International Finance and Accounting Handbook

Simplifying

In the example

The last term (the cross-product term) will be much smaller than the other two terms,
so that return to the U.S. investor is approximately the return of the security in its
home market plus the exchange gain or loss. Using this approximation, we have the
following expressions for expected return and standard deviation of return on a for-
eign security.

Expected return

Standard deviation of return

As will be very clear when we examine real data, the standard deviation of the re-
turn on foreign securities (US) is much less than the sum of the standard deviation
of the return on the security in its home country (H) plus the standard deviation of
the exchange gains and losses (x). This relationship results from two factors. First,
there is very low correlation between exchange gains (or losses) and returns in a
country (and therefore the last term Hxis close to zero). Second, squaring the stan-
dard deviations, adding them, and then taking the square root of the sum is less than
adding them directly. To see this, let

then

and

Thus, the standard deviation of the return expressed in dollars is considerably less
than the sum of the standard deviation of the exchange gains and losses and the stan-
dard deviation of the return on the security in its home currency. The reader should
be conscious of this difference in the tables that follow.
Having developed some preliminary relationships it is useful to examine some ac-
tual data on risk and return.

sUS
0.18

s^2 US
0.10^2 0.15^2

sHx
 0 1 to make the covariance zero 2

sH
0.15

sx
0.10

sUS
 3 s^2 xs^2 H 2 sHx 41 >^2

RUS
RxRH

0.200.1250.025

0.10
0.200.125 1 0.20 2  1 0.125 2

RUS
RxRHRxRH

11.3 CALCULATING THE RETURN ON FOREIGN INVESTMENTS 11 • 5