Mean-Swap Variance Hedging and Efficiency

Transcription

1 Mean-Swap Variance Hedging and Efficiency Bingxin Li a and Zhan Wang b January 15, 2018 Abstract This paper develops a new theoretical approach to calculate the optimal hedge ratio based on the mean-swap variance optimization. Using data both of the S&P 500 index futures and crude oil futures, we find that minimizing-swap variance hedging shows higher efficiency, especially during market downturns, when compared with the traditional minimizing-variance hedging. T JEL: G11, G13 Keywords: Futures Market, Hedging, Hedge Ratio, Swap Variance a Bingxin Li, Assistant Professor of Finance Department and Center for Innovation in Gas Research and Utilization (CIGRU), West Virginia University; b Zhan Wang, Finance doctoral student, West Virginia University. 1

2 I. Introduction An essential function of futures contracts is hedging, in which investors form their hedging portfolio using futures contracts and its underlying asset to reduce the risk of future price fluctuation. Previous studies on hedging mainly focus on two issues: (1) the mechanism of deriving hedge ratios; and (2) the efficiency of this hedge ratio to minimize investors risk. Hedge ratio measures the number of futures contracts being purchased or sold for each unit of underlying asset. Several objective functions have been studied in the derivation of optimal hedge ratio. The principle method is from Johnson (1960) who demonstrates that hedging through trading futures contracts is a process used to reduce risks of adverse price movements, and builds an optimal hedging model based on minimizing variance of the hedged portfolio and/or maximizing investors expected utilities. After that, Ederington (1979) and Myers and Thompson (1989) develop similar minimizing variance hedging methods using simple regression (OLS) approaches. The nature of hedging minimizes associated risks however ignores the expected return of the hedged portfolio. Therefore, this strategy in general is inconsistent with the mean-variance framework unless the individuals are infinitely risk-averse (risk averse is so big that expected returns are negligible) or the futures price follows a pure martingale process (expected return is very small and can be ignored). Literature which tries to improve overall portfolio performance incorporates both the expected return and the variance of hedged portfolio and maximizes expected utility functions (Howard and D Antonio 1984, Cecchetti, Cumby and Figlewski 1988, Hsin, Kuo and Lee 1994). However, this method still faces the same weakness as the mean-variance framework, which is based on quadratic utility function or jointly symmetric return distribution. 2

3 Other models are built to eliminate these strict assumptions regarding utility function and return distribution. Cheung, Kawn, and Yip (1990), Kolb and Okunev (1992, 1993), Lien and Luo (1993), Shalit (1995) and Lien and Shaffer (1999) apply a method based on minimizing meanextend-gini coefficient, which is consist with the concept of stochastic dominance and then in the framework of expected utility maximization. Moreover, Crum, Laugnhumm, and Payne (1981), De Jong, De Roon, and Veld (1997), Lien and Tse (1998, 2000) and Chen, Lee, and Shrestha (2001) compute hedge ratios based on the generalized semi-variance or lower partial moments minimization. They demonstrate that these hedge ratios have another attractive feature that is consistent with the risk perceived by managers who cares more about potential loss, because semivariance and lower partial moment emphasize on the return variations below the target return. Hung, Chiu and Lee (2006) propose a related hedge ratio that minimizes the Value-at-Risk of the hedged portfolio. This hedge ratio equals to minimizing variance hedge ratio if the futures price follows a pure martingale process. In this study, we develop a new framework, minimizing swap variance (SwV), to investigate the dynamics of optimal hedge ratios and the subsequent performance of hedged portfolios. Jiang and Oomen (2008) show that swap variance is a risk measure which goes beyond the second moment (variance) and combines the second moment and all higher moments. Chow and Wang (2017) formally redefine swap variance as a generalized measure of risk and uncertainty, alternative to variance. Mathematically, it equals twice of the expected difference between arithmetic and logarithmic returns and it is equivalent to a polynomial combination of all returnmoments. So swap variance is a more comprehensive risk proxy than variance, and mean-swap variance optimization is consistent with expected utility theory and stochastic dominance without restrictions on the form of utility function and return distribution. 3

4 Using S&P 500 index futures and crude oil futures data, we find that on average the difference of hedging effects under minimizing variance and minimizing swap variance methods is very small during normal time, but it can be much larger when market shock occurs, for example during the 2008 financial crisis. Specifically, the small differences of hedges ratios between minimizing-variance and minimizing-swap variance have significant negative coefficients on the market volatility index (VIX). When we incorporate expected return into hedging mechanism, the difference between the two methods becomes very large. On the equity market, mean-swap variance hedging could pursue higher expected return and higher potential gain, but it also exposes to more risks. On the crude oil market, it is more efficient in term of both returns and risks, and it is a better method for reasonable level of aggregate market risk aversion. This paper is organized as follows. Section 2 explains the application of mean-swap variance theory in hedging activities, and develops the minimizing swap variance and optimal mean-swap variance hedging methods. Section 3 explains data used in this study, tests the two new hedging methods, and compares the hedging efficiencies of these new methods with the traditional mean-variance methods. Section 4 concludes. II. Mean-Swap Variance Hedging The basic concept of forming a hedging portfolio is holding the spot underlying asset and the corresponding futures contracts at the same time to reduce the risks of the hedged portfolio. Specifically, consider a portfolio consisting of one unit of long position in spot market and h units of short position in the futures market. The return on the hedged portfolio is given by RR h = RR ss hrr ff (1) 4

5 where RR h, RR ss and RR ff are the returns of the hedging portfolio, underlying spot asset and futures contract, respectively. h is the so-called hedge ratio. In minimum-variance framework, hedge ratio can be easily solved as, h = CCCCCC(RR ss,rr ff ), by making the variance of return on hedged portfolio the smallest (Johnson 1960). VVVVVV(RR ff ) In minimum-swap variance framework, we follow Chow and Wang (2017) and use swap variance instead of variance as a more general and realistic risk measure. Swap variance is defined as twice of the expected difference of arithmetic and logarithmic returns adjusted by the mean of the arithmetic returns. SwV = 2[EE(RR rr) dddd] (2) where RR is the one-period simple rate of return, rr is the one-period log return, rr = llll(1 + RR), with RR rr 0. E(.) is the expectation operator, μμ is the expected return of RR, and dddd = μμ llll(1 + μμ). given by Therefore, the optimal hedge ratio of minimizing swap variance of the hedged portfolio is min h SSSSSS(RR h ) (3) where SSSSSS(RR h ) is the swap variance of returns on the hedge portfolio in equation (1). The first order condition, from a direct calculation, is given by dssssss(rr h ) dh = EE RR ffrr ss hrr ff RR ss hrr ff = 0 (4) Additionally, the second order condition is given by, dd dh dssssss(rr h ) = EE 2, and it is nonnegative. Therefore the set of minimum-swv hedge ratio is unique. 5 dh RR ff 2 (1+RR ss hrr ff )

6 Various studies show the importance of expected return on the hedge ratios that the optimal hedge ratio should incorporate both risk and return to be consistent with the mean-variance framework (Howard and D Antonio 1984, Cecchetti, Cumby and Figlewski 1988, Hsin, Kuo and Lee 1994). The optimal hedge ratio is given by maximizing the mean-swap variance utility function, max h EE(RR h) λλ 2 SSSSSS(RR h) (5) where λλ is the risk aversion parameter that depends on investor s risk attitude. III. Data and Empirical Results 1. Data and summary statistics We use daily spot and futures prices of S&P 500 index and WTI Crude Oil. S&P 500 index spot price is downloaded from CRSP. S&P 500 index futures price, crude oil spot price, and crude oil futures price are obtained from the Chicago Mercantile Exchange (CME). The future contracts used in this study are the contracts with the nearest delivery dates. We make adjustments for rollover based on the principal on the CME website 1. The sample period of S&P 500 index covers from April 21, 1982 to August 12, 2016, and the sample period of Crude Oil futures covers from March 29, 1990 to August 12, The summary statistics of S&P 500 index futures/spot and WTI Crude Oil futures/spot are reported in Table 1. [Insert Table 1 here] From Table 1 it is obvious that the swap variances of all return series are very close to the variances of the returns. This is because the higher moments or the tails of the return distribution 1 According to CME principal, the rollover date is 8 days before the announced expiration date. 6

7 are much smaller in magnitude than variance and variance dominants the swap variance, with more than 98.5% of the swap variance. Although the equity market and crude oil market have similar magnitude as of average returns, the variance on the crude oil market is much larger than that on the equity market, suggesting the crude oil market is more risker. And the absolute values of skewness and kurtosis are both larger on the equity market than those on the crude oil market. The kurtosis of the S&P 500 futures return is twice of that of the S&P 500 spot returns, however the kurtosis of the crude oil futures and spot returns are similar in magnitude. The above characteristics shows the distinct natures of these two markets, which will also have an impact on the hedging performance. 2. Minimizing-variance Hedge and Minimizing-swap variance Hedge To investigate the optimal hedge ratios, we develop a two-step optimization process. In the first step, we calculate the minimizing-variance and minimizing-swap variance hedge ratios using returns from previous 60 consecutive trading days. [Insert Table 2 here] Table 2 reports the average hedge ratios of the S&P 500 index and Crude Oil index over the entire sample period. From Table 2, we can tell that the difference between the hedge ratios of minimizing-variance and minimizing-swap variance over a long period is very small, and the correlation between the hedge ratios is also very high (0.9983). This again confirms that variance and swap variance are very close on average. Figure 1 and Figure 2 show the moving patterns of hedge ratios on the S&P 500 and Crude Oil markets over time. Panel A plots the minimizing-variance hedge ratio, Panel B plots the minimizing-swap variance hedge ratio, and Panel C plots the difference between the above two 7

8 hedge ratios. From Panel A and Panel B, we observe that hedge ratios increase when there were market shocks, such as in 1998 and Although Panel A and Panel B show very similar patterns, we find that in Panel C the difference between hedge ratios of minimizing-variance and minimizing-swap variance are time varying and volatile. To further test the dependence of this difference of hedge ratios on market conditions, we run regression of the difference of hedge ratios on the market volatility index, the VIX index. [Insert Figure 1 and Figure 2 here] Table 3 report the regression results of the difference between the minimizing-swap variance hedge ratio and the minimizing-variance hedge ratio over the CBOE VIX index. The beta coefficients for the S&P 500 index and the crude oil are both negative and significant, with the coefficient for the S&P 500 index is , and for the Crude Oil. It means the minimizing-swap variance hedge ratio is positively deviated from the minimizing-variance hedge ratio when market is stable (small VIX), and the minimizing-swap variance hedge ratio is negatively deviated from the minimizing-variance hedge ratio when market is volatile (large VIX). It suggests the traditional minimizing-variance hedge is over hedged when market is volatile. When considering the generalized risk measures, swap variance, the optimal hedge ratio suggests to hedge less than the minimizing-variance hedge ratio under bad market conditions. [Insert Table 3 here] Then we analyze the post hedging performance for the hedged portfolios. Table 4 reports the average post one-day return of hedged portfolios of S&P 500 index and WTI crude oil formed by minimizing-variance and minimizing-swap variance. From Table 4, the difference of post daily returns between minimizing-variance and minimizing-swap variance hedge portfolios of S&P 500 8

9 index is very small. The main reason is that S&P 500 index is well diversified, and its return distribution is approximately normal. So the swap variance tends to be very close to variance, and the hedge ratios under these two processes are almost the same. The difference of subsequent daily returns between the minimizing-variance hedged portfolio and the minimizing-swap variance hedged portfolio in the crude oil market is also very small. The reason again is that the asymmetry component on the crude oil market is very small. Its influence on hedging may be dominated by variance if we minimize swap variance as a single variable. [Insert Table 4 here] To further compare the relationship of the optimal hedge ratio and post hedging performance with market conditions, in Table 5 we compare annual averages of optimal hedge ratios and annual averages of subsequent one-day returns of the hedged portfolio on the equity market and the crude oil market. We report results for both the minimizing-variance and minimizing-swap variance process each year. From Table 5, the annual differences of both hedge ratios and post hedging returns between minimizing-swap variance and minimizing-variance are still very small in most years, however in 2008 when financial market crashed and the crude oil prices were volatile, the changes for both optimal hedge ratio and post hedge returns are large. This observation is stronger for the crude oil market. The relatively large difference in 2008 suggests minimizing-swap variance hedging can capture market asymmetries of return distribution, especially during market downturns when higher moments of return distribution become obvious. When market is volatile, the difference between variance and swap variance become larger. Minimizing-variance hedging omitted the risks in the tails of return distribution and thus leaves the portfolio not efficiently hedged. [Insert Table 5 here] 9

10 Another observation of Table 5 is that although minimum-swap variance hedging is more efficient, it does not necessarily lead to higher subsequent returns of the hedged portfolios. This is due to the nature of hedging which is to reduce risks of the portfolio and not to maximize expected returns. Next subsection we will discuss the optimal utility hedging and post hedging performance. 3. Optimal Mean-Variance Hedge and Mean-Swap Variance Hedge To incorporate both risk and return into the hedge ratio, we use mean-variance and meanswap variance hedging to compare hedging efficiencies between the two frameworks. We use three different values of risk aversion parameter, 1, 2 and 3, to calibrate the effects of hedging. Table 6 reports the average hedge ratios of the S&P 500 index and WTI crude oil under the above two methods. [Insert Table 6 here] From Table 6, we notice that the magnitude of hedge ratios is much larger than the magnitude of hedge ratios of minimizing risk hedging reported in Table 2. It means investors may hedge more to pursue higher expected returns. In Table 2, all hedge ratios are positive and slightly smaller than one, suggesting it is optimal to short futures contracts to hedge the underlying spot market. However, in Panel A of Table 6 all hedge ratios are negative, suggesting in order to achieve higher risk adjusted returns, one should long both underlying asset and futures contracts. This is not consistent with the traditional purpose of hedging which cares reducing risks only, however it is reasonable given the fact that the overall equity market realizes positive returns (historical average 12% per year since 1926). Moreover, the differences of hedge ratios between optimal mean-variance and optimal mean-swap variance are much larger than the difference when only consider risks, while the 10

11 correlation between the hedge ratios become a little smaller. Comparing hedge ratios with different risk aversion parameters, investors tend to long more (short less) future contract if they have lower risk tolerance. On crude oil market, investors short more future contracts than they do in equity market, mainly due to the characteristics of the commodity market where hedging is one of the primary functions. [Insert Table 7 here] We also report the subsequent one-day return for the hedged portfolios. Table 7 reports the summary statistics of the post one-day return of hedged portfolios on the S&P 500 index and the WTI crude oil formed by optimal mean-variance and optimal mean-swap variance hedging. From Panel A of Table 7, we observe that on the equity market optimal mean-swap variance hedged portfolios have higher returns, standard deviation, skewness, and kurtosis than the optimal meanvariance hedged portfolios, which suggests mean-swap variance hedging could pursue higher expected return and higher potential gain, but it also exposes the portfolio to more risks. The results on the crude oil market are different from those on the equity market. When the risk aversion parameters are set to 2 or 3, optimal mean-variance hedging seems more efficient than optimal mean-swap variance hedging in terms of both higher returns and lower risks. Li (2017) estimates the risk aversion coefficient on the crude oil market using both WTI crude oil futures and options data and find in general the risk aversion coefficient with the nearest maturity is close to one, which is much lower than the risk aversion coefficient estimated on the equity market. 2 This is mainly due to the fact that a very large portion of traders on the crude oil market are 2 For example, Guo and Whitelaw (2001) estimate a coefficient of relative risk aversion of 3.52 on the equity market; Bliss and Panigirtzoglou (2004) find that the relative risk aversion implied by a power utility function using S&P 500 data is between 3.37 to

12 financial speculators who do not require physical crude oil commodity but only trade for profits or portfolio diversification purposes. They have high risk tolerance and would like to accept lower risk premium to hold the opposite positions of the hedgers. So the result with λλ = 11 in Panel B of Table 7 is more plausible and important for the crude oil market. When the risk aversion coefficient is 1, optimal mean-swap variance hedged portfolio shows higher expected return, lower standard deviation and higher skewness. So optimal mean-swap variance hedging should be a better method to maximize expected utility of investors on the crude oil market. V. Conclusion This paper develops two new hedging methods based on the mean-swap variance portfolio theory. The minimizing-swap variance method shows that investor pursue the lowest total risks through hedging activities. The optimal mean-swap variance method is consistent with investors preference that maximize their expected utility, which is consist of expected return and swap variance. Using spot and futures contracts of S&P 500 index and WTI crude oil, we compare the new hedging methods with the traditional mean-variance hedging methods. The differences of hedging performance between minimizing-variance and minimizing-swap variance methods are very small during normal time, however it is much larger when market shock occurs. Moreover, these small differences have significant negative correlation with market condition, which is measure by the CBOE volatility index. When we incorporate expected return into hedging methods, the difference between optimal mean-variance hedging and optimal mean-swap variance hedging becomes obvious. 12

13 Mean-swap variance hedging could pursue higher expected return and higher potential gain, although the hedged portfolio is exposed to more risks on the equity market. For the crude oil market, mean-swap variance hedging is superior than mean-variance hedging when consider realistic risk aversion coefficient. Overall the swap variance risk measure condenses the risks in one number and is superior than the variance risk measure. Therefore, minimizing-swap variance hedging is more efficient than minimizing-variance hedging. This is more obvious for the crude oil market than for the equity market since hedging is one primary objective on the commodity markets. 13

14 Reference Bliss, Robert R. and Panigirtzoglou, Nikolaos. "Option-implied risk aversion estimates. " The Journal of Finance 1 (2004), Cecchetti, Stephen G., Robert E. Cumby, and Stephen Figlewski. "Estimation of the optimal futures hedge." The Review of Economics and Statistics (1988): Chang, Chiao-Yi, Jing-Yi Lai, and I-Yuan Chuang. "Futures hedging effectiveness under the segmentation of bear/bull energy markets." Energy Economics 32, no. 2 (2010): Chen, Sheng-Syan, Cheng-few Lee, and Keshab Shrestha. "Futures hedge ratios: a review." In Encyclopedia of Finance, pp Springer US, Ederington, Louis H. "The hedging performance of the new futures markets." The Journal of Finance 34, no. 1 (1979): Figlewski, Stephen. "Hedging performance and basis risk in stock index futures." The Journal of Finance 39, no. 3 (1984): Ghosh, Asim. "Hedging with stock index futures: Estimation and forecasting with error correction model." Journal of Futures Markets 13, no. 7 (1993): Ghosh, Asim. "Cointegration and error correction models: Intertemporal causality between index and futures prices." Journal of futures markets 13, no. 2 (1993): Guo, Hui and Whitelaw Robert. "Uncovering the risk-return relation in the stock market. " The Journal of Finance 61 (2006), Herbst, Anthony F., Dilip D. Kare, and John F. Marshall. "A time Varying, Convergence Adjusted Hedge Ratio Model." Advances in Futures and Options Research 6 (1993): Holmes, Phil. "Stock index futures hedging: hedge ratio estimation, duration effects, expiration effects and hedge ratio stability." Journal of Business Finance & Accounting 23, no. 1 (1996): Howard, Charles T., and Louis J. D'Antonio. "A risk-return measure of hedging effectiveness." Journal of Financial and Quantitative Analysis 19, no. 1 (1984): Li, Bingxin. "Speculation, risk aversion, and risk premiums in the crude oil market. " Working paper (2017). Johnson, Leland L. "The theory of hedging and speculation in commodity futures." The Review of Economic Studies 27, no. 3 (1960): Junkus, Joan C., and Cheng F. Lee. "Use of three stock index futures in hedging decisions." Journal of Futures Markets 5, no. 2 (1985): Kenourgios, Dimitris, Aristeidis Samitas, and Panagiotis Drosos. "Hedge ratio estimation and hedging effectiveness: the case of the S&P 500 stock index futures contract." International Journal of Risk assessment and management 9, no. 1-2 (2008): Lien, Donald, Keshab Shrestha, and Jing Wu. "Quantile Estimation of Optimal Hedge Ratio." Journal of Futures Markets36, no. 2 (2016): Pennings, Joost ME, and Matthew TG Meulenberg. "Hedging efficiency: a futures exchange management approach." Journal of Futures Markets 17, no. 5 (1997):

15 Table 1. Summary statistics of S&P 500 index and WTI Crude Oil spot/futures returns This table reports the mean, variance, swap variance, skewness, kurtosis of the spot and futures returns of S&P 500 index and WTI Crude Oil. The correlations between spot returns and futures return for each index is also reported. The sample period of S&P 500 index covers from April 21, 1982 to August 12, 2016, and the sample period of Crude Oil futures covers from March 29, 1990 to August 12, Mean Variance SwV Skewness Kurtosis Correlation (%) ( 10 4 ) ( 10 4 ) S&P 500 Spot S&P 500 Futures WTI Crude Oil Spot WTI Crude Oil Futures

16 Table 2. Optimal Hedge Ratios under Minimizing-variance and Minimizing-swap variance This table reports the average and standard error of the optimal hedge ratios of S&P 500 index and WTI crude oil under minimizing-variance and minimizing-swap variance hedging and the correlation between two hedge ratios Panel A. S&P 500 Index Mean Standard Error h VV Correlation h SSSSSS Panel B. Crude Oil markets Mean Standard Error h VV Correlation h SSSSSS

17 Table 3. Difference of Hedge Ratio and Volatility Index The table reports the regression results of differences between minimizing-variance hedge ratio and minimizing swap variance hedge ratio on the VIX index. DDDDDDDD = αα + ββ( VVVVVV ) + εε. Panel A and Panel B 100 show the results of S&P 500 index and the WTI crude oil, respectively. Panel A: S&P 500 Index Coefficients αα (9.86) ββ (-6.84) RR Observation 6646 Panel B: Crude Oil Market Coefficients αα (21.33) ββ (-18.13) RR Observation

18 Table 4. One-day Returns of Post-hedging Portfolios under Minimizing-variance and Minimizing-swap variance This table reports the minimum, mean medium, maximum and standard deviation of post-hedging portfolio returns for S&P 500 index and WTI crude oil under minimizing-variance and minimizingswap variance hedging, as well as the correlation of returns between two hedged portfolios. Panel A. S&P 500 Index Min (%) Mean (%) Medium (%) Max (%) Std. Dev. (%) RR VV Correlation RR SSSSSS Panel B. Crude Oil market Min (%) Mean (%) Medium (%) Max (%) Std. Dev. (%) RR VV Correlation RR SSSSSS

19 Table 5. Comparison of Optimal Hedge Ratios and Post Hedge Return Every Year This table reports the comparison of annual averaged hedge ratios and post hedging returns in the S&P 500 index and Crude Oil market between minimizing-variance and minimizing-swap variance process each year. Panel A reports the annual average of hedge ratios of the S&P 500 index, Panel B reports the annual average of subsequent one-day return of the hedged portfolio (in %) of the S&P 500 index, Panel C reports the annual average of hedge ratios in the Crude Oil market, and Panel D reports the annual average of subsequent one-day return (in %) in the Crude Oil market. Panel A. Hedge Ratio of the S&P 500 index YEAR Minimizing-SwV Minimizing-MV % CHANGE % % % % % % % % % % % % 19

20 Panel B. Post Hedge Return of the S&P 500 index YEAR Minimizing-SwV Minimizing-MV % CHANGE % % % % % % % % % % % % Panel C. Hedge Ratio in the Crude Oil Market YEAR Minimizing-SwV Minimizing-MV % CHANGE % % % % % % 20

21 % % % % % % Panel D. Post Hedge Return in the Crude Oil Market YEAR Minimizing-SwV Minimizing-MV % CHANGE % % % % % % % % % % % % 21

22 Table 6. Hedge Ratios under Optimal Mean-Variance and Mean-Swap Variance This table reports the average and standard error of the optimal hedge ratios of S&P 500 index under mean-variance and mean-swap variance hedging and the correlation between two hedge ratios. Panel A. S&P 500 Index Mean Standard Error λλ = 11 h VV Correlation h SSSSSS λλ = 22 h VV Correlation h SSSSSS λλ = 33 h VV Correlation h SSSSSS

23 Panel B. Crude Oil market Mean Standard Error λλ = 11 h VV Correlation h SSSSSS λλ = 22 h VV Correlation h SSSSSS λλ = 33 h VV Correlation h SSSSSS

24 Table 7. One-day Returns of Post-hedging Portfolios under Optimal Mean-Variance and Mean-Swap Variance This table reports the mean, standard deviation, skewness and kurtosis of THE post-hedging portfolio returns for S&P 500 index and Crude Oil futures under optimal mean-variance and mean- swap variance hedging, as well as the correlation of returns between two hedged portfolios. The risk aversion parameter is 1, 2 and 3. Panel A. S&P 500 Index Mean (%) Std. Dev. (%) Skewness Kurtosis λλ = 11 RR VV Correlation RR SSSSSS λλ = 22 RR VV Correlation RR SSSSSS λλ = 33 RR VV Correlation RR SSSSSS

25 Panel B. Crude Oil market Mean (%) Std. Dev. (%) Skewness Kurtosis λλ = 11 RR VV Correlation RR SSSSSS λλ = 22 RR VV Correlation RR SSSSSS λλ = 33 RR VV Correlation RR SSSSSS

26 Figure 1. Moving Pattern of the Optimal Hedge Ratios for S&P 500 Index Panel A: Minimizing-Swap Variance Hedge Ratios Panel B: Minimizing-Variance Hedge Ratios

27 Panel C: Difference Between Hedge Ratios of Minimizing-Swap Variance and Minimizing- Variance /14/ /29/ /7/1986 2/27/2007 6/27/ Figure 1 shows the moving patterns of hedge ratios for S&P 500 index. Panel A shows the minimizingswap variance hedge ratio, Panel B shows the minimizing-variance hedge ratio, and Panel C shows the difference between the two hedge ratios. 27

28 Figure 2. Moving Pattern of the Optimal Hedge Ratios on the Crude Oil Markets Panel A: Minimizing-Swap Variance Hedge Ratios Panel B: Minimizing-Variance Hedge Ratios

29 Panel C: Difference Between Hedge Ratios of Minimizing-Swap Variance and Minimizing- Variance /7/ /7/2005 3/24/ /16/1990 6/20/ /2/ Figure 2 shows the moving patterns of hedge ratios on the WTI crude oil market. Panel A shows the minimizing-swap variance hedge ratio, Panel B shows the minimizing-variance hedge ratio, and Panel C shows the difference between the two hedge ratios. 29