwin. 1998. “Market Efficiency and Marketing to Enhance Income of Crop Producers.” Review of Agricultural Economics 20:308−31. 23 Notes 1. The variance of profit, 2 π σ , equals 2 2 (1 ) p − F σ where 2 p σ is variance of p . 24 Figure I−1. Expected utility of a crop producer as futures price 0 f p changes when α equals to β Hedge Expected Utility Hedge Ratio α = 0.5 β = 0.5 t = 3.37 3.43 0 = f p 3.40 0 = f p 3.34 0 = f p 3.31 0 = f p 25 Figure I−2. Expected utility of a crop producer as futures price 0 f p changes when α is less than β Expected Utility Hedge Ratio α = 0.1 β = 0.5 t = 3.37 3.43 0 = f p 3.40 0 = f p 3.34 0 = f p 3.31 0 = f p 26 Figure I−3. Expected utility of a crop producer as futures price 0 f p changes when α is greater than β Expected Utility Hedge Ratio α = 0.9 β = 0.5 t = 3.37 3.43 0 = f p 3.40 0 = f p 3.34 0 = f p 3.31 0 = f p 27 Table I−1. Average Prices (cents/bu) for Hedging Strategies at September 20th (19752005) Strategies Commodity Scenario Profit Margin Hedging Always Hedging Selling at Harvest KCBT Wheat No risk 330.10 319.78 311.03 Basis risk 327.13 319.78 311.03 Yield risk 329.11 318.24 311.03 Yield and basis risk 328.09 318.24 311.03 CBOT Wheat No risk 320.73 312.58 297.29 Basis risk 319.93 312.58 297.29 Yield risk 319.81 308.83 297.29 Yield and basis risk 318.99 308.83 297.29 CBOT Corn No risk 241.26 238.94 224.81 Basis risk 239.97 238.94 224.81 Yield risk 238.65 230.26 224.81 Yield and basis risk 237.32 230.26 224.81 CBOT Soybean No risk 631.02 605.66 581.52 Basis risk 618.94 605.66 581.52 Yield risk 626.50 597.92 581.52 Yield and basis risk 613.21 597.92 581.52 28 Table I−2. Paired Differences tRatios of Average Prices, (19752005) Paired Differences Commodity Scenario Profit Margin Hedging vs. Always Hedging Profit Margin Hedging vs. Selling at Harvest KCBT Wheat No risk 1.47 2.39* Basis risk 1.13 1.78 Yield risk 1.30 2.55* Yield and basis risk 1.26 2.19* CBOT Wheat No risk 1.23 2.88* Basis risk 1.10 2.79* Yield risk 1.37 2.88* Yield and basis risk 1.25 2.78* CBOT Corn No risk 0.47 3.16* Basis risk 0.13 2.86* Yield risk 1.10 3.19* Yield and basis risk 0.82 3.26* CBOT Soybean No risk 2.22* 4.22* Basis risk 1.43 2.45* Yield risk 2.26* 4.29* Yield and basis risk 1.47 2.16* Note: tcritical value with 30 degrees of freedom at 5% significance level is 2.042. * indicates significance at 5% level. 29 Table I−3. Expected Utilities for Hedging Strategies at September 20th (19752005) Strategies Commodity Scenario Profit Margin Hedging Always Hedging Selling at Harvest KCBT Wheat No risk 2.15 0.93 0.57 Basis risk 1.79 0.93 0.57 Yield risk 1.82 0.85 0.57 Yield and basis risk 1.74 0.85 0.57 CBOT Wheat No risk 1.94 0.73 0.81 Basis risk 1.75 0.73 0.81 Yield risk 1.87 0.55 0.81 Yield and basis risk 1.67 0.55 0.81 CBOT Corn No risk 0.48 0.25 2.28 Basis risk 0.03 025 2.28 Yield risk 0.20 0.86 2.28 Yield and basis risk 0.22 0.86 2.28 CBOT Soybean No risk 6.44 3.81 2.55 Basis risk 5.03 3.81 2.55 Yield risk 6.25 3.44 2.55 Yield and basis risk 4.75 3.44 2.55 Note: We use α = β = 0.5 as levels of risk preference below and above target. 30 Table I−4. Paired Differences tRatios of Expected Utilities, (19752005) Paired Differences Commodity Scenario Profit Margin Hedging vs. Always Hedging Profit Margin Hedging vs. Selling at Harvest KCBT Wheat No risk 1.64 2.51* Basis risk 1.46 1.49 Yield risk 1.28 2.47* Yield and basis risk 1.40 1.81 CBOT Wheat No risk 1.71 3.07* Basis risk 1.36 2.92* Yield risk 1.78 3.13* Yield and basis risk 1.41 2.94* CBOT Corn No risk 1.19 3.64* Basis risk 0.39 3.17* Yield risk 1.51 3.75* Yield and basis risk 0.91 2.89* CBOT Soybean No risk 2.42* 4.43* Basis risk 1.52 1.84 Yield risk 2.44* 4.38* Yield and basis risk 1.54 1.63 Note: tcritical value with 30 degrees of freedom at 5% significance level is 2.042. * indicates significance at 5% level. 31 Table I−5. Average Prices and Expected Utilities for Hedging Strategies for Multiple Crops Scenario (19752005) Strategies Item Profit Margin Hedging Independent Profit Margin Hedging Always Hedging Selling at Harvest Average Prices ($/bu) 75.46 76.46 75.39 70.53 Expected Utilities 29.57 63.99 20.99 10.58 Note: The dates of decision making are September 20th for CBOT wheat and May 10th for CBOT soybean and corn. We use α = β = 0.5 as levels of risk preference below and above target. Table I−6. Paired Differences tRatios of Average Prices and Expected Utilities for Multiple Crops Cases (19752005) Paired Difference Item Profit Margin Hedging vs. Always Hedging Profit Margin Hedging vs. Selling at Harvest Profit Margin Hedging vs. Independent Profit Margin Hedging Average Prices 0.07 3.47* 2.42* Expected Utilities 0.98 3.66* 2.99* Note: tcritical value with 30 degrees of freedom at 5% significance level is 2.042. * indicates significance at 5% level. 32 Table I−7. Variance Ratio Tests for Futures Prices (19752006) Commodity Return Horizon (kdays) Variance Ratio Zstatistic KBCT Wheat 2 1.020 0.892 5 0.984 0.232 10 0.977 0.200 20 0.938 0.377 CBOT Wheat 2 1.007 0.357 5 0.952 0.738 10 0.922 0.842 20 0.848 1.056 CBOT Corn 2 1.033 1.857 5 1.046 0.547 10 1.048 0.510 20 1.094 0.643 CBOT Soybean 2 1.001 0.068 5 0.995 0.091 10 0.963 0.398 20 0.991 0.068 Note: July observations are deleted. Standard normal distribution Z at 5% significance level is 1.96. Table I−8. Joint Variance Ratio Tests for Futures Prices (19752006) Commodity VR(2) ZV KCBT Wheat 1.020 0.892 CBOT Wheat 1.006 1.056 CBOT Corn 1.033 1.857 CBOT Soybean 1.001 0.431 Note: July observations are deleted. Studentized maximum modulus distribution with 20 and infinity degree of freedom at 5% significance level is 3.643. 33 Table I−9. Variance Ratio Tests for Futures Prices Using Jackknife Approach(1975 2006) Commodity Return Horizon (kdays) Variance Ratio tstatistic KBCT Wheat 2 1.024 1.148 5 1.000 0.004 10 1.013 0.241 20 1.012 0.153 CBOT Wheat 2 1.010 0.571 5 0.969 1.130 10 0.944 1.528 20 0.898 1.650 CBOT Corn 2 1.036 1.723 5 1.056 1.986 10 1.069 1.532 20 1.147 1.996 CBOT Soybean 2 1.005 0.270 5 1.011 0.399 10 0.999 0.037 20 1.071 1.128 Note: July observations are deleted. tcritical value with 30 degrees of freedom at 5% significance level is 2.042. * indicates significance at 5% level. 34 II. CHAPTER II CAN REAL OPTION VALUE EXPLAIN WHY PRODUCERS APPEAR TO STORE TOO LONG? Introduction Some studies show that producers store longer than is profitable (Anderson and Brorsen 2005; Hagedorn et al. 2005). One possibility is that producers store crops longer than makes economic sense due to myopic loss aversion, which means that producers get more disutility from a loss than they get utility from receiving an equally sized gain. An alternative explanation results from producers’ decisions to sell grain being irreversible. Fackler and Livingston (2002) show that this irreversibility can create a real option value from waiting to sell grain. The key to generating a real option value is for prices to follow a mean reverting process. In the case of grain as considered by Fackler and Livingston (2002), if grain prices are low it makes sense to wait to sell because prices will revert to the mean. If prices are unusually high, it is best to sell. If prices are near the mean, there can be a real option value from waiting because there is the opportunity to wait and select a time to sell when prices are higher than currently. There are some recent studies that implement real options in agriculture. Purvis et al. (1995) examine the technology adoption of freestall dairy housing under 35 irreversibility and uncertainty and find that there can be a return to waiting to adopt in some cases. Ekboir (1997), WinterNelson and Amegbeto (1998), and Khanna et al. (2000) also used real options to analyze the investment decision of producers under uncertainty. This research focuses on answering the question, “Can real option values explain why producers appear to store too long?” To answer this question, this study first models and estimates the price process. The model attempts to capture two important features of agricultural commodity prices: mean reversion and seasonality. The price process which is modeled in this study differs from Fackler and Livingston (2002). The price process used here allows price to be a random walk within a season, but mean reverting across crop years. After estimating the price process, a universal lattice model is used to determine the cutoff price at which the producer is indifferent between selling and holding a crop. Simulations using cash prices of wheat, corn, and soybean are used to determine net returns under two different price processes, which is simple mean reversion and the new seasonal mean reversion price process. This empirical work shows that real option values cannot explain why producers appear to store too long. Theory A producer who holds stocks can be viewed as holding an American option since the producer has the option to sell at any time. The optimal storage problem is equivalent to the optimal stopping problem of an American call option which is exercised at the current price. If selling stock is irreversible the producer does not just hold stocks but holds stocks and a call option which can be exercised at the current price. An American 36 option is an optimal stopping problem of determining the optimal time to exercise an option. The decision to exercise the option is the same as with financial options. The option holder exercises the option, whenever its intrinsic value, which is the value of immediately exercising the option, is greater than its total value. Because of the early exercise possibility, American options are solved as a dynamic programming problem. Our derivation is based on risk neutral valuation rather than riskless arbitrage as in Black and Scholes (1973). The typical American call option under risk neutral evaluation can be expressed in terms of a value function, t V : (2.1) [max(0, ( ) )] rh t h h T t t V E p X e− + ∈ − = − where t h p + is the price of the underlying asset at time t + h , r is the riskfree interest rate, and X is the exercise price. The optimal storage problem differs from (2.1). First, holding stocks of a commodity incurs positive holding charges, whereas holding an option does not incur holding cost. Second, the exercise price of the optimal storage problem is current market price, which is not discounted as in (2.1). Finally, the storage problem has an initial price which is the current cash price whereas a usual American option does not have an initial value and so the option value is zero if the option is not exercised. Then, the value function of the optimal storage problem can be defined as (2.2) [max(0, )] [max( , )] ( ) (r s)h t t h h T t t r s h t h h T t t t V p E p e p E p p e − + + ∈ − − + + ∈ − = + − = where t p is cash price of a commodity at time t , T is the expiration date, and s is a per period storage cost which is a percentage of price. The producer sells stocks under the condition that 37 (2.3) [max( )] (r s)h t h h T t t p E p e − + + ∈ − ≥ That is the producer sells stocks whenever the expected return to store and sell at time t + h is less than or equal to the current market price. Data The chosen agricultural commodities are corn, soybeans and wheat. Thursday cash prices of South Central Illinois corn and soybean data from the National Agricultural Statistics Service (NASS) of the United States Department of Agriculture (USDA) are obtained from a computer database compiled by Farmdoc, University of Illinois at UrbanaChampaign (2008). Thursday cash prices of wheat at Medford, Oklahoma, are obtained from the Oklahoma Market Reports of USDA. The sample period extends from October 1975 through September 2007 for corn and soybeans, and from June 1975 through May 2007 for wheat. These primary data have some missing values for Thanksgiving and Christmas season. For these missing data, the most recently observed data are used. Annual state average prices from the National Agricultural Statistics Service (NASS) are obtained from the United States Department of Agriculture (USDA) website (2008). To estimate the price processes, 5year moving averages of annual average prices for each crop are used as mean prices. Corn and soybean storage costs from 1995 through 2004 are from Irwin et al. (2006). We calculate the previous 20 years of storage costs from 1975 to 1994 using producer price index from website of United States Department of Labor, and assume that storage costs of 2005 and 2006 equal the cost of 2004. Storage costs of wheat from 1975 38 to 2006 are obtained from Oklahoma Cooperative Extension Service at Oklahoma State University. The interest cost is calculated at the prime rate for that year plus 2%. The prime rate is the prime charged by banks in June for that year, quoted from the Kansas City Federal Reserve Bank (2008). Procedures Three main procedures are used: estimation of price process parameters, determining cutoff price, and simulation of the trading rules. A universal lattice model (Chen and Yang 1999) and discrete stochastic dynamic programming are used to determine cutoff prices. Estimation of Price Process Parameters The model of prices used here attempts to capture two important features of agricultural commodity prices, mean reversion and seasonality. A number of studies documented mean reversion in commodity cash prices (Brennan 1991; Lence et al. 1993; Dixit and Pindyck 1994; Bessembinder et al.1995; Wang and Tomek 2007). Also, some other studies have found that futures prices follow a near random walk within a contract month (Bessler and Covey 1991; Yoon and Brorsen 2005), but are mean reverting when prices across multiple contract months are used (Schroeder and Goodwin 1991). Seasonality in the mean level of price has been also well documented in commodity. For example, prices of seasonally produced goods tend to rise during the marketing season to cover the cost of storage. Price process in this research is not focused on seasonal volatility but on seasonal mean reversion. While seasonal volatility 39 is statistically significant due to the large sample size, it is relatively small and is not included here to simplify the model. A price process model which represents mean reversion can be described by (2.4) t t t t t t p − p = a + β p − p +ε ln ln − (ln ln ) 1 where t p indicates the cash price at time t, t p is the seasonal mean price, t a is a seasonal function, t represents number of weeks after harvest, β is a parameter to be estimated, and t ε is a normally distributed error term with zero mean and constant variance σ 2 . We allow prices to follow a random walk within a season, but to be mean reverting across crop years. Such a price process can be rewritten as if 0 t < t (2.5) + − + + − = − t t t t t t t t a p p a p p β ε ε (ln ln ) ln ln 1 if 0 t ≥ t where 0 t is a time within a season when the mean reverting process begins. Equation (2.5) imposes a random walk with drift in the early part of the storage season. This assumption is tested by estimating a more general model: if 0 t < t (2.6) + + − + + − + − = − t t t t t t t t t t a p p a p p p p α β ε α ε ( )(ln ln ) (ln ln ) ln ln 1 if 0 t ≥ t Restricting α to be zero gives equation (2.5). We find no evidence of differences among fourth, fifth, and sixth power polynomial functional forms. Visual inspection of a fifth power polynomial seasonality function suggests that it is more realistic than the other powers or a sinusoidal function. Therefore, we adopt a fifth power polynomial functional form for the seasonal function, t a , which is 40 (2.7) Σ= = 5 i 0 i t ia γ t where the γ s are the parameters to be estimated. If we impose a continuity restriction on the seasonal function a(t) then the change of seasonality at harvest in the current year is equivalent to the change of seasonality at harvest next year. Since this study uses weekly cash price data, we can impose a continuity condition, 0 52 a = a . Using (2.7) this can be rearranged as (2.8) 52 (52) 5 2 1 Σ= − = i i i γ γ and then, 1 γ can be obtained by other estimated parameters. Equation (2.6) is estimated using cash prices of three crops – wheat, corn, and soybean and the coefficient α is not significantly different from zero (table II−1). Therefore, α is restricted to be zero and then (2.6) can be rewritten as if 0 t < t (2.9) + − + + − = − t t t t t t t a p p a p p β ε ε (ln ln ) ln ln 1 if 0 t ≥ t , and then we can also define the simple mean reversion price process as (2.10) t t t t t p − p = a + β p − p +ε ln ln − (ln ln ) 1 . The specified value of 0 t can be determined by substituting numerical values from 0 to 51 for 0 t and selecting the one which gives the highest log likelihood value. Since this study uses a 5 year moving average as mean price p , the model is not stationary. If cost of production data had been available to use instead of the 5year moving average, the model would be stationary. The standard errors on the computer print out in this case are conditional 41 standard errors and are valid conditional on the true value of 0 t being selected and no standard errors are provided for 0 t . Therefore, a nonparametric bootstrap is used to obtain estimates of standard errors of 0 t . Ten thousand samples of size 1,738 for wheat, 1,639 for corn, and 1,637 for soybean are resampled and used to estimate the parameters. A Universal Lattice Model While the terms used in the option pricing literature are quite different than the dynamic programming terminology used by Fackler and Livingston (2002), the approaches are equivalent in that pricing an American option requires solving a stochastic dynamic program. There are many models for pricing options. Black and Scholes (1973) developed an option pricing model for European options. Cox et al. (1979) developed the binomial option pricing lattice which is widely used within finance to price American type options as it is easy to implement and handles American options relatively well. However, the binomial model assumes that the option price can just either go up or down over a time step. It does not assume that the price may remain unchanged. In 1996, Boyle introduced the trinomial option pricing model, which is similar to the binomial method in that it employs a lattice type method for pricing options. The trinomial method is more accurate than the binomial one and gives the same results as the binomial one with a fewer steps. In the trinomial lattice, the branches are up, flat, and down by an increment of change in underlying value 0p . That is, (2.11) p p p i t i t = + 0 3, , , 42 i t i t p p 2, , , = p p p i t i t = − 0 1, , , Figure 1 shows an example trinomial lattice. The branches are down, flat, and up with the risk neutral probabilities 1 R , 2 R , and 3 R , respectively, which satisfy the following three equations (2.12) i t i t i t i t i t i t i t i t R p R p R p p 1, , 1, , 2, , 2, , 3, , 3, , , , + + = +μ 2 , 2 , , 2 3, , 3, , 2 2, , 2, , 2 1, , 1, , ( ) ( ) ( ) ( ) i t i t i t i t i t i t i t i t i t R p + R p + R p − p +μ =σ 1 1, , 2, , 3, , + + = i t i t i t R R R where i t p , is the ith node of p at time t, n i t p , , is the nth lowest possible node at time t + 0t , and i,t μ and 2 ,t i σ are the expected change and the variance of i t p , during the next time interval 0t , respectively. However, in the trinomial lattice, if there is mean reversion in the process, the risk neutral probabilities of all nodes in the lattice could be negative. To solve this problem, Hull and White (1990) propose four alternative branching schemes. These alternatives include the branches of the lattice to go three ups, two ups, and one up; two ups, one up, and flat; flat, one down, and two downs; and one down, two downs, and three downs. Chen and Yang (1999) argue that in the alternative trinomial lattice there seems to be no consistent way to construct the lattice in which all probabilities are guaranteed to be positive. Thus, they extend Hull and White’s (1990) model and propose a general form of alternative branching schemes. This study uses Chen and Yang’s (1999) universal lattice model to determine real option value. With Chen and Yang’s lattice model, the three branches can be written as (2.13) p p j k p i t i t = + ( + )0 3, , , 43 p p j p i t i t = + ( )0 2, , , p p j k p i t i t = + ( − )0 1, , , where the variable j and k provide flexibility for the branches to yield nonnegative probabilities with any level of mean and variance, respectively. With this branching method, the risk neutral probabilities can be obtained solving (2.12) and then the results are (2.14) 2 2 2 , , , 1, , 2 ( )(( ) ) k p j p j k p R i t i t i t i t 0 0 − + 0 − + = μ μ σ 2 2 2 , 2 , 2, , ( ) 1 k p j p R i t i t i t 0 − 0 + = − μ σ i t i t i t R R R 3, , 1, , 2, , =1− − . To guarantee the convergence of the model, the constraints of 0 1 , , ≤ ≤ n i t P translate into the following two sets of sufficient conditions: (2.15) p k p i t i t 0 ≤ ≤ 0 , , σ 2σ and p k p j p k p i t i t i t i t 0 + 0 − ≤ ≤ 0 − 0 − 2 , 2 2 , 2 , 2 2 , μ σ μ σ and (2.16) p k i t 0 > , 2σ and p k p p k j p k p p i t i t i t i t 0 0 − − 0 + − ≤ ≤ 0 0 − − 0 2 , 2 2 , 2 , 2 2 , 2 μ σ μ σ or 44 p k p p k j p k p p k i t i t i t i t 0 0 − − 0 ≤ ≤ + 0 0 − + 0 + − 2 , 2 2 , 2 , 2 2 , 4 2 4 2 μ σ μ σ or p k p p j p k p p k i t i t i t i t 0 0 − + 0 ≤ ≤ 0 0 − + 0 + 2 , 2 2 , 2 , 2 2 , 4 2 μ σ μ σ . Since this study assumes constant volatility, which means k = 1, the risk neutral probabilities and the sets of sufficient conditions for constraints of 0 1 , , ≤ ≤ n i t P can be rewritten as (2.17) 2 2 , , 1, , 2 ( )(( 1) ) p j p j p R i t i t i t 0 0 − + 0 − + = μ μ σ 2 2 2 , 2, , ( ) 1 p j p R i t i t 0 − 0 + = − μ σ i t i t i t R R R 3, , 1, , 2, , =1− − and (2.18) p k p 0 ≤ ≤ 0 σ 2σ and p p j p p i t i t 0 + 0 − ≤ ≤ 0 − 0 − 2 2 , 2 2 , μ σ μ σ . A summary of this procedure is that 0p and 0t are chosen, and then the variable j is chosen using equation (2.18). After that, the risk neutral probabilities are obtained from equation (2.17). Finally, as mentioned in the theory section, since the optimal storage problem is equivalent to an American call option, using equation (2.2) and the value function of optimal storage problem can be determined as 45 (2.19) [max{( ) , 0}] ( ) 1, 1, 2, 2, 3, 3, t r s h t h t h t h t h t h t h h T t t V = E R p + R p + R p e− + − p + + + + + + ∈ − Then the cutoff price, which is the current market price where producer sells stocks since the expected return to store and sell at time t + h is less than or equal to the current market price, can be determined by (2.20) [ : {max( ) }] ( ) 1, 1, 2, 2, 3, 3, r s h t h t h t h t h t h t h h T t t t t C p p E R p R p R p e − + + + + + + + ∈ − = = + + We use a universal lattice model on a grid of values in p and t to determine a cutoff price. The value function is computed for each time period beginning with 52 to 1. To determine a cutoff price, we use the average cash price at harvest over the 32year period as an initial value for each crop ($3.23 for wheat, $2.38 for corn, and $5.99 for soybeans), and assume that price increments are 15 cents for corn, 38 cents for soybean, and 20 cents for wheat. Selling at harvest is the expected profit maximizing strategy when full storage and interest costs are used. Hagedorn et al. (2005) also show that selling at harvest is the best strategy when they use full storage and interest costs. However we also consider lower costs of half of the storage and interest costs as well as full storage and interest costs to determine cutoff price. Some producers are net lenders and have their own storage (so only marginal costs would affect the decision), so such lower costs are relevant for some producers. Simulation Simulations are conducted to determine net returns of the optimal strategy under two different price process: mean reversion and seasonal mean reversion. For the simulations, we design two different scenarios which depend on the level of storage and 46 interest costs. One scenario includes full storage and interest costs and another one includes half of storage and interest cost. Using equation (2.19), the simulations are conducted with weekly cash price data for corn, soybean, and wheat to find the first selling date of crops and value of selling the crop. Net returns for each crop year for each crop were computed using the first price that exceeds the specified cutoff price function. The net returns are computed as the value of sales less stockholding costs, discounted to the harvest time (2.21) ( ) r (T t ) T r T t T p e sTp e π = − − − − − where T is the sales date, t is the first date of the marketing season assumed to be the first weekday in June for wheat and the first weekday in October for corn and soybean, and s is a per period storage cost that is a percentage of price (table II−2). Results The estimated nonparametric bootstrap parameters of the seasonal mean reversion process are presented in table II−3. Mean reversion occurs late July with 3.6% weekly for corn, mid or late July with 4.2% weekly for soybean, and early or mid March with 2.3% weekly for wheat. That is, the total percentages of mean reversion for a marketing year are 28.5% for corn, 41.6% for soybean, and 32.2% for wheat. Figures II−2 through II−4 show the shapes of seasonality for corn, soybean, and wheat, respectively. After harvest, prices for corn and soybeans rapidly increase until the beginning of December and then slowly decrease. For wheat, prices also increase rapidly after harvest until early August and then slowly decrease. These seasonal price changes turn negative in early June for corn, early July for soybeans, and early March for wheat. 47 Thus, the seasonal function turns negative before mean reversion begins. This clearly indicates that producers will be selling before mean reversion begins (as most of them do), so real option values do not explain why producers appear to store too long. Optimal cutoff prices are illustrated in figures II−5 through II−10. The shapes of the graphs of the model which uses a mean reversion price process are very different from the model using a seasonal mean reversion price process. Only at extremely low prices is there ever an incentive to store to the point where seasonal mean reversion begins. Since producers would rationally sell before mean reversion begins, the real option value almost always disappears. This result contrasts with Fackler and Livingston’s (2002) model, which is that if grain prices are near the mean there can be a real option valuing from waiting to sell because there is the opportunity to wait and select a time to sell when prices are unusually high. They conclude that irreversibility confers an additional return in the form of an option to sell stocks in the future. This finding of a large real option value that can explain why producers appear to store too long is not supported. The results of simulations for corn, soybeans, and wheat are presented in tables II−4, II−5 and II−6 respectively. The difference of average net returns over the 32 years between the mean reversion model and the seasonal mean reversion model is small and the result of paired difference tests in table II−7 shows that all t values are not significant at the 5% level except a case that include full storage and interest cost for corn. Therefore, we can conclude that there is little evidence that, for most scenarios, the net returns over 32 years between the mean reversion model and the seasonal mean reversion model are different. As Brorsen and Irwin (1996) argue, statistical insignificance is a 48 typical result of marketing strategy simulation studies. The difference between marketing strategies is usually small, the variation is high, and with only one observation per year, the number of observations is small. Therefore, we rely primarily on the results in table II−3 in reaching the conclusion that the seasonal mean reversion model is preferred. Summary and Conclusion Previous studies suggest that producers tend to store crops longer than is profitable (Anderson and Brorsen 2005). Since decisions to sell are irreversible, there can be a real option value from waiting to sell grain. This research focuses on determining whether real option values can explain longer storage We estimate a new seasonal mean reversion price process using a nonparametric bootstrap rather than estimating a simple mean reversion price process. After estimating the price process, a cutoff price at which the producer is indifferent between selling and holding the crop is determined using a universal lattice model. Simulations are conducted to determine net returns under simple mean reversion and the new seasonal mean reversion price process. The estimated nonparametric bootstrap parameters of the seasonal mean reversion process show that mean reversion occurs mid or late July for corn, early July for soybean, and early March for wheat. The shapes of seasonality show that the seasonal function turns negative before mean reversion begins, which suggests that real option values are relatively unimportant in determining when producers sell their grain. The graphs of cutoff price when assuming a seasonal mean reversion price process show that producers sell before mean reversion begins except when prices are 49 extremely low. This result contrasts with Fackler and Livingston’s (2002) conclusion that irreversibility confers an additional return in the form of an option to sell stocks in the future. Therefore their finding of a large real option value that can explain why producers store too long is not supported. The simulation results represent that the difference of average net returns over the 32 years between the mean reversion model and the seasonal mean reversion model is very small and the result of paired difference tests conclude that there is little evidence that the net returns over 32 years between the mean reversion model and the seasonal mean reversion model are different. Based on the nonparametric bootstrap estimation of price process, we can conclude that the seasonal mean reversion model is preferred. 50 References Anderson, R.W. 1985. “Some Determinants of the Volatility of Futures Prices.” The Journal Futures Markets 5:331–48. Anderson, R.W., and J.P. Danthine. 1983. “The Time Pattern of Hedging and the Volatility of Futures Prices.” The Review of Economic Studies 50:249–66. Anderson, K.B., and B.W. Brorsen. 2005. “Marketing Performance of Oklahoma Farmers.” American Journal of Agricultural Economics 87:1265–71. Bessembinder, H., J.F. Coughenour, P.J. Seguin, and M.M. Smoller. 1995. “Mean Reversion in Equilibrium Asset Prices: Evidence from the Futures Term Structure.” The Journal of Finance 50:361–75. Bessler, D.A., and T. Covey. 1991. “Cointegration: Some Results on U.S. Cattle Prices.” The Journal of Finances Markets 11:461–74. Black, F., and M. Scholes. 1973. “The Pricing of Option and Corporate Liabilities.” Journal of Political Economics 81: 637−59. Brennan, M.J. 1991. “The Price of Convenience and the Valuation of Commodity Contingent Claims.” Stochastic Models and Option Values. D. Lund and B. Oksendal, eds., pp. 33–71. New York: NorthHolland Publishing Co. Brennan, M.J., and E.S. Schwartz. 1985. “Evaluating Natural Resource Investments.” The Journal of Business 58:135–57. Brorsen, B.W., and S.H. Irwin. 1996. “Improving the Relevance of Research on Price Forecasting and Marketing Strategies.” Agricultural and Resource Economics Review. 25:68–75. Chen, R.R., and T.T. Yang. 1999. “A Universal Lattice.” Review of Derivatives Research 3:115–33. Cox, D.R., and H.D. Miller. 1965. The Theory of Stochastic Processes. New York: John Wiley and Sons. 51 Cox, J.C., S.A. Ross, and M. Rubinstein. 1979. “Option Pricing: A Simplified Approach.” Journal of Financial Economics 7:229−64. Dixit, A.K., and R.S. Pindyck. 1994. Investment Under Uncertainty. Princeton, NJ: Princeton University Press. Ekboir, J.M., 1997. “Technical Change and Irreversible Investment Under Risk.” Agricultural Economics 16:54–65. Fackler, P.L., and M.J. Livingston. 2002. “Optimal Storage by Crop Producers.” American Journal of Agricultural Economics 84:645–59. Fama, E.F., and K.R. French. 1987. “Commodity Futures Prices: Some Evidence on Forecast Power, Premiums, and the Theory of Storage.” The Journal of Business 60:55–73. Farmdoc, University of Illinois at UrbanaChampaign. 2008. Available at http://www.farmdoc.uiuc.edu/marketing/cash/CashTableChart.asp. Accessed on 15 April 2008. Hagedorn, L.A., S.H. Irwin, D.L. Good, and E.V. Colino. 2005. “Does the Performance of Illinois Corn and Soybean Farmers Lag the Market?” American Journal of Agricultural Economics 87: 1271–79. Hull, J., and A. White. 1990. “Valuing Derivative Securities Using the Explicit Finite Difference Method.” Journal of Financial and Quantitative Analysis 25:87–100. Hull, J., and A. White. 1993. “OneFactor InterestRate Models and the Valuation of InterestRate Derivative Securities.” Journal of Financial and Quantitative Analysis 28:235–54. Irwin, S.H., D.L. Good, and J.MartinesFilho and R.M. Batts. 2006. “The Pricing Performance of Market Advisory Services in Corn and Soybeans Over 1995 2004.” AgMAS Project Research Report 200602. Kansas City Federal Reserve. 2008. “Bank Prime Loan.” Available at http://www.federalreserve.gov/releases/h15/data/Monthly/H15_PRIME_NA.txt Kenyon, D., K. Kling, J. Jordan, W. Seale, and N. McCabe. 1987 “Factors Affecting Agricultural Futures Price Variance.” The Journal of Futures Markets 7:73–92. Khanna, M., M. Isik, and A. WinterNelson. 2000. “Investment in SiteSpecific Crop Management Under Uncertainty: Implications for Nitrogen Pollution Control and Environmental Policy.” Agricultural Economics 24:9–21. 52 Lence, S., K. Kimle, and M.L. Hayenga. 1993. “A Dynamic Minimum Variance Hedge.” American Journal of Agricultural Economics 75:1063–71. McDonald, R.L., and D.R. Siegel. 1986. “The Value of Waiting to Invest.” The Quarterly Journal of Economics 101:707–27. Oklahoma Cooperative Extension Service, Oklahoma State University, 2007. Electronic database, Stillwater, OK. Pruvis, A., W.G. Boggess, C.B. Moss, and J. Holt. 1996. “Technology Adoption Decisions Under Irreversibility and Uncertainty: An Ex Ante Approach.” American Journal of Agricultural Economics 78:541–51. Schroeder, T.C., and B.K. Goodwin. 1991. “Price Discovery and Cointegration for Live Hogs.” The Journal of Futures Markets 11:685–696. Streeter, D.H., and W.G. Tomek. 1992. “Variability in Soybean Futures Prices: An Integrated Framework.” The Journal of Futures Markets 12:705–30. Tronstad, R., and C.R. Taylor. 1991. “Dynamically Optimal AfterTax Grain Storage, Cash Grain Sale, and Hedging Strategies.” American Journal of Agricultural Economics 73:75–88. U.S. Department of Agriculture (USDA), National Agricultural Statistics Service (NASS). 2008. Available at http://www.nass.usda.gov/QuickStats/. Accessed on 12 February 2008. U.S. Department of Agriculture (USDA), Oklahoma Market Reports. 1975−2007. U.S. Department of Labor, Bureau of Labor Statistics. (2008) Producer Price Index. Available at http://data.bls.gov/cgibin/surveymost. Accessed on 30 June 2008. WinterNelson, A., and K. Amegbeto, 1998. “Option Values to Conservation and Agricultural Price Policy: Application to Terrace Construction in Kenya” American Journal of Agricultural Economics 80:409–18. Yoon, B.S., and B.W. Brorsen. 2005. “Can Multiyear Rollover Hedging Increase Mean Returns?” Journal of Agricultural and Applied Economics 37:65−78. 53 Table II−1. Parameter Estimateα of Seasonal Mean Reversion Price Process 2 Loglikelihood Commodity α Unrestricted Restricted Likelihood Ratio Test Statistic Corn 0.0057 6275.35 6273.31 2.04 Soybean 0.0021 6540.40 6539.81 0.59 Wheat 0.0005 6772.46 6772.45 0.01 Note: 2 1 χ critical value at 5% significance level is 3.841. Estimated unrestricted model is t < t0 , + + − + + − + − − = ( ) ( )(ln ln ( )) ( ) ( ) (ln ln ( )) ( ) ln ( ) ln ( 1) a t p p t t a t p p t t p t p t α β ε α ε 0 t ≥ t . Estimated restricted model is 0 t < t , + − + + − − = ( ) (ln ln ( )) ( ) ( ) ( ) ln ( ) ln ( 1) a t p p t t a t t p t p t β ε ε 0 t ≥ t . 54 Table II−2. Per Period Storage Costs (Percentage of Price) Commodities Year Corn and Soybean Wheat 1975 0.0021 0.0035 1976 0.0022 0.0035 1977 0.0024 0.0042 1978 0.0026 0.0049 1979 0.0028 0.0049 1980 0.0032 0.0053 1981 0.0036 0.0056 1982 0.0038 0.0056 1983 0.0039 0.0060 1984 0.0041 0.0060 1985 0.0042 0.0060 1986 0.0043 0.0060 1987 0.0044 0.0060 1988 0.0046 0.0060 1989 0.0048 0.0060 1990 0.0051 0.0060 1991 0.0053 0.0060 1992 0.0055 0.0060 1993 0.0057 0.0060 1994 0.0058 0.0060 1995 0.0060 0.0060 1996 0.0060 0.0060 1997 0.0060 0.0060 1998 0.0060 0.0060 1999 0.0060 0.0060 2000 0.0060 0.0060 2001 0.0060 0.0070 2002 0.0060 0.0070 2003 0.0060 0.0070 2004 0.0060 0.0070 2005 0.0060 0.0070 2006 0.0060 0.0070 55 Table II−3. Parameter Estimates of Seasonal Mean Reversion Price Process by Nonparametric Bootstrapping Corn Soybean Wheat Coefficient Standard Deviation Coefficient Standard Deviation Coefficient Standard Deviation α 0.0356 0.0117 0.0416 0.0194 0.0230 0.0115 0 γ 0.0060 0.0034 0.0037 0.0029 0.0078 0.0037 2 γ 5.7E04 1.3E04 2.7E04 1.4E04 1.4E04 1.5E04 3 γ 2.8E05 6.8E06 1.3E05 7.3E06 2.5E06 7.0E06 4 γ 5.8E07 1.5E07 2.7E07 1.6E07 2.9E09 1.5E07 5 γ 4.4E09 1.2E09 2.1E09 1.2E09 2.3E10 1.1E09 σ 2 0.0013 8.0E05 0.0011 5.9E05 0.0012 6.5E05 0 t 44 6.1469 42 7.2619 38 10.8677 Note: Estimated model is if 0 t < t , + − + + − = + + + 1 1 1 ( ) (ln ln ) ( ) ln ln t t t t t g t p p f t p p α ε ε if 0 t ≥ t . 56 Table II−4. Sales Dates and Net Returns for Corn Sale Dates (Weeks from Harvest) Per Bushel Net Returns ($/bu) Scenario 1 Scenario Year 2 Scenario 1 Scenario 2 Model 1 Model 2 Model 1 Model 2 Model 1 Model 2 Model 1 Model 2 1975 0 17 0 32 2.61 2.31 2.61 2.46 1976 15 17 20 32 2.27 2.17 2.24 2.05 1977 27 17 33 32 2.13 1.91 2.15 2.12 1978 25 16 32 19 2.03 1.98 2.19 2.03 1979 0 0 0 17 2.62 2.62 2.62 2.26 1980 0 0 0 16 3.05 3.05 3.05 3.08 1981 0 0 0 0 2.38 2.38 2.38 2.38 1982 0 0 15 0 2.00 2.00 2.14 2.00 1983 0 0 0 16 3.41 3.41 3.41 3.00 1984 0 0 17 15 2.68 2.68 2.46 2.46 1985 15 0 33 16 2.15 2.10 2.10 2.22 1986 34 0 37 0 1.41 1.42 1.50 1.42 1987 18 0 35 16 1.68 1.63 1.81 1.74 1988 0 0 0 16 2.68 2.68 2.68 2.36 1989 0 0 0 15 2.27 2.27 2.27 2.12 1990 0 0 0 15 2.19 2.19 2.19 2.19 1991 0 0 0 15 2.42 2.42 2.42 2.31 1992 0 0 25 16 2.06 2.06 1.96 1.92 1993 0 0 13 16 2.23 2.23 2.74 2.66 1994 0 0 17 15 1.95 1.95 2.07 2.08 1995 0 0 0 15 2.94 2.94 2.94 3.21 1996 0 0 0 15 2.90 2.90 2.90 2.49 1997 0 0 15 15 2.45 2.45 2.46 2.46 1998 14 0 34 15 1.85 1.81 1.68 1.90 1999 15 0 31 15 1.74 1.77 1.91 1.82 2000 12 0 35 14 1.88 1.63 1.45 1.95 2001 0 0 31 15 1.85 1.85 1.67 1.82 2002 0 0 0 16 2.46 2.46 2.46 2.12 2003 0 0 14 16 2.03 2.03 2.26 2.43 2004 0 0 33 16 1.76 1.76 1.68 1.72 2005 0 0 18 16 1.67 1.67 1.88 1.78 2006 0 0 0 15 2.41 2.41 2.41 3.30 32 year average 2.25 2.22 2.27 2.25 Note: Scenario1 includes storage and interest costs. Scenario2 includes half of storage and interest costs. Model1 assumes that price follows a mean reversion process. Model2 assumes that price follows a seasonal mean reversion process. 57 Table II−5. Sales Dates and Net Returns for Soybean Sale Dates (Weeks from Harvest) Per Bushel Net Returns ($/bu) Scenario 1 Scenario Year 2 Scenario 1 Scenario 2 Model 1 Model 2 Model 1 Model 2 Model 1 Model 2 Model 1 Model 2 1975 0 0 0 34 5.15 5.15 5.15 5.02 1976 0 0 0 29 5.95 5.95 5.95 9.37 1977 11 0 25 34 5.53 5.05 6.37 6.27 1978 0 0 0 0 6.17 6.17 6.17 6.17 1979 0 0 0 0 6.79 6.79 6.79 6.79 1980 0 0 0 0 7.51 7.51 7.51 7.51 1981 0 0 0 0 5.96 5.96 5.96 5.96 1982 0 0 15 0 5.00 5.00 5.22 5.00 1983 0 0 0 0 8.42 8.42 8.42 8.42 1984 0 0 0 0 5.77 5.77 5.77 5.77 1985 0 0 38 0 4.88 4.88 4.36 4.88 1986 0 0 36 0 4.80 4.80 4.77 4.80 1987 0 0 0 0 5.26 5.26 5.26 5.26 1988 0 0 0 0 7.89 7.89 7.89 7.89 1989 0 0 0 0 5.50 5.50 5.50 5.50 1990 0 0 0 0 6.01 6.01 6.01 6.01 1991 0 0 0 0 5.67 5.67 5.67 5.67 1992 0 0 15 0 5.17 5.17 5.31 5.17 1993 0 0 0 0 5.88 5.88 5.88 5.88 1994 0 0 0 0 5.22 5.22 5.22 5.22 1995 0 0 0 0 6.24 6.24 6.24 6.24 1996 0 0 0 0 7.25 7.25 7.25 7.25 1997 0 0 0 0 6.16 6.16 6.16 6.16 1998 0 0 40 0 4.92 4.92 3.18 4.92 1999 0 0 37 0 4.66 4.66 4.11 4.66 2000 0 0 37 0 4.73 4.73 3.80 4.73 2001 0 0 35 0 4.26 4.26 4.28 4.26 2002 0 0 0 0 5.18 5.18 5.18 5.18 2003 0 0 0 0 6.77 6.77 6.77 6.77 2004 0 0 0 0 4.98 4.98 4.98 4.98 2005 0 0 0 0 5.24 5.24 5.24 5.24 2006 0 0 0 0 5.27 5.27 5.27 5.27 32 year average 5.75 5.74 5.68 5.88 Note: Scenario1 includes storage and interest costs. Scenario2 includes half of storage and interest costs. Model1 assumes that price follows a mean reversion process. Model2 assumes that price follows a seasonal mean reversion process. 58 Table II−6. Sales Dates and Net Returns for Wheat Sale Dates (Weeks from Harvest) Per Bushel Net Returns ($/bu) Scenario 1 Scenario Year 2 Scenario 1 Scenario 2 Model 1 Model 2 Model 1 Model 2 Model 1 Model 2 Model 1 Model 2 1975 0 0 0 23 2.91 2.91 2.91 3.03 1976 0 0 0 24 3.36 3.36 3.36 2.16 1977 0 0 25 23 1.92 1.92 2.21 2.26 1978 0 0 0 0 2.90 2.90 2.90 2.90 1979 0 0 0 0 3.40 3.40 3.40 3.40 1980 0 0 0 0 3.40 3.40 3.40 3.40 1981 0 0 0 0 3.83 3.83 3.83 3.83 1982 0 0 0 0 3.44 3.44 3.44 3.44 1983 0 0 0 0 3.39 3.39 3.39 3.39 1984 0 0 0 0 3.34 3.34 3.34 3.34 1985 0 0 0 0 2.89 2.89 2.89 2.89 1986 0 0 25 0 2.23 2.23 1.85 2.23 1987 0 0 0 0 2.32 2.32 2.32 2.32 1988 0 0 0 0 3.05 3.05 3.05 3.05 1989 0 0 0 0 3.77 3.77 3.77 3.77 1990 0 0 0 0 2.94 2.94 2.94 2.94 1991 0 0 0 0 2.52 2.52 2.52 2.52 1992 0 0 0 0 3.48 3.48 3.48 3.48 1993 0 0 0 0 2.63 2.63 2.63 2.63 1994 0 0 0 0 3.10 3.10 3.10 3.10 1995 0 0 0 0 3.91 3.91 3.91 3.91 1996 0 0 0 0 5.37 5.37 5.37 5.37 1997 0 0 0 0 3.73 3.73 3.73 3.73 1998 0 0 0 0 2.70 2.70 2.70 2.70 1999 0 0 27 0 2.36 2.36 1.65 2.36 2000 0 0 0 0 2.40 2.40 2.40 2.40 2001 0 0 0 0 2.88 2.88 2.88 2.88 2002 0 0 0 0 2.85 2.85 2.85 2.85 2003 0 0 0 0 2.83 2.83 2.83 2.83 2004 0 0 0 0 3.50 3.50 3.50 3.50 2005 0 0 0 0 3.03 3.03 3.03 3.03 2006 0 0 0 0 4.54 4.54 4.54 4.54 32 year average 3.15 3.15 3.13 3.13 Note: Scenario1 includes storage and interest costs. Scenario2 includes half of storage and interest costs. Model1 assumes that price follows a mean reversion process. Model2 assumes that price follows a seasonal mean reversion process. 59 Table II−7. Paired Differences tRatios of the Mean Net Returns between Seasonal Mean Reversion and Mean Reversion Model (19752006) Commodity Include Storage and Interest Costs Include Half of Storage and Interest Rate Corn 2.30* 0.54 Soybean 1.00 1.67 Wheat N/Aa 0.04 Note: tcritical value with 30 degree of freedom at 5% significance level is 2.042. We calculate paired differences by subtracting the net return assuming a simple mean reversion model from a net return assuming seasonal mean reversion model. a A paired difference tratio for wheat of including storage and interest costs is not available since there is no difference of net returns between the two models, so variance of paired difference is zero. 60 Figure II−1. An example trinomial lattice x −2Δ x x −Δ x x −Δ x x x x x+Δ x x +Δ x x +2Δ x x +3Δ x x +2Δ x x +Δ x x x −Δ x x −2Δ x x −3Δ x 61 0.015 0.01 0.005 0 0.005 0.01 Oct N ov Dec Jan Feb Mar Apr May Jun Jul Aug Sep t % Figure II−2. Seasonality of change in corn price 62 0.01 0.008 0.006 0.004 0.002 0 0.002 0.004 0.006 Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep t % Figure II−3. Seasonality of change in soybean price 63 0.01 0.008 0.006 0.004 0.002 0 0.002 0.004 0.006 Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May t % Figure II−4. Seasonality of change in wheat price 64 0 0.5 1 1.5 2 2.5 3 Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep t $/bu Figure II−5. Cutoff price of mean reversion price process for corn using low storage and interest costs 0 0.5 1 1.5 2 2.5 3 3.5 Oct Nov Dec Jan Feb Mar Apr May Jun Jul A ug Sep t $/bu Figure II−6. Cutoff price of seasonal mean reversion price process for corn using low storage and interest costs store sell store sell store 65 5 5.5 6 6.5 7 Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep t $/bu Figure II−7. Cutoff price of mean reversion price process for soybeans using low storage and interest costs 0 1 2 3 4 5 6 7 Oct Nov Dec Jan Feb Mar Apr May Jun Jul Aug Sep t $/bu Figure II−8. Cutoff price of seasonal mean reversion price process for soybeans using low storage and interest costs store store sell sell 66 2 2.5 3 3.5 4 4.5 5 Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May t $/bu Figure II−9. Cutoff price of mean reversion price process for wheat using low storage and interest costs 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 Jun Jul Aug Sep Oct Nov Dec Jan Feb Mar Apr May t $/bu Figure II−10. Cutoff price of seasonal mean reversion price process for wheat using low storage and interest costs store store sell sell VITA HYUN SEOK KIM Candidate for the Degree of Doctor of Philosophy Thesis: MAKING GRAIN PRICING DECISIONS BASED ON PROFIT MARGIN HEDGING AND REAL OPTION VALUES Major Field: Agricultural Economics Biographical Personal Data: Born in Seoul, Korea, on March 8, 1975, the son of Ho Tak Kim and Yoon Jeung Cho. Education: Graduated from YoungDong High School, Seoul, Korea, in February 1994; received a Bachelor of Science degree in Trade from Hankuk University of Foreign Studies, Seoul, Korea, in February 2002. Received a Master of Science degree in Agricultural Economics from Seoul National University, Seoul, Korea, in February 2004. Completed the requirements for the degree of Dotor of Philosophy in Agricultural Economics at Oklahoma State University in July 2008. Experience: Graduate Research Assistant, Department of Agricultural Economics and Rural Development, Seoul National University, March 2002 − February 2004. Assistant Researcher, Market Analysis Team, Korean Food Research Institute, February 2004 − August 2004. Graduate Research Assistant, Department of Agricultural Economics, Oklahoma State University, August 2004 − July 2008. Award: Spielman Scholarship for Excellence in Academic Performance, Department of Agricultural Economics at Oklahoma State University, March 2007. Professional Membership: American Agricultural Economics Association. Name: Hyun Seok Kim Date of Degree: July, 2008 Institution: Oklahoma State University Location: Stillwater, Oklahoma Title of Study: MAKING GRAIN PRICING DECISIONS BASED ON PROFIT MARGIN HEDGING AND REAL OPTION VALUE Pages in Study: Candidate for the Degree of Dotor of Philosophy Major Field: Agricultural Economics Scope and Method of Study: This study contains two essays. The first essay is preharvest pricing decision making and the second essay is postharvest decision making. The purpose of the first essay was to determine producer’s utility function and price processes where profit margin hedging is optimal. A statistical test of mean reversion in agricultural futures prices is conducted. The simulations were also conducted to compare the expected utility of profit margin hedging strategy with the expected utility of other strategies such as always hedging and selling at harvest. The purpose of the second essay was to determine whether real option values can explain why producers appear to store too long. To determine the real option value, we modeled and estimated a seasonal mean reversion price process which allowed price to be a random walk within a season, but mean reverting across crop years. After estimation of the price process, a universal lattice model was used to determine cutoff price. This study conducted simulations using cash prices of crops to determine differences of net returns of optimal strategy under two different price processes, which are a simple mean reversion price process and a new seasonal mean reversion price process. Findings and Conclusions: Theoretical results from the first essay showed that profit margin hedging is an optimal strategy under a highly restricted target utility function even in an efficient market. Profit margin hedging is profitable if prices are mean reverting. Simulation results showed that profit margin hedging gives the highest expected utility to producers under the highly restricted target utility function. With the variance ratio test, there is little evidence that futures prices of crops follows a mean reverting process. In the second essay, the estimated nonparametric bootstrap parameters of the seasonal mean reversion process show the seasonal function turns negative before mean reversion begins, which suggests that real option values are relatively unimportant in determining when producers sell their grain. The graphs of cutoff price when assuming a seasonal mean reversion price process show that producers sell before mean reversion begins except when prices are extremely low. Therefore, Fackler and Livingston’s (2002) finding of a large real option value that can explain why producers store too long is not supported. The simulation results show that there is little evidence that the net returns between the mean reversion model and the seasonal mean reversion model are different. ADVISER’S APPROVAL: B. Wade Brorsen 



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