Finance Letters, 2003, 1, 15-20
The Sensitivity of the Optimal Hedge Ratio to Model Specification Imad A. Moosa* La Trobe University, Australia Abstract This paper investigates the effect of the choice of the model used to estimate the hedge ratio on the effectiveness of futures and cross-currency hedging using data from the stock and foreign exchange markets. Four different models are used for this purpose to estimate the hedge ratio. The results show that model specification has little effect on the hedging effectiveness. It seems that what matters most is the correlation between the prices of the unhedged position and the hedging instrument. Key words: Hedging Effectiveness, Hedge Ratio, Cross-Currency Hedging JEL classification: G15
1. INTRODUCTION Two important questions are involved in financial hedging: (i) to hedge or not to hedge; and (ii) if the decision to hedge is taken, do we hedge the full position? This paper deals with the second question, which amounts to estimating the hedge ratio or determining the size of the position on the hedging instrument that is used to hedge a spot (cash) position. In this exercise we use data on stock prices as well as exchange rates, and hence we deal with the hedging of exposure to equity risk and foreign exchange risk. It has been suggested that the optimal hedge ratio can differ significantly, depending on the model that is used to estimate the hedge ratio (Ghosh, 1993). The conventional model takes the form of an OLS regression equation in which the dependent variable is the price of (or the rate of return on) the unhedged position, whereas the explanatory variable is the price of (or the rate of return on) the hedging instrument (which could be a forward or a futures contract). This model has been criticised on the grounds that it ignores short-run dynamics (when it is specified in levels) and the long-run information embodied in the error correction term (when it is specified in first differences). Ghosh (1993) finds the hedge ratios obtained from traditional models to be underestimated because “these models are misspecified”. Lien (1996) shows analytically that the hedger makes a mistake if the hedging decision is based on the hedge ratio derived from a first-difference model that does not contain an error correction term. There is also the issue of whether or not the hedge ratio should be estimated from a first difference or a level model, which are implicitly taken to be equivalent (see, for example, Ghosh, 1993; Witt et al., 1987). And there is the problem of whether the hedge ratio should be estimated from the conditional or the unconditional moments, which has been dealt with by Kroner and Sultan (1993) as well as Brooks and Chong (2001). In this paper we do not deal with this issue, but rather concentrate on whether or not it makes any difference for hedging effectiveness if the hedge ratio is derived from (i) a levels as opposed to a first-difference model; and (ii) a first-difference as opposed to an error correction (EC) model. 2. MEASURING THE OPTIMAL HEDGE RATIO Let pU and pA be the logarithms of the prices of the unhedged (spot or cash) position and the hedging instrument respectively, such that the rates of return on these positions are pU and pA respectively. The level and first difference models are written as
*
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ISSN 1740-6242 © 2003 Global EcoFinance™ All rights reserved.
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Moosa
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pU ,t = α 1 + h1 p A,t + ε 1, t
(1)
∆pU ,t = α 2 + h2 ∆p A,t + ε 2,t
(2)
where h1 and h2 are the estimated hedge ratios, such that the R2 of the regressions measures the hedging effectiveness. This procedure for calculating the hedge ratio is based on an application of the principles of portfolio theory, as demonstrated by Johnson (1960), Stein (1961), Ederington (1979), McEnally and Rice (1979), Franckle (1980), and Hill and Schneeweis (1982). Some economists make it sound as if the two models represented by equations (1) and (2) are two alternative means for measuring the same thing. For example, Giaccotto et al. (2001, p. 148) talk about minimising “the volatility of the total cash flow or, equivalently, the change in flow”. Hill and Schneeweis (1981) recommend the use of the first-difference model only because the model in levels produces serially correlated residuals. Strictly, speaking, however, the choice would or should depend on whether the objective is to minimise the variance of the price or the rate of return. This is because the variances of the hedged positions corresponding to equations (1) and (2) are
σ 2 ( p H ) = σ 2 ( pU ) + h12σ 2 ( p A ) − 2h1σ ( pU , p A )
(3)
σ 2 (∆p H ) = σ 2 (∆pU ) + h22σ 2 (∆p A ) − 2h2σ (∆pU , ∆p A )
(4)
The minimum-risk (or the minimum-variance) hedge ratios can be obtained by differentiating equations (3) and (4) with respect to the hedge ratios and solving the first order conditions, which are written as
dσ 2 ( p H ) = 2h1σ 2 ( p A ) − 2σ ( pU , p A ) = 0 dh1
(5)
dσ 2 (∆p H ) = 2h2σ 2 (∆p A ) − 2σ (∆pU , ∆p A ) = 0 dh2
(6)
Solving the first order conditions gives
h1 =
σ ( pU , p A ) σ 2 ( pA)
(7)
h2 =
σ (∆pU , ∆p A ) σ 2 (∆p A )
(8)
Obviously, the hedge ratios h1 and h2 are not necessarily equal. We will find out what happens when the hedge ratio estimated from a model in levels is used to reduce the variance of the rate of return on the hedged position and vice versa. One problem with equations (1) and (2) is that that equation (1) ignores short-run dynamics, whereas equation (2) ignores the long-run relationship as represented by equation (1). Specifically, if pU and pA are cointegrated such that ε1,t ~I(0), then equation (2) would be misspecified, in which case the correctly specified model is an error correction model of the form
∆pU ,t = α +
n
i =1
n
β i ∆pU ,t −i + h∆p A,t + γ i ∆p A,t −i + θ ( pU ,t −1 − p A,t −1 ) + ξ t
i =1
(9)
Lien (1996) argues that the estimation of the hedge ratio and the hedging effectiveness may change sharply when the possibility of cointegration between prices is ignored. In Lien and Luo (1994) it is shown that although GARCH may characterise the price behaviour, the cointegration relationship is the only truly indispensable component when comparing the ex post performance of various hedging strategies. Ghosh (1993) concluded that a smaller than optimal futures position is undertaken when the cointegration relationship is unduly ignored. He attributed the under-hedge results to model misspefcification. Lien (1996) provides a theoretical analysis of this conjecture by assuming a cointegrating relationship of the form φt = pA,t – pU,t. A simplified error correction model, which implies that prices adjust in response to disequilibrium, can be written as
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∆pU ,t = αφ t −1 + ξ1,t
(10)
∆p A,t = − βφ t −1 + ξ 2,t
(11)
If the hedge ratio is chosen to minimise σ2 ( pU,t – h pA,t), , then the hedge ratio is calculated as
σ (∆pU ,t , ∆p A,t | φ t −1 ) σ (ξ1,t ) =ρ 2 σ (ξ 2,t ) σ (∆p A,t | φ t −1 )
h=
(12)
where ρ is the correlation coefficient between ξ1,t and ξ2,t. Alternatively, the hedge ratio can be calculated from the regression equation
∆pU ,t = α + h∆p A,t + γφ t −1 + ζ t
(13)
If the cointegrating relationship is ignored then the hedge ratio is calculated as in (8). From equations (10) and (11), we have
σ (∆pU ,t , ∆p A,t ) = σ (− βφt −1 + ξ 2,t ,αφ t −1 + ξ1,t ) = −αβσ 2 (φ t −1 ) + ρσ (ξ 1,t )σ (ξ 2,t )
(14)
and
σ 2 (∆p A,t ) = σ 2 (− βφ t −1 + ξ 2,t ) = β 2σ 2 (φ t −1 ) + σ 2 (ξ 2,t )
(15)
Hence, the hedge ratio is measured as
h=
− αβσ 2 (φ t −1 ) + ρσ (ξ1,t )σ (ξ 2,t )
(16)
β 2σ 2 (φ t −1 ) + σ 2 (ξ 2,t )
Obviously there is a difference between the expressions in equation (16) and equation (12). On the basis of these two expressions, Lien (1996) concludes that an errant hedger who mistakenly omits the cointegrating relationship (by using equation 16) always undertakes a smaller than optimal position on the hedging instrument (compared with a hedger using equation 12). This proposition is confirmed empirically by Ghosh (1993). By using a general specification of equations (10) and (11), we have
∆pU ,t = αφ t −1 +
n
i =1
∆p A,t = − βφ t −1 +
ai ∆pU ,t −i + n
i =0
n
i =0
c i ∆pU ,t −i +
bi ∆p A,t −i +ξ 1,t
n
i =1
(17)
d i ∆p A,t −i +ξ 2,t
(18)
in which case the hedge ratio calculated on the basis of the correctly specified model is given by
h=
σ (∆pU ,t , ∆p A,t | φ t −1 , ∆pU ,t −i , ∆p A,t −i ) σ 2 (∆p A,t | φ t −1 , ∆pU ,t −i , ∆p A,t −i )
σ (ξ1,t , ξ 2,t ) σ (ξ1,t ) = =ρ 2 σ (ξ 2,t ) σ (ξ 2,t )
(19)
whereas the errant hedger who does not take into account the cointegration relationship will choose a hedge ratio that is given by
h=
σ (∆pU ,t , ∆p A,t | ∆pU ,t −i , ∆p A,t −i ) σ 2 (∆p A,t | ∆pU ,t −i , ∆p A,t −i )
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ρσ (ξ1,t )σ (ξ 2,t ) − αβσ 2 (φ t −1 | ∆pU ,t −i , ∆p A,t −i ) = σ 2 (ξ 2,t ) + β 2σ 2 (φ t −1 | ∆pU ,t −i , ∆p A,t −i )
(20)
which means that the errant hedger will undertake a smaller than optimal position on the hedging instrument, incurring losses in hedging effectiveness. For the purpose of assessing the hedging effectiveness based on various models, a hedge is considered to be effective when the variance of the price or the rate of return of the unhedged position is significantly higher than that of the hedged position. This will be obtained if the variance ratio satisfies the condition
VR =
σ 2 ( pU ) > F (n − 1, n − 1) σ 2 ( pH )
(21)
where n is the sample size. The effectiveness of two hedges can be compared on the basis of variance reduction, which is calculated as
VD = 1 −
1 VR
(22)
Four models are used for the purpose of calculating the hedge ratio: (i) the levels model (equation 1); (ii) the first-difference model (equation 2); (iii) a simple error correction model (equation 9 with βi = 0 and γi = 0); and (iv) a general error correction model (equation 9). 3. DATA AND EMPIRICAL RESULTS Two sets of data are used in this empirical exercise. The first is a set of monthly observations on cash and futures prices of Australian stocks. The cash price is represented by the All Ords index, whereas the futures price is represented by the SPI index. The sample, which was obtained from the Australian Stock Exchange, covers the period 1987:1-1997:12. The second sample consists of quarterly observations covering the period 1980:1-2000:4 on the spot exchange rates of the pound and the Canadian dollar against the U.S. dollar. In this case of cross-currency hedging the base currency is the pound, the exposure currency is the U.S. dollar, and the currency used for hedging is the Canadian dollar. Thus, pU = S(USD/GBP) and pA = S(USD/CAD). The data sample was obtained from the OECD’s Main Economic Indicators. Table 1 and Table 2 report the estimated hedge ratios, the variances, variance ratios and variance reductions for stock prices. Although the estimated hedge ratios are numerically different, the results in terms of variance reduction are not that different. In all cases, hedging is effective and the reduction in the variance is close (over 97 per cent in all cases). Table 1. Estimated Hedge Ratios: Stock Prices Model Level First Difference Simple EC General EC
Hedge Ratio 0.995 0.919 0.932 0.929
R2 0.99 0.98 0.99 0.99
We now examine the results obtained by using exchange rates, which are reported in Tables 3 and 4. Again, the estimated hedge ratios are numerically different but this makes little difference to the results: in no case is the hedge effective, as the variance ratios are statistically insignificant. The reason for the difference between these results and those obtained by using stock price data is attributed to the lack of correlation between pU and pA in this case, unlike the previous case. What matters for the results seems to be not the model used to estimate the hedge ratio, but rather the correlation. Results obtained by Moosa (2002) show that for an effective hedge, the correlation coefficient between pU and pA must be at least 0.50 to produce variance reduction of about 25 per cent.
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Table 2. Hedging Effectiveness: Stock Prices Variance
Model
Estimated Value 0.040390
σ ( pU ) 2
VD
0.004251
σ 2 (∆pU )
*
VR
σ 2 (∆pU − h∆p A )
Level
0.000120
35.43
97.18
σ 2 (∆pU − h∆p A )
First Difference
0.000091
46.71
97.86
σ 2 (∆pU − h∆p A )
Simple EC
0.000093
45.71
97.81
σ 2 (∆pU − h∆p A )
General EC
0.000092
46.21
σ 2 ( pU − hp A )
Level
σ 2 ( pU − hp A )
First Difference
0.000080 0.000320
97.84 *
99.80
*
99.21
504.99 126.25
Relative to σ2( pU).
Table 3: Estimated Hedge Ratios: Exchange Rates Model
Hedge Ratio
Level First Difference Simple EC General EC
0.493 0.495 0.582 0.232
R2
0.16 0.04 0.13 0.27
Table 4: Hedging Effectiveness: Exchange Rates Variance
Model
σ ( pU ) 2
VR
VD
0.003088
σ 2 (∆pU )
*
Estimated Value 0.016500
σ 2 (∆pU − h∆p A )
Level
0.002957
1.04
4.24
σ 2 (∆pU − h∆p A )
First Difference
0.002957
1.04
4.24
σ 2 (∆pU − h∆p A )
Simple EC
0.002957
1.04
4.24
σ 2 (∆pU − h∆p A )
General EC
0.002999
1.03
2.88
σ 2 ( pU − hp A )
Level
0.013700
1.20*
16.97
σ 2 ( pU − hp A )
First Difference
0.013700
1.20*
16.97
Relative to σ2( pU).
4. CONCLUSION It has been argued that the choice of the model used to estimate the hedge ratio makes some difference for the effectiveness of hedging measured as the reduction in the variance of the unhedged position. This study investigated this issue empirically by employing four different models to estimate the hedge ratios used to cover exposure to spot positions in stocks and currencies. Although the theoretical arguments for why model specification does matter are elegant, the difference model specification makes for hedging performance seems to be negligible. What matters for the success or failure of a hedge is the correlation between the prices of the unhedged position and the hedging instrument. Low correlation invariably produces insignificant results and ineffective hedge, whereas high correlation produces effective hedge irrespective of how the hedge ratio is measured.
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REFERENCES Brooks, C. and J. Chong (2001) The Cross-Currency Hedging Performance of Implied versus Statistical Forecasting Models, Journal of Futures Markets, 21, 1043-1069. Ederington, L.H. (1979) The Hedging Performance of the New Futures Markets, Journal of Finance, 34, 157-170. Franckle, C.T. (1980) The Hedging Performance of the New Futures Markets: Comments, Journal of Finance, 35, 12731279. Ghosh, A. (1993) Hedging with Stock Index Futures: Estimation and Forecasting with Error Correction Model, Journal of Futures Markets, 13, 743-752. Giaccotto, C., S.P. Hedge and J.B. McDermott (2001) Hedging Multiple Price and Quantity Exposures, Journal of Futures Markets, 21, 145-172. Hill, J. and T. Schneeweis (1981) A Note on the Hedging Effectiveness of Foreign Currency Futures, Journal of Futures Markets, 1, 659-664. Hill, J. and T. Schneeweis (1982) The Hedging Effectiveness of Foreign Currency Futures, Journal of Financial Research, 5, 95-104. Johnson, L. (1960) The Theory of Hedging and Speculation in Commodity Futures, Review of Economic Studies, 27, 139151. Kroner, K.F. and J. Sultan (1993) Time-Varying Distributions and Dynamic Hedging with Foreign Currency Futures, Journal of Financial and Quantitative Analysis, 28, 535-551. Lien, D. (1996) The Effect of the Cointegration Relationship on Futures Hedging: A Note, Journal of Futures Markets, 16, 773-780. Lien, D. and X. Luo (1994) Multiperiod Hedging in the Presence of Conditional Heteroskedasticity, Journal of Futures Markets, 13, 909-920. McEnally, R.W. and M.L. Rice (1979) Hedging Possibilities in the Flotation of Debt Securities, Financial Management, 10, 12-18. Moosa, I.A. (2002) The Effectiveness of Cross-Currency Hedging, Unpublished Paper, La Trobe University, Australia. Stein, J.L. (1961) The Simultaneous Determination of Spot and Futures Prices, American Economic Review, 51, 12012-1025. Witt, H., T. Schroeder and M. Hayenga (1987) Comparison of Analytical Approaches for Estimating Hedge Ratios for Agricultural Commodities, Journal of Futures Markets, 7, 135-146.
