International Short-Term Financing and Investment 251
(1 + ey ) > (1 + L ) 291
(1+f ) 300
(1+f ) 302
International Long-Term Financing, Capital Structure and the Cost of Capital 284
D = Z D, 283
E = Z 283
k = Z 289
International Long-Term Portfolio Investment 296
Rj,t = b 0,j + b 1, jR m,t + b 2,j$t + e j,t (11.40) 311
dL 319
dm 1 -1'—-1'-2'"2 --v-^"3 '-3 (1149) 319
+ 2ct(S 1, S 4)w 4 +1 1 = 0 319
= 2w3 s 2 (S3 ) + 2s(S 1 , S3 )w 1 + 2s(S2 , S3 )w2 320
+ 2s(S 3 , S4 )w 4 +2 1 = 0 = 2w4 s 2 (S4 ) + 2s(S 1 , S4 )w 1 + 2s(S 2 , S4 )w2 + 2s(S 3 , S4 )w 4 + 2 2 = 0 320
= w 1 + W2 + w 3 -1 = 0 320
w' = [w 1 w 2 w 3 w 4 2 1 2 2] and 331
Foreign Direct Investment 344
i 341
References 341
i 361
Index 363
R A
or equivalently that
h = ^ (6.6)
R u - a
Consider now the special case of using forward hedging. If the unhedged position is a long position on currency y, then the hedging instrument is a short forward position on y. The rate of return on the unhedged spot position and between t - 1 and t is given by
R U=^-1 = S (6.7)
St-1
Let us for the time being define the rate of return on the forward position as
S
Ra ~-1 (6.8)
Ft-1
in the sense that return arises by buying a currency at Ft-i and selling it at St. Since Ft-i = (1 + f)St-i, where f is the forward spread, it follows that
Ra= 1+S -1 = ^ (6.9)
A 1 + f 1 + f
For a perfect hedge, we have
h = —-S = (6.10)
(S-f)/(1 + f) S-f v '
It follows from equation (6.10) that for h = 1, the condition f = 0 must be satisfied. This means that if the full hedge is to be a perfect hedge, the forward spread must be equal to zero, and this would be so if the interest rates on the two currencies are equal, which is not normally the case. If a money market hedge is used instead of a forward hedge, the same arguments hold, except that the price of the hedging instrument will in this case be the interest parity forward rate, which is the forward rate consistent with covered interest parity.
Consider now a comparison between the hedge ratio under a direct forward hedge and a cross forward hedge. Let us for this propose introduce a third currency, z, and rewrite equations (6.7) and (6.8) as
Ru= SS^rk-l = S(x/y) (6.11) St-1(x/y)
R St(x/z) -1 = S( x/z) - f (x/z) (6 12)
A Ft-!(x/z) 1+f (x/z) V ;
Therefore, the optimal hedge ratio is given by
h = ^H1+f (x/z)] (6.13)
S( x/z) - f ( x/z) V y
which is obviously equal to one only if f(x/z) = 0 and S(x/y) = S(x/z).
If cross hedging involves two spot positions, then R a = S( x/z), in which case the optimal hedge ratio is given by
h = ^ (6.14)
S(x/z)
which is equal to one only if S( x/y) = S( x/z).
6.2 MEASURING THE OPTIMAL HEDGE RATIO
It is important for firms to employ the most effective model to calculate the optimal hedge ratio. In reality it may never be possible to get a perfect hedge, in which case the objective may be to minimise the variance of the rate of return on the hedged position. This variance is given by
s2 (RH) = s2 (Ru) + h2 s2 (RA)-2hs(Ru , RA ) (6.15)
The minimum-risk hedge ratio is calculated from the first-order condition
dS
H)
= 2s
2
(R
a )h
- 2s(Ru
,
Ra)
= 0 (6.16)
dh
Hence
s(RU, Ra )
h
= v
U
A
' (6.17)
s 2(Ra)
The minimum-risk hedge ratio can be calculated from historical data by estimating the regression equation
R ut = a + hRAt + e t (6.18)
The early literature focused on estimating the hedging ratio by using OLS to estimate equation (6.18) (for example, Ederington, 1979). However, this approach is subject to two criticisms. One problem is that risk minimisation without regard to the effect of expected return cannot be optimal. Hedging away the risk must also hedge away the expected return to bearing that risk. Thus, the optimal hedge ratio should be the one that maximises utility, which is a function of risk and return. Different hedge positions can be compared directly by examining their certainty equivalent returns (for example, Cecchetti et al., 1988).
The other problem is that the joint distribution of (Ru, Ra), and therefore the hedge ratio, is estimated incorrectly since there is no adjustment for the fact that it changes over time. Thus, regression employing past data will not correctly estimate the current risk minimisation ratio.
Kanheman and Tversky (1979) and Tversky and Kanheman (1986, 1992) argue that individuals do not maximise expected utility but behave according to a set of rules known as "prospect theory". One of the suggested rules is that individuals care for changes in wealth, instead of wealth itself. Lien (2001) suggests another rule, which is loss aversion. According to him, there are two versions of loss aversion. Strong loss aversion implies that individuals tend to be more sensitive to wealth losses than to wealth gains. This approach postulates constant sensitivity to losses. The other version is that of weak loss aversion, which accounts for diminishing sensitivity to losses. Albuquerque (1999) considers futures hedging under loss aversion, showing that it simply reduces to downside risk minimisation. Lien attempts to explore the effects of loss aversion on futures hedging with constant absolute-risk aversion. Specifically, he examined the effect of loss aversion on the futures trading behaviour of a short hedger.
Because the OLS estimation method does not take into account the time- varying nature of variances and covariances that make up the hedge ratio, economists started using the ARCH model of Engle (1982) and the GARCH model of Bollerslev (1986) to calculate a time-varying hedge ratio. Myers and Thompson (1989) argued that conventional procedures for estimating the hedge ratio are based on the unconditional rather than the more appropriate conditional variances and covariances. They further argue that basing hedging decisions on deviations from the conditional moments reflect more accurately the risk faced by hedgers. Myers (1991) and Baillie and Myers (1991) also suggested that optimal hedge ratio estimation may require the conditional variance and covariance of futures and spot prices (prices of the hedging instrument and the unhedged position respectively) to vary over time, resulting in time-varying hedge ratios. Recent literature deals mostly with estimating the hedge ratio by employing bivariate GARCH models, as demonstrated by Kroner and Sultan (1991, 1993) who used currency futures as the hedging instrument. When the variances and the covariances are allowed to vary over time, we use the conditional variances and covariances. Therefore
s 2 (RH,tlWt-1 ) = s 2 (Ru,tlWt-1 ) + h2 s 2 (RA,tlWt-1 ) (6.19)
- 2hs t (Ru,t , RA,tlW t-1 )
where W t-1 is the information set available at time t -1, s 2 (-)is the conditional variance and s t (•,•) is the covariance. Hence
, s t (RU,t , RA,t|W t-1 )
h
t|W
t-1
= 2,p
— (6.20)
s 2(RA,t|W t-1 )
The conditional variances and covariances can be calculated from various models as outlined by Brooks and Chong (2001), which include EWMA, univariate GARCH (GARCH, EGARCH, GJR, GARCH-t) and multivariate GARCH (VECH, BEKK). Otherwise, the ratio can be calculated by estimating the regression in a TVP framework via state space modelling and the Kalman filter.
6.3 EMPIRICAL MODELS OF THE HEDGE RATIO
Several empirical models are used to estimate the hedge ratio. These models are described in turn.
The naïve model
The naïve model simply implies that the hedge ratio is always 1. This is exactly the assumption we used to demonstrate the hedging decision in Chapter 5. Since the naïve model is naïve, we need to go no further in elaborating on it.
The implied model
The implied model allows the estimation of the conditional covariance by employing the implied volatilities derived from currency options. These volatilities may be readily available or they can be calculated from the Black-Scholes option pricing formula.
The random walk model
(6.22) (6.23)
The conventional model
Also called the simple model and the historical model, the conventional model amounts to estimating the hedge ratio from historical data by employing a linear OLS regression model. Let pu and pa be the logarithms of the prices of the unhedged position and the hedging instrument respectively, such that R u = Dp u and R a = Dp a . The regression equation corresponding to equation (6.18) is
DP u,t = a + hDP A,t + e t (6.21)
in which case h is the estimated hedge ratio and the R2 of the regression measures the hedging effectiveness. Sometimes, the regression is written in levels rather than in first differences to give
P U,t = a + hP A,t + e t in which case the optimal hedge ratio is defined as h = s( P U, P A ) s 2 (PA )
The implication of equations (6.22) and (6.23) is that the objective of hedging is to minimise the variance of the price of (rather than the rate of return on) the hedged position.
The error correction model
One problem with the conventional OLS model is that equation (6.22) ignores short-run dynamics, whereas equation (6.21) ignores the long-run relationship as represented by (6.22). Specifically, if pu and pa are cointegrated such that e t — /(0), then equation (6.21) is misspecified, and the correctly specified model is an error correction model of the form n n
DP u,t = a +Z b i DP U,t-i + hAPA,t +Z g i DP A,t-i + qe t-1 + x t (6.24)
i=1 i=1
where 0 is the coefficient on the error correction term, which should be significantly negative for the model to be valid. This coefficient measures the speed of adjustment to the long-run value of pu, as implied by equation (6.22). In other words, it is a measure of the speed at which deviations from the long-run value are eliminated.
Lien (1996) argues that the estimation of the hedge ratio and 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 misspecification.
Lien (1996) provides a theoretical analysis of this conjecture by assuming a cointegrating relationship of the form f t= P At - P ut. A simplified error correction model, which implies that prices adjust in response to disequilibrium, can be written as
Ap u,t = af t-1 + X 1,t (6.25)
AP A,t =-bft-1 + X 2 ,t (6.26)
A hedge ratio that minimises s 2 (Apu,t -hApA t ) is calculated as
, s(DPu,t, DPA,t If t-1) ,t t .
h = ^ ^ r" = P(X 1,t , X 2 ,t )
'
s(X
1,t
)
^
2
,t
)
(6.27)
where p(X 11 , X 2 t ) is the correlation coefficient between X 11 and X 2 t. Alternatively it can be calculated from the regression equation
Ap U,t = a + hAp A,t + gf t-1 + Z t (6.28)
If the cointegrating relationship is ignored, then the hedge ratio is calculated as
, s(ApU,t , ApA,t ) (6 29) h = 2 (6.29)
(6.30)
From equations (6.25) and (6.26), we have s(Ap U,t , Ap A,t ) = s(-bf t-1 + X 2 ,t , af t-1 + X 1,t )
= -abs 2 (f t-1) + P(X 1,t , x 2 ,t )s(x 1,t )s(X 2 ,t )
and
s 2 (ApA,t ) = s 2 (-bf t-1 + X 2,t ) = b2 s 2 (f t-1) + s 2 (X 2,t ) (6.31)
Hence the hedge ratio is measured as
-«bs 2(f t-1) + P(X 1,t , X 2 ,t )s(X 1,t )s(X 2 ,t)
^ = ~—~ ' ;; ' — (6.32)
b 2 s 2(f t-1) + s 2(X 2,t)
Obviously, there is a difference between the expressions in equation (6.27) and equation (6.32). On the basis of these two expressions, Lien (1996) concludes that an errant hedger who mistakenly omits the cointegrating relationship always undertakes a smaller than optimal position on the hedging instrument.
By using a general specification of equations (6.25) and (6.26), we have
n n
DP u,t = t-1 + Z ai DP U,t-i +Z bi dPa ,t-i +x 1,t (6.33)
i=1 i=0
nn
DP A,t =-bft-1 + Z ci DP U,t-i +Z d i DP A,t - i +X 2 ,t (6.34)
i=0 i=1
in which case the hedge ratio calculated on the basis of the correctly specified model is given by
s(DPU,t, DPA,t If t-1, DPU,t-i, DPA,t-i)
h = ■
s2 (APA,t If t-1' APU,t-i ' APA,t-i ) (6 35)
s
2(X
2,t
)
h
=
■
= P(X 1't ' X 2 ,t )
S(X 2,t )
whereas the errant hedger who does not take into account the cointegration relationship will choose a hedge ratio that is given by
s(APU f, ApAft I Apu/t-i, APA,t-i )
s2 (APA,tI APU,t-i, APA,t-i) (6 36)
_ P(X 1,t , X 2 ,t)s(X 1,t)s(X 2 ,t) -2 (f t-11 AP U,t-i, APA,t-i)
s2 (X2 ,t ) + b2 s 2 (f t-11 APU,t-i, APA,t-i )
which means that the errant hedger will undertake a smaller than optimal position on the hedging instrument, incurring losses in hedging effectiveness.
Bivariate ARCH/GARCH error correction models
A bivariate ARCH/GARCH error correction model can be used to accommodate the two problems discussed so far, that of allowing for the possibility of cointegration and for the time-varying nature of the second moments and the hedge ratio. Kroner and Sultan (1993) use a bivariate GARCH error correction model of the form
,t-1 )+x 1,t ,t-1 ) + X 2,t
t-1~ N(0, Ht )
s2 (*pU,t|Wt-1) st(apU,t, ApA,t|Wt-1)
Ht =
s t (Ap U,t , Ap A,t|W t-1) s 2^p A,t|W t-1)
Ap =
a
+
b(
p
ut-1
Ap
A,t
=
c
+
d(
p
u,t-1
X
I,t
X
2,t
W
s2(ApA,t|Wt-1 ) = «2 + b2 X2,t-1 + d 1s2(^A,t- 1|Wt-2 ) s(Ap U,t , Ap A,t|W t-1 )
ht =
s 2(Ap A,t|W t-1 )
Again, the time subscripts on the hedge ratio and the second moments imply time-variation.
The Kalman filter
A time-varying hedge ratio can be estimated by applying the Kalman filter to equation (6.18), which can be written in a general form as
R u(t) = R a (t )H(t) + u(t) (6.44)
where H(t) is a vector of time-varying parameters (hedge ratios) and u(t) is normally distributed with E[u(t)] = 0 and o2 [u(t)] = V. A common specification of the parameter variation is
H(t) = AH(t-1) + w(t) (6.45)
0
"
11
0
0
0
a
22
0
an-
1,n-1
0
0
0
0
ann
A = T 22 _ 0 (6.46)
such that 0 < an <1, i = 1, 2, ..., n. This restriction on the elements of A is necessary to guarantee the stability of the generating process. The random walk model can be obtained by setting A = I, i.e. a^ = 1 "i. The estimation of the vector H(t) can be carried out recursively using the Kalman filter technique. This is because equations (6.44) and (6.45) form the state space representation of the system, in which equation (6.44) is the measurement equation and equation (6.45) is the transition equation, which allows for systematically varying
parameters. The state of the system H(t) is not directly observable, but can be
observed through Ru(t). Let
E[H(t) |RU(t-1)] = H(t|t-1) (6.47) and
E[H(t) - H(t 11 -1)][H(t) - H(t It -1]' = s 2 (t 11 -1) (6.48) The Kalman filter equations are given by
H(t It-1) = AH(t-111 -1) (6.49)
s 2 (t 11 -1) = As 2 (t-111 - 1)A' + W (6.50)
G(t) = s 2 (t 11 -1)RA (t)[Ra (t)s2 (t 11 -1)RA (t) + V]-1 (6.51)
H(t It) = H(t It -1) + G(t) [R U (t) - R A (t )H(t 11 -1)] (6.52)
s 2 (tIt) = V(t11-1)-G(t)RA(t)RA(t)s 2 (t11-1) (6.53)
The initial conditions are given by H(0I0) = H(0) and s 2 (0I0) = s 2 (0). This system of equations tells us that the optimal estimator of H(t) at time £, H(t 11) is represented by a linear combination of the previous estimator, H(t It -1) and the current observation, Ru(t). Equation (6.52) shows the recursive nature of the computation. For more detail, see Cuthbertson et al. (1992).
Nonlinear models
Broll et al. (2001) suggest that the hedge ratio should be estimated from a nonlinear model of the form
P U,t = a + hP A,t + gP A ,t + £ t (6.54)
in which case the error correction term (e t = p u t - a - hP a t - gp A t ) becomes nonlinear. By applying this model to the futures market they found (i) prevalent nonlinearities in the relationship; (ii) that the relationship is likely to be convex (positive g) rather than concave (negative g); and (iii) that the order of magnitude of the nonlinear component is by and large relatively small. They concluded that the firm should export more (less) and adopt an over (under) hedge in an unbiased currency market if the spot-futures relationship is convex (concave).
The following is a more general treatment of nonlinearity, which we will undertake by referring to equation (6.22), that is p u t = a + hp a t + e t. There are two approaches to nonlinearity in a relationship like equation (6.22): by postulating either a nonlinear attractor, or a nonlinear adjustment to a linear attractor. The first approach is discussed in Granger (1991). Let us define two nonlinear functions, /1(pu) and f2(PA), both of which represent long-memory series. If we define zt as a short-memory series given by
zt = f1(P U,t) - X/2(P A,t) (6.55) then the equation of the corresponding nonlinear attractor is
f 1(P U,t ) = 1f2( P A,t) (6.56)
Hallman (1989) shows how the functions f\(pu) andf2(PA) are estimated.
Nonlinearity in the error correction model is discussed in Escribano (1987), and the procedure is applied to a model of the demand for money in Hendry and Ericcson (1991). Nonlinearity in this case is captured by a polynomial in the error correction term. Thus the error correction model corresponding to equation (6.22) is
k
Dp u,t = A( L)Dp u,t-1 + B(L)ApA,t +Z g i e t-i + X t (6.57)
i=1
where A(L) and B(L) are lag polynomials. Hendry and Ericcson (1991) suggest that a polynomial of degree three in the error correction term is sufficient to capture the adjustment process.
Multicurrency hedge ratios
Consider a model in which the percentage change in the base currency value of a firm, Vx, is influenced by changes in the exchange rates of the base currency against other currencies. This model can be represented by the regression equation
n
Vx = £ a & x/y{ ) (6.58)
i=1
In this case, the hedge ratio corresponding to each exchange rate is equal to the regression coefficient on that exchange rate, which means that hi = ai for all i. Suppose, for example, that Vx is the US dollar value of a firm and that this value is affected by changes in the exchange rates of the US dollar against three currencies: Canadian dollar, Japanese yen and pound. Equation (6.58) can be rewritten for this special case as
VUSD = h 1S(USD/CAD) + h2 S(USD/JPY) + h 3 S(USD/GBP) (6.59)
where h1, h2 and h3 are the hedge ratios corresponding to the three currencies respectively. If, for example, h1 >0, h2 < 0 and h3 > 0, then hedging one US dollar of value requires taking a short position of h1 Canadian dollars, a long position of h2 yens and a long position of h3 pounds. Recall from Chapter 4 that if
VCAD = hi S(CAD/USD) + h2 S(CAD/JPY) + h'3S(CAD/GBP) (6.60)
then h 2 = h 2, h 3 = h 3 and h \ = 1 - ( h 2 + h 3 ). The hedge ratios applicable to the Canadian dollar value of the firm are h 1, h2 and h 3 , which correspond to the US dollar, Japanese yen and pound respectively.
6.4 EVALUATING THE EFFECTIVENESS OF HEDGING
There are three approaches to the measurement of the effectiveness of hedging. These approaches will be discussed in turn.
Correlation, variance ratio and variance reduction
To start with, we have seen that perfect correlation between the prices of the asset to be hedged and the hedging instrument leads to a perfect hedge if the hedge ratio is one. If prices are less than perfectly correlated, then the optimal hedge ratio will be different from one and the hedge will be less than perfect. Hence, hedging effectiveness is indicated by the degree of correlation of the prices. One measure of the effectiveness of hedging is, therefore, the coefficient of determination (R2) of a regression of the levels or the first differences of prices, as represented by equations (6.22) and (6.21) respectively. A perfect hedge would be indicated when the value of the coefficient of determination is one. This would be obtained, of course, when prices are perfectly positively or negatively correlated.
Hedging effectiveness can be measured by the variance of the rate of return on the hedged position compared with the variance of the rate of return on the unhedged position. To evaluate the performance of hedging resulting from using different estimated ratios (and even from using different hedging instruments) the criterion that is used is the variance of the rate of return on the hedged position, s 2 (Rh ). The smaller the variance the more effective the hedge is. Of course, a mere comparison of the numerical values of the variances is inadequate and a formal test of the equality between the two variances must be conducted.
Consider first the effectiveness of a hedge against the alternative of leaving the underlying position unhedged. In this case we test the equality of the variance of the hedged position and that of the unhedged position. The null hypothesis is
H 0: s 2( Rh) = s 2(Ru ) (6.61)
against the alternative
H1: s 2( Rh) * s 2(Ru ) (6.62)
If H1 :s (Ru ) > s (Rh ), then the null is rejected if
VR
=
s
(Ru)
>
F(n-1,
n-1) (6.63)
s 2 (Rh)
where VR is the variance ratio and n is the sample size. This test can be complemented by calculating the variance reduction, which is calculated as
1 s 2(RH) s 2(Ru - hR A )
VD = 1 -— = 1 = ^ ^ (6.64)
VR s 2(Ru) s 2(Ru) V '
The test can be conducted to compare the effectiveness of two hedging positions resulting from the use of different hedge ratios or different hedging instruments. In this case, the null hypothesis becomes H0:s 2 (Rh 1) = s2(Rh 2 ), where s 2 (Rh 1 ) and s 2 (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, variance ratio and variance reduction are related to the correlation coefficient between the rates of return. The correlation coefficient is defined as
= s(R u, R A ) (6.65)
H s(Ru)s(R a) v '
which gives
s(Ru , Ra) = ps( Ru)s(Ra) (6.66)
(6.67)
h = ps( Ru)s(R A) = r s( Ru) s 2(Ra) I s( Ra) _
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 = p.
To derive the relationship between the variance ratio and the correlation coefficient, we rewrite equation (6.63) as
VR
=
^ 2
S22(Ru
) (6.68)
s 2 (Ru) + h2 s 2 (Ra) -2hs(Ru , Ra) By substituting equations (6.66) and (6.67) into equation (6.68), we get
s 2 (Ru)
VR = -
s 2 (Ru) + P2 s 2 (Ru)s 2 (Ra) -2[(ps(Ru)s(Ra ))/(s2 (Ra ))]Ps(Ru)s(Ra)
(6.69)
which can be simplified to obtain 1
VR = -4- (6.70)
1 - P2
(6.71)
> 0
— (VR) =- „ n dp (1 - p2)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
VD = 1 -— = p2 VR
which shows the variance reduction is equivalent to the coefficient of determination 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 1
VR = 2 2 1 2 (6.73)
1 - h 2 [(s 2(Ra )) / (s 2 (Ru ))]
(6.72)
2
(6.74)
— (VR) = - dh {1 - h 2 [(s 2 (R a))/( s 2 (R u ))]} 2
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
(6.75)
s2(Ru)
1
VD = 1 -— = h 2 VR
which gives
A(VD)
= 2h
dh
s
2(RA)
s
2
(Ru).
(6.76)
>0
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, particularly 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
h 0: m(U) = m(H) (6.77)
whereas the alternative hypothesis is written as
H1: m(U) * m(H) (6.78)
The null hypothesis is rejected if
EUzEH [Z > t(2n -2) (6.79)
s V 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, n is the sample size, t(2n - 2) is the critical value of the t distribution with 2n - 1 degrees of freedom, and
s = n[s2(U) + s2(H)] (6.80)
V 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
X (U) = - X KtSt+ 1 (6.81)
nt=1
1 n _
X (H) = - X KtSt (6.82)
nt=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 associated 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 correction 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(M t -M) + ut (6.83)
where Dt = w\t - W2t, Mt = w\t + W2t, M is the mean of M, w\t is the forecasting error at time t of 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 negative, the observations of the series must be multiplied by -1 before running the regression. The estimates of the intercept term (ao ) and the slope (a j) are used to test the statistical difference between the MSEs of two different models. If the estimates of a o and a 1 are both positive, then a Wlad test of the joint hypothesis H 0: a 0 = a 1 = 0 is appropriate. However, if one of the estimates is negative and statistically significant then the test is inconclusive. But if the estimate is negative and statistically insignificant the test remains conclusive, in which case significance is determined by the upper-tail of the t-test on the positive coefficient estimate.
The empirical evidence
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 mechanism, as shown by Lien (1996). However, Moosa (2002e) examined the sensitivity 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 effectiveness 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 ineffective 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%.
6.5 STATIC AND DYNAMIC HEDGING
A static hedge involves a constant hedge ratio, whereas a dynamic hedge involves a time-varying hedge ratio. Dynamic hedging may involve changing other aspects of the specification of the hedge. In general, we could say that dynamic hedging entails the adjustment of the hedge with the passage of time.
By referring to equations (6.1) and (6.2) we can see that a perfect hedge is obtained with a hedge ratio of one only if Ru = Ra. If at a particular point in time, t, we have Ru,t = RA,t, then ht = 1. However, if things change such that
RU,t+ 1 * RA,t+ 1,then ht+1 * 1
There are several reasons why it may be necessary to adjust the hedge over time. The first of these reasons is that the hedge may require rolling over the position in the hedging instrument, and this particularly applies to long-term exposure. Suppose that the maturity of the exposure is five years whereas the maturity of the available hedging instruments is one year. In this case a position on the instrument is taken and rolled over to the second year, and so on, until the exposure matures at the end of the fifth year.
The second reason is that it is most likely that the exposure itself changes over time. A change in foreign sales means changing receivables, and the same can happen to payables. Also, the firm's attitude towards risk or its ability to withstand adverse developments may change. However, this may boil down to using hedging instruments for speculative purpose if, for example, a firm sees profit/loss on a position involving a hedge, in which case the firm may decide to realise the profit or wind down the position to cut losses.
The third reason is that management may, with the passage of time, acquire new information about the effectiveness of a hedge or about the appropriate hedge ratio. For example, the delta of an option is the change in the price of an option relative to the change in the underlying exchange rate, which means that it is the hedge ratio. The delta of an option changes with the value of the underlying exchange rate, which necessarily implies a changing hedge ratio. Hence the hedge ratio should be adjusted with changes in the delta of the option.
The fourth reason is that it is possible for the structure of the underlying asset to be changing. Suppose, for example, that a firm wants to hedge exposure arising from changes in the effective exchange rate of the base currency against a group of other currencies. The effective exchange rate is normally calculated as a weighted average of the bilateral exchange rate relatives against the currencies of the major trading partners. In this case the structure of the underlying asset changes because the currencies and/or their weights are revised every now and then. Kavussanos and Nomikos (2000) study the effectiveness of futures contracts whose underlying asset is an index when the structure of the index changes. They concluded that the effectiveness of the BFI contract (a freight futures contract), which is a hedging instrument that is traded on the BIFFEX (Baltic International Freight Futures Exchange) has increased as a result of making the index more homogenous.
Adjusting the hedge is not a free operation, as it involves transaction costs. However, it remains the case that monitoring requires an ongoing commitment. Kroner and Sultan (1993) suggest that investors will adjust the hedge only if the benefits of the operation offset its costs. Assuming that k is the percentage return that is lost due to transaction costs every time the hedge is adjusted, the return on the hedged position if there is an adjustment is
R H,t+1 = RU,t+1 - htR A,t+1 - k (6.84)
whereas the return on the hedged position without a readjustment is
R H,t+1 = RU,t+1 - h't RA,t+1 (6.85)
where h't is the hedge ratio from the most recent adjustment. This means that the expected return is -k if there is adjustment and 0 otherwise. The conditional variances with and without adjustment are given respectively by
s 2 (RH,t+1) = s 2+1(RU,t+1) -2h ts t+1(RH,t+1, RU,t+1 ) (6 86)
+ h 2 s 2+ 1(RA ,t+1)
s 2 (R H,t+1) = s 2+1( RU,t+1) - 2h'ts t+1( RH,t+1, RU,t+1 )
(6.87)
+ h'2 s 2+1( RA,t+1)
A mean-variance expected utility maximiser with an expected utility function of the form EU(Rh t+1) = E(Rh t+1) -gs 2 (Rh t+1 ), where y is the degree of risk aversion, will adjust the hedge if and only if
-k - g[s2+ 1(RU,t+1) - 2h ts t+ 1(RH,t+1, RU,t+1) + h 2 s 1(RA,t+1 )] (6 88)
> s (2+1( RU,t+1) - 2h't s t+ 1(RH,t+1, R U ,t+1 ) + h s 1
(RA
In their study, Kroner and Sultan show that an investor in the pound would have adjusted the hedge 18 times during the 263 week period they examined.
6.6 AN ILLUSTRATION USING CROSS CURRENCY HEDGING
In cross currency hedging, both the position to be hedged and the hedging instruments are currencies. The position to be hedged is a spot position on a currency in which payables or receivables are denominated. The hedging instrument is a spot or a forward position on another currency. For the purpose of this illustration we will use a spot position as the hedging instrument.
If x is the base currency, y is the currency denominating assets or liabilities and z is the currency to be used as the hedging instrument, then Ru = S(x/y), RA = S( x/z)and R H = S( x/y ) - hS( x/z). The hedge ratio is then calculated from the regression equation
S( x/y) = a + hS( x/z) (6.89)
which gives
h = a[S( x/y),S( x/z)] (6.90)
a 2 [S( x/z)]
Notice now what happens if the roles of currencies y and z are reversed by taking currency z to be the currency in which payables or receivables are denominated and y to be the hedging instrument. In this case the hedge ratio is calculated from the regression equation
S( x/z) = a' + h S( x/y) (6.91)
which gives
h , = s[S( x/y),S( x/z)]
a 2 [S(x/y)] '
By combining equations (6.92) and (6.90), we obtain
h = hs2[S( x/z)] (6.93)
a 2 [S(x/y)]
In this exercise quarterly data on the exchange rates of nine currencies against the US dollar are used, covering the period 1991:1-2000:4. In the first instance the US dollar is taken to be the base currency, and all possible combinations for the exposure currency and the hedge are tried. The estimated hedge ratios and (in parentheses) the corresponding coefficient of determination (variance reduction) are reported in Tables 6.1 and 6.2. From these two tables we can see that the best available cross hedge, when the base currency is the US dollar, involves the Danish and Swiss currencies and those involving the Norwegian and Danish currencies. Irrespective of which currency is the exposure currency and which is the hedging instrument, the same hedging effectiveness is produced (the same R2). Notice, however, that the hedge ratio depends on which currency is the exposure and which one is the hedging instrument as demonstrated by equation (6.93). For example, if the DKK is the exposure currency and the CHF is the hedging currency, then the hedge ratio is 1.131. If, the roles of the two currencies are reversed, the hedge ratio becomes 0.718. In Tables 6.3 and 6.4 the corresponding results are reported when the base currency is the Swedish krona rather than the US dollar.
The results presented in Tables 6.1-6.4 are used to represent graphically the (stochastic version of the) relationships derived in Section 6.4. Figures 6.1-6.3 are scatter diagrams (with best-fit lines) for the relationship between the hedge ratio and the correlation coefficient as described by equation (6.67). It is obvious that the relationship is linear and positive as predicted by the equation. Figures 6.4 and 6.5 relate the variance ratio and variance reduction to the correlation coefficient. The relationship in both cases positive and nonlinear. Figures 6.6-6.7 are scatter diagrams of the variance ratio and variance reduction on the hedge ratio. These graphs are the empirical counterparts of the theoretical relationships represented by equations (6.73) and (6.75) respectively.
Figures 6.8 and 6.9 illustrate the comparative effectiveness of forward hedging and cross hedging respectively by plotting the rates of return on the unhedged and hedged positions. In Figure 6.8(a) the base currency is the US dollar, whereas the exposure currency is the pound. A hedged position formed by taking an opposite position on a forward contract produces a rather stable rate of return on the hedged position. Figure 6.8(b) shows the same when the exposure currency is the Canadian dollar and Figure 6.8(c) shows the effectiveness of forward hedging when the base currency is the pound and the exposure currency is the Canadian dollar. Now, compare this performance of forward hedging with the performance of cross currency hedging, as shown in Figure 6.9. By using three different currency combinations, we can see that cross hedging is not as effective as forward hedging. In fact, it is worse than remaining unhedged when the pound is used to hedge a Canadian dollar position when the base currency is the US dollar (Figure 6.9(b)).
Table 6.5 reports the variances of the rates of return on the hedged and unhedged positions for the six cases shown in Figures 6.8 and 6.9. It is obvious that forward hedging is more powerful and that, in one particular case, cross hedging leads to inferior results compared to what is obtained under the no- hedge decision.
TABLE
6.1
Cross currency hedge ratios: 1 (USD is the base currency).3
AUD
CAD
CHF
DKK
GBP
JPY
NOK
NZD
SEK
AUD
0.883b
-0.113
-0.023
0.055
0.018
0.041
0.672b
0.074
(0.176)
(0.025)
(0.001)
(0.003)
(0.001)
(0.002)
(0.482)
(0.013)
CAD
-
-0.074
-0.061
0.046
-0.010
0.017
0.076
0.026
-
(0.048)
(0.021)
(0.010)
(0.001)
(0.001)
(0.027)
(0.007)
CHF
-
1.131b
0.785b
0.409b
1.102b
0.314
0.627b
-
(0.813)
(0.344)
(0.206)
(0.664)
(0.053)
(0.462)
DKK
-
0.658b
0.312b
0.912b
0.336b
0.526b
-
(0.381)
(0.188)
(0.716)
(0.096)
(0.511)
GBP
-
0.152
(0.051)
0.767b
(0.576)
0.288
(0.080)
0.515b
(0.558)
JPY
-
0.427
(0.081)
0.520b
(0.119)
0.168
(0.027)
NOK
-
0.338b
(0.112)
0.547b
(0.644)
NZD
-
0.151
(0.049)
SEK
-
a
NZD
SEK
TABLE 6.2 Cross currency hedge ratios: 2 (USD is |
the base |
curren |
|||||
|
AUD |
CAD |
CHF |
DKK |
GBP |
JPY |
NOK |
AUD |
- |
|
|
|
|
|
|
CAD |
0.199b |
- |
|
|
|
|
|
|
(0.176) |
- |
|
|
|
|
|
CHF |
-0.224 |
-0.651 |
- |
|
|
|
|
|
(0.025) |
(0.048) |
- |
|
|
|
|
DKK |
-0.029 |
-0.340 |
0.718b |
- |
|
|
|
|
(0.001) |
(0.021) |
(0.813) |
- |
|
|
|
GBP |
0.061 |
0.224 |
0.438b |
0.579b |
- |
|
|
|
(0.003) |
(0.010) |
(0.344) |
(0.381) |
- |
|
|
JPY |
0.044 |
-0.109 |
0.502b |
0.603b |
0.335 |
- |
|
|
(0.001) |
(0.001) |
(0.206) |
(0.188) |
(0.051) |
- |
|
NOK |
0.044 |
0.081 |
0.602b |
0.785b |
0.752b |
0.190 |
- |
|
(0.002) |
(0.001) |
(0.663) |
(0.716) |
(0.576) |
(0.081) |
- |
NZD |
0.717b |
0.360 |
0.169 |
0.286 |
0.278 |
0.229b |
0.334b |
|
(0.482) |
(0.027) |
(0.053) |
(0.096) |
(0.080) |
(0.119) |
(0.113) |
SEK |
0.171 |
0.265 |
0.736b |
0.973b |
1.084b |
0.161 |
1.177b |
|
(0.012) |
(0.007) |
(0.462) |
(0.511) |
(0.558) |
(0.027) |
(0.644) |
0.329 (0.049)
aRows represent the hedging currencies, whereas columns represent the exposure currencies. Hedge ratios are calculated from the regression Sj = a + hSj, where i and j represent rows and columns respectively. The coefficient of determination, which measures hedging effectiveness, is placed in parentheses. bStatistically significant at the 5% level.
aRows represent the exposure currencies, whereas columns represent the hedging currencies. Hedge ratios are calculated from the regression Sj = a + hSj, where i and j represent rows and columns respectively. The coefficient of determination, which measures hedging effectiveness, is placed in parentheses. bStatistically significant at the 5% level.
6.6 AN ILLUSTRATION USING |
CROSS CURRENCY HEDGING |
177 |
||||
TABLE 6.3 Cross currency hedge ratios: 3 (SEK is |
the base |
currency) |
a |
|
||
AUD CAD CHF |
DKK |
GBP |
JPY |
NOK |
NZD |
SEK |
AUD - 0.936b 0.316 |
0.754b |
0.979b |
0.484b |
1.014b |
0.927b |
- |
(0.740) (0.043) |
(0.200) |
(0.305) |
(0.285) |
(0.272) |
(0.802) |
- |
CAD - 0.448b |
0.788b |
1.061b |
0.499b |
1.095b |
0.761b |
- |
- (0.103) |
(0.285) |
(0.425) |
(0.362) |
(0.376) |
(0.640) |
- |
CHF - |
0.938b |
0.533b |
0.363b |
0.922b |
0.278b |
- |
- |
(0.715) |
(0.209) |
(0.372) |
(0.520) |
(0.167) |
- |
DKK |
- |
0.619b |
0.357b |
0.954b |
0.353b |
- |
|
- |
(0.347) |
(0.444) |
(0.686) |
(0.332) |
- |
GBP |
|
- |
0.274b |
0.796b |
0.362b |
- |
|
|
- |
(0.287) |
(0.527) |
(0.384) |
- |
JPY |
|
|
- |
1.256 |
0.745b |
- |
|
|
|
- |
(0.341) |
(0.424) |
- |
NOK |
|
|
|
- |
0.334b |
- |
|
|
|
|
- |
(0.394) |
- |
NZD |
|
|
|
|
- |
- |
SEK |
|
|
|
|
|
- |
aRows represent the exposure currencies, whereas columns represent the hedging currencies. Hedge ratios are calculated from the regression Sj = a + hSj, where i and j represent rows and columns respectively. The coefficient of determination, which measures hedging effectiveness, is placed in parentheses. b Statistically significant at the 5% level.
TABLE 6.4 Cross currency hedge ratios: 4 (SEK is the base currency).4
AUD CAD CHF DKK GBP JPY NOK NZD SEK
AUD -
CAD 0.790b
(0.740) CHF 0.137
(0.043) DKK 0.265b
(0.200) GBP 0.312b
(0.306) JPY 0.591b
(0.285) NOK 0.268b
0.230b |
- |
|
(0.103) |
- |
|
0.328b |
0.762b |
- |
(0.259) |
(0.715) |
- |
0.401b |
0.392b |
0.562b |
(0.425) |
(0.209) |
(0.348) |
0.723b |
1.025b |
1.244b |
(0.361) |
(0.372) |
(0.444) |
0.343b |
0.564b |
0.719b |
(0.378) |
(0.520) |
(0.687) |
0.841b |
0.560b |
0.939b |
(0.640) |
(0.167) |
(0.332) |
1.049b |
- |
|
(0.287) |
- |
|
0.662b |
0.272b |
- |
(0.528) |
(0.342) |
- |
1.060b |
0.569b |
1.178b |
(0.384) |
(0.424) |
(0.394) |
(0.802) SEK -
Correlation coefficient
FIGURE 6.1 The hedge ratio as a function of the correlation coefficient (all data).
Correlation
coefficient
FIGURE
6.2
The hedge ratio as a function of the correlation coefficient (USD is
the base currency).
Correlation
coefficient
FIGURE
6.3
The hedge ratio as a function of the correlation coefficient (SEK is
the base currency).
0.0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Correlation
coefficient
FIGURE
6.4
The variance ratio as a function of the correlation coefficient (all
data).
0.0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Correlation
coefficient
FIGURE
6.5
Variance reduction as a function of the correlation coefficient (all
data).