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Hedging downside risk: futures vs. options
International Review of Economics and Finance 10 (2001) 159 ± 169
Hedging downside risk: futures vs. options Donald Liena,*, Yiu Kuen Tseb a
Department of Economics, University of Kansas, 213 Summerfield Hall, Lawrence, KS 66045-2113, USA b National University of Singapore, Singapore Received 6 July 1999; received in revised form 3 February 2000; accepted 5 May 2000
Abstract In this paper, we compare the hedging effectiveness of currency futures vs. currency options on the basis of the lower partial moments (LPMs). The LPM measures an individual hedger's downside risk, as opposed to the two-sided risk measure. Two estimation methods are applied to estimate the optimal hedge ratio: the empirical distribution function method and the kernel density estimation method. We consider both methods for three currencies: the British pound, the Deutsche mark, and the Japanese yen. Currency futures are found to be a better hedging instrument than currency option. D 2001 Elsevier Science Inc. All rights reserved. JEL classification: F31; G11 Keywords: Currency futures; Currency options; Lower partial moments; Hedging
1. Introduction Given different derivative products, a hedger will choose the instrument with the highest hedging effectiveness. Using the mean±variance criterion, Chang and Shanker (1986) found that currency futures are better hedging instruments than currency options. Hancock and Weise (1994), however, showed that the optimal hedge positions for the S&P 500 index options and those for the S&P 500 futures lead to similar mean returns. In contrast, Benet and Luft (1995) found that the S&P 500 futures outperform the S&P 500 options in variance
* Corresponding author. Tel.: +1-785-864-3501; fax: +1-785-864-5270. E-mail address: [email protected] (D. Lien). 1059-0560/01/$ ± see front matter D 2001 Elsevier Science Inc. All rights reserved. PII: S 1 0 5 9 - 0 5 6 0 ( 0 0 ) 0 0 0 7 4 - 5
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reduction. Moreover, upon taking into account the transaction cost, futures have a larger excess return per unit risk than options. As options are designed to eliminate extreme downside risk, which is one-sided, comparisons based upon variance, which is a two-sided risk measure, are biased against options. In a recent survey, Adams and Montesi (1995) showed that corporate managers are mostly concerned about downside risk. Earlier, Petty and Scott (1981) found that many Fortune 500 firms identified risk as the probability of falling below a target return. To measure downside risk, Lee and Rao (1988) and Mao (1970) adopted the LPM approach. Hereby, downside risk is measured by a probability-weighted power function of the shortfall from a specific target return. In this regard, one tends to expect options to perform better in reducing downside risk. This conjecture, however, was rejected by Ahmadi, Sharp, and Walther (1986). Assuming a one-to-one hedge ratio, Ahmadi et al. showed that for a nontradable spot position, futures provide significantly more effective hedging than options when the target return is zero. Using simulated data and maintaining the one-to-one hedge ratio assumption, Korsvold (1994) demonstrated that if quantity risk (i.e., the uncertainty in the spot position) is incorporated, options dominate over futures. This result, however, is not convincing as it is based on simulated data. Adams and Montesi (1995) also found that corporate managers prefer to hedge the downside risk using futures rather than options, citing the large transaction cost in option trading as the main reason. In this paper, we compare the hedging strategy of currency futures and options based upon the lower partial moments (LPMs). While most previous works assume a naive oneto-one hedge ratio, this paper estimates the optimal hedge ratio that minimizes the LPM.1 Comparisons of hedging effectiveness are based upon these optimal hedge ratios. Following Lien and Tse (1999), nonparametric methods are adopted to estimate the optimal hedge ratio.2 The results are compared against the minimum-variance (MV) hedge strategy. The remaining part of this paper is organized as follows. In Section 2, we review the use of the LPM in the hedging framework. We highlight the possibility that futures may outperform options in reducing downside risk. Section 3 discusses two alternative methods for the estimation of the optimal hedge ratio that minimizes the LPM. These are the empirical distribution function method and the nonparametric kernel method. Data construction for the currency futures, currency options, and spot rates is discussed in Section 4. Three currencies are considered: the British pound, the Deutsche mark, and the Japanese yen. The empirical results are summarized in Section 5, where the hedging performances of the futures and options are compared. It is shown that currency futures are a better instrument for hedging either two-sided or downside risk. The paper is concluded in Section 6.
1 A notable exception is the recent paper by de Jong, de Roon, and Veld (1997). Their study, however, considered futures only. In addition, their results are based on the optimal hedge ratios estimated by the empirical distribution function method. In this paper, we consider the more efficient kernel estimation method as well. 2 Lien and Tse (1998) adopted a parametric approach to estimate time-varying LPM hedge ratios.
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2. Lower partial moments Consider a portfolio with a random return X. Let F(.) denote the distribution function of X and assume an individual has a target return c. An outcome larger than c is desirable. Thus, the individual faces only a one-sided risk (the downside risk), which occurs when X falls short of c. This risk can be measured by the LPM L(c,n,X ) defined in Eq. (1): Zc 1 L c; n; X c ÿ xn dF x; ÿ1
where n is a positive integer.3 Suppose that a hedger is endowed with an initial wealth W0 and a given nontradable spot position Q at time 0. At time 1, his wealth is W1 = W0 + DpQ, where Dp is the change in the spot price from time 0 to time 1. As Dp is unknown at time 0, the hedger faces a downside risk in W1. We measure this risk by L(c,n,W1). Suppose the hedger buys kQ futures contracts to hedge this risk. Then, his wealth Wf 1 at time 1 is: Wf 1 W0 Dp kDf Q;
2
where Df is the change in the futures price from time 0 to time 1. Let p0 and f0 denote the spot and futures prices, respectively, at time 0. Then, rp = Dp/p0 is the spot return and rf = Df/f0 is the futures return. Consequently, Eq. (2) can be rewritten as Eq. (20): 20 Wf 1 W0 rp qrf p0 Q; where ÿ q = ÿ kf0/p0 is the (adjusted) hedge ratio. Formulated in this way, the hedger faces a downside risk measured by L(c,n,Wf 1). Note that Wf 1 is a function of q. The hedger will choose q to minimize L(c,n,Wf 1). Apart from futures, options also serve as a hedging instrument. Assume the hedger sells qQ units of call options, the strike price of which is S, at a premium C. He has a short position if q > 0 and a long position if q < 0. If time 1 coincides with the maturity of the option, the hedger's wealth Wc1 at time 1 is expressed as Eq. (3): Wc1 W0 fDp qmin S ÿ p1 ; 0 qCgQ;
3
where p1 is the spot price at time 1. Let pc denote the market profit Dp + q min(S ÿ p1, 0) + qC. When q = 1, pc S ÿ p0 + C. Otherwise, there is no upper or lower bounds for pc. Thus, Wc1 incurs a downside risk. The above analysis applies when the option expires at time 1. Otherwise, the hedger's wealth Wd1 at time 1 is expressed as Eq. (4): Wd1 W0 rp grc p0 Q;
4
where rc is the return of the call option over the period from time 0 to time 1, and ÿ g = ÿ qC/ p0. The hedger will choose g to minimize L(c,n,Wd1). Note that there is a one-to-one monotonic transformation between Wf 1 (or Wd1) and mf (or md) where mf = rp + qrf and 3
In general, n can be any positive number as adopted in the Fishburn (1977) risk measure. For example, Korsvold (1994) considered the case of n = 1.5. In this paper, we only consider integer values of n.
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md = rp + grc. Thus, we can equivalently consider the LPMs of mf and md. This approach removes the effects of the initial wealth W0 and the nontradable spot position Q. 3. Estimation of the optimal hedge ratios We now consider the estimation of the optimal hedge ratio of futures contracts. Similar method applies to options. Note that the optimal hedge ratio is chosen to minimize L(c,n,mf), which can be rewritten as E{[max(0, c ÿ mf)]n}. The optimal hedge ratio q*, therefore, satisfies the following first-order condition: ÿnEfmax 0; c ÿ rp ÿ q rf nÿ1 rf g 0:
5
It can be shown that the second-order condition is always satisfied when n 2. Although Eq. (5) indicates that q* is a function of n and c, the effects of n and c on q* are undetermined (see Lien & Tse, 2000 for the details). Whenever the joint distribution of rp and rf is known, we can apply numerical methods to find the optimal hedge ratio. In empirical studies, however, the true distribution of rp and rf is unknown and must be estimated. In this paper, we adopt an indirect approach. We estimate the distribution of the portfolio return for any given q. Specifically, for a given q, we construct the data series for mf from the data of rf and rp, and then apply nonparametric methods to estimate the distribution of mf. The details are as follows. Let g(.) be a smooth probability density function, called the kernel function. Suppose we have a random sample of N observations of mf, say, m1, m2, . . ., mN, calculated from a given q. Using kernel method, the probability density of mf at a given point y, denoted by f ( y), is estimated using Eq. (6): f^ y 1=Nh
N X
g y ÿ mi h;
6
i1
where h (>0) is the bandwidth (Hardle, 1990). Silverman (1986) suggested that the choice of the kernel function has minimal effects on the density estimates. In this paper, we choose g(.) to be the density function of the standard normal random variable so that Eq. (7) is defined as: g z 2pÿ1=2 exp ÿz 2 =2;
7
and the LPM is estimated using Eq. (8): ^ n; mf L c;
Zc
c ÿ yn 1=Nh
N X
g y ÿ mi =hdy:
8
i1
ÿ1
After a change of variable from y to z = ( y ÿ mi)/h, we have: ^ n; m 1=N L c; f
N X i1
Q c; n; mi ;
9
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where cÿm Z i =h
Q c; n; mi
c ÿ hz ÿ mi n g zdz:
10
ÿ1
It is well known that the bandwidth h is critical in estimating the probability density. Following Silverman (1986), we determine the optimal size of h by the cross-validation method (see Lien & Tse, 2000 for details). Since we apply the kernel method to the return data, the optimal bandwidth depends upon the value of the given q. To reduce computational burden, we calculated a sample of optimal bandwidth for selected values of q that are expected to cover the optimal hedge ratio. The optimal bandwidth is then taken as the average of the sample. To sum up, beginning with a given hedge ratio, we construct the data series for the portfolio return. The kernel method and the optimal bandwidth are then applied to calculate the estimated LPM using Eqs. (9) and (10). We then adopt grid search method to find the optimal hedge ratio that produces the smallest estimated LPM. Kernel method is computationally time consuming. Empirical distribution function method provides an alternative (see Harlow, 1991; Price, Price, & Nantell, 1982). Herein, the LPM is estimated as follows (Eq. (11)): X ~ 1=N c ÿ mi n : 11 L c; n; mf mi c
When the bandwidth is very small, the kernel method and the empirical distribution function method provide similar estimates. Thus, the latter is a special case of the former. Reiss (1981) demonstrated that the kernel method has better statistical performance than the empirical distribution function method. In this paper, we consider both methods. 4. Data description We consider futures and options of the British pound, the Deutsche mark, and the Japanese yen. Daily closing prices of the currency futures contracts traded on the Chicago Mercantile Exchange were obtained from the Futures Industry Institute Data Center. For the options, daily trading data were provided by the Philadelphia Stock Exchange (PHLX). The spot exchange rate data were also extracted from the PHLX data files. In the empirical study, we consider a hedging horizon of one week, or more specifically 5 working days. This hedging horizon was chosen to alleviate the problem of thin trading for the options, and to make the selection of options more manageable. For each currency, we constructed a futures series with intervals of 5 working days. The nearest contract was selected, with roll-over occurring at approximately 10 days before the expiry of the contract. The return on the futures over the hedge interval was calculated as the differenced (logarithmic) futures price over the price of the futures contract at the time of the initialisation of the hedge. The same formula was used in computing the return of the spot and the options.
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At the roll-over, care was taken to ensure that the same contract was used to calculate the return over the hedge interval. The option files were compiled to form a reduced data set with trading dates that matched those of the futures. To form a hedged portfolio, we selected American call options with exercise prices close to the spot rates. Additional criteria were imposed. First, we gave preference to the nearby contracts. Secondly, when there were more than one contract satisfying the first criterion, we selected the one with the highest volume. In computing the returns of the options, care was taken to ensure that the same contract was used at the times when the hedge was initialised and lifted. Due to the problem of a thin market, however, there were circumstances in which no suitable options were available for some intervals. In such cases, no hedging was considered and the corresponding futures data were ignored. Apart from thin trading, data errors also contributed to the loss of observations. It should be noted that as the return distributions are assumed to be time invariant, the estimation in this paper applies to random samples of return observations rather than serially dependent observations. Thus, apart from the loss of efficiency due to missing observations, the existence of missing data should have no effects on the statistical properties of the estimates. We compiled the returns on the futures, options, and spots of the selected currencies. For the British pound, the data cover the period from January 1984 to October 1994, with 444 observations. For the Deutsche mark, the data cover the period from August 1983 to September 1994, with 526 observations. As for the Japanese yen, there are 424 observations covering the period from January 1984 to January 1997. Table 1 provides the summary statistics of the data. For all currencies, the spot and futures returns display similar statistical behaviour. The option returns, however, are much more volatile. While the variances of the spot and futures returns lie between 2 and 4, the variances of the option returns are much larger. None of the returns is significantly different from zero. Except for the spot and futures returns of the Deutsche mark, all series exhibit excessive skewness and kurtosis. The normality assumption can be easily rejected based upon the Bera±Jarque statistic.
Table 1 Summary statistics of returns on spot, futures, and option Currency
Security
Mean
Variance
Skewness
Kurtosis
BP DM JY BP DM JY BP DM JY
spot spot spot futures futures futures option option option
0.0474 0.0971 0.1252 0.0388 0.1207 0.1272 7.3047 2.3470 2.2618
3.1998 2.8656 2.3631 3.4516 2.9138 2.3923 7912.9 3890.0 6069.4
ÿ 2.2327 0.4298 5.3484 ÿ 3.5557 ÿ 0.3572 6.0286 37.458 27.599 22.221
11.804 4.635 16.331 16.044 3.872 18.259 125.44 84.61 43.15
Returns are expressed in percentage. The currency codes are: BP = British pound, DM = Deutsche mark, and JY = Japanese yen. The skewness and kurtosis are standardised.
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5. Empirical results We consider a list of possible target returns ranging from 0.0% to 2.0%.4 The empirical distribution function and the kernel density methods were applied to evaluate the LPMs with order n equal to 1, 2, and 3. The results from the kernel density method are summarized in Table 2. The empirical distribution function method generated qualitatively similar results, which are not reported here. While we report only the results of the kernel method using the sample average of optimal bandwidths,5 we also experimented with different bandwidths. We found that the optimal hedge ratios are robust to the choices of the bandwidth, although the resulting LPMs may not be. For each currency, the optimal hedge ratios of the futures and options display similar properties. However, across different currencies, the optimal hedge ratios behave differently. In the case of the British pound, we found that the optimal futures hedge ratio always decreases as the target return increases. That is, if the hedger becomes more optimistic (so that, for example, while initially 1% was not considered as a loss, now it is), fewer futures contracts will be sold against a unit of spot position. For the Deutsche mark, the optimal futures hedge ratio decreases with increasing target return when n = 1 and increases with increasing target return when n = 3. The ratio increases and then decreases when n = 3. The Japanese yen demonstrates yet another different case. Herein, the optimal futures hedge ratio decreases as the target return increases when n = 1. For the other two cases, the ratio first increases and then decreases. For the Deutsche mark and Japanese yen, the MV hedge ratio is almost always greater than the LPM hedge ratios. That is, a hedger concerned about downside risk will overhedge if the MV hedge ratio is followed. In the case of the British pound, the relationship between the MV and LPM hedge ratios depends upon n. When n = 3, the MV strategy almost always underhedges the downside risk. Due to its high volatility, currency options do not appear to be a useful hedging vehicle. In each case, the optimal hedge ratio is rather small. When n = 2 or 3, the optimal hedge ratio of the options always increases as the target return increases. That is, a more optimistic hedger will sell more call options. The effects of n on the optimal futures hedge ratio also vary across currencies. For the British pound, the optimal hedge ratio increases as n increases. On the other hand, the optimal hedge ratio for the Deutsche mark futures decreases when n increases except when c 1%. In the case of the Japanese yen, the optimal hedge ratio increases with increasing n except when
4
Most empirical works assume a zero target return. Here, we allow the target return to be positive. A positive target return seems intuitive if the derivatives are investment outlays. On the other hand, as the hedger attempts to reduce risk through derivative positions, he is willing to pay a cost. That is, he may be willing to accept a negative return in order to reduce the downside risk. In the empirical study, we considered some cases with negative targets. Though not reported here, the results pertaining to negative target returns are consistent with our main results. 5 We calculated the optimal bandwidths over a range of hedge ratios that are likely to cover the optimal value. The kernel estimates of the lower partial moments were then recalculated based on the maximum and minimum values of the range of optimal bandwidths.
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Table 2 Optimal LPM hedge ratios Currency
Target return (%)
Derivative
n=1
n=2
n=3
BP
0.0
futures options futures options futures options futures options futures options futures options futures options futures options futures options futures options futures options futures options
0.920 0.018 0.872 0.017 0.822 0.013 0.577 0.006 0.942 0.026 0.904 0.026 0.739 0.025 0.440 0.014 0.936 0.018 0.891 0.019 0.716 0.019 0.444 0.015
0.936 0.010 0.924 0.018 0.909 0.011 0.896 0.011 0.867 0.019 0.893 0.020 0.875 0.021 0.829 0.021 0.913 0.013 0.919 0.014 0.894 0.015 0.844 0.015
0.967 0.008 0.946 0.009 0.930 0.009 0.916 0.010 0.756 0.017 0.807 0.018 0.846 0.018 0.849 0.019 0.880 0.012 0.903 0.013 0.905 0.013 0.887 0.014
0.4 1.0 2.0 DM
0.0 0.4 1.0 2.0
JY
0.0 0.4 1.0 2.0
The figures in the table are the hedge ratios that give the lowest LPMs. They are estimated by the kernel method for the three currencies: BP = British pound, DM = Deutsche mark, and JY = Japanese yen. n is the order of the LPM. For comparison, the MV hedge ratios (i) using futures are: 0.917 (BP), 0.907 (DM), and 0.948 (JY), and (ii) using options are: 0.013 (BP), 0.022 (DM), and 0.016 (JY).
c = 0%. Thus, a hedger who cares more about large losses may sell more or fewer futures contracts to hedge the downside risk.6 There are more common features across currency options. For the British pound, the optimal hedge ratio for options decreases as n increases except for c = 2%. Similarly, the optimal hedge ratio for the Deutsche mark and Japanese yen options decreases with increasing n. Thus, a hedger who is more concerned about large losses will sell fewer call options. Table 3 compares the LPMs obtained with optimal futures positions to those obtained with optimal options positions. Except for four cases (out of 63), currency futures produce smaller LPMs than currency options. We conclude that, when hedging downside risk, futures 6
Across the three different currencies, there is no uniform relationship between the optimal hedge ratio and the power term n. As the value of n is related to an individual hedger's risk attitude or characteristics, we conclude that risk characteristics do not solely determine the optimal LPM strategy. This is not unexpected as the distribution functions of the underlying assets, particularly the tails of the distribution, help determine the downside risk a hedger may incur.
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Table 3 Comparison of hedging performance: futures vs. options Target return (%)
Currency
n=1
n=2
n=3
0.0
BP DM JY BP DM JY BP DM JY BP DM JY
0.4063 0.6710 0.5140 0.6743 0.9202 0.8614 0.8740 1.0136 1.0269 0.9665 1.0088 1.0175
0.1100 0.4671 0.2141 0.2282 0.5856 0.3945 0.4493 0.7821 0.6864 0.7078 0.9315 0.9054
0.0318 0.3533 0.0932 0.0682 0.4196 0.1692 0.1739 0.5600 0.3626 0.4109 0.7679 0.6759
0.4 1.0 2.0
The figures are the ratios of the LPM of the hedged portfolio associated with the optimal futures position to the LPM of the hedged portfolio associated with the optimal option position. A figure less than 1 shows that futures is more effective than option in reducing downside risk.
contracts are better instruments than options. This conclusion contrasts the popular belief. However, as n increases or c decreases the performance of options improves relative to the futures. That is, the (relative) hedging effectiveness of options increases when the hedger becomes more concerned about large losses or more pessimistic. For comparisons of the mean and variance (evaluated at the optimal LPM hedge ratios), we found that for each currency, futures always dominate options in producing smaller variances. The mean return for the British pound futures is always positive, whereas the mean return for the British pound options is almost always negative. Thus, regardless of which criterion is applied, the futures contract is a better hedging vehicle for the British pound. The Japanese yen presents a different case. Herein, the mean returns for the futures and options are always both positive with the former being smaller than the latter. If one compares the two instruments by the mean return, options ought to be preferred. Nonetheless, for hedging purpose, when the risks are compared the futures is again a better instrument. Similar arguments apply to the Deutsche mark where the mean return of the options is always positive and larger than that of the futures. The above analyses rely upon the whole sample. Post-sample comparisons are considered below. First, we designate the most recent 80 data points as ``post samples.'' We utilize the remaining data points to estimate the optimal hedge ratios for the futures and options. These ratios are then applied to the post samples to calculate the post-sample variances and LPMs. The results are shown in Table 4. To conserve space, only the cases for c = 0% and 2% are presented. For both British pound and Japanese yen, futures dominate options in both variance and LPMs. The same conclusion holds for the Deutsche mark based on the estimation sample. Post-sample comparisons, however, strongly favor the options. This result casts some doubt on the conclusions based upon within-sample comparisons. Nonetheless, one should be careful in interpreting the post-sample comparisons. First, the data series may be subject to structural change. As a consequence, the optimal hedge ratios
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Table 4 Post-sample performance comparison: futures vs. options n
Currency
1
BP DM JY
2
BP DM JY
3
BP DM JY
Target return (%)
Variance (within-sample)
Variance (post-sample)
LPM (within-sample)
LPM (post-sample)
0.0 2.0 0.0 2.0 0.0 2.0 0.0 2.0 0.0 2.0 0.0 2.0 0.0 2.0 0.0 2.0 0.0 2.0
0.1373 0.4580 0.1882 0.6282 0.2939 1.1542 0.1457 0.1555 0.2180 0.2382 0.2830 0.3565 0.1385 0.1453 0.2071 0.2198 0.2899 0.3056
0.2007 0.4486 3.0146 1.7785 0.1533 0.8424 0.1638 0.1731 3.3074 3.1522 0.2076 0.2016 0.1506 0.1593 3.0856 3.1931 0.2140 0.1994
0.4138 0.9737 0.4538 0.9958 0.5701 1.0200 0.1130 0.7344 0.1867 0.8405 0.2538 0.9452 0.0331 0.4560 0.0853 0.5958 0.1233 0.7415
0.3767 0.9462 2.0224 1.0727 0.4958 1.0115 0.1014 0.6071 3.8244 1.4463 0.1398 0.8198 0.0322 0.2854 6.5410 2.0171 0.0291 0.8075
The figures are the ratios of the variance/LPM associated with the optimal futures position to the variance/LPM associated with the optimal option position. LPM denotes the LPM, and n is the order of the LPM. The postsample size is 80 for all cases.
derived from the estimation sample may deviate from the optimal hedge ratios of the post samples. In our case, the Deutsche mark has a MV hedge ratio of 0.907 for the whole sample and 0.956 for the reduced estimation sample in the post-sample analysis. Further examination of the data indicates the prevalence of possible structural change. The problem is aggravated in this study as the post-sample size must be sufficient for the density estimation. Secondly, to calculate the post-sample performance we need to estimate the density function. A small sample size such as eighty is deemed to be rather small. Thus, we encounter two problems: one can be alleviated by a small post sample, but the other requires a large post sample. In any case, the post-sample results ought to be treated with caution. 6. Concluding remarks In this paper, we compare the hedging effectiveness of currency futures vs. currency options on the basis of the LPMs. The LPM measures an individual hedger's downside risk, as opposed to the two-sided risk measure. Two estimation methods are applied to estimate the optimal hedge ratio: the empirical distribution function method and the kernel density estimation method. While the former is easier to implement computationally, the latter is more efficient statistically. We consider both methods for three currencies: British pound, Deutsche mark, and Japanese yen. We find that the currency futures is almost always a better hedging instrument than the currency options. The only situation in which options outper-
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form futures occurs when the individual hedger is optimistic (with a large target return) and not too concerned about large losses (so that large losses do not impose greater weights than small losses). Acknowledgments The research of the first author was in part supported by a general research grant from the University of Kansas. Helpful comments from the editor, Carl Chen, and two anonymous referees are gratefully acknowledged. Any remaining errors are, of course, the sole responsibility of the authors. References Adams, J., & Montesi, C. J. (1995). Major issues related to hedge accounting. Newark, CT: Financial Accounting Standard Board. Ahmadi, H. Z., Sharp, P. A., & Walther, C. H. (1986). The effectiveness of futures and options in hedging currency risk. Advances in Futures and Options Research, 1, 171 ± 191. Benet, B. A., & Luft, C. F. (1995). Hedge performance of SPX index options and S&P 500 futures. Journal of Futures Markets, 15, 691 ± 717. Chang, J. S. K., & Shanker, L. (1986). Hedging effectiveness of currency options and currency futures. Journal of Futures Markets, 6, 289 ± 305. de Jong, A., de Roon, F., & Veld, C. (1997). Out-of-sample hedging effectiveness of currency futures for alternative models and hedging strategies. Journal of Futures Markets, 17, 817 ± 837. Fishburn, P. J. (1977). Mean-risk analysis with risk associated with below-target returns. American Economic Review, 67, 116 ± 126. Hancock, G. D., & Weise, P. D. (1994). Competing derivative equity instruments: empirical evidence on hedged portfolio performance. Journal of Futures Markets, 14, 421 ± 436. Hardle, W. (1990). Applied nonparametric regression. Cambridge: Cambridge Univ. Press. Harlow, W. V. (1991). Asset allocation in a downside-risk framework. Financial Analysts Journal, 47, 28 ± 40. Korsvold, P. E. (1994). Hedging Efficiency of Forward and Option Currency Contracts. Working Papers in Economics, No. 195, Department of Economics, University of Sydney. Lee, W. Y., & Rao, R. (1988). Mean lower partial moment valuation and lognormally distributed returns. Management Science, 34, 446 ± 453. Lien, D., & Tse, Y. K. (1998). Hedging time-varying downside risk. Journal of Futures Markets, 18, 705 ± 722. Lien, D., & Tse, Y. K. (2000). Hedging downside risk with futures contracts. Applied Financial Economics, 10, 163 ± 170. Mao, J. (1970). Models of capital budgeting, E ± V vs. E ± S. Journal of Financial and Quantitative Analysis, 4, 657 ± 675. Petty, J. W., & Scott, D. F. (1981). Capital budgeting practices in large American firms: a retrospective analysis and update. In: G. J. Derkinderen, & R. L. Crum (Eds.), Readings in strategies for corporate investment. Boston: Pitman Publishing. Price, K., Price, B., & Nantell, T. J. (1982). Variance and lower partial moment measures of systematic risk: some analytical and empirical results. Journal of Finance, 37, 843 ± 855. Reiss, R. D. (1981). Nonparametric estimation of smooth distribution functions. Canadian Journal of Statistics, 8, 116 ± 119. Silverman, B. W. (1986). Density estimation for statistics and data analysis. New York: Chapman & Hall.
Hedging downside risk: futures vs. options Donald Liena,*, Yiu Kuen Tseb a
Department of Economics, University of Kansas, 213 Summerfield Hall, Lawrence, KS 66045-2113, USA b National University of Singapore, Singapore Received 6 July 1999; received in revised form 3 February 2000; accepted 5 May 2000
Abstract In this paper, we compare the hedging effectiveness of currency futures vs. currency options on the basis of the lower partial moments (LPMs). The LPM measures an individual hedger's downside risk, as opposed to the two-sided risk measure. Two estimation methods are applied to estimate the optimal hedge ratio: the empirical distribution function method and the kernel density estimation method. We consider both methods for three currencies: the British pound, the Deutsche mark, and the Japanese yen. Currency futures are found to be a better hedging instrument than currency option. D 2001 Elsevier Science Inc. All rights reserved. JEL classification: F31; G11 Keywords: Currency futures; Currency options; Lower partial moments; Hedging
1. Introduction Given different derivative products, a hedger will choose the instrument with the highest hedging effectiveness. Using the mean±variance criterion, Chang and Shanker (1986) found that currency futures are better hedging instruments than currency options. Hancock and Weise (1994), however, showed that the optimal hedge positions for the S&P 500 index options and those for the S&P 500 futures lead to similar mean returns. In contrast, Benet and Luft (1995) found that the S&P 500 futures outperform the S&P 500 options in variance
* Corresponding author. Tel.: +1-785-864-3501; fax: +1-785-864-5270. E-mail address: [email protected] (D. Lien). 1059-0560/01/$ ± see front matter D 2001 Elsevier Science Inc. All rights reserved. PII: S 1 0 5 9 - 0 5 6 0 ( 0 0 ) 0 0 0 7 4 - 5
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reduction. Moreover, upon taking into account the transaction cost, futures have a larger excess return per unit risk than options. As options are designed to eliminate extreme downside risk, which is one-sided, comparisons based upon variance, which is a two-sided risk measure, are biased against options. In a recent survey, Adams and Montesi (1995) showed that corporate managers are mostly concerned about downside risk. Earlier, Petty and Scott (1981) found that many Fortune 500 firms identified risk as the probability of falling below a target return. To measure downside risk, Lee and Rao (1988) and Mao (1970) adopted the LPM approach. Hereby, downside risk is measured by a probability-weighted power function of the shortfall from a specific target return. In this regard, one tends to expect options to perform better in reducing downside risk. This conjecture, however, was rejected by Ahmadi, Sharp, and Walther (1986). Assuming a one-to-one hedge ratio, Ahmadi et al. showed that for a nontradable spot position, futures provide significantly more effective hedging than options when the target return is zero. Using simulated data and maintaining the one-to-one hedge ratio assumption, Korsvold (1994) demonstrated that if quantity risk (i.e., the uncertainty in the spot position) is incorporated, options dominate over futures. This result, however, is not convincing as it is based on simulated data. Adams and Montesi (1995) also found that corporate managers prefer to hedge the downside risk using futures rather than options, citing the large transaction cost in option trading as the main reason. In this paper, we compare the hedging strategy of currency futures and options based upon the lower partial moments (LPMs). While most previous works assume a naive oneto-one hedge ratio, this paper estimates the optimal hedge ratio that minimizes the LPM.1 Comparisons of hedging effectiveness are based upon these optimal hedge ratios. Following Lien and Tse (1999), nonparametric methods are adopted to estimate the optimal hedge ratio.2 The results are compared against the minimum-variance (MV) hedge strategy. The remaining part of this paper is organized as follows. In Section 2, we review the use of the LPM in the hedging framework. We highlight the possibility that futures may outperform options in reducing downside risk. Section 3 discusses two alternative methods for the estimation of the optimal hedge ratio that minimizes the LPM. These are the empirical distribution function method and the nonparametric kernel method. Data construction for the currency futures, currency options, and spot rates is discussed in Section 4. Three currencies are considered: the British pound, the Deutsche mark, and the Japanese yen. The empirical results are summarized in Section 5, where the hedging performances of the futures and options are compared. It is shown that currency futures are a better instrument for hedging either two-sided or downside risk. The paper is concluded in Section 6.
1 A notable exception is the recent paper by de Jong, de Roon, and Veld (1997). Their study, however, considered futures only. In addition, their results are based on the optimal hedge ratios estimated by the empirical distribution function method. In this paper, we consider the more efficient kernel estimation method as well. 2 Lien and Tse (1998) adopted a parametric approach to estimate time-varying LPM hedge ratios.
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2. Lower partial moments Consider a portfolio with a random return X. Let F(.) denote the distribution function of X and assume an individual has a target return c. An outcome larger than c is desirable. Thus, the individual faces only a one-sided risk (the downside risk), which occurs when X falls short of c. This risk can be measured by the LPM L(c,n,X ) defined in Eq. (1): Zc 1 L c; n; X c ÿ xn dF x; ÿ1
where n is a positive integer.3 Suppose that a hedger is endowed with an initial wealth W0 and a given nontradable spot position Q at time 0. At time 1, his wealth is W1 = W0 + DpQ, where Dp is the change in the spot price from time 0 to time 1. As Dp is unknown at time 0, the hedger faces a downside risk in W1. We measure this risk by L(c,n,W1). Suppose the hedger buys kQ futures contracts to hedge this risk. Then, his wealth Wf 1 at time 1 is: Wf 1 W0 Dp kDf Q;
2
where Df is the change in the futures price from time 0 to time 1. Let p0 and f0 denote the spot and futures prices, respectively, at time 0. Then, rp = Dp/p0 is the spot return and rf = Df/f0 is the futures return. Consequently, Eq. (2) can be rewritten as Eq. (20): 20 Wf 1 W0 rp qrf p0 Q; where ÿ q = ÿ kf0/p0 is the (adjusted) hedge ratio. Formulated in this way, the hedger faces a downside risk measured by L(c,n,Wf 1). Note that Wf 1 is a function of q. The hedger will choose q to minimize L(c,n,Wf 1). Apart from futures, options also serve as a hedging instrument. Assume the hedger sells qQ units of call options, the strike price of which is S, at a premium C. He has a short position if q > 0 and a long position if q < 0. If time 1 coincides with the maturity of the option, the hedger's wealth Wc1 at time 1 is expressed as Eq. (3): Wc1 W0 fDp qmin S ÿ p1 ; 0 qCgQ;
3
where p1 is the spot price at time 1. Let pc denote the market profit Dp + q min(S ÿ p1, 0) + qC. When q = 1, pc S ÿ p0 + C. Otherwise, there is no upper or lower bounds for pc. Thus, Wc1 incurs a downside risk. The above analysis applies when the option expires at time 1. Otherwise, the hedger's wealth Wd1 at time 1 is expressed as Eq. (4): Wd1 W0 rp grc p0 Q;
4
where rc is the return of the call option over the period from time 0 to time 1, and ÿ g = ÿ qC/ p0. The hedger will choose g to minimize L(c,n,Wd1). Note that there is a one-to-one monotonic transformation between Wf 1 (or Wd1) and mf (or md) where mf = rp + qrf and 3
In general, n can be any positive number as adopted in the Fishburn (1977) risk measure. For example, Korsvold (1994) considered the case of n = 1.5. In this paper, we only consider integer values of n.
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md = rp + grc. Thus, we can equivalently consider the LPMs of mf and md. This approach removes the effects of the initial wealth W0 and the nontradable spot position Q. 3. Estimation of the optimal hedge ratios We now consider the estimation of the optimal hedge ratio of futures contracts. Similar method applies to options. Note that the optimal hedge ratio is chosen to minimize L(c,n,mf), which can be rewritten as E{[max(0, c ÿ mf)]n}. The optimal hedge ratio q*, therefore, satisfies the following first-order condition: ÿnEfmax 0; c ÿ rp ÿ q rf nÿ1 rf g 0:
5
It can be shown that the second-order condition is always satisfied when n 2. Although Eq. (5) indicates that q* is a function of n and c, the effects of n and c on q* are undetermined (see Lien & Tse, 2000 for the details). Whenever the joint distribution of rp and rf is known, we can apply numerical methods to find the optimal hedge ratio. In empirical studies, however, the true distribution of rp and rf is unknown and must be estimated. In this paper, we adopt an indirect approach. We estimate the distribution of the portfolio return for any given q. Specifically, for a given q, we construct the data series for mf from the data of rf and rp, and then apply nonparametric methods to estimate the distribution of mf. The details are as follows. Let g(.) be a smooth probability density function, called the kernel function. Suppose we have a random sample of N observations of mf, say, m1, m2, . . ., mN, calculated from a given q. Using kernel method, the probability density of mf at a given point y, denoted by f ( y), is estimated using Eq. (6): f^ y 1=Nh
N X
g y ÿ mi h;
6
i1
where h (>0) is the bandwidth (Hardle, 1990). Silverman (1986) suggested that the choice of the kernel function has minimal effects on the density estimates. In this paper, we choose g(.) to be the density function of the standard normal random variable so that Eq. (7) is defined as: g z 2pÿ1=2 exp ÿz 2 =2;
7
and the LPM is estimated using Eq. (8): ^ n; mf L c;
Zc
c ÿ yn 1=Nh
N X
g y ÿ mi =hdy:
8
i1
ÿ1
After a change of variable from y to z = ( y ÿ mi)/h, we have: ^ n; m 1=N L c; f
N X i1
Q c; n; mi ;
9
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where cÿm Z i =h
Q c; n; mi
c ÿ hz ÿ mi n g zdz:
10
ÿ1
It is well known that the bandwidth h is critical in estimating the probability density. Following Silverman (1986), we determine the optimal size of h by the cross-validation method (see Lien & Tse, 2000 for details). Since we apply the kernel method to the return data, the optimal bandwidth depends upon the value of the given q. To reduce computational burden, we calculated a sample of optimal bandwidth for selected values of q that are expected to cover the optimal hedge ratio. The optimal bandwidth is then taken as the average of the sample. To sum up, beginning with a given hedge ratio, we construct the data series for the portfolio return. The kernel method and the optimal bandwidth are then applied to calculate the estimated LPM using Eqs. (9) and (10). We then adopt grid search method to find the optimal hedge ratio that produces the smallest estimated LPM. Kernel method is computationally time consuming. Empirical distribution function method provides an alternative (see Harlow, 1991; Price, Price, & Nantell, 1982). Herein, the LPM is estimated as follows (Eq. (11)): X ~ 1=N c ÿ mi n : 11 L c; n; mf mi c
When the bandwidth is very small, the kernel method and the empirical distribution function method provide similar estimates. Thus, the latter is a special case of the former. Reiss (1981) demonstrated that the kernel method has better statistical performance than the empirical distribution function method. In this paper, we consider both methods. 4. Data description We consider futures and options of the British pound, the Deutsche mark, and the Japanese yen. Daily closing prices of the currency futures contracts traded on the Chicago Mercantile Exchange were obtained from the Futures Industry Institute Data Center. For the options, daily trading data were provided by the Philadelphia Stock Exchange (PHLX). The spot exchange rate data were also extracted from the PHLX data files. In the empirical study, we consider a hedging horizon of one week, or more specifically 5 working days. This hedging horizon was chosen to alleviate the problem of thin trading for the options, and to make the selection of options more manageable. For each currency, we constructed a futures series with intervals of 5 working days. The nearest contract was selected, with roll-over occurring at approximately 10 days before the expiry of the contract. The return on the futures over the hedge interval was calculated as the differenced (logarithmic) futures price over the price of the futures contract at the time of the initialisation of the hedge. The same formula was used in computing the return of the spot and the options.
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At the roll-over, care was taken to ensure that the same contract was used to calculate the return over the hedge interval. The option files were compiled to form a reduced data set with trading dates that matched those of the futures. To form a hedged portfolio, we selected American call options with exercise prices close to the spot rates. Additional criteria were imposed. First, we gave preference to the nearby contracts. Secondly, when there were more than one contract satisfying the first criterion, we selected the one with the highest volume. In computing the returns of the options, care was taken to ensure that the same contract was used at the times when the hedge was initialised and lifted. Due to the problem of a thin market, however, there were circumstances in which no suitable options were available for some intervals. In such cases, no hedging was considered and the corresponding futures data were ignored. Apart from thin trading, data errors also contributed to the loss of observations. It should be noted that as the return distributions are assumed to be time invariant, the estimation in this paper applies to random samples of return observations rather than serially dependent observations. Thus, apart from the loss of efficiency due to missing observations, the existence of missing data should have no effects on the statistical properties of the estimates. We compiled the returns on the futures, options, and spots of the selected currencies. For the British pound, the data cover the period from January 1984 to October 1994, with 444 observations. For the Deutsche mark, the data cover the period from August 1983 to September 1994, with 526 observations. As for the Japanese yen, there are 424 observations covering the period from January 1984 to January 1997. Table 1 provides the summary statistics of the data. For all currencies, the spot and futures returns display similar statistical behaviour. The option returns, however, are much more volatile. While the variances of the spot and futures returns lie between 2 and 4, the variances of the option returns are much larger. None of the returns is significantly different from zero. Except for the spot and futures returns of the Deutsche mark, all series exhibit excessive skewness and kurtosis. The normality assumption can be easily rejected based upon the Bera±Jarque statistic.
Table 1 Summary statistics of returns on spot, futures, and option Currency
Security
Mean
Variance
Skewness
Kurtosis
BP DM JY BP DM JY BP DM JY
spot spot spot futures futures futures option option option
0.0474 0.0971 0.1252 0.0388 0.1207 0.1272 7.3047 2.3470 2.2618
3.1998 2.8656 2.3631 3.4516 2.9138 2.3923 7912.9 3890.0 6069.4
ÿ 2.2327 0.4298 5.3484 ÿ 3.5557 ÿ 0.3572 6.0286 37.458 27.599 22.221
11.804 4.635 16.331 16.044 3.872 18.259 125.44 84.61 43.15
Returns are expressed in percentage. The currency codes are: BP = British pound, DM = Deutsche mark, and JY = Japanese yen. The skewness and kurtosis are standardised.
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5. Empirical results We consider a list of possible target returns ranging from 0.0% to 2.0%.4 The empirical distribution function and the kernel density methods were applied to evaluate the LPMs with order n equal to 1, 2, and 3. The results from the kernel density method are summarized in Table 2. The empirical distribution function method generated qualitatively similar results, which are not reported here. While we report only the results of the kernel method using the sample average of optimal bandwidths,5 we also experimented with different bandwidths. We found that the optimal hedge ratios are robust to the choices of the bandwidth, although the resulting LPMs may not be. For each currency, the optimal hedge ratios of the futures and options display similar properties. However, across different currencies, the optimal hedge ratios behave differently. In the case of the British pound, we found that the optimal futures hedge ratio always decreases as the target return increases. That is, if the hedger becomes more optimistic (so that, for example, while initially 1% was not considered as a loss, now it is), fewer futures contracts will be sold against a unit of spot position. For the Deutsche mark, the optimal futures hedge ratio decreases with increasing target return when n = 1 and increases with increasing target return when n = 3. The ratio increases and then decreases when n = 3. The Japanese yen demonstrates yet another different case. Herein, the optimal futures hedge ratio decreases as the target return increases when n = 1. For the other two cases, the ratio first increases and then decreases. For the Deutsche mark and Japanese yen, the MV hedge ratio is almost always greater than the LPM hedge ratios. That is, a hedger concerned about downside risk will overhedge if the MV hedge ratio is followed. In the case of the British pound, the relationship between the MV and LPM hedge ratios depends upon n. When n = 3, the MV strategy almost always underhedges the downside risk. Due to its high volatility, currency options do not appear to be a useful hedging vehicle. In each case, the optimal hedge ratio is rather small. When n = 2 or 3, the optimal hedge ratio of the options always increases as the target return increases. That is, a more optimistic hedger will sell more call options. The effects of n on the optimal futures hedge ratio also vary across currencies. For the British pound, the optimal hedge ratio increases as n increases. On the other hand, the optimal hedge ratio for the Deutsche mark futures decreases when n increases except when c 1%. In the case of the Japanese yen, the optimal hedge ratio increases with increasing n except when
4
Most empirical works assume a zero target return. Here, we allow the target return to be positive. A positive target return seems intuitive if the derivatives are investment outlays. On the other hand, as the hedger attempts to reduce risk through derivative positions, he is willing to pay a cost. That is, he may be willing to accept a negative return in order to reduce the downside risk. In the empirical study, we considered some cases with negative targets. Though not reported here, the results pertaining to negative target returns are consistent with our main results. 5 We calculated the optimal bandwidths over a range of hedge ratios that are likely to cover the optimal value. The kernel estimates of the lower partial moments were then recalculated based on the maximum and minimum values of the range of optimal bandwidths.
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Table 2 Optimal LPM hedge ratios Currency
Target return (%)
Derivative
n=1
n=2
n=3
BP
0.0
futures options futures options futures options futures options futures options futures options futures options futures options futures options futures options futures options futures options
0.920 0.018 0.872 0.017 0.822 0.013 0.577 0.006 0.942 0.026 0.904 0.026 0.739 0.025 0.440 0.014 0.936 0.018 0.891 0.019 0.716 0.019 0.444 0.015
0.936 0.010 0.924 0.018 0.909 0.011 0.896 0.011 0.867 0.019 0.893 0.020 0.875 0.021 0.829 0.021 0.913 0.013 0.919 0.014 0.894 0.015 0.844 0.015
0.967 0.008 0.946 0.009 0.930 0.009 0.916 0.010 0.756 0.017 0.807 0.018 0.846 0.018 0.849 0.019 0.880 0.012 0.903 0.013 0.905 0.013 0.887 0.014
0.4 1.0 2.0 DM
0.0 0.4 1.0 2.0
JY
0.0 0.4 1.0 2.0
The figures in the table are the hedge ratios that give the lowest LPMs. They are estimated by the kernel method for the three currencies: BP = British pound, DM = Deutsche mark, and JY = Japanese yen. n is the order of the LPM. For comparison, the MV hedge ratios (i) using futures are: 0.917 (BP), 0.907 (DM), and 0.948 (JY), and (ii) using options are: 0.013 (BP), 0.022 (DM), and 0.016 (JY).
c = 0%. Thus, a hedger who cares more about large losses may sell more or fewer futures contracts to hedge the downside risk.6 There are more common features across currency options. For the British pound, the optimal hedge ratio for options decreases as n increases except for c = 2%. Similarly, the optimal hedge ratio for the Deutsche mark and Japanese yen options decreases with increasing n. Thus, a hedger who is more concerned about large losses will sell fewer call options. Table 3 compares the LPMs obtained with optimal futures positions to those obtained with optimal options positions. Except for four cases (out of 63), currency futures produce smaller LPMs than currency options. We conclude that, when hedging downside risk, futures 6
Across the three different currencies, there is no uniform relationship between the optimal hedge ratio and the power term n. As the value of n is related to an individual hedger's risk attitude or characteristics, we conclude that risk characteristics do not solely determine the optimal LPM strategy. This is not unexpected as the distribution functions of the underlying assets, particularly the tails of the distribution, help determine the downside risk a hedger may incur.
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Table 3 Comparison of hedging performance: futures vs. options Target return (%)
Currency
n=1
n=2
n=3
0.0
BP DM JY BP DM JY BP DM JY BP DM JY
0.4063 0.6710 0.5140 0.6743 0.9202 0.8614 0.8740 1.0136 1.0269 0.9665 1.0088 1.0175
0.1100 0.4671 0.2141 0.2282 0.5856 0.3945 0.4493 0.7821 0.6864 0.7078 0.9315 0.9054
0.0318 0.3533 0.0932 0.0682 0.4196 0.1692 0.1739 0.5600 0.3626 0.4109 0.7679 0.6759
0.4 1.0 2.0
The figures are the ratios of the LPM of the hedged portfolio associated with the optimal futures position to the LPM of the hedged portfolio associated with the optimal option position. A figure less than 1 shows that futures is more effective than option in reducing downside risk.
contracts are better instruments than options. This conclusion contrasts the popular belief. However, as n increases or c decreases the performance of options improves relative to the futures. That is, the (relative) hedging effectiveness of options increases when the hedger becomes more concerned about large losses or more pessimistic. For comparisons of the mean and variance (evaluated at the optimal LPM hedge ratios), we found that for each currency, futures always dominate options in producing smaller variances. The mean return for the British pound futures is always positive, whereas the mean return for the British pound options is almost always negative. Thus, regardless of which criterion is applied, the futures contract is a better hedging vehicle for the British pound. The Japanese yen presents a different case. Herein, the mean returns for the futures and options are always both positive with the former being smaller than the latter. If one compares the two instruments by the mean return, options ought to be preferred. Nonetheless, for hedging purpose, when the risks are compared the futures is again a better instrument. Similar arguments apply to the Deutsche mark where the mean return of the options is always positive and larger than that of the futures. The above analyses rely upon the whole sample. Post-sample comparisons are considered below. First, we designate the most recent 80 data points as ``post samples.'' We utilize the remaining data points to estimate the optimal hedge ratios for the futures and options. These ratios are then applied to the post samples to calculate the post-sample variances and LPMs. The results are shown in Table 4. To conserve space, only the cases for c = 0% and 2% are presented. For both British pound and Japanese yen, futures dominate options in both variance and LPMs. The same conclusion holds for the Deutsche mark based on the estimation sample. Post-sample comparisons, however, strongly favor the options. This result casts some doubt on the conclusions based upon within-sample comparisons. Nonetheless, one should be careful in interpreting the post-sample comparisons. First, the data series may be subject to structural change. As a consequence, the optimal hedge ratios
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Table 4 Post-sample performance comparison: futures vs. options n
Currency
1
BP DM JY
2
BP DM JY
3
BP DM JY
Target return (%)
Variance (within-sample)
Variance (post-sample)
LPM (within-sample)
LPM (post-sample)
0.0 2.0 0.0 2.0 0.0 2.0 0.0 2.0 0.0 2.0 0.0 2.0 0.0 2.0 0.0 2.0 0.0 2.0
0.1373 0.4580 0.1882 0.6282 0.2939 1.1542 0.1457 0.1555 0.2180 0.2382 0.2830 0.3565 0.1385 0.1453 0.2071 0.2198 0.2899 0.3056
0.2007 0.4486 3.0146 1.7785 0.1533 0.8424 0.1638 0.1731 3.3074 3.1522 0.2076 0.2016 0.1506 0.1593 3.0856 3.1931 0.2140 0.1994
0.4138 0.9737 0.4538 0.9958 0.5701 1.0200 0.1130 0.7344 0.1867 0.8405 0.2538 0.9452 0.0331 0.4560 0.0853 0.5958 0.1233 0.7415
0.3767 0.9462 2.0224 1.0727 0.4958 1.0115 0.1014 0.6071 3.8244 1.4463 0.1398 0.8198 0.0322 0.2854 6.5410 2.0171 0.0291 0.8075
The figures are the ratios of the variance/LPM associated with the optimal futures position to the variance/LPM associated with the optimal option position. LPM denotes the LPM, and n is the order of the LPM. The postsample size is 80 for all cases.
derived from the estimation sample may deviate from the optimal hedge ratios of the post samples. In our case, the Deutsche mark has a MV hedge ratio of 0.907 for the whole sample and 0.956 for the reduced estimation sample in the post-sample analysis. Further examination of the data indicates the prevalence of possible structural change. The problem is aggravated in this study as the post-sample size must be sufficient for the density estimation. Secondly, to calculate the post-sample performance we need to estimate the density function. A small sample size such as eighty is deemed to be rather small. Thus, we encounter two problems: one can be alleviated by a small post sample, but the other requires a large post sample. In any case, the post-sample results ought to be treated with caution. 6. Concluding remarks In this paper, we compare the hedging effectiveness of currency futures vs. currency options on the basis of the LPMs. The LPM measures an individual hedger's downside risk, as opposed to the two-sided risk measure. Two estimation methods are applied to estimate the optimal hedge ratio: the empirical distribution function method and the kernel density estimation method. While the former is easier to implement computationally, the latter is more efficient statistically. We consider both methods for three currencies: British pound, Deutsche mark, and Japanese yen. We find that the currency futures is almost always a better hedging instrument than the currency options. The only situation in which options outper-
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form futures occurs when the individual hedger is optimistic (with a large target return) and not too concerned about large losses (so that large losses do not impose greater weights than small losses). Acknowledgments The research of the first author was in part supported by a general research grant from the University of Kansas. Helpful comments from the editor, Carl Chen, and two anonymous referees are gratefully acknowledged. Any remaining errors are, of course, the sole responsibility of the authors. References Adams, J., & Montesi, C. J. (1995). Major issues related to hedge accounting. Newark, CT: Financial Accounting Standard Board. Ahmadi, H. Z., Sharp, P. A., & Walther, C. H. (1986). The effectiveness of futures and options in hedging currency risk. Advances in Futures and Options Research, 1, 171 ± 191. Benet, B. A., & Luft, C. F. (1995). Hedge performance of SPX index options and S&P 500 futures. Journal of Futures Markets, 15, 691 ± 717. Chang, J. S. K., & Shanker, L. (1986). Hedging effectiveness of currency options and currency futures. Journal of Futures Markets, 6, 289 ± 305. de Jong, A., de Roon, F., & Veld, C. (1997). Out-of-sample hedging effectiveness of currency futures for alternative models and hedging strategies. Journal of Futures Markets, 17, 817 ± 837. Fishburn, P. J. (1977). Mean-risk analysis with risk associated with below-target returns. American Economic Review, 67, 116 ± 126. Hancock, G. D., & Weise, P. D. (1994). Competing derivative equity instruments: empirical evidence on hedged portfolio performance. Journal of Futures Markets, 14, 421 ± 436. Hardle, W. (1990). Applied nonparametric regression. Cambridge: Cambridge Univ. Press. Harlow, W. V. (1991). Asset allocation in a downside-risk framework. Financial Analysts Journal, 47, 28 ± 40. Korsvold, P. E. (1994). Hedging Efficiency of Forward and Option Currency Contracts. Working Papers in Economics, No. 195, Department of Economics, University of Sydney. Lee, W. Y., & Rao, R. (1988). Mean lower partial moment valuation and lognormally distributed returns. Management Science, 34, 446 ± 453. Lien, D., & Tse, Y. K. (1998). Hedging time-varying downside risk. Journal of Futures Markets, 18, 705 ± 722. Lien, D., & Tse, Y. K. (2000). Hedging downside risk with futures contracts. Applied Financial Economics, 10, 163 ± 170. Mao, J. (1970). Models of capital budgeting, E ± V vs. E ± S. Journal of Financial and Quantitative Analysis, 4, 657 ± 675. Petty, J. W., & Scott, D. F. (1981). Capital budgeting practices in large American firms: a retrospective analysis and update. In: G. J. Derkinderen, & R. L. Crum (Eds.), Readings in strategies for corporate investment. Boston: Pitman Publishing. Price, K., Price, B., & Nantell, T. J. (1982). Variance and lower partial moment measures of systematic risk: some analytical and empirical results. Journal of Finance, 37, 843 ± 855. Reiss, R. D. (1981). Nonparametric estimation of smooth distribution functions. Canadian Journal of Statistics, 8, 116 ± 119. Silverman, B. W. (1986). Density estimation for statistics and data analysis. New York: Chapman & Hall.