Relaxing Standard Hedging Assumptions in the Presence of Downside Risk

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1 Relaxing Standard Hedging Assumptions in te Presence of Downside Risk Fabio Mattos Pilip Garcia Carl Nelson * Paper presented at te NCR-134 Conference on Applied Commodity Price Analysis, Forecasting, and Market Risk Management St. Louis, Missouri, April 18-19, 2005 Copyrigt 2005 by Fabio Mattos, Pilip Garcia and Carl Nelson. All rigts reserved. Readers may make verbatim copies of tis document for non-commercial purposes by any means, provided tat tis copyrigt notice appears on all suc copies. * Fabio Mattos (fmattos@uiuc.edu) is a researc assistant, Pilip Garcia is a professor, and Carl Nelson is an associate professor at te Department of Agricultural and Consumer Economics, University of Illinois at Urbana-Campaign

2 1 Relaxing standard edging assumptions in a downside risk framework Te purpose of tis study is to analyze ow te introduction of a downside risk measure and less restrictive assumptions can cange te optimal edge ratio in te standard edging problem. Based on a dataset of futures and cas prices for soybeans in te U.S., te empirical findings indicate tat optimal edge ratios cange dramatically wen a one-sided risk measure is adopted and standard assumptions are relaxed. Furter, te results suggest tat in a downside risk framework wit realistic edging assumptions tere is little or no incentive for farmers to edge. Keywords: downside risk, edging, futures markets INTRODUCTION Hedging models traditionally adopt variance as a measure of risk, and minimum-variance edge ratios are calculated for agents to follow. Wile te minimum-variance edge ratios are tractable and easy to estimate, te underlying assumptions may not be consistent wit edgers observed beavior, wic may explain wy estimated and observed edge ratios differ (Peck and Namias, 1989; Collins, 1997). One problem wit te traditional edge ratios is tat te measure of risk implies tat agents consider positive and negative deviations from te expected return as equally undesirable events. However, risk is frequently perceived by agents as a failure to acieve a certain level of return (Unser, 2000). In tis context, downside risk measures, wic assume tat returns below a certain target involve risk and returns above tis target represent better investment opportunities, can be igly relevant (Grootveld and Hallerbac, 1999). In recent survey papers, Lien and Tse (2002) argued tat a one-sided measure is more relevant in a edging context tan te traditional twosided measure represented by te variance, and Cen, Lee and Sresta (2003) empasized tat one-sided risk measures like te semivariance are consistent wit te risk perceived by individuals. Anoter problem wit standard edging models is te restrictive set of assumptions about agents beavior. Cen, Lee and Sresta (2003) point out tat almost all work to calculate edge ratios fails to incorporate transaction costs and te possibility of investments in assets oter tan cas and futures positions. In general, minimum-variance edge ratios are calculated assuming tat borrowing, lending and investing in alternative activities are not allowed, tere are neiter initial margin deposits nor brokerage fees in futures trading, and production is deterministic. But tese assumptions are excessively restrictive and may not reflect agent s situations. By definition, tere are margin deposit requirements and brokerage fees in futures markets. Furter, opportunities to borrow and lend, as well as to invest in alternative investments, are at times available to edgers. Lence (1996) conducted a study in wic several assumptions were relaxed and is results sowed tat edge ratios can cange dramatically under more realistic assumptions. But as Cen, Lee and Sresta (2003) empasize Lence s findings are based on a specific utility function, a given set of return distributions, and it remains to be seen if te findings old for downside risk edge ratios. In tis paper, we analyze ow estimated edge ratios and te opportunity cost of edging are affected wen risk is measured in a downside framework, and te beavioral assumptions of

3 2 te standard model are relaxed by allowing alternative investments and te introduction of brokerage fees. Hedge ratios are calculated under utility maximization based on a constant relative risk aversion (CRRA) utility function wic allows te absolute level of risk aversion to cange wit wealt. A data set of U.S. soybean futures and cas returns, and te S&P500 returns between 1990 and 2004 are used in te analysis. REVIEW OF LITERATURE Te notion of one-sided risk as been discussed since te early 50 s, and as evolved into several downside-risk measures. Recently, te lower partial moment (LPM) is te measure tat as been mostly used in te literature. Te lower partial moment of order α wit target δ is defined as: α δ α ( R δ) = ( R δ) dg( R) LPM ;, (1) were R is te investment return, δ is te target return, and G(R) is te cumulative distribution function of R. Te parameter α reflects te order of te partial moment, and can be seen as a measure of risk aversion. A value of α < 1 implies a risk-preference beavior, wile α > 1 imply a risk-aversion beavior (Grootweld and Hallerbac, 1999). For α > 1, iger values of α indicate tat te agent is more concerned wit te magnitude of te deviation below te target, wereas small values of α indicate tat te agent is not particularly interested in te amount of loss incurred by te deviation below te target (Fisburn, 1977). Several risk measures commonly adopted are special cases of te LPM. For α = 0, te term in parentesis in expression (1) becomes 1, and te measure is te probability of falling below te target. If te target is set to zero ( δ = 0), ten te measure is just te probability of loss. Wen α = 1, te lower partial moment represents te expected deviation of returns below te target. For α = 2, te measure is similar to te variance, but wit deviations computed only for observations below te target return. If te target is set to te mean return, ten te lower partial moment of order two (LPM 2 ) is te semi-variance. Moreover, if te target is set to te mean return and returns are symmetrically distributed, te LPM 2 is proportional to te variance, i.e., bot risk measures would lead to te same ordering of risky assets (Eftekari, 1998). In contrast to te variance, te LPM offers flexibility in modeling risk beavior. Wile te variance as a measure of risk imposes tat any deviation from te expected return is considered an undesirable event, te LPM assumes tat only deviations below a certain target is taken as risk, and tis target can be te expected return or any oter one defined by te edger. Moreover, te LPM allows for different levels of risk aversion, wile tis is not an explicit issue wit te variance. Altoug te idea of downside risk as been identified for some time, not many studies ave been performed calculating edge ratios in a mean-downside risk framework, and even less using agricultural commodities as te asset to be edged. Still, many studies ave been developed using te concept of downside risk in pricing models and in te estimation of minimum-downside risk edge ratios. Since tere is no analytic solution for te minimum-downside risk edge ratio,

4 3 various researcers ave adopted different metods to calculate tis ratio. Several studies ave focused on calculating optimal edge ratios by minimizing downside risk and comparing tem to minimum-variance edge ratios. Eftekari (1998) minimizes te lower partial moment of order two (LPM 2 ) wit target set to zero to calculate te optimal edge ratio for te FTSE-100 stock index from 1985 to Using continuously compounded returns on spot and nearby futures prices, e adopted two edging orizons (one- and two-week) and a dynamic strategy based on rolling windows. Te general result is tat minimum-lpm edge ratios are sligtly smaller and tend to yield a better risk/return combination tan te minimum-variance edge ratios. In terms of edging effectiveness, te LPM approac usually led to somewat smaller risk tan in te variance approac. 1 Similarly, Lien and Tse (2000) calculated te minimum-lpm and te minimum-variance edge ratios for te Nikkei Stock Average index over 1-week edging orizons from January 1988 to August Tree orders of te LPM were used (1, 2, and 3), and te target returns ranged from 1.5% to +1.5%. In general te optimal edge ratio increased as te order of te LPM increased, as well as wen te target return increased. Teir findings suggest tat te minimum-lpm and te minimum-variance edge ratios may differ sarply, particularly wen te edger is willingly to absorb small losses and very cautious about large losses, i.e., wen te target return is small and te order of te LPM is large. Turvey and Nayak (2003) calculated minimum-semivariance 2 edge ratios for Kansas weat edged on te Cicago Board of Trade weat futures contract, and Texas steers edged on te Cicago Mercantile Excange live cattle futures contract using several targets. Daily price observations were used for weat ( ) and for steers/live cattle ( ). Teir results were consistent wit previous studies in te sense tat minimum-semivariance edge ratios were usually smaller tan te minimum-variance edge ratios, but te difference between te two ratios varied depending on te target and te distribution of risk. Moreover, te minimum-semivariance edge was found to offer a better protection against downside risk tan te minimum-variance edge. A different approac was followed by Cen, Lee and Sresta (2001), wo adopted a mean-downside risk framework to estimate optimal edge ratios. Tey argued tat edge ratios obtained by simple minimization of te generalized semivariance (GSV) 3 migt not be consistent wit te concept of stocastic dominance, since tey are usually dependent on te target return. Consequently, tey argued tat tese edge ratios sould be calculated using utility maximization in a mean-risk framework. Using an empirical distribution-based tecnique as te estimation procedure, te autors calculated te mean-gsv edge ratios for te S&P500 index wit two targets (zero, and te sample average of te S&P500 spot price canges) and a range of values for te parameter α from 1.25 to 60, and compared tem wit te minimum-gsv edge ratio. 4 Teir results sowed tat, as te order of te GSV increased, bot te mean-gsv and te minimum- GSV edge ratios tended to become smaller and converge to a level close to 0.7 under bot targets. Furter, te mean-gsv edge ratios were usually smaller tan te minimum-gsv edge ratios, and sowed less variability for lower levels of risk. Anoter area of recent investigation as focused on identifying te implications of relaxing 1 Tese results do not necessarily old wen te sample size is small and wen edges are adjusted frequently. 2 Teir definition of semivariance is basically te second-order lower partial moment defined in equation (1). 3 Tey adopt te same definition of te lower partial moment for te GSV. 4 Four oter models were examined: minimum variance edge ratio, mean-extended-gini edge ratio, Sarpe ratiobased edge ratio, and mean-variance edge ratio

5 4 assumptions of te traditional mean-variance edging model. Based on an expected utility maximization framework, Lence (1996) incorporated te possibility of lending, borrowing, and investing in alternative investments. Transaction costs in futures markets (initial margins and brokerage fees) and stocastic production were also included. Using a CARA utility function and calibrating te model for grain storage, Lence considered tree levels of risk aversion and tree edging orizons in te simulations. His results sowed tat te maximum-expected-utility edge ratios obtained from relaxing te conventional assumptions can differ substantially from te standard minimum-variance edge ratios, and in some cases optimal edges were close to zero. His findings suggested tat te mean-variance edge ratios can be far from optimum in te presence of alternative investments and stocastic production. Optimal edge ratios were also found to be very sensitive to transaction costs. In te present study, we combine te notion of downside risk and relaxation of te standard assumptions. A downside risk framework is adopted wit low target returns, and several of te standard edging assumptions are relaxed. Te researc metod and te data are discussed in te next sections. RESEARCH METHOD Te analysis is based on a risk-averse farmer, wo takes a sort position in te futures market to edge stored soybeans. Te farmer is assumed to maximize te expected utility of final wealt W 1 = W 0 r, were W 1 and W 0 are final and initial wealt, respectively, and r is te return from te farmer s edged portfolio. Two standard edging assumptions are relaxed as brokerage fees are introduced and an alternative investment is allowed. Tese two assumptions are relaxed one at a time and ten togeter, wic yields te four different models in equations (2) troug (5). r r r r ( r ) = r + 1 (2) c f ( r b) = r + 1 (3) c f ( s A ) [ rc + ( r f ) ] + sa ra = 1 1 (4) ( s A ) [ rc + ( rf b) ] + sa ra = 1 1 (5) were r c is te return on te cas position 5, r f is te return on te futures position, r A is te return on te alternative investment, s A is te sare of te farmer s wealt invested in te alternative investment, is te edge ratio, and b is te brokerage fee as a proportion of te initial futures price. 6 5 C 1 r c =, were C 0 and C 1 are te respective cas prices at te beginning and at te end of te edging period. Te C 0 returns on te futures position and on te alternative investment are calculated in te same way. 6 B, were B is te brokerage fees in US$/contract and F b = 0 is te futures price at te beginning of te edging F 0 period, suc tat F1 + B 1 r f b = 1. F 0

6 5 Equation (2) represents te standard edging model. Te first assumption to be relaxed is te absence of transaction costs (equation 3). Five levels of brokerage fees (0.0005, 0.001, , , and 0.005) are introduced in te model, wic are taken as a proportion of te initial futures prices. Brokerage fees ave declined during te period of tis study. In 1990 brokerage fees would commonly be between and 0.005, wile by 2004 tese values would be between and Te second assumption relaxed allows an alternative investment (equation 4), wic means tat part of te farmer s wealt can be invested in assets oter tan soybeans. Te returns on te S&P500 index are used to reflect returns on alternative investments available to te farmer. Tree values for te sare of farmer s wealt invested in oter assets are assumed: 0.10, 0.25, and Finally, te fourt model relaxes te two assumptions simultaneously (equation 5); five levels of brokerage fees and tree investment scenarios are used in tis case. Te optimal edge ratio is calculated assuming utility-maximization of te farmer s final wealt. Since te joint distribution of cas, futures, and S&P500 returns is elliptically symmetric, and final wealt satisfies te location-scale condition, expected utility can be written as a function of te first two moments of te return distribution (Camberlain, 1983; Meyer, 1987). A constant relative risk aversion (CRRA) location-scale objective function is used (Nelson and Escalante, 2004): E 1 [ U ( W 1 )] = V ( µ, LPM ) = (6) 2 µ γ LPM ( r, δ ) were µ is te mean return on te edged portfolio, ã is te coefficient of relative risk aversion, and LPM 2 ( r ;δ ) is te second-order lower partial moment of te portfolio return r wit target δ. Te CRRA utility function is consistent wit agents observed beavior since it exibits constant relative risk aversion and decreasing absolute prudence. Unlike te constant absolute risk aversion utility function, te CRRA utility function also exibits risk vulnerability wic Gollier and Pratt (1996) argue is a natural restriction of utility functions. Risk vulnerability means tat te addition of an unfair background risk to initial wealt causes risk-averse decision makers to become more risk averse toward any oter independent risk. In a price edging context, tis implies tat an increase in revenue variability caused by stocastic production sould increase te optimal edge. Te coefficient of relative risk aversion ã is specified to be 3 wic is sligtly more risk averse tan average estimates of farmer risk preferences. 8 Nelson and Escalante (2004) found coefficients of relative risk aversion derived from istorical financial attributes of Illinois farms to range from 0.27 to Te order two of te lower partial moment is cosen because it is most comparable to te traditional measure of variance. Te targets are arbitrarily set at five levels: zero and four s of te return distribution: 50 t, 25 t, 10 t, and 5 t. A target equal to zero means tat te edger is only concerned wit negative returns, wile targets set to lower s imply te edger is mainly concerned wit extreme losses. Te estimation of te optimal edge ratio in te presence of downside risk follows Eftekari (1998). First te edge ratio is set to = 0, and te values of expected return, lower 2 7 A edger is assumed to pay between US$15 and US$25 per contract in brokerage fees currently wic is about alf te fees tat existed in Qualitatively similar findings were found for simulations using relative risk aversion ranging from 1 to 5.

7 6 partial moment and expected utility are calculated. Ten te edge ratio is increased by a small fraction and tese values are calculated again for te new edge ratio. Tis process is repeated until te edge ratio reaces a large enoug number, wic is arbitrarily set to = Te value of wic yields te igest expected utility is considered te optimal edge ratio. Mean-variance edge ratios are estimated by tis metod using te CRRA utility function presented in equation (6) were te variance of te portfolio return σ is used as te second moment of te distribution. 2 Finally, a minimum-variance edge ratio is used for comparison, and is obtained by dividing te covariance between cas and futures returns by te variance of futures returns: ( Rc, R f ) ( R ) Cov =. (7) Var f Opportunity costs of placing sub-optimal edge ratios are also calculated. Tese costs represent te minimum return required by te edger to accept placing a sub-optimal edge, and can be estimated as follows: opt sub [ ( R )] E[ U ( R OC )] E U = (8) + opt were R is te return provided by te optimal edge ratio, sub-optimal edge ratio, and OC is te opportunity cost. sub R is te return provided by te DATA Te empirical simulations are conducted using futures and cas prices of U.S. soybeans, and quotes of te S&P500 index from January 1990 troug June Tree edging orizons are adopted: 4, 12, and 24 weeks. Te soybean prices and te S&P500 quotes were obtained from te Commodity Researc Bureau (CRB), and correspond to midweek (Wednesday) closing prices. Te cas prices refer to soybeans in Central Illinois. Te futures prices refer to te contracts traded at te Cicago Board of Trade (CBOT). Contract monts are January, Marc, May, July, August, September and November, and te nearby futures contract tat corresponds to te lengt of te edging orizon was used. Te selected contract permits te edger to maintain te position witout aving to roll over to a new contract. For example, if te agent wit a 12-week edging orizon placed a edge on September 8, 1993, te date to lift te edge would be February 23, 1994 and so te Marc contract is used to place te edge. Following tis procedure, te edger avoids potential risk in rolling te edge forward at te expiration of te November contract. RESULTS Te discussion of te results focuses on te 4-week orizon. Results for te 12- and 24- week orizons are qualitatively similar and are not presented for brevity. Summary statistics for futures, cas, and S&P500 returns are presented in Table 1. It is assumed tat futures markets are unbiased, i.e., te empirical distribution of te futures returns were adjusted suc tat E ( r f ) = 0. All empirical distributions are somewat leptokurtic. However, Jarque-Bera fails to reject normality for te cas and S&P500 returns. Te p-value of te test statistic for futures returns is

8 , wic is not strong evidence against normality. A normal Q-Q plot of te futures returns data reveals tat te distribution is fat tailed. Table 1. Summary Statistics for Futures, Cas and S&P500 Returns (4-week orizon) futures cas S&P500 Mean 0.00% 0.58% 1.04% Median 0.10% 1.16% 0.99% Std. deviation 6.81% 6.66% 4.99% Kurtosis Skewness Maximum 21.67% 19.60% 13.95% Minimum % % % Sample size JB test (p-value) ( ) LPM r;δ 2 ä = % 4.56% 3.04% ä = 50 t 4.78% 5.16% 3.49% ä = 25 t 2.95% 3.14% 2.22% ä = 10 t 1.50% 1.30% 1.35% ä = 5 t 0.92% 1.12% 0.88% Correlation futures cas S&P Correlation-LPM 2 cas futures cas S&P500 futures S&P500 ä = ä = 50 t ä = 25 t ä = 10 t ä = 5 t As a first assessment of te differences between two-sided and one-sided risk measures, te standard deviation is greater tan te square root of te second-order lower partial moment 9 (LPM 2 ), particularly as te target is reduced (Table 1). Te square of te second-order lower partial moment declines monotonically wit a lowering of te target for all te returns. However, te correlations among te returns beave differently. Te sample correlation between cas and futures returns decline modestly as te target is lowered, but te sample correlation between cas- S&P500 and futures-s&p500 drops to zero at te lowest targets. Te canging correlations at different targets suggest tat te ordering of risky assets and consequently te edge ratios may cange at te lower targets. Te traditional minimum-variance edge ratio is 0.89 in te 4-week edging orizon (Table 2). Based on te CRRA utility function, te standard model 10 yields an optimal edge ratio of 0.90 wen te variance is adopted as te risk measure. Allowing for te presence of downside risk in te standard model, te optimal edge ratio becomes smaller as te target return is set at 9 Te square root of te second-order lower partial moment in a downside risk context is equivalent to te standard deviation in a variance context. 10 Te standard model assumes no borrowing and lending, no transaction costs, and deterministic production.

9 8 lower levels of te distribution of returns. 11 In te extreme situation were te edger is concerned only wit losses below te 10 t and 5 t s of te return distribution, te optimal edge ratios are 0.59 and 0.68 respectively. Table 2. Optimal Hedge Ratios at te 4-week Horizon Minimum CRRA CRRA downside risk wit target set to: variance Meanvariance zero 50 t 25 t 10 t 5 t Standard model Standard model + brokerage fees b = b = b = b = b = Standard model + investment in alternative asset s A = s A = s A = Standard model + brokerage fees + investment in alternative asset s A = 0.10 b = b = b = b = b = s A = 0.25 b = b = b = b = b = s A = 0.50 b = b = b = b = b = Hedge ratios become smaller as te standard assumptions are relaxed. Wen brokerage fees are introduced edge ratios drop quickly, turning to zero at iger fees and lower targets. Bot iger fees and lower target returns cause edge ratios to drop, but it appears tat lower targets ave a greater impact tan iger fees. Using te variance as te risk measure, te optimal edge 11 In previous studies, te optimal edge ratios in a downside risk framework ave been estimated between 0.7 and 1.0 wit target returns set to zero wic corresponds to our estimate, 0.86.

10 9 ratio reaces a low of 0.54 wit te igest level of brokerage fee. In te presence of downside risk, te optimal edge ratio can reac about 0.6 in te standard model, and quickly approaces zero even wen low brokerage fees are combined wit lower targets. Te introduction of an alternative investment in te model at lowest level (S A = 0.10) does not markedly cange te edge ratios developed under te standard model. However, as te level of te alternative investment increases LPM edge ratios increase, particularly at lower target returns. For example, wit te target return set to te 10 t, te optimal edge ratio increases from 0.69 to 1.12 as te farmer s sare of wealt invested in an alternative asset increases from 10% to 50%. Increases in te optimal futures positions may reflect a lowering of portfolio risk wen te sare of te alternative investment increases, particularly as te correlation between futures-s&p500 and cas-s&p500 returns decline. Te opportunity costs of not edging are presented in Table 3 for te standard model. Under bot te variance and te LPM risk measures, opportunity costs are small for edge ratios close to te optimal level, indicating tat minor departures from te optimal edge ratio are not penalized severely. Comparing te risk measures, opportunity costs of not edging are lower in te presence of downside risk, wic is expected since downside risk edge ratios are lower tan mean-variance edge ratios. For example, at =0 in a variance context, te igest opportunity cost exists, 7.28% of te initial wealt, but tis drops off quickly to 0.24% at te 5 t in te downside risk framework. Relatively low opportunity costs for downside risk measures are also observed wen low brokerage fees are introduced in te model (Figure 1). At te 25 t, wen brokerage fees are equal to or smaller tan 0.125%, te opportunity costs of placing sub-optimal edge ratios in te presence of downside risk barely surpass 1%. More generally, te downside risk framework implies very low opportunity costs of placing sub-optimal edges wen target returns and brokerage fees are low, suggesting tat farmers are not penalized by edging at a sub-optimal ratio or not edging at all. In te presence of iger brokerage fees, te opportunity costs of edging cange substantially. Wit brokerage fees of 0.5%, opportunity costs increase monotonically and farmers are penalized eavily wen teir edge deviates from its optimal zero value. Table 3. Opportunity Costs of Hedging at te 4-week orizon Standard Model (annual return as a percentage of initial wealt) CRRA CRRA downside risk wit target set to: Meanvariance zero 50 t 25 t 10 t 5 t = = = = = = = = = = =

11 10 Figure 1. Opportunity Costs of Hedging at te 4-week Horizon (annual return as a percentage of initial wealt) brokerage fees target = 25 t target = 5 t opportunity cost 7% 6% 5% 4% 3% 2% b= b= b=0.005 opportunity cost 7% 6% 5% 4% 3% 2% b= b= b= % 1% 0% edge ratio 0% edge ratio In te presence of an alternative investment, te opportunity costs of placing sub-optimal edges are small. Altoug te introduction of alternative assets in te farmer s portfolio often leads to iger optimal edge ratios, te opportunity cost of not edging (=0) never reaces more tan 3% (Figure 2). Wen 50% of te farmer s wealt is invested in alternative assets te case tat yields te igest edge ratios and te target return is set to eiter te 50 t or te 25 t s, te opportunity cost of not edging is 0.97% and 0.33% respectively. For lower targets, te opportunity cost of not edging is almost zero. Wile te opportunity cost can reac nearly 3.0% at te lowest level of investment and te 50 t - target, te opportunity cost of edging barely reaces 1% at te 25 t and lower targets. Figure 2. Opportunity Costs of Hedging at te 4-week Horizon (annual return as a percentage of initial wealt) alternative investment S A = 0.10 S A = % 3.0% opportunity cost 2.5% 2.0% 1.5% 1.0% 0.5% 50t perc. 25t perc. opportunity cost 2.5% 2.0% 1.5% 1.0% 0.5% SA = 0.1 SA = % edge ratio 0.0% edge ratio

12 11 Wen brokerage fees are introduced togeter wit te alternative investment, optimal edge ratios are quickly driven towards zero, and similar to te previous discussion about te effects of brokerage fees on optimal edge ratios, te opportunity cost of not edging becomes zero and te opportunity cost of actually edging becomes relatively ig. Wen 50% of te farmer s wealt is invested in alternative assets and te targets are set to te 25 t and te 5 t s, te opportunity cost of not edging is zero in bot cases, wile te opportunity cost of edging increases as iger edge ratios are adopted (Figure 3). Figure 3. Opportunity Costs of Hedging at te 4-week Horizon (annual return as a percentage of initial wealt) brokerage fees and alternative investment (S A = 0.50) target = 25 t target = 5 t opportunity cost 3.5% 3.0% 2.5% 2.0% 1.5% 1.0% b= b= b=0.005 opportunity cost 3.5% 3.0% 2.5% 2.0% 1.5% 1.0% b= b= b= % 0.5% 0.0% edge ratio 0.0% edge ratio SUMMARY, DISCUSSION AND IMPLICATIONS Tis paper analyzed ow edge ratios and te opportunity costs of placing sub-optimal edges vary as a downside risk measure is introduced in te presence of transaction costs and alternative investments. Te findings indicate tat downside risk edge ratios can differ substantially from te standard mean-variance edge ratios at low targets and wen te standard assumptions are relaxed. Altoug it migt ave been expected tat low targets would automatically reduce edge ratios our findings indicate tat tis is not te case. Hedge ratios in te standard model can increase at lower targets as te correlations among te variables in te downside portion of te return distribution cange. Tese results support Lien and Tse (2002), and Demirer and Lien s (2003) discussion of te conceptual properties of te minimum-lpm edge ratio, and Lien and Tse (2000) and Turvey and Nayak s (2003) empirical evidence using te minimum-lpm model tat optimal edge ratios do not necessarily decrease as target returns are set to lower levels. Te introduction of transaction costs appears to ave te largest effect on te optimal edge ratio, particularly at lower target return levels. Using transaction costs tat existed near te beginning of te sample period, edge ratios decline quickly to zero as te target return declines. Wit more current transaction costs, edge ratios decline towards zero but not as rapidly. In te presence of alternative investment opportunities, edge ratios are not strongly affected

13 12 relative to te standard model at lower levels of investment. However, edge ratios increase as te level of te alternative investment increases and te target declines. Wen alternative investments and transaction costs are introduced simultaneously, te effect on te edge ratios is dramatic and te optimal edge ratios are driven to zero most quickly, particularly as te target returns decline. Wit regards to te opportunity cost of not edging te results are clear. In te presence of non-zero optimal edges, te opportunity cost of not edging is small. In te presence of a zero optimal edge, wic is primarily driven by iger transaction costs, te opportunity cost of actually edging increases quickly, particularly at lower target returns. In tese situations, tere appears to be little incentive for farmers to edge. Recall Lence (1996) demonstrated tat edge ratios for a farmer can cange dramatically wen te assumptions used to estimate minimum-variance edge ratios are relaxed. He found tat te inclusion of transaction costs, alternative investments and stocastic production can cause minimum-variance and maximum-utility optimal edges to differ substantially, and reduce te level of farmer edging. Cen, Lee and Sreta (2003) questioned te robustness of Lence s findings due to te utility function, te specifics of te return distributions, and te definition of risk measure used in te analysis. How do our findings add to tis dialogue? Our results, based on a CRRA utility function, te most recent returns for cas, futures and te S&P500, te LPM measure of downside risk, are quite compatible wit Lence s conclusions and implications. Even toug we do not consider te stocastic production case, our results clearly demonstrate te sensitivity of edge ratios to deviations from te minimum- variance assumptions. Furter, in our simulations te farmer as little incentive to edge, and even wen te incentive to edge emerges te opportunity cost of not edging is relatively small. Our findings also support te notion meanvariance utility-maximizing edge ratios sould be used wit caution in te presence of downside risk. Variance-based edge ratios are close to downside risk-based edge ratios only under specific conditions. In our analysis, tis occurs wen te target return is set to eiter zero or te 50 t of te distribution and te standard assumptions old. Wile variance-based edge ratios are easier to calculate tan downside-risk-based edge ratios, teir results tend to differ dramatically wen more realistic models are used. Finally, our findings migt elp explain te observed limited use of futures contracts for edging by soybean and grain farmers. If producers are intensely concerned wit downside risk and transaction costs are not negligible, ten te opportunity costs of not edging using futures contracts may be small and edge ratios may be close to zero. Interestingly, transaction costs wic ave been declining in importance only measure brokerage fees. Clearly, oter costs including initial and maintenance margin deposits and te opportunity of following futures markets exist wic can furter reduce te motivation of producers to edge. However, te recent increase in ig-volume, larger grain producers may reduce per unit transaction costs and make edging more attractive for tese producers. REFERENCES Camberlain, G. (1983). A caracterization of te distributions tat imply mean-variance utility functions. Journal of Economic Teory, 29,

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