CURRENCY ARBITRAGE
6.4 EVALUATING THE EFFECTIVENESS OF HEDGING
The test can be conducted to compare the effectiveness of two hedging posi
tions resulting from the use of different hedge ratios or different hedging instru
ments. In this case, the null hypothesis becomes H0:s2(RH ,1) =s2(RH ,2 ), where s2(RH ,1 ) and s2(RH ,2 ) are the rates of return on the hedged positions resulting from hedge number one and hedge number two respectively.
These tests can be done within-sample or out of sample. Suppose that the historical data that is used to estimate the hedge ratio is available for the period t = 1, 2, ..., n. If the hedge ratio and the test statistics are calculated on the basis of the full sample, then this is an in-sample test. If, however, the hedge ratio is estimated over the sub-sample period t= 1, 2, ..., k, where k< n, then the test statistics are calculated from the observations t= k+1, k+ 2, ..., n, which makes it an out-of-sample test.
Since correlation between the rates of return, RU and RA, is crucial for the success of hedging, we will now attempt to find out how the hedge ratio, vari
ance ratio and variance reduction are related to the correlation coefficient between the rates of return. The correlation coefficient is defined as
s(RU, RA )
(6.65) r=
s(RU)s(RA) which gives
s(R U , RA) =rs(RU)s(R A) (6.66)
By substituting equation (6.66) in equation (6.17), which defines the hedge ratio, we obtain
rs(RU)s(RA) æs(RU) ö÷ (6.67)
h = s2(RA) =rççès(RA) ÷ø
It is obvious from equation (6.67) that the hedge ratio is linearly related to the correlation coefficient. If s(RU) =s(RA ), then h=r.
To derive the relationship between the variance ratio and the correlation coefficient, we rewrite equation (6.63) as
s2(RU )
(6.68) VR=
s2(RU) +h2 s2(RA) -2hs(RU , RA)
By substituting equations (6.66) and (6.67) into equation (6.68), we get s2(RU)
VR =
s2(RU) +r2 s2(RU)s2(RA) -2[( rs(RU)s(RA ))/(s2(RA ))]rs(RU)s(RA) (6.69) which can be simplified to obtain
VR = 1 (6.70)
1 -r2
implying a positive relationship between VR and r, because
d (VR) = 2 r >0 (6.71)
dr (1 -r2 2 )
As we can see from equations (6.70) and (6.71), the relationship between the variance ratio and the correlation coefficient is nonlinear. It also follows from equation (6.70) that
1
VD= - 1 =r2 (6.72)
VR
which shows the variance reduction is equivalent to the coefficient of determi
nation of the underlying regression.
To derive the relationship between the variance ratio and the hedge ratio, we substitute (6.67) into (6.70) to obtain
VR = 1 (6.73)
1 -h 2 [(s 2(R A )) / (s 2(RU ))]
which gives
d (VR) = 2h[(s2(R )) / (s2(RU ))]
dh {1 -h 2 [(s2(R
A
)) / (s2(R ))]} 2 (6.74)
A U
As we can see from equations (6.73) and (6.74) the variance ratio is a nonlinear positive function of the hedge ratio. Finally, we have
1 =h2 é
ês2(RA)ù VD= -1
VR ês2(RU)úú (6.75)
ë û
which gives
d (VD) =2hêés2(RA)ù
>0 (6.76)
dh ês2(RU)úú
ë û
which again shows a positive and nonlinear relationship between variance reduction and the hedge ratio.
The base currency value of payables and receivables
Another criterion that is used for evaluating hedging effectiveness, particu
larly with contingent exposures involving foreign currency payables and receivables, is the base currency value of the payables or receivables. The hedger may wish to optimise the domestic currency values of payables and receivables (maximising receivables and minimising payables). If this is the case then one hedge will be preferred to another if the mean domestic currency value of payables (receivables) is lower (higher) than under the other hedge (see, for example, Moosa, 2002c). Let m( )U and m( )H be the population
means of the domestic currency values of the payables under no-hedge and the hedge decisions. The null hypothesis is
H0: (U) =m m(H) (6.77)
whereas the alternative hypothesis is written as
H1: (U) ¹m m(H) (6.78)
The null hypothesis is rejected if ( ) -X H) n2
X U (
>t n(2 -2) (6.79)
s 2n
where X U( ) and X H( ) are respectively the sample means of the domestic currency values of the payables under no-hedge and hedge decisions, nis the sample size, t(2n– 2) is the critical value of the tdistribution with 2n– 1 degrees of freedom, and
[ 2(U) +s2(H)]
(6.80) s = n s
2n-2
where s2 is the estimated sample variance. The sample mean domestic currency values of the payables or receivables are defined respectively as
1 n
( ) = K S (6.81)
X U
å
t t+ 1nt=1 1 n
( ) = K S (6.82)
X H n
å
t tt=1
where S is the conversion rate implicit in the hedge (which is equal to the forward rate in forward hedging and the interest parity rate in money market hedging). Naturally, the hedger may not only be interested in optimising the mean value of the payables or receivables but also in the variability of the cash flows. In this case, testing the equality of the variances must also be used. A final choice decision can be made, depending on the risk–return trade-off. Of course, the same procedure can be used to test the effectiveness of hedging under different hedge ratios.
Out-of-sample forecasting
Finally, the effectiveness of hedging may be tested by comparing the out-of- sample forecasting power of the underlying models that are used to calculate the hedge ratio. For example, Ghosh (1993) compares the first-difference model with the error correction model. Specifically, he calculates the root mean square error (RMSE) of the two models and concludes that the error correction model is associ
ated with about 15% reduction in the RMSE of the first-difference model. The two models can be alternatively estimated on the basis of a likelihood ratio test between
the restricted and unrestricted models. He also concludes that (i) the error correc
tion model provides better estimates of the optimal hedge ratio, which reduces risk as well as the cost of hedging; and (ii) that the hedge ratios from traditional models are underestimated.
The problem here is that economists tend to follow the faulty procedure of deriving inference on the superiority of the forecasting power of a model over another simply by comparing the numerical values of the mean square errors of the forecasts, without testing whether or not the difference between the mean square errors is statistically significant. Such tests are actually available, the simplest of which is the AGS test, designed by Ashley, Granger and Schmalensee (1980). The test requires the estimation of the linear regression
Dt =a0 +a1(Mt-M) +ut (6.83)
where Dt = w1t – w2t, Mt = w1t + w2t, Mis the mean of M, w1t is the forecasting error at time tof the model with the higher MSE, w2t is the forecasting error at time t of the model with the lower MSE. If the sample mean of the errors is nega
tive, the observations of the series must be multiplied by –1 before running the regression. The estimates of the intercept term (a0 ) and the slope (a1) are used to test the statistical difference between the MSEs of two different models. If the estimates of a0 and a1 are both positive, then a Wlad test of the joint hypothesis H 0: a0 =a1 = 0 is appropriate. However, if one of the estimates is negative and statistically significant then the test is inconclusive. But if the estimate is nega
tive and statistically insignificant the test remains conclusive, in which case significance is determined by the upper-tail of the t-test on the positive coeffi
cient estimate.
Theempiricalevidence
The issue of estimating the hedge ratio and assessing the effectiveness of hedging has been the focus of considerable empirical research in the literature. A large number of studies have been carried out to evaluate the hedging effectiveness based on various methods for measuring the hedge ratio. Initially, hedge ratios were calculated as the slope coefficient from an OLS regression (Ederington, 1979). Kroner and Sultan (1993) estimated time-varying hedge ratios using a bivariate error correction model with a GARCH error structure. They showed that this model provides greater risk reduction than the conventional models.
Brooks and Chong (2001) examined the ability of several models to generate optimal hedge ratios when cross currency hedging is used. By using four currency pairs they found that an exponentially weighted moving average model leads to lower portfolio variances than any of the GARCH-based, implied or time-invariant approaches.
Ghosh (1993) demonstrated that less than optimal hedging would result if the hedge ratio is estimated from a model that ignores the error correction mecha
nism, as shown by Lien (1996). However, Moosa (2002e) examined the sensi
tivity of the optimal hedge ratio estimates to four different model specifications
and found no evidence for any relationship between model specification and hedging effectiveness. What matters most for hedging effectiveness, he concludes, is the correlation between the prices of the unhedged position and the hedging instrument. In another paper, Moosa (2002d) examined the effec
tiveness of cross currency hedging compared with that of forward and money market hedging. He demonstrated that cross currency hedging is not only less effective than forward and money market hedging, but also that it is totally inef
fective unless the exchange rate of the exposure currency and that of the third currency (the hedging instrument) are highly correlated. The results indicate that for effective cross currency hedging a correlation coefficient of 0.5 is required to reduce the variance of the rate of return on the unhedged position by 25%.