19.3. THE INTERNATIONALCAPM 725
from various countries have different views on the distributions of real returns on
any given asset or portfolio.
Example 19.8
A clear example is the return on the two countries’ T-bills. Suppose that there is
no inflation. While to aus investor, thecadT-bill is one of the available risky
assets, it is risk-free to a Canadian investor. On the other hand, theusdT-bill is a
risky asset to a Canadian investor but risk-free to ausinvestor. Thus, the perceived
distribution of (real) returns depends on the nationality of the investor.
Thus, we need to derive aCAPMthat takes into account the heterogeneous view-
points of investors from various countries. In keeping with our discussion of the
standardCAPM, we initially ignore inflation. To simplify the analysis, we shall ini-
tially assume there are just two countries, theusand Canada. Once you understand
the two-country model, you can easily generalize to the case of many countries.
The problem is that the Canadian investor’s portfolio choice is based on how
each asset contributes to the variance and expected excess return on the portfolio
measured incad, while theusinvestor’s portfolio choice is based on the assets’
contributions to a portfolio whose risk and return are expressed inusd. Let, as
usual, the asterisk refers to the foreign country (say, theus);p∗refers to the portfolio
held by theusinvestor; ̃r∗jrefers to a return infcon stockj(whose nationality, if
any, we do not really need to know);r∗, unsubscripted, as usual refers to theusd
risk-free rate; and ̃r∗p∗denotes the return, inusd, on theusmarket portfoliop∗.
Then what we know about portfolio choice can be summarized as follows:^11
Canadians choosepsuch that E( ̃rj−r) =λcov( ̃rj, ̃rp), (19.26)
Americans choosep∗such that E( ̃r∗j−r∗) =λcov( ̃rj∗, ̃r∗p∗). (19.27)What, then, is the relation between expected excess returns and the world market
portfolio, which is the sum ofpandp∗? To identify that link, we have to translate
[19.27] into the same currency as [19.26], thecad. Using a trick called Ito’s Lemma
(see Technical Note 19.2), [19.27] can be translated intocadas follows:
Americans choosep∗such that E( ̃rj−r) =λcov( ̃rj,r ̃p∗)+(1−λ) cov( ̃rj, ̃s), (19.28)where ̃sis the percentage change in the exchange rate (cadperusd). What is going
on here is thatusinvestors really care about their wealth expressed inusd,Wus∗,
because the consumption prices relevant to them are (almost) constant inusdand
far less so incad. We can always re-expressWus∗ ascad-measured wealth divided
by thecad/usd exchange rate,Wus∗ = Wus/S. So people who care aboutW∗
will act as if they care about wealth incadsure enough—because, everything else
(^11) It would not have been very painful to allow for different risk aversions across countries too,
but little additional insight would have been gained, so we setλ∗=λ.