Risk Aversion, Prudence, and the Three-Moment Decision Model for Hedging

Transcription

1 Risk Aversion, Prudence, and the Three-Moment Decision Model or Hedging Xiaomei Chen Graduate Research Assistant School o Economic Sciences Washington State University P.O. Box Pullman, WA cxm9531@yahoo.com (509) H. Holly Wang Associate Proessor School o Economic Sciences Washington State University Ron C. Mittelhammer Regents Proessor Director o the School o Economic Sciences Washington State University Selected Paper prepared or presentation at the American Agricultural Economics Association Annual Meeting, Long Beach, Caliornia, July 3-6, 006 Copyright 006 by Xiaomei Chen, H. Holly Wang, and Ron C. Mittelhammer. All rights reserved. Readers may make verbatim copies o this document or non-commercial purposes by any means, provided that this copyright notice appears on all such copies.

2 Risk Aversion, Prudence, and the Three-Moment Decision Model or Hedging Linear moment preerence unctions have been widely used in decision analysis as approximations o the Von Neumann-Morgenstern expected utility (EU). The twomoment mean-variance model is the most popular one among them. It was originated by Markowitz (195) as a portolio selection tool, extended by Tobin (1958) to include riskree assets, and applied in equilibrium analysis by Sharpe (1964) and Lintner (1965) to the pricing o capital assets. Comparing to EU models, the two-moment model requires less inormation rom decision makers and rom random distributions. However, challenges (Borch, 1969; Feldstein, 1969) to the appropriateness o the approximation have caused the deenders o mean-variance to either modiy the conditions or improve the model by adding more moments. Theoretically, the two-moment model can yield a consistent optimal decision with EU i 1) the decision maker s utility unction is quadratic, or ) the stochastic return is normally distributed (Tobin, 1958), or 3) the random variables satisy the location-scale constraint (Meyer, 1987). However, Arrow and Hicks denounced a quadratic unction as absurd because o its limited range o applicability and highly implausible implication o increasing absolute risk-aversion. On the other hand, the assumption o normal distribution o all risky outcomes is not realistic since returns typically are not normally distributed. Since all o these constraints are very restrictive, many studies have expanded the model to incorporate higher moments. Samuelson (1970) noted that including more than the irst two moments can improve the solution or any arbitrarily short, inite time interval. Tsiang (197) pointed out skewness preerence may be prevalent in investors 1

3 behavior because modern inancial institutions provide a number o devices or investors to increase the positive skewness o the returns o their investments. The skewness preerence has received special attention in the asset pricing and portolio theory (Kraus and Litzenberger, 1976; Friend and Westerield, 1980; Sears and Wei, 1988; Lim 1989). The three-moment model will be important or analytical studies on risk management decision modeling when the distribution is skewed. Poitras and Heaney (1999) compared the optimal demand or put options derived rom the two-moment and three-moment models. It is shown theoretically the optimal demand or put options was reduced with positive skewness preerence. However, they did not compare their results to the expected utility model, and their derived moment models require a speciic utility orm. Further development and application o three-moment model in agricultural risk management using derivatives are very limited. In this paper, we will develop a general three-moment model and compare it and the traditional two-moment model to the expected utility in the setting o an individual producer hedging in the utures market. Speciic objectives include (1) to theoretically develop the linear three-moment model analogue to the existing mean-variance model, () to apply it in the context o hedging and derive the optimal solution as well as comparative statics, and (3) to numerically compare the optimal hedge ratios (OHR) derived rom the two-, three-moment models and the ull expected utility model under alternative preerence parameters. Only the second and third moments are concerned in this paper because higher moments add little, i any, inormation about the distribution s physical eatures (Arditti, 1967).

4 Model A decision maker s preerence can be represented by a Von Neumenn- Morgenstern utility unction U(π). Using Taylor s expansion, EU( π) = EU( μ+ ε ) EU + E U + E ε u + E ε u 3 ( μ) [ ε '( μ)] [ ''( μ)]/ [ '''( μ)]/3! where E() is the expected value operator,π is the random proit, μ is the expected proit, and ε is the error term with zero expected value. Because maximizing the utility unction o the certainty equivalent is equivalent to maximizing the expected utility unction (Robinson and Barry, 1987) and the utility unction is monotonically increasing, the three-moment model in terms o mean, variance and skewness are derived as (Appendix): (1) Max CE Max E( ) λ λη π = π V ( π) + S( π ) 6 where π CE is the certainty equivalent o proit; E(), V() and S() are the expectation, variance and skewness 1 operators; λ is Arrow-Pratt s absolute risk aversion coeicient, i.e., U''/ U' ; η is Kimball (1990) s absolute prudence level, i.e., U'''/ U'' which is isomorphic to Arrow-Pratt s absolute risk aversion. According to Kimball (1990), the absolute prudence measures the sensitivity o the optimal choice o a decision variable to risk. This term suggests the propensity to prepare and orearm onesel in the ace o uncertainty while risk aversion measures how much one dislikes uncertainty and would turn away rom uncertainty i possible. πη is a measure o relative prudence, just as πλ is a measure o relative risk aversion. According to Arrow (1971), the essential properties or an investor s utility unction are: (1) positive marginal utility or wealth, i.e., U > 0, () decreasing marginal 3

5 utility or wealth, i.e., U < 0, and (3) non-increasing absolute risk aversion, i.e. U 0. Thus both λ and η should be positive, i.e., the decision maker is risk averse and prudent. He/she would always desire positive skewness o return π when the mean and variance o the return remain constant. The two-moment model is equation (1) without the third term, orη = 0. Assuming no transaction costs or trading utures contracts, the return π in the utures market or an individual armer is speciied as: () π = π 0 + px c+ ( 0 ) y where π 0 is the initial wealth; p is the cash price at harvest; x is the nonstochastic production level; c is the cost o producing x; y is the hedging level in the utures market to be determined; 0 is the price at planting time; is the utures price at harvest. Denoting σ p, σ, σ p, p s, s, σ p, σ p as the variances, covariance, skewnesses and coskewness o the cash and utures prices respectively, the expected value, variance and skewness o the return rom hedging become: (.1) E( π ) = π 0 + xe( p) c+ [ 0 E( )] y (.) V ( π ) = x + y p σ σ xyσ p 3 3 (.3) S( π ) = 3 3 x y x yσ p+ xy σ p s p s Substituting the speciic expected value, variance and skewness o return in equation (1), the irst order condition o the model yields: (3) E( ) σ η y = + x [ s ( y ) + x y ] * 0 p * * MVS MVS σ p x σ p MVS λσ σ σ 4

6 where y * MVS is the optimal hedging levels or the three-moment models. The two-moment optimal hedge level * y MV equals the irst two terms o equation (3). As expected, the optimal hedge levels rom the two models are the same, or the three-moment model does not add any inormation upon the two-moment model, i the decision maker is prudence neutral, i.e. η = 0. The closed orm solution or equation (3) is 3 : (4) y * xσ p σ MVS = + s + Δ ηs where Δ= ( σ ) [ ηxσ p ηs ηx σ / ( p xσ p λ+ E )/ λ]. It suggests that the solutions rom the two models can be equal when the decision maker is not prudence neutral only iσ p = σ, σ p = s and in the unbiased utures market ( 0 = E()) The armer would ully hedge, namely, he or she will hedge the same amount as the production level in that case. Thereore, we have the ollowing proposition. 0 Proposition 1: The optimal hedge levels o mean-variance and mean-variance-skewness models are equal i: (i) the decision maker is prudence neutral, i.e. η = 0 ; (ii) σ p = σ, σ p = s and the utures market is unbiased; or (iii) σ 0, and = 0. = p We reerσ p s = σ as cash and utures prices are perectly correlated in the two moment, andσ p = s as perectly correlated in the third moment, assuming the variance and skewness o the two prices are the same or convenience. Only when the two prices 5

7 are perectly correlated, the mean-variance model yields a ull hedge or risk averse armers in the unbiased market. (ii) says i the two prices are urthermore perectly correlated in the third moment, the mean-variance-skewness model yields a ull hedge or risk averse and prudent armers. The two cases are implied in a more strong condition when there is no basis risk, ie. p =. Then a decision maker will always make a ully hedge in either model (and in the ull expected utility model). (iii) says i the two prices are un-coskewed, and the utures price is not skewed, then the means-variance-skewness model also yields the same optimal hedging levels as the mean-variance model, because the hedging will not aect the skewness o the return and thereore the prudent preerence will not aect hedging. Corollary 1: The risk averse and prudent armer will make a ull hedge in an unbiased market i there is no basis risk. The ollowing comparative static propositions can be derived by partially dierentiating the two optimal hedge levels with respect to each parameter. Proposition : The short position will be increased (or decreased) and the long position will be decreased (or increased) i current utures price goes up (or down) in both meanvariance and mean-variance-skewness models, while holding all other parameters constant. Proo: Partially dierentiate the two optimal hedge levels with the initial utures price: (5) y * MV 0 1 = > 0 λσ (6) * y MVS 0 1 = > 0 1/ λδ 6

8 The values o the optimal hedge levels are monotonically increasing with the initial utures price. A higher optimal hedge level means hedge more or a short position hedger and hedge less or a long position hedger because the absolute value is decreased. This is a speculating eect because a higher current utures price means more expected proit gain (loss) or a short (long) hedger. For both models, the response is a smaller or a more risk averse hedger because the speculating position deviates rom the optimal risk reducing position, and the more risk averse hedger chooses to deviate less. Proposition 3: The short position will be increased (or decreased) and the long position will be decreased (or increased) i the covariance between the cash and utures prices increases (or decreases) in both mean-variance and mean-variance-skewness models, while holding all other parameters constant. Proo: The ollowing are obtained by partially dierentiating with the covariance o cash and utures prices. (7) y σ * MV p x = > σ 0 (8) y σ * MVS p x = > Δ 1/ 0 This is a risk reducing eect because a higher covariance means lower basis risk and the risk reducing eect gives more incentive on short hedging but less incentive on long hedging. The responses rom both models are proportional to the production level. Proposition 4: The decision maker hedges more (or less) i the current utures price is lower (or higher) than the expected utures price as the decision maker becomes more 7

9 risk averse in both mean-variance and mean-variance-skewness models, while holding all other parameters constant. The risk aversion coeicient will not aect the hedging position when the utures market is unbiased, when other parameters remain constant. Proo: Dierentiate with the risk aversion level and obtain: (9) y λ * MV = 0 E( ) λ σ (10) * MVS y 0 E( ) = 1/ λ λ Δ According to (9) and (10), the optimal hedge levels rom the two models change in the same direction as risk aversion increases. Both equations have a positive sign as 0 < E() and a negative sign as 0 > E(). Risk averters will make a ull hedge when there is no basis risk in the unbiased utures market. This result will not change with the risk aversion level. The ull hedge minimizes risk. When the current utures price is lower than the expected maturity price, both models advise the decision maker to underhedge, namely, to sell less than their production level. As they become more risk averse they will increase their hedging levels toward the ull level, because their risk reducing incentive increases relative to their loss reducing incentive. I the current uture price is suiciently low the decision maker would be likely to take a long position, namely, buy now and sell in the uture rom the utures market. In that case, the armers would hedge less as they are more risk averse. On the other hand, when the current utures price is higher than the expected maturity price, both models recommend over hedging, and more risk-averse armers will over hedge less so as to be closer to the ull hedge level. 8

10 The comparative statics o the three-moment optimal hedge level on the other parameters are: (11) (1) * 4 1/ [ ( MVS σ ηxσ σ p ηx σ p σ η 0 E( )/ λ)] Δ = + η η η y s s s s y x = + * MVS σ p σ s s ηs 1/ {( σ ηxσ p) ηs [ ηx σ p xσ p 0 / λ E( ) / λ]/ } ηs Δ + + * y MVS x (13) = [1 + 1/ ( 1 Δ σ + ηxσ p ηxs )] σ p s The signs or equation (11), (1) and (13) are ambiguous. These signs will be examined or the ollowing empirical example. Simulation and Numerical Results Numerical analysis o an example examines the level o approximation o the two-moment and three-moment models to the expected utility model by comparing the optimal hedge ratios (OHR). The hedge ratio is the ratio o hedging to the production level. Assume the hedger has the commonly used CRRA (constant relative risk aversion) utility unction: (14) U( π) = (1 θ) π 1 (1 θ ) where θ is the relative risk aversion coeicient. The optimal hedge ratio or the expected utility model is solved numerically. For two- and three-moment models, the optimal hedge ratios are obtained by (3) ignoring the third term and (4). The values o θ range rom 1 to 4 ollowing Dynan (1993). Six levels o relative risk aversion coeicient (θ ), 9

11 speciically 1.5,,.5, 3, 3.5, 4, are analyzed. The six levels o the absolute risk aversion coeicient λ and absolute prudence coeicient η are calculated based on the relative risk aversion levels ( λ = θ / π, η = (1 + θ) / π ) 4. The analysis assumes a representative armer who grows wheat in U.S. Paciic Northwest region. The initial wealth determined by average per acre is $550 per acre. Production cost is $30 per acre and production level is bushels per acre. Bivariate gamma distribution is chosen to simulate the wheat cash and utures prices or the 00 harvest period because (1) it s positively skewed; () gamma random variables (cash and utures prices in this case) are positive; (3) it acilitates including the skewness parameter in the simulation. The approach o Law and Kelton (198) is used to simulate the correlated bivariate gamma distribution. The correlation between the wheat cash and utures prices is The scale and location parameters or the gamma distribution are calculated rom the variances (0.37 and 0.56 or wheat cash and utures prices) and skewnesses (0.1 and 0.9 or wheat cash and utures prices) 5. The mean values are adjusted to $3.7 and $3 per bushel respectively ater the simulation. These parameter levels are determined based on the weekly Portland spot market cash price and CBOT utures price data rom September 1998 to August 001. The descriptive statistics or the simulated cash and utures prices are shown on Table 1. The skewness o the simulated cash and utures prices are signiicant, although they appear small. 6 The initial uture price 0 is set at three levels ($3.0, 3.00 and.80 per bushel). The utures market is unbiased when 0 equals $3.00 per bushel. The hedger is likely to take a short (or long) position i 0 equals to $ 3.0 (or.80) per bushel. 10

12 Table shows the optimal hedge ratios (OHR) rom three models under six levels o relative risk aversion and three levels o initial utures prices. The results show that OHRs rom the three-moment model are closer to those rom expected utility model than two-moment model OHRs in all situations. The evidence rom this example strongly avors the three-moment model over the two-moment model. Comparing the absolute OHR values, the armer hedges more (or less) in the MVS model than in the MV model when he is in a long (short) position. Based on equation (3), the optimal hedge level o the MVS model has one more term than that o MV model, which is due to skewness o proit. I a armer with a short (long) position hedges more, the skewness o proit which he desires will be decreased (increased) according to the deinition o skewness o proit. Thus compared to MV model, the armer would hedge less (more) in a short (long) position. When the initial utures price changes rom $.8 to $3 and $3. per bushel, the OHR values o both models increase, consistently with Proposition. The OHRs rom the MV model increase at a constant rate or each relative risk aversion level. But the values rom MVS do not have the same pattern with each level o initial utures price increase, which is also consistent to Proposition as in equations 5 and 6. The absolute values o OHRs rom the MV model do not change in the unbiased utures market while they drop with the relative risk aversion in the biased utures market. This is consistent with Proposition 4 (equation 9). The absolute values o MVS OHRs decrease in biased and unbiased utures as relative risk aversion increases. This seeming inconsistency arises because the particular CRRA utility we choose is not constant in prudence, because the prudence level is related to the risk aversion level. In 11

13 order to compare the MVS results to the true utility maximization results, we allow the prudence to vary accordingly. Thereore, the conditions in Proposition 4, i.e., holding all other parameters constant, is violated, and the OHR changes or MVS model in Table is a result o a joint increase in both risk aversion and prudence. In both models, the armer hedges more when he is in a short position than in a long one. This is because the minimum risk position is short. When the market goes biased or the same level in both directions, the short hedge is enhanced and the long hedge is only a residual ater the short position has been ully reduced. The comparative statics are also numerically checked so as to illustrate the ambiguous signs o equation (11) and (1). Equation (13) is not checked because the coskewness can not be controlled in the simulation because it changes with the skewness. First, we examine how the MVS OHR changes with the skewness o utures prices. The cash price skewness is ixed because it is not directly related to OHR (Equation 4). Three skewness levels (0.5, 1.0 and 1.5 times o the original skewness) are chosen or the utures prices so that the bivariate gamma distributed cash and utures prices could be simulated. 7 Two more bivariate gamma distributed cash and utures prices are simulated based on the change o skewness. According to the simulated data, the coskewness, σ p decreases and σ p increases as the skewness o utures goes up and vice versa. The comparative static results o MVS OHRs on utures price skewness are demonstrated in Table 3. Both unbiased and biased (long and short positions) are considered. The armer takes a longer position when the utures price is more skewed. The intuition is that the armer uses hedging to both reduce variance and increase 1

14 skewness o the proit, and i utures price is more skewed the long position can ampliy the proit skewness more eectively. The increasing skewness motivates the armer to increase his long hedge position at a cost o decreased variance. The same reasoning can be used to explain the smaller short position when skewness increases in the biased utures market. The opposite behavior occurs under the unbiased market because the short positions are much smaller than in the other two cases. The variance reducing eect dominates the skewness increasing eect comparing the variance equation (.) and skewness equation (.3) o proit. This causes the armer to take a larger short position. Thereore, the comparative static on utures skewness cannot be simply determined in sign. The inluences o risk aversion and prudence on the OHRs in the MVS model is shown in Figure 1 (a) and (b), respectively. The relative risk aversion and prudence range rom 1 to 5. Empirical research on prudence levels is not available. The range is chosen at the same level as risk aversion because the two are oten close in commonly used utility unctions such as exponential (constant absolute risk aversion preerence), log or power unctions (constant relative risk aversion preerence). The impact on OHR rom relative risk aversion has consistent pattern as in proposition 4, when the relative prudence level is ixed at. When relative risk aversion is ixed at, the hedging position decreases as the armer becomes more prudent so that all three lines in Figure 1(b) are downward sloping. The downward slope in the unbiased market is so small that the line looks horizontal. The inluence o the prudence on the market is trivial in this case. The decreasing position means hedging less in short and more in long. We have also set risk aversion at 13

15 other levels, but the OHRs show the same pattern, i.e., decreasing with prudence. This means the sign o equation (11) is not sensitive to the preerence parameters. Compared to Figure 1(a), risk aversion makes a big dierence in OHR than prudence in the biased utures market. Figure demonstrates how relative risk aversion and prudence aect the certainty equivalent in the MVS model in an unbiased utures market. The certainty equivalent is the certain amount o money which leaves the decision maker equally well-o as the speciied risky hedging opportunity. The higher certainty equivalent means higher utility achieved with hedging. The results show that changes o certainty equivalent brought by dierence prudent levels are small relative to the changes brought by dierent risk aversion levels. The certainty equivalent always decreases as the risk aversion increases in both biased and unbiased markets, because the armer requires higher compensation or risk. For the same reason, one might expect the certainty equivalent to increase as prudence increases. However, it decreases as the prudence increases when in a very large short position (See Figure (b)). This occurs because the long position increases very ast (Figure 1(b)) which reduces the proit skewness enough to oset the increased prudence. Summary and Conclusion A linear mean-variance-skewness (three-moment) model is developed and applied to the hedging decision in the utures market. The optimal hedge ratio (OHR) and associated comparative statistics are derived and compared theoretically rom both threemoment and two-moment models. The term prudence introduced by Kimball is 14

16 included in the three-moment model. The OHRs rom the two models are equal only when: 1) the decision maker is prudence neutral or; ) with unbiased utures markets, assuming perect correlation o cash and utures prices in both second and third moments. The OHR o the three-moment model changes in the same direction as that o the twomoment model when the initial utures price, covariance o cash and utures prices and risk aversion coeicient change respectively. Otherwise, eects on OHRs are not deinite theoretically. The signs on the comparative statics o the three-moment OHR on the other parameters such as the prudence level, skewness o utures prices and coskewness o utures and cash prices are ambiguous. The two and three moment models are also compared against the expected utility model or a numerical example so as to examine which model provides a closer approximation to expected utility. We assume the hedger is a typical armer, with the common CRRA utility unction, who grows wheat in the Paciic Northwest. The results show the OHRs rom the MVS model is closer to those rom expected utility model than those rom MV model in all situations considered. This strong evidence suggests that the MVS model is superior to the MV model. The armer hedges more (less) in the MVS model than in the MV model when he/she is in a long (short) position. This results rom the additional term skewness o return in the MVS model. The comparative statics o MVS OHRs on utures price skewness indicates the armer takes a longer position so as to increase the beneit rom increased positive proit skewness when the utures price is more skewed. The opposite behavior or the unbiased market is primarily because the short positions are much smaller than in the other two cases, and the variance reducing 15

17 eect dominates the skewness increasing eect. There s no clear pattern when the armer is in a short position. The inluences o risk aversion and prudence on OHRs or the MVS model are also examined. The ranges o relative risk aversion and prudence are extended a little based on the common CRRA utility unction. The numerical results show the armer ull-hedges in the unbiased market and hedges less as risk aversion increases in the biased utures market. The hedging position decreases as the armer becomes more prudent. Risk aversion has a greater inluence on OHR than prudence in the biased utures market. The certainty equivalent consistently decreases as the risk aversion increases in both biased and unbiased market, because the armer requires his/her certain compensation or the risky hedge. Similarly, the certainty equivalent should be expected to increase with prudence; however, it decreases in a very large short position. This is because the long position increases quickly which reduce the proit skewness thereby osetting the eect o the increased prudence. 16

18 Endnote: 1 Skewness reers to the third moment instead o standardized third moment in this paper. σ =, σ = E{[ E( )] [ p E( p)]} E{[ E( )][ p E( p)] } p p 3 The irst order condition equation has two roots and result in two closed orms o * y actually. The sign MVS operator beore the square root could be add or subtract. The add operator is chosen in order to achieve the maximum by the second order condition (SOC). 4 For the particular CRRA preerence, the relative prudence is determined once the relative risk aversion is set at a certain level. U = π π θ 3, U (1 + θ ) θπ, = = θ(1 + θ) π 3 π π U (+ θ ) λ = U ''( μ) /[ U '( μ)] = θ / μ,η = U'''( μ)/[ U''( μ)] = (1 + θ)/ μ. 5 3 Location parameter α = 4( σ ) / S and scale parameter β S /σ =. 6 Formal hypothesis test is conducted. H 0 : S = 0 vs. H 1 S 0 where S is the skewness. Then the statistic z, z = S/ 6/ n, where n =10,000 the number o observations, ollows the standard normal distribution under the null hypothesis. Here, z is and or cash and utures prices, respectively, and both are larger than the critical value at 5%. Thereore, the null hypothesis o zero skewness is rejected or both. 7 I the skewness is less than 0.5 times, the utures price would be almost normally distributed which is not the interest o this paper. I the skewness is larger than 1.5 times, the bivariate gamma distributed cash and utures prices would not be able to be simulated. 17

19 Reerence Arditti, Fred D. Risk and the Required Return on Equity. The Journal o Finance (March 1967): Arrow, K.J. Essays in the Theory o Risk-Bearing, Chicago: Markham Publishing Co., Borch, Karl. A Note on Uncertainty and Indierence Curves. Review o Economic Studies 36 (Jan. 1969):1-4. Diacogiannis, George P., Three-Parameter Asset Pricing. Managerial and Decision Economics 15 (1994): Dynan, Karen E. How Prudent are Consumeres? The Journal o Political Economy 101 (1993): Feldstein, Martin. Mean-Variance Analysis in the Theory o Liquidity Preerence and Portolio Selection. Review o Economic Studies 36 (Jan. 1969): 5-1. Friend, I., and R. Westerield. Co-Skewness and Capital Asset Pricing. The Journal o Finance 35 (1980): Kimball, Miles S. Precautionary Saving in the Small and in the Large. Econometrica 61(Jan. 1990): Kimball, Miles S. Standard Risk Aversion. Econometrica 61(May, 1993): Kraus, Alan, and Robert H. Litzenberger. Skewness Preerence and the Valuation o Risk Assets. The Journal o Finance 31 (1976): Law, A.M. and W. David Kelton. Simulation Modeling and Analysis, McGraw-Hill Book Company,

20 Lim, Kian-Guan. A New Test o the Three-Moment Capital Asset Pricing Model. The Journal o Financial and Quantitative Analysis 4 (1989): Lintner, John. The Valuation o Risk Assets and the Selction o Risky Investments in Stock Portolios and Capital Budgets. Review o Economics and Statistics 47 (1965): Markowitz, Harry. Portolio Selection. The Journal o Finance 7 (195): Meyer, Jack. Two-Moment Decision Models and Expected Utility Maximization. The American Economic Review 77 (1987): Poitras, Georey, and John Heaney. Skewness Preerence, Mean-Variance and the Demand or Put Options. Managerial and Decision Economics 0 (1999): Robinson, L. J., and Peter J. Barry. The Competitive Firm s Response to Risk, Macmillan Publishing Company, New York, Samuelson, Paul A. The Fundamental Approximation Theorem o Portolio Analysis in terms o Means, Variances and Higher Moments. The Review o Economic Studies 37 (1970): Sears, R.S., and K.C. J. Wei. The Structure o Skewness Preerences in Asset Pricing Models with Higher Moments: An Empirical Test. The Financial Review 3 (1988): Sharpe, William F. Capital Asset Prices: A Theory o Market Equilibrium Under Conditions o Risk. Journal o Finance 19 (1964): Tobin, James. Liquidity Preerence as Behavior Towards Risk. Review o Economic Studies 5 (1958):

21 Tsiang, S.C. The Rationale o the Mean-Standard Deviation Analysis, Skewness Preerence, and the Demand or Money. The American Economic Review 6 (197): Wang, H. H., L. D. Makus and X. Chen. The Impact o US Commodity Programmes On Hedging In The Presence O Crop Insurance. European Review o Agricultural Economics 31 (004):

22 Table 1: Descriptive Statistics or Simulated Cash and Futures Prices (Units: $/bushel) Variable N Mean Skewness StDev Minimum Median Maximum cash uture

23 Table : Optimal Hedge Ratios Comparison under Six Relative Risk Aversion Levels and Three Levels o Initial Futures Prices θ unbiased utures market (0 = E = $3/bushel) Mean-Variance Model Mean-Variance-Skewness Model Expected Utility Model = $3./bushel Mean-Variance Model Mean-Variance-Skewness Model Expected Utility Model = $.8/bushel Mean-Variance Model Mean-Variance-Skewness Model Expected Utility Model Note: For 0 = $.8/bushel, the negative hedge ratios mean the hedger takes a long position.

24 Table 3: Impacts o Futures Price Skewness on Three-Moment Optimal Hedge Ratios theta unbiased utures market (0 = E = $3/bushel) 0.5*S S *S = $3./bushel 0.5*S S *S = $.8/bushel 0.5*S S *S Note: For 0 = $.8/bushel, the negative hedge ratios mean the hedger takes a long position. 3

25 OHR π λ OHR 0.5 π η =.8 0 = 3 0 = 3. (a) =.8 0 = 3 0 = 3. (b) Figure 1: Comparative Statics o Optimal Hedge Ratio (OHR) by Relative Risk Aversion Level and Relative Prudence level. Note: (1) For 0 = $.8/bushel, the negative hedge ratios mean the hedger takes a long position. πλ is relative risk aversion and π * () * η is relative prudence. 4

26 CEMVS π λ 0=3 0=3. 0=.8 CEMVS π η 0=3 0=3. 0=.8 (a) (b) Figure : Certainty Equivalent Changes with Relative Risk Aversion Level and Relative Prudence Level respectively. Note: * πλ is relative risk aversion and π * η is relative prudence. 5

27 Appendix: Derivation o the three-moment model in terms o mean, variance and skewness. U ( π CE) = U( μ m) = EU( μ + ε) where μ is the expected proit return, m is premium and ε is the error term with zero expected value. The proit returnπ is a random variable which is equal to μ + ε. U( μ m) U( μ) mu '( μ) EU EU E U E ε u E ε u 3 ( μ+ ε) ( μ) + [ ε '( μ)] + [ ''( μ)]/ + [ '''( μ)]/3! = U + σ U + S U ( μ) ''( μ) / k '''( μ)]/ 3! m= σ U''( μ)/[ U'( μ)] + S ku'''( μ)/[3! U'( μ )] μ π CE = m = μ λ σ / + S ku'''( μ) /[6 U'( μ )] = μ λ σ / + S kλη /6 where λ = U ''( μ) /[ U '( μ)], η = U '''( μ) /[ U ''( μ)] 6