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Dynamic hedging with non-martingale futures prices and time-varying volatilities Marlowe, D. J. 1996

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D Y N A M I C H E D G I N G W I T H N O N - M A R T I N G A L E F U T U R E S PRICES A N D T I M E - V A R Y I N G VOLATILITIES D J . Marlowe B.Sc, The University of Toronto, 1991 M.A., The University of Toronto, 1992 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE (BUSINESS ADMINISTRATION) in THE FACULTY OF GRADUATE STUDIES (Faculty of Commerce and Business Administration) We accept this thesis as conforming to the required standard THE UNIVERSITY OF BRITISH COLUMBIA September 1996 ® D J . Marlowe, 1996 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of C O M a c e f B U s m e c s . A o M i K i t T f t A t ' a H The University of British Columbia Vancouver, Canada Date Sepre^sca. 27. DE-6 (2/88) Abstract Conventional hedging theory fails to take into account a number of stylized facts about exchange rate dynamics, most importantly the time-varying nature of volatility and the cointegration between spot and futures prices. In an effort to address this, recent studies have re-examined the hedging problem using both error correction models and GARCH error structures. These studies, however, rely on the questionable assumption of martingale futures prices. This implies that a strategy of holding futures contracts should never produce a non-zero expected profit, since the expectation of any subsequent futures price will always equal today's futures price. This paper derives a hedging model which does not impose this assumption, and uses out-of-sample testing to assess its performance relative to both the time-varying martingale model and other static models. Comparisons reveal that the martingale assumption is non-trivial, and that the hedge ratios derived under this assumption will differ greatly from those derived without it. Moreover, those derived without the martingale assumption can produce a higher expected utility for the hedger under certain circumstances. Table of Contents Abstract ii Table of Contents iii List of Tables iv List of Charts v Section I. Introduction 1 Section II. Hedging Theory 2 Section III. Mathematical Derivation of a General Model 8 Section IV. Methodology for Model Estimation 20 Section V. Data 24 Section VI. Model Estimation 32 1. Tests for Unit Roots and Cointegration 32 2. Bivariate GARCH Estimation 33 3. Hedge Ratio Formulas to be Examined 37 4. Monte Carlo Simulation 40 5. Calculations 42 6. Performance Measures 44 7. Artificial Neural Networks 52 8. Monetary Performance Measures 63 Section VII. Conclusions 67 References 71 Charts 74 iii List of Tables Table 6.1: Phillips-Perron Unit Root Tests Table 6.2: Maximum Likelihood Estimates of Bivariate GARCH Error Correction Model 33 35 Table 6.3a: Expected Utility of Wealth Changes Among Hedging Strategies, Low Risk Aversion (7 = 1) Table 6.3b: Expected Utility of Wealth Changes Among Hedging Strategies, Medium Risk Aversion (7=4) Table 6.3c: Expected Utility of Wealth Changes Among Hedging Strategies, High Risk Aversion (7=10) Table 6.3d: Skewness (S), Kurtosis (K) and Normality (BJ) of Wealth Changes Table 6.4: Maximum Likelihood Estimates of Bivariate GARCH-ANN Error Correction Model 48 49 50 51 56 Table 6.5a: Expected Utility of Wealth Changes Among Hedging Strategies, Low Risk Aversion (7 = 1) Table 6.5b: Expected Utility of Wealth Changes Among Hedging Strategies, Medium Risk Aversion (7=4) Table 6.5c: Expected Utility of Wealth Changes Among Hedging Strategies, High Risk Aversion (7=10) Table 6.6: Maximum Monetary Value of Hedging Strategies 60 61 62 65 IV List of Charts Chart 6.1: Weekly Out-of-Sample Hedge Ratios, Japanese Yen (GARCH) 74 Chart 6.2: Weekly Out-of-Sample Hedge Ratios, Swiss Franc (GARCH) 74 Chart 6.3: Weekly Out-of-Sample Hedge Ratios, Deutschmark (GARCH) 75 Chart 6.4a: Hedging Effectiveness, Japanese Yen 75 Chart 6.4b: Weekly Out-of-Sample Changes in Wealth, Japanese Yen (GARCH) 76 Chart 6.5a: Hedging Effectiveness, Swiss Franc 76 Chart 6.5b: Weekly Out-of-Sample Changes in Wealth, Swiss Franc (GARCH) 77 Chart 6.6a: Hedging Effectiveness, Deutschmark 77 Chart 6.6b: Weekly Out-of-Sample Changes in Wealth, Deutschmark (GARCH) 78 Chart 6.7: Actual vs Estimated GARCH Volatility, Japanese Yen, 1990-95 79 Chart 6.8: Actual vs Estimated ANN Volatility, Japanese Yen, 1990-95 79 Chart 6.9: Weekly Out-of-Sample Hedge Ratios, Japanese Yen (ANN) 80 Chart 6.10: Weekly Out-of-Sample Hedge Ratios, Swiss Franc (ANN) 80 Chart 6.11: Weekly Out-of-Sample Hedge Ratios, Deutschmark (ANN) 81 Chart 6.12: Weekly Out-of-Sample Changes in Wealth, Japanese Yen (ANN) 81 Chart 6.13: Weekly Out-of-Sample Changes in Wealth, Swiss Franc (ANN) 82 Chart 6.14: Weekly Out-of-Sample Changes in Wealth, Deutschmark (ANN) 82 V I. Introduction The objective behind hedging theory is to minimize the amount of uncertainty associated with an exposed (risky) spot position by taking a position in another risky asset whose price movements will largely offset any adverse price movements in the first asset. By convention, the optimal hedge ratio can be found by regressing price movements in the spot position on those of the hedging asset — typically a futures contract. By the properties of ordinary least squares (OLS), shorting this number of futures contracts for every long spot contract of the same size will represent a portfolio with the least residual variance. Dynamic hedging theory relaxes one of the key assumptions behind this model. Namely, in a dynamic model the relationship between the underlying and hedging assets is no longer assumed to be static. As new information becomes available, hedgers can update their model to arrive at a new optimal hedge ratio within the time frame of the original hedge. Kroner and Sultan (1991; 1993) develop a dynamic model which uses a GARCH variance to model time-varying hedge ratios, the underlying assumption being that GARCH can reasonably approximate the dynamic nature of the spot-futures relationship. In their model, Kroner and Sultan assume that futures prices follow a martingale (i.e. E(F t + 1) = FJ. This greatly simplifies the utility maximization problem, since it not only eliminates additional covariance terms which are difficult to estimate, but also produces a hedge ratio which is independent of the hedger's risk tolerance. The question remains, however, how one would go about constructing an optimal hedge if this assumption is violated — and it often is violated, especially in the short-run. As will be shown, when the martingale assumption does not hold, the optimal hedge ratio contains additional terms which are a function of the hedger's level of risk aversion. One of these terms allows for a speculative gain incurred by holding futures contracts from the first period to the next, while the others provide a hedge for the speculative gains to be realized in subsequent periods. It will be shown that the optimal hedge will be a function of next period's expectations — expectations which are not known with certainty in this period, but which can be estimated based on today's information set through Monte Carlo simulation. This paper is organized as follows. Section II presents an analysis of the hedging literature, paying particular attention to GARCH hedging models. Section III derives an optimal hedge ratio mathematically when the assumption of martingale prices is abandoned, and shows how the martingale model can be regarded as a special case of this general model. The estimation procedure is outlined in Section IV, Section V discusses the data and Section VI presents empirical findings. Section VII concludes. II. Hedging Theory Traditionally, the optimization problem used to find the most efficient hedge ratio has been structured in a portfolio context, with no attention paid to the feature which distinguishes futures from forwards — marking to market. The reasoning behind this deliberate oversight is usually that these daily transactions have a zero (or at least negligible) expected value. While this may be true under a martingale assumption, it need not hold in all cases or for all currencies — especially in the short-run, during which exchange rates can follow pronounced upward or downward trends in response to anticipated or observable economic events and monetary policies. In a one-period portfolio optimization problem, it is assumed that a hedger holds one unit of spot currency, and must find some quantity, 6, of futures contracts to maximize utility, which is described by the mean and variance of changes in wealth: where W„ St and F t are wealth, the spot price and the futures price in time t, and 7 is a measure of the investor's risk tolerance. This framework can be found in several papers, including Hill and Schneeweis (1982), Chang and Shanker (1986), Baillie and Myers (1991), Castelino (1992) and Kroner and Sultan (1993). Many authors choose to drop the mean term from (2.2) for various reasons — some regard the hedger's problem as purely risk minimization, while others assume that spot and futures prices are martingales, and therefore the expected change in wealth will be zero. Another convenient property of hedges constructed without the mean term in (2.2) is that the solution turns out to be independent of utility, and can therefore be applied to any individual or firm wishing to hedge in the futures market. While a typical portfolio optimization problem uses returns instead of changes in wealth, several authors specify the (2.1) dW, = (SrS0) + 9(FrFo), (2.2) Max U(dW,) = E(dW,) - 7Var(dW0, problem as above in terms of first differences (Hill and Schneeweis (1982); Anderson and Danthine (1983); Duffie and Jackson (1990)). The primary reason for doing this is that it is consistent with marking to market transactions, which are based on level changes rather than returns. Often, (2.1) and (2.2) are expressed in terms of terminal wealth (W,) rather than changes (dW,), but since initial wealth is a known value at time 0, these two forms are functionally equivalent. To solve (2.1) and (2.2), substitute the wealth constraint into (2.2) to get: (2.3a) Max E[(St-S0) + 0(Ft-F^] - 7Var[(St-S0) + 6(Ft-F0)]. Since time 0 prices are known, (2.3a) reduces to: (2.3b) Max E(S.) - S0 + 0[E(F,)-Fo] - 7[Var(S.) + 92Var(F,) + 29Cov(St,F,)]. Maximizing with respect to 6 and solving the first order condition solves for the optimal hedge ratio: (2.3c) [E(Ft)-F0] - 27eVar(F,) - 27Cov(St,F,) = 0, or, (2.3d) 6* = [E(Ft)-F0]/27Var(Ft) - Cov(St,Ft)/Var(Ft). If futures prices are martingales (Kroner and Sultan (1993)), the first term drops out of -4-(2.3d), and the minimum-variance hedge ratio becomes: (2.3e) 6* = -Cov(St,Ft)/Var(Ft), which is independent of 7. Perhaps the most significant observation to be made about hedge ratios since they started to appear in the literature is that the optimal solution depicted in (2.3d) implicitly assumes that the covariance between spot and futures prices remains constant over time (Grammatikos and Saunders (1983), Cecchetti et al (1988)). As such, holding 0* futures contracts as a hedge for every unit of spot currency will no longer be optimal if the covariances change before the hedge expires, or if the hedging horizon extends beyond one period. To address this inherent instability, modifications to (2.3d) which have been suggested are overlapping regression procedures (Grammatikos and Saunders (1983)), or using conditional moments (Cecchetti et al (1988), Baillie and Myers (1990), Myers (1991), Kroner and Sultan (1991; 1993)). This latter approach defines the optimal hedge ratio as a function of the conditional moments at time t rather than the unconditional moments: (2.3e) 9t* = [E(F t + 1 j Q,)-FJ/27Var(Ft+110.) - Cov(S l + 1,F t + 1 j 0,)/Var(Ft+11Q,). Under this framework, the hedge ratio is updated every period, using all available information at any given time. To obtain the conditional distributions, many authors have relied on the ARCH and GARCH estimation techniques of Engle (1982) and Bollerslev (1986), which characterize the conditional variances in any period as a function of past variances and/or residuals (Baillie and Myers (1991), Myers (1991), Kroner and Sultan (1991; 1993)). This procedure takes advantage of the consensus in the literature that observes that foreign exchange prices — or more generally, prices of virtually any publicly traded financial asset — exhibit significant ARCH effects in the second moment (McCurdy and Morgan (1987), Bollerslev et al (1992), De Vries (1994)). Because spot and futures prices will invariably have non-zero conditional covariances, the price dynamics are typically modelled through a bivariate model, such as the one depicted in (2.4a) through (2.4g): (2.4a) dSt = E(dSt|Q,1) + est, (2.4b) dF t = E(dF t|fi,1) + 6FT, (2.4c) [e^epjfi,!] ~ N(0,H,), (2.4d) H t = hSf,t hff.t (2.4e) hSSjt = Uos + tflshss,t.i + JWs,t-i2, (2.4f) hff,t = tf0F + B 1 Fh f f i t r l + I W F , ^ 2 , (2.4g) h s f ; t = p(hSSit)1/2(hffit)1/2. As before, S and F denote the spot and futures prices, respectively. The conditional means in (2.4a) and (2.4b) represent functions which can take one of many forms — they may be constants (or zero), moving averages of some sort, lagged dependent variables, and/or any -6-other exogenous or pre-determined variables. Equation (2.4c) states that the residuals in the mean equations are jointly normal with zero mean and covariance matrix H — a constant correlation model with variance given by (2.4e) and (2.4f) and covariance based on a time-invariant correlation coefficient, p. The information set at time t is given by fit. The terms dSt and dFt are first-differences (St - St.,), although log-differences (log(S,) - log^.,)) and returns (St/Sn - 1) can be found in the literature. Strictly speaking, however, log-differences and returns are inconsistent with the underlying utility maximization solution, which explicitly expresses wealth as a function of first-differences in spot and futures prices. Log-differences and first-differences produce identical hedge ratios, however, if one makes the simple assumption that the initial spot position consists of one unit of foreign currency (as opposed to one dollar's worth of foreign currency). While the above model describes a general formula, a number of modifications have been suggested in other studies. Kroner and Sultan (1993) specify the mean equations (2.4a) and (2.4b) with a constant and an error correction term of the form (c^/Su - Ft.,)) to capture the effects of cointegration between spot and futures prices. Baillie and Myers (1991) find that while spot and futures prices of commodities do not tend to be cointegrated, prices of most financial variables do. As such, an error correction term should be included in the model to impose long-run equilibrium constraints on two financial price series, as per Engle and Granger (1987), though not necessarily for two commodity price series. Since currency prices exhibit excess kurtosis (see, for instance, De Vries (1994)), another modification to the above GARCH model replaces the normal distribution in (2.4c) with a Student's t distribution, as in Baillie and Myers (1991). When the conditional spot and futures price distributions are given by (2.4a) through (2.4g), the optimal hedge in (2.3e) becomes: III. Mathematical Derivation of a General Model Despite the widespread use of hedging models such as those described by (2.3d), (2.3e) and (2.5), they are all based on a questionable foundation. Futures contracts are distinct from forward contracts in that they generate uncertain cashflows from marking to market, yet wealth as defined by (2.1) disregards these daily transactions when t is defined to be more than one day. An investor's wealth consists of both the value of assets in his or her portfolio and the cumulative cashflows generated by that portfolio. In this vein, a number of authors have analyzed the problem by defining wealth as the sum of the spot position and the cumulative marking to market transactions (Anderson and Danthine (1983), Duffie and Jackson (1990), Duffie and Richardson (1991)). In these studies, wealth is defined (in discrete time) as: (2.5) et* = E(dF t + 1!fi l)/27h f f > t-h s fyh f f ) t. (3.1) Wt = St + EjG/Fv.-Fj), j=0,..t-l, while utility is mean-variance in terminal wealth, as in (2.2). Thus, in these studies, the investor chooses the hedge ratio which maximizes a utility function increasing in expected wealth and decreasing in variance of wealth. Strictly speaking, the cumulative marking-to-market cashflows should also include net interest charges, but this modification is not explored here. An important distinction which should be made on a theoretical level is that utility in this framework no longer applies to wealth in every period, but rather the change in wealth between the beginning and end of the hedging horizon. Since the objective is to hedge a cashflow to be received at some future date, T, the value of the spot/futures portfolio is irrelevant between now and date T — the hedger is interested only in designing a hedge which makes time T wealth less risky than it would otherwise be, as traditional hedge accounting defers derivative profits and losses from being reported on the balance sheet until the hedge has been lifted. It can be shown through the mathematical proof below, however, that under the martingale assumption (and only under this assumption), the hedge ratios are identical for both the one-period and multi-period utility maximization problems. Due to the cumbersome mathematics involved, the optimal hedge derived here will be restricted to a two-period model. The extension to three periods and beyond is a relatively straightforward procedure involving one additional covariance term for each additional period. In a two-period model (starting from time T), terminal wealth at time T+2 is equal to the sum of the terminal spot position and the cumulative marking to market transactions on the futures contracts held as a hedge: (3.1 ) W T + 2 = S x + 2 + 0T+I(FT+2~FT+I) + ®T(FT+1"FT)J where 0 T + j are the hedge ratios determined at the beginning of time T+j and held until the beginning of time T+j + 1. The agent is assumed to have a mean-variance utility function in terminal wealth: (3.2) U T + 2 . t(W T + 2) = F™_ t (W T + 2 ) - 7VarT+2. t(WT+2), where t e [1,2] and 7 is a coefficient of risk aversion. Since a conditional variance comprises two components — the variance of next period's expectation and the expectation of next period's variance, equation (3.2) can be expanded into: (3.3) U T + 2 . t (W T + 2 ) = E T + 2 . t (W T + 2 ) - 7VarT + 2. l[ET + 2. t + 1(WT + 2)] - 7^+2JVarx+2.t+1(AVT+2)]. This is the objective function which will be maximized in the two-period model. Proposition 1 In a two-period model with mean-variance utility, the optimal hedge in the first period will comprise both a hedge for the spot position and a hedge for the anticipated marking to market profits or losses in the second period. The optimal hedge in the second period will be identical to the standard one-period hedge based on conditional moments. -10-Proof In a two-period setting, the objective is to select 0 T and 9 T + 1 which will maximize U T(W T + 2) and U T + 1 (W T + 2 ) , respectively. Because future realizations of the utility function depend on the stochastic price variables, U T + 2 . t (W x + 2 ) may not be independent across time periods. In other words, maximizing the function for t=l and t=2 separately will not necessarily generate the optimal solution. Because the optimal 0 T turns out to be dependent on the optimal 0 X + 1 (as will be shown below), the two maximization problems cannot be solved independently of one another except under special conditions. In the first period, T, the objective function is given by (3.3): Because the formulas become cumbersome, each of the three right-hand side components will be solved separately, then put back together at the end. Substituting (3.1) into each components of (3.3a): (3.4b) 7VarT[ET + 1(WT + 2)] = 7Var T{E r + I[S T + 2 + 0 T + 1 (F T + 2 -F T + 1 ) + 0T(FT+1-FT)]}, (3.4c) 7Er[VarT + 1(WT + 2)] = 7Er{VarT + 1[ST + 2 + 0 T + i ( F T + 2 - F T + 1 ) + ex(FT + rFT)]}. (3.3a) U T(W T + 2) = Er(WT + 2) - 7VarT[ET + 1(WT + 2)] - 7^[Var T + 1(W T + 2)]. (3.4a) E T(W T + 2) — E T [S T + 2 + 0T+I(FT+2"F T +I) + 0T(FT+I"FT)L Looking first at (3.4a), if expectations are taken through the right-hand side of the equation, everything after time T will be stochastic: (3.4a') E T ( W T + 2 ) = ^(ST+2) + E T [0 T + 1 (F T + 2 -F T + 1 )] + eT[E r(FT + 1)-FT]. Equation (3.4b) is equal to the variance of next period's expectation of terminal wealth, which becomes: (3.4b') 7 V a r x [ E T + 1 ( W X + 2 ) ] 7Var x[E T + 1(S T + 2)] + 7Var x{0T + 1[ET + 1(FT + 2)-FT + 1]} + 7VarT[0T(FT + 1-FT)] + 27Cov T{E T + 1(S T + 2),0 T + 1[E T + 1(F T + 2)-F T +i]} + 27CovT[ET + 1(ST + 2),0T(FT + 1-F x)] + 27Cov T{0 T + 1[E T + 1(F x + 2)-F x + 1],0 x(F x + 1-F T)} 7VarT[ET +i(ST + 2)] + 7Var x{0 x + 1[E T + 1(F x + 2)-F T + 1]} + 70 x 2Var T[F T + 1] + 27Cov T{E T +,(S T + 2),0 T + 1[E x + 1(F x + 2)-F x + 1]} + 270 TCov x[E T +i(S x + 2),F T + 1] + 270 xCov T{0 x + ,[E T +i(F T + 2)-F T + 1],F x + 1}. Finally, equation (3.4c) is the time T expectation of the conditional variance of terminal wealth at time T + l , which reduces as follows: (3.4c') 7Er[Var X + 1 (W X + 2 ) ] = 7Er{Var x + 1[S x + 2 + 9 x + i ( F x + 2 - F x + ] ) + 9x(Fx+j-Fx)]} - 1 2 -7Er{VarT + 1(ST + 2) + V a r T + 1 [ G X + 1 ( F T + 2 - F T + ] ) ] + Var T + 1[e T(F T +,-F T)] + 2Cov x + 1 [S T + 2 , 9 T + i (F T + 2 -F T + 1 ) ] + 2Cov x + 1 [S T + 2 ,0 T (F T + i-F T )] + 27CovT+j [0T +1 ( F X + 2 - F T + ] ) , 0 T (F T + , -F X ) ] } 7ET[VarT + 1(ST + 2)] + 7 E T l © T + i 2 V a r X + , ( F X + 2 ) ] + 27E T [0 x + 1 Cov x + 1 (S T + 2 ,F T + 2 ) ] . Putting (3.4a'), (3.4b') and (3.4c') together in equation (3.3a): (3.5) U x (W x + 2 ) E x (W x + 2 ) - 7Var x[E x + 1(WT + 2)] - 7Er[Var x + 1 (W x + 2 )] - Er(S x + 2) + E x [ 0 X + 1 ( F T + 2 - F X + I ) ] + © ^ ( F T + ^ - F T ] - 7VarT[Er+i(ST+2)] 7Var T {0 x + 1 [E T + 1 (F T + 2 ) -F T + 1 ]} - 7© T 2 Var T [F T + 1 ] -27Cov T {E T + 1 (S T + 2 ) ,0 x + 1 [E T + 1 (F T + 2 )-F T + j]}- 270TCovT[Er + 1(S x + 2),F x +i] 270 T Cov T {0 x + 1 [E T + ] (F X + 2 ) -F X + 1 ] ,F T + , } - 7 E r[Var x + 1(S x + 2)] -7E T [0 x + 1 2 Var X + 1 (F T + 2 ) ] - 27E T [0 x + 1 Cov x + 1 (S T + 2 ,F X + 2 )]. Maximizing (3.5) with respect to 0X gives the following first order condition: -13-(3.5a) rErvFr+O-FT] - 2 76 TVar T[F T + 1] -27Cov x[ET + 1(ST + 2),FT + 1] - 27Cov T{0 x + 1[E T + 1(F T + 2)-F T + 1],F x +i} = 0. Isolating for 6T, the optimal hedge ratio is: (3.5b) 9X* = [E x(FT + 1)-FT]/2 7VarT[FT + I] - Cov T[E T + 1(S x + 2),F T + 1]/Var T[F T + 1]-CovT { 9 T +j [ET+1 (FT + 2) - F T + J ], F T + , }/Var1[FT+1]. The first term is a risk-adjusted function of expected speculative profit from holding a futures contract from period T to period T+ l (specifically, going long if the expected change is positive and going short if the expected change is negative), the second term is a hedge ratio for the T + l expectation of the terminal spot price, and the third term is a hedge ratio for the expected profit from hedging in T+l . The optimal hedge ratio in period T+l is much less complicated, since only time T+2 variables are stochastic. Substituting equation (3.1) into equation (3.3) in time T+l reduces as follows: (3.6) U T + 1 (W T + 2 ) Fr+i(Wx+2) - 7Var x + 1[E T + ,(W x + 2)] - 7Fr+i[VarT+1(Wx+2)] = &r+l(WT+2) -7Var x + 1(W T + 2) -14-ET+I [S t + 2 + 9 X + 1 ( F T + 2 - F X + 1 ) + 9 T ( F T + R F X ) ] - 7Var T + 1 [S T + 2 + ©T+I(FT+2"FT+I) 9 X (FT+1"FT)] — E T + 1 ( S x + 2 ) + 9 T + 1 [E T + ] (F x + 2 ) -F x + 1 ] + 6 T (F T + rF T ) - 7VarT + 1(S x + 2) -79 T + 1 2 Var T + 1 (F T + 2 ) - 270 T + 1 Cov T + 1 (S T + 2 ,F T + 2 ) . Maximizing (3.6) with respect to 9 T + 1 gives the first order condition: (3.6a) [Er + 1(F T + 2)-F T + 1] - 279 T + 1 Var T + ! (F x + 2 ) - 27Cov x + 1(S T + 2 ,F x + 2) = 0. Solving for 9 X + 1 gives the optimal hedge ratio in T+l as: (3.6b) 9X + 1* = [E x + 1 (F x + 2 )-F T + I ]/27Var x + 1 (F T + 2 )- Cov x + 1 (S x + 2 ,F x + 2 ) /Var x + 1 (F T + 2 ) . Analogous to the optimal hedge ratio in (3.5b), the optimal hedge in (3.6b) consists of a risk-adjusted speculative profit term and a hedging term. Since there are no subsequent profits from marking to market at time T+l , (3.6b) does not have the third term of (3.5b) which hedges expected net profits from next period's transaction. Because there are no stochastic variables extending beyond time T+2, the problem can be viewed as a one-period maximization problem: past hedge ratios are irrelevant, and the only concern is finding a hedge which maximizes utility over this last period. -15-Proposition 2 Under the assumption of martingale spot and futures prices, (3.5b) reduces to a one-period hedge analogous to (3.6b). The one-period hedge is the optimal solution to both a one-period and a multi-period problem if and only if this assumption holds. Proof Under the assumption of martingale prices, the conditional expectations terms drop out of the formula, since the expected value j periods ahead will always be equal to today's value: E T ( S T +j) = S T , and, ET(FT+J) = FT-Making these substitutions in (3.5b), the formula reduces to: (3.5c) GT* = [FT-FT]/2 7VarT[FT + 1] - CovT[ST +i,FT + 1]/VarT[FT +i] -Cov T{e T + 1 [F T + 1 -F T + 1 ] ,F x + 1 }/Var T [F T + 1 ] ,or 9X* = -Cov x[S T + 1 ,F T + 1]/Var T[F T + 1], -16-which is the same hedge ratio found in Kroner and Sultan (1991; 1993) and Duffie and Jackson (1990). Under this assumption, the optimal hedge ratio in any time period is simply a one-period hedge based on the conditional expectation of the next period's covariances. The most interesting observation is that under a martingale assumption, the one-period hedge solves the multi-period problem. In other words, (3.5c) is the optimal solution for two distinct utility maximization problems: max0UT(Wx + 2) and max eUT(W x + 1). It must be emphasized that the result in (3.5c) applies only if spot and futures prices follow a martingale. If a hedger can expect to realize non-zero net profits from holding either spot currency or futures contracts, then (3.5c) is no longer optimal, since there will be non-zero expected gains or losses in the next period which can also be hedged. While the martingale assumption makes the hedge ratio easy to estimate, it is not necessarily valid for all currencies and hedging horizons. Proposition 3 The extension of this model to three periods and beyond adds one additional covariance term for each additional period being hedged. The optimal hedge at time T+3-j (j > 1) includes a speculative component, a hedge for the tenninal spot position, and j-1 hedges for each anticipated marking to market transaction between time T+3-j and T+2. -17-P r o o f The extension to three periods can be derived in a similar manner as the two-period model: conditional variances at time T+3 can be split into the variance of the time T+2 expectations and the expectation of time T+2 variances. For a three period model, the objective is to maximize (3.2), where W T + 3 is given by: (3.8) W T + 3 = S T + 3 + 9 T + 2 (F x + 3 -F T + 2 ) + ©T+I(FT+2"FT+I) + QT(FT+1_FT)-Looking only at the variance component, the conditional variance of W X + 3 at time T-3 breaks down as follows: (3.9) V a r T ( W T + 3 ) VarT(ST + 3) + Var x [9 T + 2 (F x + 3 -F x + 2 )] + Var T [9 T + 1 (F T + 2 -F T + i ) ] + 9 x 2Var T(F T + 1) + 2Cov x[S x + 3 ,9 x + 2(F T + 3-F T + 2)] + 2Cov T[S T + 3 ,9 T + ,(F T + 2-F T + 1)] + 29TCov[ST + 3,FT + 1] + 2Cov T [9 T + 2 (F T + 3 -F x + 2 ) ,9 T + 1 (F x + 2 -F T + 1 )] + 29 T Cov[9 T + 2 (F T + 3 -F x + 2 ) ,F T + 1 ] + 29 T Cov[9 T + 1 (F x + 2 -F x + , ) ,F T + 1 ] . The mean component is given by: (3.10) Er(WT + 3). = F-T[S t + 3] + Br[e T + 2 (F T + 3 -F T + 2 )] + ET[©T+I(FT+2:FT+I)] + © T [ E T ( F T + 1 ) - F T ] . Maximizing Er(W T + 3 ) - 7VarT(WT+3) with respect to 0 T gives the optimal solution: (3.11) 0T* = [E T(F T + 1)-F T]/2 7Var T(F T + 1) - C o v ^ ^ ^ / V a r ^ F ^ ) -Cov T [G T + i (F T + 2 -F T + i ) ,F T + i ] /Var T (F x + 1 ) - Cov T [9 T + 2 (F T + 3 -F T + 2 ) ,F T + i ] /Var X (F T + 1 ) . From (3.3), the conditional variance at time T will equal the variance of the time T + l expectation plus the expectation of the time T+l variance. Noting that all the covariance terms have F T + 1 in them, however, means that the time T+l variance will always be zero. This leaves: (3.12) eT* = [E T(F T + 1)-F T]/2 7Var T(F T + 1) - Cov T [E T + 1 (S T + 3 ) ,F T + 1 ] /Var T (F T + 1 ) -Cov x {e T + 1 [E T + 1 (F x + 2 )-F T + 1 ] ,F x + 1 }/Var 1 (F T + 1 ) -Cov T {E T + 1 [e T + 2 (F T + 3 -F T + 2 )] ,F x + 1 }/Var 1 (F T + 1 ) . Analogous to the two-period hedge, the first term is a risk-adjusted function of the expected speculative profit over the period, the second term is a hedge for the expected terminal spot position, the third term is a hedge for the expected marking to market profits next period, and the last term is a hedge for the expected marking to market profits in the final period. Again, under the assumption of martingale prices, (3.12) will reduce to a function analogous -19-to (3.5c), since all expected futures profits will be zero. A discernable pattern emerges as the hedging horizon increases: every additional period adds one covariance term for the additional marking to market transaction between now and the end of the hedge. An interesting observation is that multi-period hedges include terms for expected marking to market profits given all available information today; whether the actual marking to market profit meets these expectations appears to be irrelevant. IV. Methodology for Model Estimation The first step in formulating a model is testing for unit roots and cointegration. Unit root tests are necessary to ensure that the first differences of the price series are stationary. Cointegration tests are used to determine whether or not an error correction term should be included, as discussed in Section II. If spot and futures prices are cointegrated, an error correction term should be included as an independent variable in the mean equations to impose long-run equilibrium constraints, i.e. spot and futures prices may differ in the short-run, but will ultimately converge in the long-run (Engle and Granger (1987)). Recent studies have examined more complex conditional volatilities — higher-order GARCH, GARCH-in-mean and exponential GARCH — but all conclude that these more convoluted models perform no better than the simple GARCH(1,1) model (McCurdy and Morgan (1987); Baillie and Bollerslev (1990); Baillie and Myers (1990); Kroner and Sultan (1991; 1993); Bollerslev et al (1992); De Vries (1994)). Due to the widespread consensus that GARCH(1,1) is adequate to model conditional volatilities of exchange rates, this -20-functional form will be assumed for the purposes of this analysis, bypassing tests of other ARCH and GARCH specifications. The log likelihood function of a bivariate normal distribution is given by equation (4.1): (4.1) ln(L) = E t[-0.51n(|H l|)-0.56 t ,H t"16j + C, where et is the 2x1 vector of (jointly normal) residuals in the mean equation, [est epj', jH t | is the determinant of the conditional covariance matrix, and C is a constant equal to -Tln(2ir)/2 (which drops out when taking first-order conditions). Since the information matrix is block-diagonal, i.e. the mixed partial derivatives are zero between all mean and variance parameters, the variance parameters can be estimated using a consistent estimate of the mean parameters, and vice versa (Engle (1982); Bollerslev (1986)). This greatly simplifies the estimation, since it reduces the number of parameters which have to be jointly estimated. Since OLS estimates are consistent (but inefficient) in the presence of heteroskedasticity, the mean equations are estimated using OLS, and their residuals are used to find the variance parameters which maximize (4.1) using the Berndt, Hall, Hall and Hausman (1974) optimization algorithm. Once the optimal variance parameters have been found, they are used to find the mean parameters which maximize the likelihood function, using the same algorithm. The standard errors are calculated using the outer product of the gradient (OPG) estimator (Davidson and MacKinnon (1993)). The one drawback of maximum likelihood estimation is that it restricts the model to assume normally distributed residuals, which, as -21-mentioned previously, may hot be the best assumption for series which display a certain degree of excess kurtosis. The most significant limitation in calculating non-martingale hedges lies in the third term of (3.5b). Because the optimal hedge ratio is a function of the futures price, the covariance between next period's hedge ratio and next period's futures price cannot be assumed to be zero ex ante. Unfortunately, because this term is the covariance between a futures price and a distribution of variances (which are themselves a function of the futures price), the term cannot be solved through simple algebra. Using Monte Carlo simulation, however, a distribution of futures prices and corresponding hedge ratios can be generated, which can then be used to estimate the third term using a simple OLS regression. Assuming a hedger wishes to construct a two-period hedge beginning today (time T), the first step is to generate a distribution of all possible spot and futures prices next period (time T+l). Since GARCH variances are strictly backward-looking, the distribution of time T + l residuals is known at time T, provided this distribution is defined entirely by its mean and variance (i.e. normal). Residuals are randomly drawn from this distribution (with replacement) and added to the time T expectation of time T+l spot and futures prices to produce a distribution of time T+l prices. Because the spot and futures residuals are assumed to be jointly normal, one must be drawn from its marginal distribution with the other drawn from its conditional distribution. The spot price is therefore drawn from a normal distribution with mean Er(S T + 1 ) and variance h s s T + 1 , while the corresponding futures price is drawn from a distribution with conditional mean ET(FT+I) + p(ST+rET(ST+i))(hff,T+i/hssT+i)1 / 2and variance (l-p2)hff T + 1 . The choice of which -22-variable to draw conditional upon the other is purely arbitrary, and can just as easily be reversed. In period T + l , the optimal hedge ratio will depend on the expected change in the futures price from time T + l to T+2 and the conditional covariances. Since GARCH covariances for time T+2 prices depend on time T+1 residuals, the time T + l hedge will be a function of the time T+l prices. The expected change in the futures price from time T + l to time T+2 may also depend on time T+1 prices, as is the case when an error correction term is used. In short, at time T the optimal time T+l hedge ratio is a distribution rather than a single point, and is dependent on time T+l prices. The objective in period T is to estimate the optimal hedge in period T + l based on the estimated distribution of time T+l prices. Using this estimate of the optimal time T + l hedge, an estimate of the optimal time T hedge can be constructed. In this sense, the problem is being solved backwards. Recall from Section III the optimal time T + l and time T hedge ratios: (4.2a) e T + 1* = [E T + 1(F T + 2)-F x + 1]/27Var T + 1(F T + 2) - Cov T + 1 (S T + 2 ,F T + 2 ) /Var T + 1 (F T + 2 ) , and (4.2b) 9T* = [Er(FT+i)-FT]/27VarT[FT + 1] - Cov^Er+^ST+^.F^J/VarT-fF^,]-Cov T{e T + I*[E x + 1(F x + 2)-F T + 1],F T + 1}/Var T[F T + 1]. The first step is therefore to generate a distribution of time T + l prices. The second step is to construct a time T + l hedge ratio for each spot-futures pair. The third step is to calculate the second term in (4.2b) using the distribution of time T+l prices, and the last step is to estimate the third term in (4.2b) using the distribution of time T + l hedge ratios (step two) and futures prices (step one). This last step is done by regressing the (estimated) distribution 0 t + 1 * [ E T + 1 ( F T + 2 ) T F t + 1 ] on the (estimated) distribution F T + i . In order to evaluate the hedging performance of each model, out-of-sample tests are required. The models are therefore estimated using only the first four years of a five-year (weekly) data set. The models are then re-estimated moving the data set forward one period, such that the same number of observations are used, i.e. "rolling GARCH". This procedure is repeated for an entire year. At the end of the year, there are 52 models, each producing its own set of two-week hedges. Such a scenario could apply, for instance, for a firm whose foreign exchange receivables were always two weeks hence, such that each week called for a new two-week hedge to be constructed. The change in wealth over a two-week hedging horizon is then calculated assuming each of these models is used for the entire year. Expected utility from each model is calculated using the mean and variance of changes in wealth over the year. V. Data Finding an adequate data set entails some difficulties which stem from the fact that futures contracts expire, while the spot price does not. It is therefore important to understand how to perform time-series analysis with futures prices when few futures contracts trade continuously for more than six to nine months. The standard — yet conceptually flawed — method for addressing this problem in the literature is known as "chaining". This procedure defines the price of the contract with the closest maturity as the futures price, switching from one contract to the next as each one expires (see, for example, Chang and Shanker (1986)). There is a serious problem with this procedure, arising from the fact that as the maturity date draws near, the futures price converges to the spot price. Not only does the price level converge, but the price dynamics also converge: futures price changes more closely resemble spot price changes around the maturity date of the contract (Herbst et al (1989)). By far, the most common way of addressing this problem is constructing a chained data set which proceeds to the second-nearest contract before the first contract expires, usually at least three weeks prior to expiration (Kroner and Sultan (1993); Castelino (1992); Cecchetti et al (1988); Benet (1992)). The implicit consensus is that the "expiration effect" characterized by convergence of the basis (F t-S,) and higher correlation with the spot price does not manifest itself until the last few weeks of trading, and as such, chaining the data such that this period is systematically excluded will circumvent econometric problems resulting from this effect. Although this chaining procedure is widely used in the literature, it cannot be used in the context of hedging a specific cash flow: the maximization of the utility function is based on the covariance between the spot price and the futures price for a fixed expiration. Estimating hedge ratios with a chained data set imposes a declining expiration, which is inconsistent with the optimization problem. Going back to the fundamentals in (3.1), a more explicit notation would include a subscript r on the futures price to denote the expiration date on the contract, which is constant regardless of the time period t: (5.1) W t = St + E t e , ( F t / F , . u ) . r > t . Working this into the optimal hedge formulas, the optimal G will depend on r: (5.2) 6 T > ; = [ET(FT + 1 > r)-FT > T]/2 7VarT[FT + 1,T] - CovT[T^+1(ST+7),VT+hl/VaiT[FT+u] -Cov T{e T + 1[E T + 1(F x + 2 i T)-F T + l T],F x + 1}/VarT[F T +,> T], and (5.3) G T +i> T* = [Er+i(F T + 2 ; T)-F x + l i T]/27Var T + l(F T + 2 ? T) - Cov T + 1 (S x + 2,F x + 2 ^/Var x +,(F T + 2 > T) To estimate (5.2) correctly, one would need to find the covariance between the expected time T+2 spot price and the time T+l price of-futures contract with (T-T-1) days to expiration, the (cross-sectional) variance of this futures contract, the expected speculative profit of this contract, and the time T+l hedge ratio using this contract. Using a chained data set would provide, for example, the covariance between the expected time T+2 spot price and the time T + l price of futures contracts with expiration dates three weeks to several months away. There is no reason to believe that the covariance would be identical for T=3 weeks and T=6 months, yet this is the implicit assumption behind this methodology. The only instance in which chained futures prices would be appropriate would be for "perpetual hedging", i.e. finding an optimal hedge ratio for an on-going position with no maturity date, such as the dollar value of a foreign stock portfolio or a subsidiary. Perpetual hedging, however, does not lend itself well to this analysis, since the utility function in (3.2) is characterized as a function of a terminal wealth equation, implying the hedge wilj ultimately be lifted. Ideally, r should be held constant when calculating the hedge ratios; unfortunately, that leaves only four data points per year, since there are only four maturity dates per year in the futures market. The only way to circumvent this problem is to synthesize futures prices with a fixed time to expiration for each observed spot price. Herbst et al (1989) use a weighted average of existing contracts to estimate the price of a "perpetual contract", or a contract which always has r days to expiration. To synthesize the price of a contract with 117 days to expiration (Pn 7), for example, they would find the price of a "near" contract (PN, where N < 117) and the price of a "far" contract (PF, where F > 117), and construct a weighted average such that P 1 1 7 would lie on the line connecting PN and PF: (5.4) P U 7 = PF + (PF - PN)(F - 117)/(F - N). This method allows them to use futures prices with declining maturities, since it adjusts the weighting scheme with each observation — the closer N or F is to 117 on each observation date, the higher its weight; if N or F equals 117 on any given day, it will be afforded a 100% weight. The most significant drawback of (5.4) is that it implies that the futures price is a linear function of the time to maturity. If interest rates are low and the time horizon is relatively short, this may be a reasonable approximation. In general, however, as the maturity date moves further and further ahead, the forward and futures premium/discount is an increasing function of relative interest rates — even when the term structure of interest rates in both countries is flat. When applied to actual data, this formula produces some sizeable errors. Using (5.4) and futures quotes from the newspaper, one should be able to find the price of one from the prices of two others. The error was relatively low on the Canadian dollar (0.01%), but was as high as 0.5% on the Japanese yen. The errors had no consistent sign across currencies. Another drawback is that this formula cannot generate a futures price for every spot price if T is low — only 24 values per year could be calculated for a contract expiring in five days, for example (i.e. there will be an observable PN only for N = 0,..,5). Only if T > 91 will there always be a near contract trading on any given date. A less data-intensive alternative is to proxy futures prices with forward prices. Since both forwards and futures are contracts which lock in a price today for delivery at some later date, and since both ultimately converge to the spot price as the expiration date approaches, they will inevitably be highly correlated. The only distinction lies in the fact that futures contracts entail daily cash flows which can be borrowed or lent at some non-zero interest rate, unlike the one-time payment from a forward contract at the expiration date. Today's forward price is therefore a function of the known yield on a multi-period bond, while today's futures price is a function of unknown one-day interest rates. The futures price will equal the forward price only under two scenarios: if interest rates are non-stochastic, or if interest rates are stochastic but independent of the futures price. Cox, Ingersoll and Ross (1981) examine the relationship between forward and futures prices in the absence of institutional factors such as taxes and bid-ask spreads. They demonstrate that the forward price is always equal to the present value of a contract which will pay: (5.5) ST( 1 + r0 T) at time T, where r 0 T is the T-period yield on a default-free bond. The futures price can be expressed as the present value of a contract which will pay: at time T, where r; are the one-period interest rates prevailing each day throughout the life of the contract. If interest rates are non-stochastic, then to avoid arbitrage the T-period return must equal the product of the one-period returns, thus equating (5.5) and (5.6). Cox et al (1981) also derive a useful proof which demonstrates that the difference between forward and futures prices is a function of the covariance between futures prices and interest rates. They show that the difference between the futures and forward prices (ftT -F t T ) is equal to the current value of the payment flow: (5.6) ST(l+r0)(l+r1)...(l+rT.1) (5.7) S j[f j + 1,T-f j,T][P j/(P j +,- l)],j = t, . . ,T-l , where Pt is the price at time t of a riskless discount bond paying one dollar at time T. In continuous time, (5.7) reduces to: (5-8) - J sTfu,T[Cov(fu,T,Pu)]du. From (5.8), if the covariance between futures prices and bond prices (interest rates) is positive (negative), then the forward price will exceed the futures price. A positive covariance would mean that people long in futures would expect interest rates to decline when they make money from marking to market, and would expect interest rates to rise when they lose money from marking to market. Investors would expect to make more money on average in the forward market than in the futures market under these conditions, and would therefore demand a discount to compensate for the higher interest cost of holding futures. A number of papers have tried to estimate the difference between forward and futures prices for various commodities (Cornell and Reinganum (1981); French (1983)). While the basis is statistically significant for Treasury Bills and copper, for instance, there is no statistically significant difference between forward and futures prices in the foreign exchange market (Cornell and Reinganum (1981)), with the discrepancy typically being less than the average bid-ask spread in the forward market. This does not mean that marking-to-market cashflows themselves are trivial, but rather that the interest rate risk in the foreign exchange market is not large enough to warrant a statistically significant premium or discount on the futures price relative to the forward price. -30-The foregoing has established that without loss in generality, futures prices can be proxied by forward prices. While forward and futures prices may be different in practice, this difference is both small and statistically insignificant as far as currencies are concerned. Moreover, since interest rates are assumed to be zero in defining the optimal hedge ratios above, and since zero interest rates imply that forward prices equal futures prices, this proxy is consistent with the rest of the analysis. Weekly spot and 30-day forward data for the Deutschmark (DM), Japanese yen (JPY) and Swiss franc (CHF) from January 3, 1990 to December 27, 1995 are used in the ensuing analysis (313 observations). The prices are the New York foreign exchange selling rates applicable to interbank transactions of $1 million or more as quoted by Bankers Trust at 3pm E T , obtained through Datastream and the Wall Street Journal. Prices are sampled each Wednesday to circumvent holiday effects, and on the rare occasion that a holiday falls on a Wednesday, the Tuesday rate is used. The choice of a 30-day forward contract as a proxy for the futures price is purely arbitrary, and could easily be replaced with forward prices of other maturities (either quoted or synthetic). However, since hedgers typically use the nearest contract when constructing a hedge, the 30-day price seems most appropriate. During the sample period, there is only one outlier which had to be excluded from the data set. On June 28, 1995, Japan and the US entered into a last minute agreement which would open up Japanese markets to US auto exports. This agreement led to a sharp increase in the 30-day forward rate, as analysts predicted this agreement would give a short-run boost to the dollar vis-a-vis the yen. However, since the agreement was unenforceable, and analysts had their doubts as to whether or not it would succeed, the 90-day forward rate -31-remained the same. On June 21, 1995, the 30-day forward rate (JPY/USD) was trading at a premium of ¥0.37. One week later, it traded at a discount of ¥0.85, followed by a premium of ¥0.34 the week after that. To smooth out this blip, the June 28 forward rate was replaced by the spot rate less the average premium on either side, i.e. (spot - ¥0.355). No other outliers were excluded for any of the three currencies. VI. Model Estimation 1. Tests for Unit Roots and Cointegration The first step in formulating a bivariate model is determining whether the two price series are a) non-stationary, and b) cointegrated. If the series are non-stationary, then to avoid spurious regression, the variables must be transformed to make them stationary either through differencing or log-differencing (returns). If the series are non-stationary but cointegrated, then a bivariate model of their differences or returns should include an error correction term in order to impose long-run equilibrium constraints on the model (Engle and Granger (1987)). As can be seen in Table 6.1, both spot and forward rates are difference-stationary. The stationarity of the basis implies that spot and forward rates are cointegrated, with cointegrating vector [1,-1]. Table 6.1 Phillips-Perron Unit Root Tests Levels DM JPY CHF Spot -2.3770 -2.2548 -2.0711 Forward -2.3541 -2.2900 -2.0419 Differences Spot -18.208 -17.356 -17.595 Forward -18.281 -17.374 -17.556 Basis Spot - Forward -4.9067 -6.1796 -4.0854 Numbers represent the Phillips-Perron test statistic for a unit root, with a constant and trend term included. Asymptotic critical value, 90% significance level: -3.13 2 . Bivariate G A R C H Estimation The bivariate model of spot and futures prices regresses price changes on an intercept and an error correction term. The mean equations are given by: (6.1a) dSt = a o s .+ als(S^-Ful) + est and (6.1b) dFt = a 0 F + a1F(S t.rF t.,) + eFt. -33-Both spot and forward prices are in levels rather than logarithms, since the utility maximization problem from which the hedge is derived is solved in terms of differences rather than returns. The residuals, eSt and eF„ are jointly normal with zero mean and covariance H t , where H t is the two-by-two matrix: (6.1c) hSs,i hsf, hSf,t n f l " , t in which the elements are defined by: (6.Id) (6.1e) and (6. If) nss,t — ft as "I" Gis€st-i "I" ^ 2shss,t-) hff. = ftop + filF£Ft-l + ^2Fhfft-l, h,f.t = p(hSS)lhffjt) 1/2 The maximum likelihood estimates of the mean and variance parameters are given in Table 6.2. Asymptotic standard errors are in parentheses. Despite the insignificant intercepts in the mean equations for the Deutschmark and the Swiss franc, the martingale assumption is violated for all three currencies if the basis is non-zero in the previous period: (6.2) E,,(Ft) - F„ = c*0F + a.pCS,.,-^.,),-which is not zero provided a,F ^ 0 and S,., ^ F t.,, and/or a 0 F ^ 0. -34-Table 6.2 Maximum Likelihood Estimates of Bivariate GARCH Error Correction Model DM CHF JPY Mean Parameters 0.0015 -0.0013 -0.1539 (0.0220) (0.0097) (0.0367) 0.0940 0.0631 -0.5802 (0.0260) (0.0481) (0.1131) 0.0021 -0.0011 -0.1483 (0.0222) (0.0101) (0.0296) 0.2882 0.1701 -0.4354 (0.0384) (0.0471) (0.1009) Variance Parameters Bos 0.0001 (0.0691) 0.0000 (0.0303) 0.0977 (0.0337) B.s 0.0569 (0.0550) 0.0257 (0.0525) 0.0287 (0.0018) BM 0.7919 (0.0623) 0.7229 (0.0458) 0.9412 (0.0135) B0F 0.0001 (0.0548) 0.0000 (0.0474) 0.1495 (0.0356) B,F 0.0511 (0.0638) 0.0278 (0.0396) 0.0322 (0.0020) B2F 0.7827 . (0.0541) 0.7236 (0.0426) 0.9199 (0.0140) P 0.9958 (0.0067) 0.9938 (0.0077) 0.9987 (0.0000) log(L) 2396.0 2537.6 68.8 -35-Table 6.2 cont'd DM CHF JPY C O R R 6 C O R R 2 4 Q 2 24 (S) Q 2 24 (F) LR1 LR2 33.4 21.8 4.90 26.90 14.57 39.96 122.2 121.0 2.03 18.37 19.72 32.34 501.4 144.2 2.11 12.13 12.27 12.02 (95% c.v. = 12.59) (95% c.v. = 9.488) (95% c.v. = 12.59) (95% c.v. = 36.42) (95% c.v. = 36.42) (95% c.v. = 36.42) The test statistics reported at the bottom of Table 6.2 are as follows. LR1 is the likelihood ratio test of the null hypothesis that all slope coefficients (in both mean and variance equations) are zero. LR2 is the likelihood ratio test of the null hypothesis that the slope coefficients in the variance equation are zero (i.e. homoskedastic). CORR6 and CORR24 are Box-Pierce-Ljung tests on vstvft, where vst and vft are the standardized residuals of the spot and futures equations, respectively. CORR6 tests for serial correlation up to the 6th order, and CORR24 tests up to the 24th order. Q224 is the Box-Pierce-Ljung test for up to 24th order serial correlation in the squared standardized residuals from each price series. All tests are rejected at the 95% level, with the exception of up to the Box-Pierce-Ljung test for the futures variance for the Deutschmark. The Box-Pierce-Ljung test actually rejects the null of uncorrected variances in the futures price when only one lag is included — a particularly troubling result, although it may explain why the hedge ratios and wealth computations for this currency tend to be so different from those of the other two. -36-3 . Hedge Ratio Formulas to be Examined As derived previously, Kroner and Sultan's conditional hedge is given by the formula: (6.3) G t K S = -Cov t(S l + 1,F t + 1)/Var t(F t + 1), and the non-martingale hedge is given by the formula: (6.4) G™ = E l(AF t + I)/2 7Var l(F t + 1) - Cov t(E l + 1(S t + 2),F t + 1)/Var t(F t + 1) -Cov t(* t + 1,F t + 1)/Var t(F t + 1), % = 6 , ™ ^ ^ , ) . Comparing (6.3) and (6.4), there are three key differences. The first is the inclusion of the speculative term, Et(dF t+i)/27Var t(F t+1), to account for expected speculative gains to be realized from buying or selling futures contracts today. The second is the inclusion of a term which hedges expected gains to be realized from holding futures contracts in subsequent periods, Cov t(^ t +i,F t +,)/Var t(F t + 1). Finally, there is a slight distinction in the definition of the hedging component: for the conditional hedge, it is the conditional covariance between spot and futures prices, while for the new hedge, it is the conditional covariance between expected spot and futures prices. Equations (6.3) and (6.4) denote the time t hedge ratios for a spot position maturing two periods hence, in period t+2. In period t+l, there will be no subsequent profits or losses arising from a hedge in period t+2, and as such, the third term in (6.4) disappears in -37-period t+l. The formula in (6.3) holds for both periods. In other words, a two-period hedging strategy using the conditional hedge in (6.3) will use the same formula in both periods, whereas a hedging strategy using the non-martingale hedge in (6.4) will use all three terms in the first period, and only the first two terms in the second. Furthermore, the second term in (6.4) will lose the expectation operator in period t+l, such that it becomes identical to (6.3). To see how hedging performance changes as these new terms are added to the conditional model, the "KS" model depicted in (6.3) will be compared to three different non-martingale hedges. The first hedge (hereinafter referred to as Hedge A) is simply the conditional hedge with the speculative term included. The second hedge (Hedge B) is the preceding hedge with the additional term to hedge for next period's speculative profit. Finally, the third hedge (Hedge C) will represent the complete non-martingale model as given in (6.4): (6.4a) E l(AF t + 1)/2 7Var t(F, + 1) - Cov t(S t +i,F t +,)/Var t(F t + 1) , (6.4b) E t(AF t + 1)/2 7Var t(F l + 1) - Cov t(S t + 1,F t + 1)/Var t(F t + 1) -C o v ^ ^ F ^ / V a r ^ , ) , * , = e t AE t(AF t + 1), and (6.4c) E t(AF t + 1)/2 7Var t(F l + l) - C o v ^ ^ + 2 ) , F t + 0 / V a r t ( F t + 1 ) -Cov t(* t + 1,F t + 1)/Vai .r t(F l+1), % = G t AE t(AF t + 1). -38-In all three cases, the time t.+1 hedge will be given by: (6.4) 9 t + 1 A , e i + 1B , 0 t + 1 c = E t + 1 (AF t + 2 ) /2 7 Var t + 1 (F t + 2 ) -Cov t + I (S t + 2 ,F t + 2 ) /Var t + 1 (F t + 2 ) , which is equal to: (6.4e) E t + I (AF t + 2 )/2 7 Var t + 1 (F t + 2 ) + 6 t + 1 K S . To simplify Hedge C in (6.5c), recall from (6.1a) that changes in the spot rate are given by dSt = otos + aiS(S t.,-F t_,) + es t. Thus, at time t+l, the expected spot rate one period hence will be equal to: (6.5) E t +,(S t + 2) = S t + 1 + E t + 1(dS t + 2) = a o s + (a l s+l)S t +, - a i S F t + 1 . Substituting (6.5) into the second term of (6.4c) and solving for the covariance yields: (6.6) -Cov t (E l + 1 (S t + 2 );F t + 1 ) = -[(a ls+ l)Cov t(S t + 1,F t + 1) - a l sVar t(F t + I)]. Dividing (6.6) by Var t(F t + 1), the second term in (6.4c) can be expressed as: (6.7) - (a l s +l )Cov t (S t + „F t + I ) /Var t (F t + 1 ) + a l s = ( « l s + l ) ( 9 t K S ) + a l s . -39-4. Monte Carlo Simulation The first stage in the Monte Carlo simulation is generating a distribution for spot and futures prices one period ahead as described in the methodology. This is done by drawing residuals with replacement from the joint distribution with zero mean and variance given by (6.1c), and adding these residuals to the conditional expectations of (6.1a) and (6.1b). A total of 1000 spot-futures pairs are constructed in this manner for each out-of-sample period. The third term in Hedges B and C is the one which poses the greatest difficulty in estimation, and which requires these simulated prices. Unlike the time T conditional hedge, the time T + l conditional hedge will not be uniform across time T+l observations (recall Section IV). Moreover, the covariance between the optimal time T + l hedge ratio and the time T + l futures price will not necessarily be zero, since the hedge ratio is itself a function of the futures price and futures variance. The -$rT+] term in (3.5b) is equal to the time T+l hedge ratio multiplied by the expected change in the futures price from time T + l to time T+2, given by (6.1b): (6.8) ¥ T + 1 = 9 T + 1 E T + 1 (dF T + 2 ) = 9 t + 1 |>OF + aiF(ST+rFT+i)]-For each time T + l simulation, there will be a corresponding ^ T + I . The third term in Hedges B and C is equal.to the OLS B coefficient in the model: -40-(6.9) * T + 1 = a + BF T + 1 + e, which can be estimated over the 1000 simulated observations. Since 6 x + i is a function of the risk aversion coefficient, 7, there will be one estimated ^ T + 1 series for each 7. Throughout the analysis, three different coefficients of risk aversion are considered — low (7=1), medium (7=4) and high (7 = 10). The first term in Hedges A, B, and C is equal to the expected net gain from holding one futures contract (long or short) from period T to period T+l , adjusted for risk. This is found by taking the expectation of (6.1b) conditional upon information at time T (which will be uniform for all simulated time T+1 observations) and dividing by the 7 times the conditional variance of the futures price, where 7 is the coefficient of risk aversion. Equation (6.3) and the second term in (6.4a) and (6.4b) is found by calculating the conditional variances and covariance for time T+1 prices. These will also be uniform for all simulated time T + l observations, since the conditional variances depend only on time T variances and residuals. The preceding describes the procedure for calculating the time T and T + l hedges. To assess their relative performance, out-of-sample tests need to be performed. The GARCH model is estimated using four years of weekly data up to 1994, from which the hedge ratios for the first week of 1995 are generated (using Monte Carlo simulations as described above). The model is then re-estimated by moving the sample forward one period, from which the hedge ratios for the second week of 1995 are generated. This "rolling window" procedure is repeated for each week of 1995 until there is a set of hedge ratios for -41-all 52 weeks in 1995. Changes in wealth at the end of each two-week hedging horizon are given by: (6.10) dWT + 2 = S T + 2 + GX*(FT+1-FT) + e x + 1*(FT + 2-FT + 1) - ST. The mean and variance of (6.10) over the 52 out-of-sample periods can then be substituted into the mean-variance utility function for different levels of risk aversion. Because utility has a mean component, the optimal hedge need not be the one with the smallest variance; however, as risk aversion increases, the variance becomes the dominant consideration. 5 . Calculations Table 6.2 above presents the estimated coefficients over the entire five-year sample. For out-of-sample testing, the models are estimated using four-year rolling windows. Since there are 52 such models for each currency, and since the parameter estimates do not differ dramatically from the five-year estimates, the coefficients from each out-of-sample estimation will not be presented. Over the out-of-sample period, the average estimate for B in (6.9) is 0.002 for the yen, -0.04 for the franc and 0.008 for the Deutschmark. The estimates are all statistically significant for the franc, but nine (17%) are insignificant for the yen and 29 (56%) are insignificant for the Deutschmark. Given the magnitude and inconsistency of these results, this term can probably be dropped from hedge ratios B and C without severe consequences. For the purposes of this analysis, this term is included in the hedge ratios when it is statistically significant; otherwise it is omitted. Charts 6.1 to 6.3 compare the estimated hedge ratios for each currency. Only Hedge B is plotted, as it represents a middle ground among the three non-martingale hedge ratios: Hedge A will be further away from Hedge KS relative to Hedge B, while Hedge C will be closer to Hedge KS relative to Hedge B. All three charts assume average risk aversion (7=4). As the third term is found to be typically small, the main distinction between Hedge B and Hedge KS lies in the speculative component (recall that Hedge B is equal to the sum of the speculative term, Hedge KS and the third term for hedging next period's expected profit). In all three cases, Hedge B is much more volatile than Hedge KS, particularly for the Deutschmark. The large hedge ratios for the franc and the Deutschmark can be explained by their relatively low volatilities — both of these currencies have volatilities of roughly 10%, whereas the yen's volatility is roughly 100% (recall that this is in terms of first differences, not levels). As such, dividing the expected change by its variance will produce a much larger number for currencies with low volatilities. This effect is largely subdued when a higher risk aversion coefficients is used. Another interesting observation is that in four periods, Hedge B is positive for the Deutschmark, implying that the optimal hedge for a long spot position is a long futures position. In comparison to these dynamic hedges, the conventional static hedge is given by the estimated 13 in the regression: (6.12) St = a + BFt + et. -43-Since spot and forward prices are cointegrated, (6.12) will not be spurious. All three currencies have an optimal 15 close to one: 0.99452 for the JPY, 1.011 for the DM and 1.0039 for the CHF. Since the conventional hedge is assumed to be constant over time, it is not re-estimated for each out-of-sample period. 6. Performance Measures Charts 6.4a, 6.5a and 6.6a plot the changes in wealth from an unhedged position versus that achieved using Hedge KS. The purpose of these charts is to show the extent to which hedging eliminates risk. Charts 6.4b, 6.5b and 6.6b plot the changes in wealth achieved for an average investor (7=4) using Hedge B versus that achieved for the same investor using Hedge KS. Again, since there is little difference between Hedges A, B and C, only Hedge B is used in the charts. These charts show that the relative performance of the non-martingale hedges depends on which currency is being examined. For the yen, for instance, there is little difference in wealth changes between strategies for the most part, whereas Hedge B produces much more volatile changes for the other two currencies. In fact, comparing the 'a' charts with the 'b' charts for the franc and Deutschmark, Hedge B produces changes in wealth which are of the same magnitude but opposite sign as those of an unhedged position. The most interesting observation is that while Hedge B produces large changes in wealth, positive changes tend to be larger than the negative changes, such that changes in wealth on average will be positive. This result raises another interesting point: if hedgers are assumed to be utility -44-maximizers rather than simply variance minimizers, this increase in variation need not imply that the hedger would always prefer Hedge KS to Hedge B. In fact, it can be shown that hedgers with low risk-aversion will always prefer Hedge B to Hedge KS, and even those with medium risk-aversion will prefer Hedge B in the case of the Swiss franc. Tables 6.3a to 6.3c show the mean and variance of changes in wealth for each hedging strategy, calculated across the 52 out-of-sample periods. Table 6.4 shows the same calculations for the conventional hedge, for the naive hedge (0 = -1) and for an unhedged spot position. For each level of risk aversion, the last column is calculated from the identity: (6.13) E[U(dW)] = E(dW) - 7Var(dW). Looking at the first column in all three tables, the non-martingale hedges, A, B, and C all produce a higher expected value of terminal wealth than any other hedging strategy. The only instance in which a higher mean could be attained would be through an unhedged yen position. The second column clearly shows why this would be undesirable: the variance of an unhedged yen position is over 100 times the magnitude of the mean. This second column presents three other interesting findings. First, the conventional and naive hedges for all currencies appear to eliminate between 96.4% (DM) and 99.8% (JPY) of the variance of terminal wealth, such that the incremental variance reduction attributable to Hedge KS is negligible. Second, the non-martingale hedges for the franc and Deutschmark entail higher variances than unhedged positions, suggesting that a hedger. concerned solely with variance minimization would prefer an unhedged position to any of Hedges A, B, or C. Finally, the -45-higher variance in the second column is associated with higher means in the first column, consistent with the risk-return trade-off inherent to portfolio theory: what started out as a hedging model has evolved into a portfolio allocation model. The last column in each table calculates the expected utility from each hedging strategy. For hedgers with low risk-aversion, Hedge C — the complete non-martingale model — generates a higher expected utility than any other strategy, consistent with the mathematical derivation in Section III. As risk aversion is increased, however, the results are less consistent. For medium risk aversion, the conventional hedge is optimal for the yen, any of Hedges A, B, or C is equally optimal for the franc, and Hedge KS is optimal for the Deutschmark. For high risk aversion, the conventional hedge is again optimal for the yen, while Hedge KS is optimal for the franc and the Deutschmark. These findings suggest that non-martingale hedges are optimal only for low coefficients of risk aversion. However, this is not quite the end of the story. The third and fourth columns of Tables 6.3a to 6.3c show the minimum and maximum values of changes in wealth. For the unhedged, naive, conventional and KS models, the maximum and minimum appear to be virtually symmetric about zero, implying that the upside risk as equal to the downside risk, and that the distribution of dW is approximately normal. For Hedges A, B, and C, however, the maximum values are much greater than the absolute values of the minimum values. This suggests that for the non-martingale hedges, the distribution of dW is skewed, and that these large positive values are inflating the variance. This in itself complicates comparisons, since mean-variance utility implicitly assumes that the distribution of returns is normal and can be defined by its first two moments. Charts 6.4b, 6.5b and -46-6.6b clarify the point: for the yen, large increases in wealth from weeks nine through 17 are driving the variance; for the franc, there are only two outcomes for which the loss exceeds 0.1 CHF/USD, compared to seven outcomes for which the gain exceeds the same amount; for the Deutschmark, three outcomes lose more than 0.2 DM/USD, compared to six which gain more than 0.2 DM/USD. Table 6.3d provides more formal evidence of non-normality. For both the yen and franc, changes in wealth produce Bera-Jarque statistics which strongly reject the null hypothesis of normality at the 5% level when non-martingale hedges are used, yet fail to reject the null when the conventional, naive or KS hedges are used. The Deutschmark produces some perplexing results, in that the conventional, naive and KS hedges produce wealth distributions which are highly leptoktirtic (much more so than any of the non-martingale hedges, in fact). This can be explained by the presence of unusually large changes in wealth, relatively speaking, occurring during the second half of the out-of-sample period. When the out-of-sample period is truncated to the first half of 1995, the Deutschmark displays the same pattern as the other two currencies, with insignificant Befa-Jarque statistics for all of the same three hedging strategies. -47-Table 6.3a Expected Utility of Wealth Changes Among Hedging Strategies Low Risk Aversion (7 = !) JPY E(dW) Var(dW) Min(dW) Max(dW) E[U(dW)] Hedge A 0.04018 0.04897 -0.32031 0.75397 -0.00879 Hedge B 0.04059 0.05026 -0.32050 0.76164 -0.00967 Hedge C 0.04758 0.051.79 -0.32087 0.75591 -0.00421 Hedge KS -0.00385 0.01348 -0.31125 0.34382 -0.01733 Conventional 0.00605 0.01376 -0.33696 0.34044 -0.00771 Naive 0.00569 0.01381 -0.33000 0.34000 -0.00812 Unhedged 0.07118 7.24710 -6.98000 6.78000 -7.17592 CHF Hedge A 0.02454 0.00494 -0.11715 0.26795 0.01960 Hedge B 0.02455 0.00494 -0.11715 0.26792 0.01961 Hedge C 0.02455 0.00494 -0.11710 0.26784 0.01962 Hedge KS 0.00000 0.00001. -0.00778 0.00774 0.00000 Conventional -0.00040 0.00001 -0.02080 0.00770 -0.00041 Naive -0.00002 0.00001 -0.00770 0.00760 -0.00002 Unhedged -0.00595 0.00104 -0.08500 0.07250 -0.00699 DM Hedge A 0.07393 0.05713 -0.56712 0.66239 0.01680 Hedge B 0.07373 0.05672 -0.56676 0.66222 0.01701 Hedge C 0.07402 0.05693 -0.56726 0.66254 0.01708 Hedge KS 0.00027 0.00002 -0.01128 0.01802 0.00025 Conventional -0.00035 0.00004 -0.02180 0.02065 -0.00038 Naive -0.00002 0.00003 -0.02120 0.02030 -O.O0OO5 Unhedged -0.00460 0.00110 -0.07260 0.08000 -0.00570 -48-Table 6.3b Expected Utility of Wealth Changes Among Hedging Strategies Medium Risk Aversion (y=4) JPY E(dW) Var(dW) Min(dW) Max(dW) E[U(dW)] Hedge A 0.02321 0.02901 -0.31891 0.67708 -0.09284 Hedge B 0.02362 0.02965 -0.31910 0.68390 -0.09497 Hedge C 0.03062 0.02971 -0.31125 0.67817 -0.08822 Hedge KS -0.00385 0.01348 -0.31125 0.34382 -0.05777 Conventional 0.00605 0.01376 -0.33696 0.34044 -0.04899 Naive 0.00569 0.01381 -0.33000 0.34000 -0.04953 Unhedged 0.07118 7.24710 -6.98000 6.78000 -28.9172 CHF Hedge A • 0.01504 0.00269 -0.10064 0.23692 0.00427 Hedge B 0.01504 0.00269 -0.10062 0.23690 0.00427 Hedge C 0.01505 0.00269 -0.10066 0.23683 0.00427 Hedge KS 0.00000 0.00001 -0.00778 0.00774 -0.00002 Conventional -0.00040 0.00001 -0.02080 0.00770 -0.00046 Naive -0.00002 0.00001 -0.00770 0.00760 -0.00004 Unhedged -0.00595 0.00104 -0.08500 0.07250 -0.01011 DM Hedge A 0.04390 0.03323 -0.45028 0.59718 -0.08902 Hedge B 0.04383 0.03323 -0.45017 0.59713 -0.08907 Hedge C 0.04412 0.03324 -0.45067 0.59744 -0.08885 Hedge KS 0.00027 0.00002 -0.01128 0.01802 0.00018 Conventional -0.00035 0.00004 -0.02180 0.02065 -0.00050 Naive -0.00002 0.00003 -0.02120 0.02030 -0.00014 Unhedged -0.00460 0.00110 -0.07260 0.08000 -0.00901 -49-Table 6.3c Expected Utility of Wealth Changes Among Hedging Strategies High Risk Aversion (7=10) JPY E(dW) Var(dW) Min(dW) Max(dW) E[U(dW)] Hedge A 0.01981 0.02768 -0.31863 0.66171 -0.25703 Hedge B 0.02023 0.02821 -0.31882 0.66835 -0.26189 Hedge C 0.02722 0.02798 -0.31919 0.66262 -0.25261 Hedge KS -0.00385 0.01348 -0.31125 0.34382 -0.13865 Conventional 0.00605 0.01376 -0.33696 0.34044 -0.13154 Naive 0.00569 0.01381 -0.33000 0.34000 -0.13236 Unhedged 0.07118 7.24710 -6.98000 6.78000 -72.39980 CHF Hedge A 0.01314 0.00259 -0.10941 0.23071 -0.01278 Hedge B 0.01314 0.00259 -0.10940 0.23070 -0.01277 Hedge C 0.01315 0.00259 -0.10944. 0.23063 -0.01277 Hedge KS 0.00000 0.00001 -0.00778 0.00774 -0.00002 Conventional -0.00040 0.00001 -0.02080 0.00770 -0.00054 Naive -0.00002 0.00001 -0.00770 0.00760 -0.00007 Unhedged -0.00595 • 0.00104 -0.08500 0.07250 -0.01635 DM Hedge A Hedge B Hedge C Hedge KS Conventional Naive Unhedged 0.03790 0.03785 0.03814 0.00027 -0.00035 -0.00002 -0.00460 0.03331 0.03331 0.03329 0.00002 0.00004 0.00003 0.00110 -0.42691 -0.42685 -0.42735 -0.01128 -0.02180 -0.02120 -0.07260 0.58414 0.58411 0.58443 0.01802 0.02065 0.02030 0.08000 -0.29516 -0.29530 -0.29480 0.00005 -0.00073 -0.00032 -0.01563 -50-Table 6.3d Skewness (S), Kurtosis (K) and Normality (BJ) of Wealth Changes Low Risk Aversion Hedge A 1.137 Hedge B 1.230 Hedge C 1.058 Medium Risk Aversion Hedge A 1.000 Hedge B 1.019 Hedge C 0.931 High Risk Aversion Hedge A Hedge B Hedge C Risk Invariant Hedge KS Conventional Naive None 0.979 0.983 0.922 0.295 0.495 0.484 -0.114 JPY K 2.146 2.326 1.739 3.471 3.373 3.159 3.644 3.571 3.437 2.344 1.876 1.765 0.473 BJ 12.536 13.828 12.897 8.978 9.123 7.422 9.024 8.901 7.632 1.655 4.768 5.235 13.676 1.697 1.697 1.699 1.443 1.443 1.442 1.354 1.354 1.352 0.125 0.150 0.140 0.109 CHF K 4.366 4.361 4.370 6.088 6.088 6.074 6.175 6.175 6.160 3.426 3.556 3.479 1.181 BJ 28.457 28.405 28.511 37.979 37.968 37.753 36.993 36.993 36.754 0.464 0.462 0.463 0.339 0.339 0.337 0.368 0.368 0.365 D M K 1.166 1.181 1.177 1.523 1.522 1.526 1.205 1.203 1.211 BJ 8.974 8.848 8.886 5.611 5.620 5.582 7.993 8.010 7.933 1.96 -0.210 0.518 0.849 0.654 -0.298 7.129 0.408 6.594 60.117 10.009 104.754 10.054 106.505 1.006 9.865 S = Skewness: E[dW3]/ad W 3 K = Kurtosis: E[dW4]/<xdW4 BJ = Bera-Jarque statistic for normality: T[S2/6 + (K-3)2/24], 5% c.v. = 5.99 7 . Artificial Neural Networks While the analysis in the preceding section demonstrates that non-martingale hedges are optimal only for hedgers with low risk-aversion, there is a possibility that this result is not independent of the volatility estimate. In other words, would these results change if a better estimate of exchange rate volatility can be found? As Chart 6.7 shows for the yen, GARCH is able to capture broad trends quite adequately, yet fails to display the same magnitudes and sharp movements of the actual series. What remains to be seen is whether a volatility estimate which is capable of approximating these large, sharp movements will have any bearing on the hedging effectiveness of any of the dynamic models. A modification which .has yet to be applied to hedging theory is the use of artificial neural networks in the specification of the means and variances. As recent work by Donaldson and Kamstra (1996a; 1996c) applies ANN techniques to international stock index volatility and fundamental asset pricing, the logical progression would be to apply ANN models to currency markets. Artificial neural networks, simply put, are non-linear input-output models which have their origins in the cognitive sciences. Originally designed to crudely approximate the learning process of the brain, they can be applied to econometrics to "learn" the data generating process by observing a sufficiently large number of outcomes. To get the output from the input variables, the data passes through a series of "transfer functions". The most common transfer function is the sigmoid curve: (6.14a) f(x) = a + (1+exp[-(b+cx)])-1. - 5 2 -When a series of these transfer functions are added together, almost any non-linear function can be approximated. The parameters (a,b,c) determine the shape of the curve. In general, f(x) will be a logistic curve confined to a specified range, with bounds given by [a,a+l]. The parameter b determines the value of x for which the output variable, y = f(x), reaches its midpoint. High values of b will skew the output, while b=0 will produce a symmetric function (i.e. f(x) reaches its midpoint when x reaches its midpoint). The parameter c determines the rate at which the function transfers from a to a+1 (if c > 0) or from a+1 to a (if c < 0). Input variables into the ANN model take the form of standardized variables, with zero mean and unit variance: (6.14b) Zt = (yt - Hy)/oy, where /Xy is the in-sample mean of y and ay is its in-sample standard deviation. Observations are therefore afforded more or less weight depending on how many standard deviations they are from the unconditional mean in a given time period. These normalized forecasts are plugged into a series of i transfer functions of the form: (6.14c) n^,yd = (1.+ exp[-( 7 o i + Ej7jizJ)])-\ which range from 0 to 1, since the "a" term is 0. The values used for y0 and y} are purely arbitrary. Different models can be constructed by selecting these parameter values using a -53-random number generator, then performing some test to figure out which combination best fits the observed data. In applying ANN techniques to the hedging model in (2.4), the mean equations of (2.4a) and (2.4b) remain the same, as do (2.4c), (2.4d) and (2.4g). The two variance equations, however, can be replaced by non-linear functions of the form: (6.15a) h s M = 60S + flishss,t-i + IWt-12 + ^sfe.tAs.h), (6.15b) hff>t = 80F + J31Fhrf,t-i + JW.t-i2 + 6 3 F*F(zF i t,XF | h), (6.15c) * s = [l+exp(X,zs>tr, + X 2z s t . , 2 + X 3zS | t. 2 + X4zs>t.22)]-\ (6.15d) * F = [l+exp(X,zFjt., + X2zF t_i2 + X 3zF ) t. 2 + X4zF>t.22)]"1, (6.15e) z s t. d = [eSl.d-E(es)]/as, (6.15f) z F > d = [eF.,.d-E(eF)]/crP, and, (6.15g) 0.5Xj ~ U[-l,l], where <xs and aF are the unconditional standard deviations of the spot and futures rate, respectively, and the X are non-parametric. The optimal model is selected by choosing randomly drawn sets of X terms and finding the specification with the best fit. Note that the ANN model does not replace the linear GARCH terms, but rather augments the familiar model. Similar ANN terms can be added to the mean equations, although this possibility is not explored here. Equations (6.15a) and (6.15b) imply that the conditional volatility depends not only on last period's volatility (clustering), but also on last period's residual (asymmetry). -54-The first step in estimating a GARCH-ANN model is randomly drawing a set of Xj from a uniform distribution. The z terms in (6.15e) and (6.15f) are functions of the residuals from the mean equations, and can be calculated after estimating the mean equations by OLS. Since the number of lags and moments is restricted to be two, this means that (6.15c) and (6.15d) will be of the form: (6.16) % = [l+exp(X1zit.1 + Xaz,,.,2 + X3zi>t_2 + X4zi>t.22)])]1. The optimal X set is found by maximizing the likelihood function for each model, then selecting the five which produce the lowest.Schwarz Criterion (or highest log likelihood function, since the number of parameters is held constant in this case). Of these five models, the one which produces the in-sample spot and futures estimates which most closely resemble the actual realizations (similar means and unconditional variances) is selected as the optimal model. Table 6.4 shows the GARCH(1,1)-ANN( 1,2,2) parameter estimates for each currency over the full sample (1990-95). Chart 6.8 plots the conditional volatility of the yen. Unlike the GARCH volatility in Chart 6.7, the ANN volatility follows actual volatility (defined as squared deviations of spot changes around their mean) much more precisely, with estimated peaks coinciding almost exactly with actual peaks. Like the GARCH model, however, the ANN model does not adequately capture the magnitude of the peaks. As can be seen from Table 6.4, the intercepts decrease significantly when the ANN terms are included. This may be due to the fact that the ANN terms have an intercept embedded in them — if all the z -55-terms are zero, the ANN assumes a value of (1 +exp(0))"1 = 0.5. Since this property alone could be responsible for the insignificant intercepts and ANN terms, not too much emphasis will be placed on the insignificant t-statistics. Moreover, the likelihood ratio tests of the null hypothesis that the ANN model is insignificantly different from the GARCH model is rejected in all cases. The out-of-sample simulation is performed in the same manner as before using the ANN models: parameter estimates are generated using four years of data, a one-period-ahead estimated is calculated, then the models is re-estimated after rolling the data set forward one period, until an entire year of one-period-ahead estimates are obtained. Hedge ratios are again calculated using the ANN conditional volatilities and expected changes in the futures price, and a distribution of changes in wealth is constructed. Tables 6.5a to 6.5c provide the means, variances and expected utilities from each hedging strategy under different assumptions about risk aversion. Table 6 .4 Maximum Likelihood Estimates of Bivariate GARCH-ANN Error Correction Model DM CHF JPY Mean Parameters -0.0006 -0.0013 -0.1494 (0.0017) (0.0011) (0.0593) 0.0612 -0.0646 0.3706 (0.3024) (0.3185) (0.3652) -0.0001 -0.0012 -0.1542 (0.0018) (0.0011) (0.0597) 0.2420 0.0367 0.4141 (0.2988) (0.3203) (0.3641) -56-Table 6.4 cont'd DM CHF JPY Variance Parameters Bos 6.9E-05 3.4E-05 0.0116 (1.4E-04) (1.1E-04) (0.4817) Bis 0.0537 0.0286 0.1035 (0.8018) (0.0470) (0.0585) B K 0.8010 0.7661 0.8828 (0.1541) (0.1536) (0.1392) B3s 9.8E-05 6.5E-05 0.0154 (2.0E-04) (1.9E-04) (0.7368) B 0F 7.3E-05 3.5E-05 0.0110 (1.5E-04) (1.1E-04) (0.5633) BIF 0.0495 0.0282 0.1097 (0.0518) (0.0592) (0.0672) B 2 F 0.8018 0.7485 0.8750 (0.1305) (0.1581) (0.1499) B 3 F 1.0E-04 7.7E-05 0.0256 (1.7E-05) (1.6E-04) (0.8526) P 0.9956 0.9933 0.9985 (0.0006) (0.0013) (0.0002) log(L) 2433.8 2557.3 90.7 LR1 109.0 161.6 545.2 (95% c.v. = 12.59) LR2 97.4 160.4 188.0 (95% c.v. = 9.488) LR3 75.6 39.4 43.8 (95% c.v. = 5.991) CORR6 5.97 5.85 2.02 (95% c.v. = 12.59) CORR24 23.23 19.59 15.24 (95% c.v. = 36.42) Q224 (S) 11.83 18.92 13.98 (95% c.v. = 36.42) Q224 (F) 38.39 25.64 17.19 (95% c.v. = 36.42) -57-The test statistics reported at the bottom of Table 6.4 are for the same tests as in Table 6.2, with the exception of LR3. This is a likelihood ratio test of the null hypothesis that the neural network coefficients are zero (i.e. the alternate hypothesis is a GARCH(1,1) model). Despite insignificant t-statistics on B3s and G3f, the LR test rejects the null hypothesis that these coefficients are jointly zero for all currencies. As mentioned above, this is most likely due to some form of multicollinearity between the neural network terms and the intercepts. Again, all null hypotheses are rejected at the 95% level with the exception of up to 24th order serial correlation in the futures variance for the Deutschmark. Once again, the Box-Pierce-Ljung test for first-order serial correlation in the futures variance finds significant serial correlation for the Deutschmark, which in may be causing the peculiar results for this currency. The results from this simulation are inconclusive. The only clear pattern which emerges is that the ANN model always increases utility for the non-martingale models relative to GARCH, for any level of risk aversion. This can be due to a higher mean (CHF), a lower variance (DM), or both (JPY). Expected utility from Hedge KS increases for the yen, decreases for the franc, and stays roughly the same for the Deutschmark. All of these results appear to be invariant with respect to risk tolerance. With only one exception, the non-martingale models which outperform Hedge KS using GARCH still outperform using ANN, and Hedges A, B, and C remain optimal only for hedgers with low risk-aversion. Within the class of non-martingale hedges, the relative rankings are not preserved for all currencies: for the yen, for example, Hedge C generates the highest expected utility among the three non-martingale hedges using GARCH, while Hedge A generates the highest -58-expected utility using ANN. Charts 6.9 to 6.11 plot the hedge ratios derived from the ANN model, and charts 6.12 to 6.14 plot the changes in wealth. Despite what may appear to be significant changes in the hedge ratios — notably for the franc during the first two months out-of-sample — there appears to be little change in wealth relative to the GARCH model. All of these findings suggest that the performance of non-martingale hedges is not primarily driven by the variance, but rather by the mean. To test this hypothesis, a number of neural network specifications were used on the yen: one composed of standardized spot and futures prices, one composed of standardized error correction terms, and one hybrid of the previous two models. As none of these models produced fitted values which differed noticeably from the basic error correction model, results are not reported, and the out-of-sample forecasting exercise was not attempted. To summarize, despite the fact that this simple ANN model does appear to improve variance estimates, this improvement alone is not sufficient to alter the initial findings, and snon-martingalemodels remain optimal only for hedgers with low risk aversion. The most probable means of improving their performance would be to derive a model which can more accurately approximate the means rather than the variances, since the speculative term, E t(dF t+1)/27Var t(F t+I), is what ultimately distinguishes the non-martingale model from the martingale model. -59-Table 6.5a JPY E(dW) Expected Utility of Wealth Changes Among Hedging Strategies Low Risk Aversion (y=l) Var(dW) Min(dW) Max(dW) E[U(dW)] GARCH Hedge A Hedge B Hedge C Hedge KS ANN Hedge A Hedge B Hedge C Hedge KS CHF 0.04018 0.04059 0.04758 -0.00385 0.04912 0.05282 0.04920 0.02222 0.04897 0.05026 0.05179 0.01348 0.02926 0.03518 0.02971 0.01392 -0.32031 -0.32050 -0.32087 -0.31125 -0.23345 -0.25717 -0.23854 -0.23507 0.75397 0.76164 0.75591 0.34382 0.52209 0.66919 0.52365 0.36567 -0.00879 -0.00967 -0.00421 -0.01733 0.01986 0.01764 0.01950 0.00830 GARCH Hedge A Hedge B Hedge C Hedge KS ANN Hedge A Hedge B Hedge C Hedge KS 0.02454 0.02455 0.02455 0.00000 0.04275 0.04100 0.04126 -0.00138 0.00494 0.00494 0.00494 0.00001 0.01296 0.01177 0.01194 0.00005 -0.11715 -0.11715 -0.11710 -0.00778 -0.13805 -0.13954 -0.13802 -0.01622 0.26795 0.26792 0.26784 0.00774 0.42250 0.41864 0.41969 0.01703 0.01960 0.01961 0.01962 0.00000 0.02979 0.02923 0.02931 -0.00143 DM GARCH Hedge A Hedge B Hedge C Hedge KS 0.07393 0.07373 0.07402 0.00027 0.05713 0.05672 0.05693 0.00002 -0.56712 -0.56676 -0.56726 -0.01128 0.66239 0.66222 0.66254 0.01802 0.01680 0.01701 0.01708 0.00025 ANN Hedge A Hedge B Hedge C Hedge KS 0.05437 0.05437 0.05444 0.00028 0.03638 0.03637 0.03639 0.00002 -0.39171 -0.39171 -0.39169 -0.01616 0.52715 0.52712 0.52719 0.01853 0.01799 0.01800 0.01805 0.00025 -60-Table 6.5b Expected Utility of Wealth Changes Among Hedging Strategies Medium Risk Aversion (7=4) JPY E(dW) Var(dW) Min(dW) Max(dW) E[U(dW)] GARCH Hedge A Hedge B Hedge C Hedge KS 0.02321 0.02362 0.03062 -0.00385 0.02901 0.02965 0.02971 0.01348 -0.31891 -0.31910 -0.31125 -0.31125 0.67708 0.68390 0.67817 0.34382 -0.09284 -0.09497 -0.08822 -0.05777 ANN Hedge A Hedge B Hedge C Hedge KS 0.03895 0.04251 0.03889 0.02222 0.02207 0.02616 0.02213 0.01392 -0.23311 -0.25629 -0.23767 -0.23507 0.49470 0.49449 0.49395 0.36567 -0.04934 -0.06212 -0.04963 -0.03345 CHF GARCH Hedge A Hedge B Hedge C Hedge KS 0.01504 0.01504 0.01505 0.00000 0.00269 0.00269 0.00269 0.00001 -0.10064 -0.10062 -0.10066 -0.00778 0.23692 0.23690 0.23683 0.00774 0.00427 0.00427 0.00427 -0.00002 ANN • Hedge A Hedge B Hedge C Hedge KS 0.02454 0.02455 0.02455 0.00000 0.00719 0.00715 0.00717 0.00005 -0.19222 -0.19507 -0.19354 -0.01622 0.37813 0.37681 0.37786 0.01703 -0.00171 -0.00211 -0.00194 -0.00159 DM GARCH Hedge A Hedge B Hedge C Hedge KS 0.04390 0.04383 0.04412 0.00027 0.03323 0.03323 0.03324 0.00002 -0.45028 -0.45017 -0.45067 -0.01128 0.59718 0.59713 0.59744 0.01802 -0.08902 -0.08907 -0.08885 0.00018 ANN Hedge A Hedge B Hedge C Hedge KS 0.03240 0.03239 0,03246 0.00028 0.02081 0.02081 0.02081 0.00002 -0.31899 -0.31899 -0.31897 -0.01616 0.47376 0.47373 0.47381 . 0.01853 -0.05082 -0.05083 -0.05078 0.00019 -61-Table 6.5c JPY E(dW) Expected Utility of Wealth Changes Among Hedging Strategies High Risk Aversion (7=10) Var(dW) Min(dW) Max(dW) E[U(dW)] GARCH Hedge A Hedge B Hedge C Hedge KS ANN Hedge A Hedge B Hedge C Hedge KS CHF 0.01981 0.02023 0.02722 -0.00385 0.03691 0.04045 0.03683 0.02222 0.02768 0.02821 0.02798 0.01348 0.02108 0.02481 0.02108 0.01392 -0.31863 -0.31882 -0.31919 -0.31125 -0.23304 -0.25612 -0.23749 -0.23507 0.66171 0.66835 0.66262 0.34382 0.49131 0.49111 0.49056 0.36567 -0.25703 -0.26189 -0.25261 -0.13865 -0.17391 -0.20769 -0.17393 -0.11695 GARCH Hedge A Hedge B Hedge C Hedge KS ANN Hedge A Hedge B Hedge C Hedge KS 0.01314 0.01314 0.01315 0.00000 0.02389 0.02358 0.02384 -0.00138 0.00259 0.00259 0.00259 0.00001 0.00705 0.00707 0.00706 0.00005 -0.10941 -0.10940 -0.10944 -0.00778 -0.20450 -0.20617 -0.20464 -0.01622 0.23071 0.23070 0.23063 0.00774 0.36926 0.36845 0.36950 0.01703 -0.01278 -0.01277 -0.01277 -0.00002 -0.04663 -0.04710 -0.04678 -0.00190 DM GARCH Hedge A Hedge B Hedge C Hedge KS 0.03790 0.03785 0.03814 0.00027 0.03331 0.03331 0.03329 0.00002 -0.42691 -0.42685 -0.42735 -0.01128 0.58414 0.58411 0.58443 0.01802 -0.29516 -0.29530 -0.29480 0.00005 ANN Hedge A Hedge B Hedge C Hedge KS 0.02800 0.02800 0.02807 0.00028 0.02067 0.02067 0.02067 0.00002 -0.31038 -0.31037 -0.31041 -0.01616 0.46309 0.46305 0.46313 0.01853 -0.17868 -0.17870 -0.17866 0.00006 -62-8 . Monetary Performance Measures While the utility measures in sub-sections 6 and 7 above demonstrate that non-martingale hedges produce higher expected utility than any other hedge for low levels of risk aversion, this concept is somewhat abstract. In order to put these results in a more meaningful context, they must be translated into monetary terms. A useful gauge of a hedge's value is the monetary amount that someone with a given risk tolerance would be willing to pay in order to acquire the hedge. A rational hedger would pay some amount X to purchase the hedge only if the expected benefit was at least sufficient to recover the cost. The upper bound on X is that value which equates the utility of hedged wealth less X with the utility of unhedged wealth: a hedger would be willing to pay for the hedge provided his/her utility from doing so is greater than his/her utility from leaving the position unhedged. The hedger therefore wants to find X such that: where W H is wealth from hedging and W 0 is unhedged wealth. Since the utility function is mean-variance, this means that X is found by subtracting the utility of unhedged wealth from the utility of hedged wealth. To see this, recall the original utility function: (6.17) U(WH - X) = U(Wu), (6.18) U(W) = E(W) - 7*Var(W). -63-Substituting (6.18) into (6.17): (6.19) E(W„-X) - 7*Var(WH-X) = UQNV), or, (6.20) E(WH) - E(X) - 7*[Var(WR) + Var(X)-2Cov(WH,X)] = UCvVu) Since X is a constant, E(X) = X and Var(X) = Cov(WH)X)=0, such that equation (6.20) reduces to: Table 6.6 presents the maximum amounts (denominated in units of foreign currency per USD) that a hedger with a given risk tolerance (low, medium or high) would pay to acquire each hedge, based on the formula in (6.21'). Negative outcomes have been assigned a value of zero, implying that no rational hedger would ever consider purchasing these hedges. The fourth and fifth rows, for instance, indicate that a hedger with low risk-aversion would pay up to 7.17171 JPY to hedge 1 USD worth of yen-denominated cashflows using Hedge C , but would pay a maximum of only 7.15859 JPY to hedge the same cashflows using Hedge KS. In other words, a Hedge C is worth 0.01312 JPY/USD more than Hedge KS to this hedger. (6.21) U(WH) - X = u w , or (6.21') x = u(w„) - urwu). -64-Table 6.6 Maximum Monetary Value of Hedging Strategies 7= 1 7 = 4 7=10 Japanese Yen (JPY/USD). GARCH ANN STATIC ANN STATIC Hedge A 7.16713 28.82436 72.14277 Hedge B 7.16625 28.82223 72.13791 Hedge C 7.17171 28.82898 72.14719 Hedge KS 7.15859 28.85943 72.26115 Hedge A 7.19578 28.86786 72.22589 Hedge B 7.19356 28.85508 72.19211 Hedge C 7.19542 28.86757 72.22587 Hedge KS 7.18422 28.88375 72.28285 Conv 7.16821 28.86821 72.26826 Naive 7.16780 28.86767 72.26744 Swiss Franc -(CHF/USD) Hedge A 0.02659 0.01438 0.00357 Hedge B 0.02660 0.01438 0.00358 Hedge C 0.02661 0.01438 0.00358 Hedge KS 0.00699 0.01009 0.01633 Hedge A 0.03678 0.00840 0.00000 Hedge B 0.03622 0.00800 0.00000 Hedge C 0.03630 0.00817 0.00000 Hedge KS 0.00556 0.00852 0.01445 Conv 0.00658 0.00965 0.01581 Naive 0.00697 0.01007 0.01628 -65-Table 6.6 cont'd 7=1 ( = 4 7=10 GARCH ANN STATIC Hedge A Hedge B Hedge C Hedge KS Hedge A Hedge B Hedge C Hedge KS Conv Naive Deutschmark (CM/USD) 0.02250 0.02271 0.02278 0.00595 0.02369 0.02370 0.02375 0.00595 0.00532 0.00565 0.00000 0.00000 0.00000 0.00919 0.00000 0.00000 0.00000 0.00920 0.00851 0.00887 0.00000 0.00000 0.00000 0.01568 0.00000 0.00000 0.00000 0.01569 0.01490 0.01531 Table 6.6 clearly shows that for hedgers with low risk-aversion, hedges A, B, and C are clearly worth more than Hedge KS or either of the static strategies. While the per unit amounts do not appear to be particularly large, the differences can become substantial when applied to multi-million dollar cashflows. As the monetary valuations are a linear function of the utility valuations, the basic findings remain unchanged: non-martingale models are preferable to the static and martingale models, and ANN models are prefable to GARCH models only for low coefficients of risk aversion; for hedgers with medium and high risk aversion, the preferred hedge depends on which currency is being hedged and which variance specification (ANN, GARCH or static) is being assumed. -66-VI. Conclusions Exchange rates, like most other financial data, exhibit periods of volatility clusters. In light of this feature, theory dictates that a time-varying hedge should be superior to a static hedge when an economic agent has a mean-variance utility function in wealth. Another feature of exchange rate data is that it need not follow a martingale as is commonly assumed, and as such, ignoring this feature may lead to sub-optimal hedging strategies even if time-varying volatility is accounted for. Although the resulting hedge has the convenient property of being independent of risk tolerance (not to mention easy to calculate) under a martingale assumption, it will be inconsistent with the first-order conditions for maximizing the utility function if the data violates this assumption. Assuming that futures prices are non-martingales, the utility function is maximized to arrive at a new hedge ratio with a speculative component. This component in isolation dictates whether one goes long or short in the futures market, depending on whether futures prices are rising or falling. The magnitude of this component is determined jointly by the size of the expected change, the variance of the futures price, and the hedger's risk tolerance: large expected changes, low variances and/or low coefficients of risk aversion all contribute to a large speculative component. Since this speculative component enters into the model not only in the initial period, but also in the subsequent period by way of the expected hedging strategy next period, a third term must be added to a two-period hedge ratio to provide a hedge for the speculative profits incurred from hedging in the subsequent period. This term is not easily calculated, and must be estimated using forecasted price data. As was -67-found in the previous section, when this term is estimated using Monte Carlo simulations, it is found to be relatively small. Moreover, as it is not always statistically significant, it can probably be disregarded completely without serious adverse consequences. The utility function used to derive an optimal hedge ratio is typically of the mean-variance variety. The use of such a functional form implies two things: that a utility . maximizer considers both properties in formulating a strategy rather than the variance on its own, and that both properties alone define the distribution of wealth. Non-martingale hedges are derived from a mean-variance utility function, and as such must be evaluated on the basis of the mean and variance of resulting wealth. It has been demonstrated, however, that this strategy produces a wealth distribution which can no longer be described solely by its mean and variance, and as such, comparisons with strategies which produce normal wealth distributions may be misleading. For the yen and the franc, the distribution of wealth turns out to be both skewed and leptokurtic; for the Deutschmark, the distribution turns out to be unskewed and platykurtic. This analysis has found a number of interesting properties of the various hedging strategies. First, the conditional hedge of Kroner and Sultan (1991; 1993) does produce the minimum-variance hedge when compared to other static models and the dynamic non-martingale models. Second, despite the fact that this hedge has a minimum variance, the reduction in total variance beyond that accounted for by the naive and conventional hedges is very small — less than a one per cent reduction in variance at most, with the static hedges eliminating between 97 and 99 per cent of the risk of an unhedged position. Third, non-martingale strategies add risk (in some cases more risk than an unhedged position), but also -68-increase the expected return on a spot-futures portfolio. For low levels of risk aversion, this increase in return is sufficient to compensate for the higher risk. Fourth, non-martingale hedges produce a higher expected change in wealth than any other hedging strategy, regardless of the hedger's risk tolerance. Finally, these findings appear to be driven by the mean equations, as a more complex variance specification fails to significantly alter the basic results and utility rankings. While the foregoing analysis has striven to show how the martingale assumption can lead to spurious results, a number of factors have been overlooked. First, utility is compared using only a two-week hedging model. It is not clear from these results whether extending the hedge to three or more periods would improve or worsen the performance of these models. Second, the entire analysis has been constructed on the assumption that there are 30 days remaining to expiration on the futures contract. The results may differ if a longer or shorter duration is used. Third, a key institutional feature of futures contracts (albeit a feature which is seldom activated) is the limit on daily movements in the futures price (Cornell and Reinganum (1981)), which brings into question the use of forward prices as a proxy during periods of high volatility. This would be particularly important when applying this kind of model to volatile currencies such as the Mexican peso (which incidentally does trade in the futures market). Finally j the hedges constructed here have not attempted to incorporate non-zero interest rates or bid-ask spreads. In summary, the non-martingale hedges derived here are optimal according to the underlying theory, but may not be optimal in practice if current econometric techniques cannot accurately approximate the mean equations. Simple artificial neural networks improve the conditional volatility estimates for at least two of the three currencies, although -69-not significantly enough to offset the mean effects. Given the limitations of simple error-correction models in accurately predicting exchange rate movements over the short-run, these hedges are optimal only for hedgers who are willing to assume a certain degree of risk above that which they would endure through other hedging strategies. -70-References Anderson, R.W. and J-P. Danthine: The Time Pattern of Hedging and the Volatility of Futures Prices, Review of Economic Studies 50 (1983) Baillie, R.T. and R J . Myers: Bivariate GARCH Estimation of the Optimal Commodity Futures Hedge, Journal of Applied Econometrics 6 (1991) Benet, B.A.: Hedge Period Length and Ex-Ante Futures Hedging Effectiveness: The Case of Foreign-Exchange Risk Cross Hedges, Journal of Futures Markets 12 (1992) Bollerslev, T.: Generalized Autoregressive Conditional Heteroskedasticity, Journal of Econometrics 31 (1986) Bollerslev, T., R.Y. Chou, and K.F. Kroner: ARCH Modelling in Finance, Journal of Econometrics 52 (1992) Bollerslev, T. and J.M. 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Granger: Cointegration and Error Correction: Representation, Estimation and Testing, Econometrica 55 (1987) French, K.R.: A Comparison of Futures and Forward Prices, Journal of Financial Economics 12 (1983) Grammatikos, T. and A. Saunders: Stability and the Hedging Performance of Foreign Currency Futures, Journal of Futures Markets 3 (1983). Herbst, A.F. , D.D. Kare and S.C. Caples: Hedging Effectiveness and Minimum Risk Hedge Ratios in the Presence of Autocorrelation: Foreign Currency Futures, Journal of Futures Markets 9 (1989) Hill, J. and T. Schneeweis: The Hedging Effectiveness of Foreign Currency Futures, Journal of Financial Research 5 (1982) Judge, G.G., R.C. Hill, W.E. Griffiths, H. Lutkepohl and T-C. Lee: Introduction to the Theory and Practice of Econometrics. New York: John Wiley & Sons (1988) Kroner, K.F. and J. Sultan: Exchange Rate Volatility and Time Varying Hedge Ratios, in S.G. Rhee and R.P. Chang (eds.), Pacific-Basic Capital Markets Research. Volume II. North-Holland (1991) Kroner, K.F. and J. Sultan: Time-Varying Distributions and Dynamic Hedging with Foreign Currency Futures, Journal of Financial and Quantitative Analysis 28 (1993) Kuan, C M . and H. White: Artificial Neural Networks: An Econometric Perspective, Econometric Reviews 13 (1994) McCurdy, T.H. and I.G. Morgan: Tests of the Martingale Hypothesis for Foreign Currency Futures with Time-Varying Volatility, International Journal of Forecasting 3 (1987) Myers, R . J . : Estimating Time-Varying Optimal Hedge Ratios on Futures Markets, Journal of Futures Markets 11 (1991) Sercu, P. and R. Uppal: International Financial Markets and the Firm. Cincinnati: South-western (1995) -73-Chart 6.1: Weekly Out-of-Sample Hedge Ratios Japanese Yen (GARCH) - 0 . 9 0 - 0 . 9 5 ^ -1.00-F - 1 . 0 5 - 1 . 1 0 -f - 1 . 1 5 I i i i i I I I I I I I I I I M I I I ! I I I I II Week Chart 6.2: Weekly Out-of-Sample Hedge Ratios Swiss Franc (GARCH) - 0 . 9 0 -- 1 . 1 0 - 1 . 3 0 - 1 . 5 0 -- 1 . 7 0 -1.90-4 - 2 . 1 0 I i i i I I i i i i i i I I i i i I I i i i i i i i i , i M i i i i i i i i M M i i i i i i i i Week -74-Chart 6 . 3 : Weekly Out-of-Sample Hedge Ratios Deutschmark (GARCH) 3.00 -5.00 I i I I i i i i i i I I I I i l I I i i i i i I I I I i i i i I I i i i i i i i i i i i I I i i i i i Week Chart 6 . 4 a : Hedging Effectiveness Japanese Yen J P Y / U S D 8.00 - i Week -75-C h a r t 6 . 4 b : W e e k l y O u t - o f - S a m p l e C h a n g e s i n W e a l t h J a p a n e s e Y e n ( G A R C H ) Chart 6 . 5 a : Hedging Effectiveness Swiss Franc -76-C h a r t 6 . 5 b : W e e k l y O u t - o f - S a m p l e C h a n g e s i n W e a l t h S w i s s F r a n c ( G A R C H ) Chart 6 . 6 a : Hedging Effectiveness Deutschmark D M / U S D 0.10-1 - 0 . 0 8 \ I I i M i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i i Week -77-C h a r t 6 . 6 b : W e e k l y O u t - o f - S a m p l e C h a n g e s i n W e a l t h D e u t s c h m a r k ( G A R C H ) DM/USD O .8O - 1 Week -78-Chart 6.7: Actual vs Estimated GARCH Volatility, Japanese Yen, 1990-95 4 5 . 0 - • 4 0 . 0 -Chart 6.8: Actual vs Estimated ANN Volatility, Japanese Yen, 1990-95 -79-Chart 6.9: Weekly Out of Sample Hedge Ratios Japanese Yen (ANN) Chart 6.10: Weekly Out of Sample Hedge Ratios Swiss Franc (ANN) 0.00 - 3 . 5 0 I i i i i i i i i i i i i i i i i i I ' . I i I I i i i i i i i i i i i I I M i i i i i i i i i i i i i Week -80-Chart 6.11: Weekly Out of Sample Hedge Ratios Deutschmark (ANN) Chart 6.12; Weekly Out of Sample Changes in Wealth Japanese Yen (ANN) -81-Chart 6.13: Weekly Out of Sample Changes in Wealth Swiss Franc (ANN) C H F / U S D 0 . 4 0 -i i i i t i i i i t i i i i r r T i T i i i i i i i i i \ i i i i i i i i i i i i i i i i i i i i i ] i i Week Chart 6.14: Weekly Out of Sample Changes in Wealth Deutschmark (ANN) D M / U S D 0 . 6 0 -i Week -82-


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