3.7 Uncovered arbitrage when the cross rates are stable

If the interest rate on currency x is lower than the interest rate on currency y, then profitable uncovered arbitrage can be executed by going long on y and short on x, provided that y does not depreciate by more than the interest rate differential before unwinding the positions. Recall that the uncovered margin in this case would be p = iy -i_x + S, which means that if S = 0 the uncovered margin will be equal to the interest rate differential. Of course, it is unlikely that S = 0 under a system of flexible exchange rates, so we could reformulate the proposition to the following: p »iy -i_x if S » 0.

It may be possible to find a pair of currencies whose cross exchange rate is so stable that the condition S » 0 may be satisfied. Since major floating curren­cies have a general tendency to move against the US dollar in the same direc­tion, the cross exchange rate between two currencies tends to be stable if the two currencies move proportionately against the dollar. The proposition that is put forward here is that if the exchange rates of two currencies, x and y, against a numeraire, z, are highly correlated, then the cross exchange rate between x and y tends to be stable. The following is a proof of this proposition.

At points in time t and t +1, we have

St (x/y) = ^ (3.31) St (y/z)

St+1 (x/y) = (3.32)

^St+1⁽ y^/z)

Therefore

S(x/y) = ^St+1^(x/y) -1 = ^[St+ 1^{(x/z)] / [S}t+ 1^(y/z)] -1 (3.33)

^V " St (x/y) [St (x/z)]/[St (y/z)] ^V '

which gives

S( x/y) = ¹ + ^{S( x/z)} -1 = 0 (3.34)

1 + S y/z)

because perfect correlation implies that S( x/z) = S( y/z). In a less extreme case, high correlation implies that S( x/z) » S(y/z), and hence S( x/y) » 0.

An illustration

In this example, the numeraire currency is the US dollar, and there are nine currencies under consideration. Table 3.1 is a correlation matrix of the exchange rates against the US dollar, calculated from quarterly data covering the period 1990:1-2000:4. In general it mostly shows high and significant posi­tive correlations among the exchange rates, except some negative correlations involving the Japanese yen. The indication here is that, with the exception of the yen, currencies have been moving against the dollar in the same direction. The highest correlation is found between the exchange rates of the Norwegian krone and the Danish kroner (which is not surprising). Other high correla­tions are found between the Australian dollar and each of the following

TABLE 3.1 Correlations of the US dollar exchange rates.

AUD	CAD	CHF	DKK	GBP	JPY	NOK	NZD	SEK
AUD 1.00	0.68	0.65	0.78	0.36	-0.03	0.86	0.86	0.79
CAD	1.00	0.23	0.47	0.48	-0.39	0.71	0.28	0.82
CHF		1.00	0.92	0.22	0.29	0.81	0.77	0.60
DKK			1.00	0.37	0.08	0.94	0.77	0.77
GBP				1.00	-0.62	0.52	0.18	0.70
JPY					1.00	-0.10	0.17	-0.36
NOK						1.00	0.73	0.91
NZD							1.00	0.53
SEK								1.00

TABLE 3.2 Interest rate differentials (end of 2000).^a

	AUD	CAD	CHF	DKK	GBP	JPY	NOK	NZD	SEK
AUD	0.00	-0.46	-2.84	-0.87	-0.32	-5.62	1.22	0.51	-2.13
CAD		0.00	-2.38	-0.41	0.14	-5.16	1.68	0.97	-1.67
CHF			0.00	1.97	2.52	-2.78	4.06	3.35	0.71
DKK				0.00	0.55	-4.75	2.09	1.38	-1.26
GBP					0.00	-5.30	1.54	0.83	-1.81
JPY						0.00	6.84	6.13	3.49
NOK							0.00	-0.71	-3.35
NZD								0.00	-2.64
SEK									0.00

^aEach cell represents the interest rate on the currency in the row minus the interest rate on the currency in the column. For example, -5.62 is the interest rate on the yen minus the interest rate on the Australian dollar.

currencies: Norwegian krone, New Zealand dollar, Swedish krona and Danish kroner.

High correlations are a necessary but not sufficient condition for executing a profitable uncovered arbitrage operation along the lines suggested in this section. The necessary condition is a wide interest rate differential. Table 3.2 is a matrix of (three-month) interest rate differentials as at the end of 2000.

By combining the information in the two tables we can pick some currency pairs that are suitable for this operation. These are shown in Table 3.3. For example, by going short on the Swiss franc and long on the Norwegian krone, an arbitrager can earn over four percentage points in interest rate differential. With a correlation coefficient of 0.81, it is unlikely that the Swiss franc would appreciate against the Norwegian currency by an amount that would wipe out this differential.

TABLE 3.3 Some examples of	uncovered arbitrage operations.
Operation Short position	Long position	Correlation	Interest rate
on	on	coefficient	differential
1 AUD	SEK	0.79	2.13
2 CHF	DKK	0.92	1.97
3 CHF	NOK	0.81	4.06
4 CHF	NZD	0.77	3.35
5 DKK	NOK	0.94	2.09
6 SEK	NOK	0.91	3.35