CURRENCY ARBITRAGE
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 xis the base currency, yis the currency denominating assets or liabilities and zis the currency to be used as the hedging instrument, then RU =S x y), RA =S x z( / )and RH =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= S x y S x z
S x z s
s
[( / ), ( / )]
[( / )]
2 (6.90)
Notice now what happens if the roles of currencies y and zare reversed by taking currency z to be the currency in which payables or receivables are denominated and yto be the hedging instrument. In this case the hedge ratio is calculated from the regression equation
( / h S x y) (6.91)
S x z) =a¢ + ¢( / which gives
( / ), ( /
(6.92) s[S x y S x z)]
h¢ =
s2[S x y)]
By combining equations (6.92) and (6.90), we obtain hs2[S x z)]
h¢ = ( /
(6.93) s2[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 combi
nations 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 equa
tion. 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 reduc
tion 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).a
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 CAD
– (0.176) –
(0.025) –0.074
(0.001) –0.061
(0.003) 0.046
(0.001) –0.010
(0.002) 0.017
(0.482) 0.076
(0.013) 0.026 CHF
– (0.048) –
(0.021) 1.131b
(0.010) 0.785b
(0.001) 0.409b
(0.001) 1.102b
(0.027) 0.314
(0.007) 0.627b DKK
– (0.813)
–
(0.344) 0.658b
(0.206) 0.312b
(0.664) 0.912b
(0.053) 0.336b
(0.462) 0.526b GBP
– (0.381)
–
(0.188) 0.152
(0.716) 0.767b
(0.096) 0.288
(0.511) 0.515b JPY
– (0.051)
–
(0.576) 0.427
(0.080) 0.520b
(0.558) 0.168 NOK
– (0.081) –
(0.119) 0.338b
(0.027) 0.547b – (0.112) (0.644)
NZD – 0.151
SEK
– (0.049) – –
aRows represent the exposure currencies, whereas columns represent the hedging currencies.
Hedge ratios are calculated from the regression Si = +hSj, where i and j represent rows and columns respectively. The coefficient of determination, which measures hedging effective
ness, is placed in parentheses.
bStatistically significant at the 5% level.
a
TABLE 6.2 Cross currency hedge ratios: 2 (USD is the base currency).a
AUD CAD CHF DKK GBP JPY NOK NZD SEK
AUD –
–
CAD 0.199b –
CHF
(0.176) –0.224
–
–0.651 – DKK
(0.025) –0.029
(0.048) –0.340
–
0.718b – GBP
(0.001) 0.061
(0.021) 0.224
(0.813) 0.438b
–
0.579b – JPY
(0.003) 0.044
(0.010) –0.109
(0.344) 0.502b
(0.381) 0.603b
–
0.335 – NOK
(0.001) 0.044
(0.001) 0.081
(0.206) 0.602b
(0.188) 0.785b
(0.051) 0.752b
–
0.190 – NZD
(0.002) 0.717b
(0.001) 0.360
(0.663) 0.169
(0.716) 0.286
(0.576) 0.278
(0.081) 0.229b
–
0.334b – SEK
(0.482) 0.171
(0.027) 0.265
(0.053) 0.736b
(0.096) 0.973b
(0.080) 1.084b
(0.119) 0.161
(0.113) 1.177b
–
0.329 – (0.012) (0.007) (0.462) (0.511) (0.558) (0.027) (0.644) (0.049) –
aRows represent the hedging currencies, whereas columns represent the exposure currencies.
Hedge ratios are calculated from the regression Sj = +hSi, where i and j represent rows and columns respectively. The coefficient of determination, which measures hedging effective
ness, is placed in parentheses.
bStatistically significant at the 5% level.
a