Dynamic futures hedging in currency markets

Transcription

1 University of Massachusetts Boston From the SelectedWorks of Atreya Chakraborty 1999 Dynamic futures hedging in currency markets Atreya Chakraborty, University of Massachusetts, Boston Available at:

2 The European Journal of Finance 5, (1999) Dynamic futures hedging in currency markets ATREYA CHAKRABORTY 1 and JOHN T. BARKOULAS 2 1 Graduate School of International Economics and Finance, Brandeis University, Waltham, MA USA 2 Department of Economics and Finance, Louisiana Tech University, Ruston, LA USA The hedging effectiveness of dynamic strategies is compared with static (traditional) ones using futures contracts for the ve leading currencies. The traditional hedging model assumes time invariance in the joint distribution of spot and futures price changes thus leading to a constant optimal hedge ratio (OHR). However, if this timeinvariance assumption is violated, time-varying OHRs are appropriate for hedging purposes. A bivariate GARCH model is employed to estimate the joint distribution of spot and futures currency returns and the sequence of dynamic (time-varying) OHRs is constructed based upon the estimated parameters of the conditional covariance matrix. The empirical evidence strongly supports time-varying OHRs but the dynamic model provides superior out-of-sample hedging performance, compared to the static model, only for the Canadian dollar. Keywords: dynamic hedging, optimal hedge ratio, bivariate GARCH model, currency futures 1. INTRODUCTION The broadening of world trade and increasing sophistication of corporate funding strategies has caused many corporate treasurers to search for effective and ef cient ways to hedge foreign exchange risk. In addition to the transaction and translation exposures related to current operations, multinational corporations must also consider the potential competitive advantage that effective treatment of foreign exchange risk may yield (Lewent and Kearney, 1990). 1 A relatively inexpensive and reliable strategy for hedging foreign exchange risk involves the use of foreign currency futures markets. The static hedging model with futures contracts (Johnson, 1960; Stein, 1961; Ederington, 1979) assumes that the joint distribution of spot and futures returns is time-invariant and therefore the optimal hedge ratio (OHR), de ned as the optimal number of futures holdings per unit of spot holdings, is constant over time. The minimum variance OHR corresponding to the most effective hedge, may be derived from an ordinary-least-squares (OLS) regression of spot price 1 Geczy et al. (1997), for example, nd that, among the 41% Fortune 500 rms in 1990 that actively manage nancial risk, 70% of these rms use currency derivatives to control foreign exchange-rate exposures. The European Journal of Finance ISSN X print/issn online 1999 Taylor & Francis Ltd

3 300 A. Chakraborty and J.T. Barkoulas changes on futures price changes, analogous to the estimation of the market risk measure beta in the well-known Capital Asset Pricing Model. With respect to foreign currencies, extensive empirical research (Dale, 1981; Hill and Schneeweis, 1981; Grammatikos and Saunders, 1983 to mention a few) has shown that currency futures contracts have been very effective in reducing foreign exchange risk. The underlying assumption of the static hedging model of time-invariant asset distributions has recently been challenged. The Autoregressive Conditional Heteroscedastic (ARCH) framework of Engle (1982) and its extension to a generalized ARCH (GARCH) structure by Bollerslev (1986) have proven to be very successful in modelling asset price second-moment movements. Bollerslev (1987), Bailie and Bollerslev (1989), and Diebold (1988) have shown that the GARCH (1,1) model is effective in explaining the distribution of exchange rate changes. McCurdy and Morgan (1987, 1988) have also shown that currency futures price changes can be adequately modelled by a GARCH (1,1) process. The dynamic hedging model accounts for the time variation in the joint distribution of the spot and futures price changes thus resulting in a sequence of time-varying optimum or risk-minimizing hedge ratios. Given the distinct theoretical advantages of the dynamic hedging method over the static one, a number of studies have employed the multivariate GARCH framework to examine its hedging performance for various assets. The performance of the dynamic hedging strategy is usually evaluated on the basis of the ex-ante or outof-sample percentage reduction in the unconditional variance of the dynamically hedged portfolio relative to the variance of the statically hedged portfolio. The collective evidence shows that the GARCH-modelled dynamic hedging strategies are empirically appropriate but the risk-reduction improvements over constant hedges vary across markets and may be sensitive to the sample period employed in the analysis. More speci cally, Baillie and Myers (1991) using the multivariate GARCH speci cation to model the conditional covariance matrix for six commodities, documented superior hedging effectiveness for the dynamic model. Myers (1991) however found that for wheat the GARCH hedge only marginally outperformed the constant hedge in terms of out-of-sample risk reduction. Park and Switzer (1995) found that the GARCH-based dynamic hedging strategy obtains improvements in forecasting accuracy over the static hedge for three types of stock index futures. Similar evidence was reported by Lypny and Powalla (1998) for the German stock index DAX futures. 2,3 Cecchetti et al. (1988) did not document superior hedging effectiveness for the dynamic model applied to the 20-year Treasury bonds. Regarding currency futures, Kroner and Sultan (1991) applied the bivariate GARCH model for spot and futures returns for the Japanese yen and found it to be inferior to the static OLSbased model in terms of out-of-sample forecasting performance. In a subsequent study, Kroner and Sultan (1993) found that for four out of the ve leading 2 Holmes (1996) found that the FTSE-100 stock index futures contract provides a very effective hedge (compared to an unhedged position). The hedging strategy employed by Holmes is a static one though he allows for GARCH(1,1) effects in the disturbance term of the regression equation. 3 See Figlewski (1984) for an analysis of the different sources of basis risk and their implications for the hedging effectiveness of the stock index futures contract.

4 Dynamic futures hedging in currency markets 301 currencies (the exception was the British pound) GARCH risk-minimizing hedge ratios provide greater, though marginally so, out-of-sample risk reductions and utility gains compared to standard alternative models. In this paper we implement dynamic hedging strategies in the currency markets using futures contracts and their relative hedging effectiveness in terms of attained in-sample and out-of-sample risk reductions is evaluated. More speci cally, dynamic hedging strategies using the nearest maturity futures contract for the Canadian dollar, Deutsche mark, British pound, Swiss franc, and Japanese yen are empirically assessed and compared to their traditional counterparts. The joint distribution of spot and futures price changes is modelled as a bivariate GARCH process. The model s parameters are used to estimate the time-dependent covariance matrix, and in turn calculate the sequence of time-varying OHRs. These OHRs are then compared to those implied by the static, naive, and no-hedge models in terms of in-sample and outof-sample hedging performance. The present analysis differs from Kroner and Sultan (1993) both in terms of the sample period under investigation and the speci cation of the GARCH error structure in the bivariate model. Our evidence suggests that (i) time-varying as opposed to constant OHRs are empirically relevant, and (ii) dynamic hedging reduces risk on an out-of-sample basis more than the alternative strategies considered only for the Canadian dollar. Explanations of these empirical ndings as well as extensions to the methodological framework are also offered. The rest of the paper is constructed as follows. Section 2 presents the traditional (static) and dynamic hedging models. The econometric methodology is described in Section 3. Empirical estimates of the bivariate GARCH model for each currency and comparison of in-sample and out-of-sample hedging effectiveness of the no-hedge, naive-hedge, static-hedge, and dynamic-hedge strategies are reported in Section 4. Finally, the paper concludes in Section 5 with a summery of the empirical ndings and suggestions for future work. 2. HEDGING MODELS In this section, we brie y describe the two main competing hedging models of foreign exchange risk: the static or traditional and dynamic futures hedging models The static or traditional hedging model Assume an investor who faces a two-period investment decision and can use only futures contracts to hedge against foreign risk exposure. Let P 1 s and P2 s be the cash or spot prices at times t 1 and t 2, respectively. Similarly, let P 1 f and P 2 f be the futures prices at times t 1 and t 2, respectively. Let R represent the stochastic return on the hedged portfolio consisting of one unit of spot holdings and b units of futures holdings, that is, R 5 P 2 s 2 P1 s 1 b(p2 f 2 P 1 f ). The expected return, E(R), on this portfolio is given by E(R) 5 E(P 2 s 2 P1 s ) 1 b(p2 f 2 P 1 f ) 5 E(s) 1 be(f) (1)

5 302 A. Chakraborty and J.T. Barkoulas where s and f represent spot and futures price changes, respectively. The variance on this portfolio, VAR(R), is given by VAR(R) 5 VAR(s) 1 b 2 VAR(f) 1 2b COV(s,f) (2) Assume that the investor has a utility function which depends only on the rst two moments of the distribution of her portfolio returns and takes the form U(R) 5 E(R) 2 r VAR(R) (3) where r is the investor s coef cient of risk aversion (r. the following maximization problem: 0). The investor solves Max U(R) 5 E(s) 1 be(f) 1 r(var(s) 1 VAR(f) 1 2b COV(s,f)) (4) with respect to b. Setting the partial derivative of (4) with respect to b equal to zero and solving for b provides with the OHR, which is given by b* 5 E(f) 2 2r COV(s,f) 2r VAR(f) If the sequence of futures prices is assumed to follow a martingale process, that is, E(F t 1 1 ) 5 F t, then expression (5) reduces to 4 b* 5 2 COV(s,f) VAR(f) This solution represents the minimum-risk hedge coef cient, namely, the proportion of currency spot position that is hedged to provide the maximum amount of reductions in price change variance. In other words, expression (6) is the solution to the problem of minimizing (2) with respect to b. Extending the analysis to a multiperiod framework with a time-separable utility function and assuming that the variance covariance matrix of the spot and futures price changes is time invariant results in a sequence of optimal hedge ratios, [b 1,...,b t ], whose elements are the same as the two-period OHR. Therefore, the OHR in (6) is equivalent to the negative of the slope coef cient of a regression of spot price changes on futures price changes and can be estimated by ordinary least squares. In the special case where spot and futures prices are perfectly negatively correlated, the covariance between the two series equals the variance of futures price changes and b* 5 1 is obtained. In this case the investor achieves a (5) (6) 4 More formally, a stochastic process [X(t),t [ T] is said to be a martingale process relative to an increasing sequence of s - elds Z 1, Z 2,...Z t,... if (i) X(t) is a random variable relative to Z t, for all t [ T; (ii) E(ï X(t)ï ), (i.e. its mean is bounded) for all t [ T; and (iii) E(X(t)ï Z t 2 1 ) 5 X(t 2 1), for all t [ T.

6 Dynamic futures hedging in currency markets 303 perfect hedge in the sense that the change in basis risk is zero. 5 This hedging strategy of matching spot and futures holdings one-to-one will be referred to as the naive strategy. The static hedging model can be criticized on two grounds. First, it does not allow for a time-varying bivariate distribution of spot and futures price changes. However, an extensive body of research has shown that the variances and covariances of spot and futures prices are time-varying. Accounting for the time variation in the joint distribution of spot and futures price changes in the estimation process gives rise to time-varying hedging coef cients thus rendering the use of a constant OHR methodologically incorrect. Second, the static model employs unconditional moments as a risk measure. Since we can presume that market participants make use of all available information at the time of the hedging decision, the conditional moments of spot and futures price changes should be used in constructing OHRs, which should dominate those formed from unconditional moments in terms of performance. These two quali cations lead to the development of the dynamic model which we present next The dynamic hedging model Let s t 1 1 and f t 1 1 represent the spot and futures price changes from time t to t 1 1. Also de ne b t as the number of futures contracts purchased as of time t. Then R t s t b t f t 1 1 represents the stochastic return at time t 1 1 on a portfolio consisting of one unit of spot holdings and b t units of futures contracts formed at period t. Then in a multi-period framework, the investor at each time t maximizes the utility function U(R t 1 1 ï V t ) 5 E(R t 1 1 ï V t ) 2 r VAR(R t 1 1 ï V t ) (7) with respect to b t, conditional on all relevant information available at time t, represented by V. The rst-order condition yields b * t 5 t E(f t 1 1 ï V t ) 2 2r COV(s t 1 1,f t 1 1 ï V t ) 2r VAR(f t 1 1 ï V t ) (8) Again, assuming that the futures rate sequence is a martingale-difference operator, the above expression reduces to b * t 5 2 COV(s t 1 1,f t 1 1 ï V t ) VAR(f t 1 1 ï V t ) (9) The difference between expressions (6) and (9) is that conditional moments in (9) have substituted unconditional moments in (6). Therefore, expression (9) produces a sequence of time-varying OHRs as the investor re-evaluates his/her hedging strategy conditional upon the information set available at each point in 5 Basis is de ned as the difference between the futures and spot prices so that the change in the basis is [(P 2 f 2 P 2) s 2 (P1 f 2 P 1)] or s 2 [(P2 s 2 P1) s 2 (P2 f 2 P 1)]. f

7 q 304 A. Chakraborty and J.T. Barkoulas time. This dynamic speci cation of the investor s hedging strategy does away with the shortcomings of the static model and should, in principle, lead to superior hedging performance ECONOMETRIC METHODOLOGY Estimation of the sequence of OHRs in the dynamic model requires a speci cation of the joint distribution of the spot and futures price changes which allows for time-dependent conditional variances. Multivariate GARCH(p,q) models, as presented in Bollerslev et al. (1988), Baba et al. (1989), and Bolleslev (1990) can be used for that purpose. A bivariate GARCH(p,q) model to represent the joint distribution of spot and futures price changes is speci ed as follows: y t 5 a X t 1 h t (10a) h t ï V t ~ N(0,H t ) (10b) H ijt 5 C ij0 1 O p l5 1 h t 2 l D ijl h t 2 l 1 O q n5 1 G ijn H ijt 2 n (10c) where y t 5 (s t,f t ) is a (2 3 1) vector containing spot and futures price changes; X t 5 (X s,x f ) is a (2 3 k) vector containing the explanatory variables for the spot and futures price equations; a 5 (a s,a f ) is a (2 3 k) coef cient vector for the spot and futures price equations; and h t 5 (h st,h ft ) is a (2 3 1) vector containing the innovations in the spot and futures price equations. The variance covariance parameterization followed here for the bivariate GARCH model is the one presented in Baba et al. (1987). It is speci ed as H t 5 C C 1 O p l5 1 A l h t 2 l h t 2 1 A l 1 O q l5 1 B l H t 2 l B l (11) h where C is a symmetric (2 2) parameter matrix, and A l and B l are either unrestricted (2 2) parameter matrices (full positive de nite parameterization) or diagonal (2 2) parameter matrices (diagonal positive de nite parameterization). The parameterization in equation (11) guarantees positive de niteness of H t for all values of t in the sample space under very weak conditions.7 Letting be the vector representing the parameters of the model and T the sample size, the log-likelihood function is given by 6 It must be noted that differential equations involving time-invariant parameters can be solved from a spectral form from bounded operators applied to an entire function f(m) of a bounded operator M or from Sylvestor s well-known theorem. But when time variance is introduced, the method of variation of parameters or the use of the special case of Dirichlet series can be more appropriately employed. We thank an anonymous referee for bringing these methods to our attention. 7 A suf cient condition for H t to be positive de nite for all possible values of h t is that H 0,H 2 1,..., H 2 p 1 1 are all positive de nite and either C or any B l has full rank (Baba et al., 1987, p.10).

8 Dynamic futures hedging in currency markets 305 L(q ) 5 2 T log 2p O T i5 1 logï H t ï h t H t h t (12) where q 5 (a,c,a 1,B 1 ). The model can be estimated with maximum likelihood numeric techniques. The parameter values can be used to estimate the timevarying covariance matrix. The sequence of OHRs, [b t ], can then be calculated according to expression (9) using the estimated conditional variances and covariances at each point in time t. 4. EMPIRICAL ESTIMATES 4.1. Data and preliminary tests Exchange rates are weekly spot and futures rates from 6 January 1982 to 27 December 1991 (a total of 520 returns observations) obtained from the Data Resources Incorporated (DRI) and the Center of Research for Futures Markets, respectively. Both rates are opening dollar prices for the Canadian dollar (CD), Deutsche mark (DM), British pound (BP), Swiss franc (SF), and Japanese yen (JY) in New York and the IMM of the Chicago Mercantile Exchange, respectively. The futures contract used is the nearest to expiration that is not in the delivery month. The data correspond to observations on each Wednesday. In the event the market was closed on a Wednesday, the observation for the next day the market was open is taken. The subsequent analysis is performed on the rst differences of the logarithmic transformation of spot and futures prices (returns series). 8 Table 1 reports preliminary test statistics for our sample series. Application of the Jarque Bera normality test indicates that the unconditional distribution of spot and futures returns is non-normal for the CD, BP and JY. Consistent with previous literature, leptokurtosis is the main reason for rejecting the normality hypothesis. We fail to reject the normality null in spot and futures returns for the DM and SF. The serial independence tests suggest that all returns series are serially uncorrelated but not independent. There is temporal dependence in the second moments of all returns (with the possible exception of futures returns 8 Phillips Perron unit root tests (Phillips, 1987; Phillips and Perron, 1988 were performed on the (log) spot and futures prices and their rst differences in order to assess the low-frequency properties of the series. We cannot reject the null hypothesis of a unit root, though the null hypothesis of a second unit root is strongly rejected for all price series. To ensure the robustness of the evidence that all price series contain a single unit root, the Kwiatkowskl Phillips Schmidt Shin unit-root tests (KPSS; Kwiatkowski, Phillips, Schmidt and Shin, 1992 were also performed on the levels and the rst differences of the price series. The innovation of the KPSS test is that, contrary to the standard unit-root tests, the null hypothesis of stationarity (level or trend) is tested against the alternative of a unit root. The results from applying the KPSS tests strongly indicate the existence of a single unit root in all price series. To conserve space, these results are not reported here but are available upon request.

9 O 306 A. Chakraborty and J.T. Barkoulas Table 1. Test statistics for the logarithmic differences (returns) of weekly spot and futures currency series Statistic CD DM BP SF JY Spot Currency Returns Mean (m ) 0.391E E E E E-2 t(m = 0) Std. Dev E E E E E-1 m (0.107) (0.107) (0.107) (0.107) (0.107) m (0.214) (0.214) (0.214) (0.214) (0.214) JB *** *** 4.753* *** ABP(1) * ABP(2) * ABP(6) ABP(12) ARCH(1) *** 8.101*** 8.043*** 8.367*** *** ARCH(2) *** *** 9.868*** *** *** ARCH(6) *** *** *** *** *** ARCH(12) *** *** *** *** *** Futures Currency Returns Mean (m ) 0.337E E E E E-2 t(m = 0) Std. Dev E E E E E-1 m (0.107) (0.107) (0.107) (0.107) (0.107) m (0.214) (0.214) (0.214) (0.214) (0.214) JB *** *** 5.947* *** ABP(1) ABP(2) ABP(6) ABP(12) ARCH(1) *** * *** ARCH(2) *** 7.465** * *** ARCH(6) *** ** *** *** ARCH(12) *** ** *** * The sample period is 1/6/82 to 12/27/91 for a total of 520 weekly returns observations. m 3 and m 4 are the sample skewness and excess-kurtosis coef cients, respectively. Standard errors are given in parentheses. The standard error for the coef cient of skewness (kurtosis) in each series is computed as Ï 6 T (Ï 24 T ), where T is the number of observations. JB is the Jarque Bera normality test statistic. ABP(k) is the adjusted Box Pierce Q-test statistic for autocorrelation of order k which corrects for the presence of conditional heteroscedasticity in the data, it is given by k 2 H r(t)/s(t)j, t=1 where r(t) is the autocorrelation parameter for order t, s(t) =Î S 1 T D S 1 + g 2 (t) s 4 D, g 2 (t) is the tth autocovariance of the squared data, s is the sample standard deviation of the data, and T is the number of observations (Diebold, 1986). ARCH(k) is the Engle s (1982) LM test for ARCH effects of order k. Signi cance levels: *=10%, **=5%, ***=1%.

10 Dynamic futures hedging in currency markets 307 for the SF) as evidenced by Engle s (1982) Lagrange-multiplier test statistics for the presence of ARCH effects Estimation of bivariate GARCH models for spot and futures currency returns To capture time-varying second-moment effects in the joint distribution of spot and futures currency returns, a bivariate GARCH (1,1) model is tted to the data. This model speci cation is obtained by setting p 5 q 5 1 in (10c) and (11). Table 2 reports the empirical estimates of the bivariate GARCH(1,1) model for all ve currencies with the full positive de nite parameterization in the conditional covariance matrix. The hypothesis of a constant covariance matrix is strongly rejected for all currencies. The model was also estimated with the diagonal positive parameterization in the conditional covariance matrix but the crossequation restrictions were rejected. 10 The statistical soundness of the bivariate GARCH(1,1) model is additionally substantiated by means of residual diagnostic tests. As Table 2 shows, these tests indicate that standardized residuals in each equation are serially uncorrelated and there are no remaining GARCH effects as evidenced by uncorrelatedness in the vector of squared standardized residuals. The bivariate GARCH model reduces but does not eliminate excess kurtosis in the standardized residuals of the returns series for which the normality assumption was rejected In-sample and out-of-sample hedging performance The static and dynamic hedging models call for constant and time-varying OHRs, respectively. In this section, the OHRs under these two hedging strategies are estimated, their statistical properties are investigated, and the in-sample and out-of-sample hedging performance they provide is evaluated. According to the static model, the OHR is constant and can be obtained as the slope coef cient of a regression of spot returns on futures returns. Table 3 reports the results from running such an OLS regression for each currency. These results are consistent with the previous work (see Grammatikos and Saunders, 1983, for example). As evidenced in the previous section, however, the conditional second moments of spot and futures returns are time-varying thus rendering the static hedging model inappropriate. Given the bivariate GARCH model of spot and futures returns, the sequence of time-varying OHRs can be constructed using 9 To model the time-varying conditional second moments, we tted a univariate GARCH(1,1) model to each of the returns series. This model provides a good explanation of the data based on residual diagnostic tests. However, the assumption of conditional normality density for the error process fails to account for the observed leptokurtosis in the series for which the normality hypothesis was rejected. The estimation results for the univariate GARCH(1,1) model for all the returns series are not reported here but are available upon request. 10 The likelihood-ratio test statistics for the null hypothesis of diagonal positive de nite parameterization versus the full positive de nite parameterization are (0.000), (0.000), (0.000), (0.030), and (0.001) for the CD, DM, BP, SF, and JY, respectively (marginal signi cance levels are given in parentheses).

11 308 A. Chakraborty and J.T. Barkoulas Table 2. Estimation of the bivariate GARCH models with full positive de nite parameterization 1000 y t = a +h t ; y t = (s t,f t ) h t ï W t 1 ~ N(0, H t ) H t = C C+A h t 1 h t 1 A+B H t 1 B CD DM BP SF JY a a (1.134) (0.509) ( 1.462) (0.567) (0.722) (1.091) (0.565) ( 1.585) (0.673) (0.694) C (9.483) (2.635) (8.221) (6.848) (22.420) C (11.294) (0.547) (4.434) (4.933) (26.438) C (9.139) (7.106) (4.687) (13.567) (2.531) A ( 0.743) (1.260) ( 5.575) (5.182) (1.425) A ( 2.033) ( 0.864) (9.072) ( 3.188) (1.335) A ( 2.791) ( 0.991) (0.408) (1.792) (3.923) A ( 0.215) (1.105) (1.866) ( 0.092) ( 1.502) B (14.308) ( ) (12.872) (5.134) (0.105) B (0.024) (11.386) ( 0.863) ( 0.961) (0.362) B ( 2.369) ( ) (5.397) (5.647) ( 0.027) B (19.365) (12.689) (10.255) ( 0.625) ( 0.201) log-likelihood lr test: A = B = (0.000) (0.000) (0.000) (0.001) (0.001) Spot equation m m JB (0.000) (0.661) (0.014) (0.324) (0.000) Q(12) (0.371) (0.325) (0.342) (0.241) (0.051) Q 2 (12) (0.458) (0.031) (0.429) (0.582) (0.585)

12 Dynamic futures hedging in currency markets 309 Table 2. Continued CD DM BP SF JY Futures equation m m JB (0.000) (0.795) (0.000) (0.201) (0.000) Q(12) (0.919) (0.509) (0.401) (0.517) (0.273) Q 2 (12) (0.824) (0.007) (0.194) (0.602) (0.955) The sample period is 1/6/82 to 12/27/91 for a total of 520 weekly returns observations. s t and f t are the spot and futures currency returns series, respectively. t-statistics are given in parentheses for the coef cient estimates. Marginal signi cance levels are given in parentheses for the rest of the test statistics. m 3 and m 4 are the sample skewness and excess kurtosis in the standardized residuals, respectively. JB is the Jarque Bera normality test statistic in the standardized residuals. Q(12) and Q 2 (12) are the Box Pierce Q-test statistic for autocorrelation of order 12 in the standardized and squared standardized residuals, respectively. the empirical estimates of the conditional covariance matrix. At each point in time t, the model s parameter values are used to estimate the time-dependent conditional covariance matrix, and in turn calculate the OHR according to expression (9) (without the negative sign), that is, Table 3. Estimation of optimal hedge ratios from OLS regressions (static hedging model) 1000 s t = b 0 +b 1 (1000 f t ) Currency b 0 b 1 Adj. R 2 CD (0.102) (0.016) DM (0.201) (0.012) BP (0.191) (0.011) SF (0.228) (0.013) JY (0.190) (0.012) The sample period is 1/6/82 to 12/27/91 for a total of 520 weekly returns observations. s t and f t are the spot and futures currency returns series, respectively. Standard errors are shown in parentheses. Adj. R 2 is the adjusted coef cient of determination. Signi cance levels: * = 10%, ** = 5%, *** = 1%.

13 310 A. Chakraborty and J.T. Barkoulas Table 4. Summary statistics for the estimated time-varying (GARCH-based) optimal hedge ratios Statistic CD DM BP SF JY Mean Std Dev m m Minimum Maximum m 3 and m 4 are the sample skewness and excess-kurtosis coef cients, respectively. b * t 5 H 21,t 1 1 H 22,t 1 1 (13) where H ij,t 1 1 is the element in the ith row and jth column of the estimated conditional covariance matrix H t. Repeating this at each point in time results in a sequence of time-varying OHRs over the sample period. Table 4 reports summary statistics for the estimated time-varying OHRs for each currency. The time-varying OHRs show considerable uctuation and none appears to be consistent with the assumption of constant OHR. They are skewed and leptokurtic as indicated by their third and fourth sample moments. Application of the Phillips Perron and KPSS unit root tests on the estimated time-varying OHRs suggest that they are stationary processes with the possible exception of the CD for which the series is not suf ciently informative to distinguish between the unit root and stationarity hypotheses. 11 Thus the evidence strongly supports the hypothesis that OHRs are time-varying but stationary. The ultimate test of superiority of a hedging modelling strategy however lies in its ability to achieve the maximum possible reduction in portfolio variance. This is what matters from an investor s viewpoint. We therefore proceed to measure and compare both the in-sample and out-of-sample variance of portfolio returns (s t 2 b t f t ) under the following four hedging strategies: no hedge (b 5 0), naive hedge (b 5 1), static hedge (b 5 b OLS ), and dynamic hedge (b 5 b* t ). Table 5 presents the in-sample hedging performance of the different hedging approaches considered. Not surprisingly, the no-hedge strategy has the poorest performance (maximum portfolio return variance) followed by the naive-hedge strategy. The dynamic GARCH-based hedge provides the most effective hedge 11 The apparent stationarity of the dynamic OHRs provides indirect evidence that the estimated series of conditional covariances and variances for futures returns are stationary. The dynamic OHRs could be stationary, however, even if the conditional covariances and variances for futures returns contain a unit root insofar as their logarithms were cointegrated with the cointegrating factor being equal to one. The evidence of stationary OHRs for foreign currencies stands in sharp contrast to the unit-root evidence for OHRs for several commodities as reported in Baillie and Myers (1991).

14 Dynamic futures hedging in currency markets 311 Table 5. In-sample hedging performance Currency No hedge Naive hedge Static hedge Dynamic hedge CD Mean Variance DM Mean Variance BP Mean Variance SF Mean Variance JY Mean Variance The sample period is 1/6/82 to 12/27/91 for a total of 520 weekly returns observations. The mean and variance statistics are calculated for the portfolio return series s t b t f t, where b t is the series of the ex-post computed hedge ratios under different hedging strategies. b t = 0 for the no-hedge strategy, b t = 1 for the naive strategy, b t = b OLS for the static hedge, and b t = b t * (GARCH-based) for the dynamic hedge. for DM, BP, and SF. For the CD and JY the traditional OLS-based hedge achieves the lowest portfolio variance. However, the differences in portfolio variances between the static and dynamic hedges are very small. Therefore, the additional complexity introduced by the bivariate GARCH model may not be justi ed in terms of superior hedging effectiveness. Realistically, however, an investor needs to determine his/her hedging position ex ante and the ability of a hedging strategy to reduce the out-of-sample portfolio variance is the desirable feature. Even an inadequate model could have a satisfactory in-sample performance as it possesses the bene t of hindsight. The out-of-sample hedging performance of the static and dynamic models is evaluated as follows. The last 52 observations are withheld from the sample, so that OLS and bivariate GARCH models are estimated using only the rst 468 observations. For the static hedge, the in-sample estimate of the hedged position is used to forecast the one-week-ahead OHR. For the dynamic hedge, the one-week-ahead forecasted OHR is the ratio of the one-week forecast of conditional covariance to the one-week forecast of the variance. The same experiment is repeated for the following week, with a new observation added to the data set. Updating of the model and forecasting of the optimal futures hedge under the two models continues until the data set is exhausted, giving a series of 52 forecasted OLS- and GARCH-based OHRs, and optimal portfolios for each model. At each point in time, only information available at that time is used in order to form forecasts of the hedging coef cients. The OLS-based OHRs remain virtually constant over the forecast period while the GARCH-based OHRs exhibit substantial variability. The variances of the outof-sample portfolio returns under the no-hedge, naive, static, and dynamic strategies are reported in Table 6. The results are striking but mixed. For the CD, the dynamic hedge reduces out-of-sample portfolio variance by % relative

15 312 A. Chakraborty and J.T. Barkoulas Table 6. Out-of-sample hedging performance (variances of portfolio returns) to static hedge. However, the opposite result is obtained for the rest of the currencies. The dynamic hedge results in an increase of the out-of-sample portfolio variance relative to static hedge by %, %, 2.894% and 6.818% for the DM, BP, SF and JY, respectively. Actually, for these four currencies, even the naive hedge dominates the dynamic hedge in terms of portfolio variance reduction. Use of forecasted dynamic OHRs destabilize, in effect, the investor s ex ante variance of portfolio returns. The evidence therefore rejects the dynamic hedging model for foreign currencies with the only exception for the CD. 5. CONCLUSIONS CD DM BP SF JY No hedge Naive hedge Static hedge Dynamic hedge The full sample period is 1/6/82 to 12/27/91 for a total of 520 weekly returns observations; the rst 468 observations are used for in-sample estimation (test set) with the remainder of 52 observations being used for out-of-sample forecasting purposes. The variance statistics are calculated for the portfolio return series s t b t f t, where b t is the series of the ex-ante computed hedge ratios under different hedging strategies. b t = 0 for the no hedge strategy; b t = 1 for the naive strategy, b t = b OLS for the static hedge, and b t = b* t (GARCH-based) for the dynamic hedge. This paper has empirically assessed the appropriateness of the dynamic hedging model with foreign currency futures. The bivariate GARCH model for spot and futures returns is estimated for the ve leading currencies. This model represents well the data and indicates substantial time-dependency in the conditional covariance matrix. OHRs are constructed from the empirical estimates of the conditional covariance matrix and are found to be time-varying, thus rendering the static hedging model methodologically inappropriate. Despite its statistical soundness, however, a dynamic hedge does not result in meaningful gains in hedging effectiveness relative to a constant hedge retrospectively and produces superior hedging performance only for the CD prospectively. A case may therefore be made in favour of a constant hedge since the extra computational and rebalancing costs associated with the dynamic hedge do not compensate with signi cant and consistent reductions in the investor s foreign exchange risk. A number of explanations can be offered for the inability of the GARCH model to achieve superior hedging performance. First, given the persistence in the conditional covariance matrix, the presence of an outlier can erroneously affect

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