International Finance: Putting Theory Into Practice

716 CHAPTER 19. SETTING THE COST OF INTERNATIONAL CAPITAL

The first formula is pretty obvious. To interpret the second one, it helps to derive
it in two steps, as follows:^5

var( ̃rp) = cov( ̃rp, ̃rp) = cov(

∑N

j=1

xj ̃rj, ̃rp) =

∑N

j=1

xjcov( ̃rj,r ̃p), (19.14)

where cov( ̃rj, ̃rp) = cov( ̃rj,

∑N

k=1

xkr ̃k) =

∑N

k=1

xkcov( ̃rj, ̃rk). (19.15)

This tells you that the portfolio variance is a weighted average of each asset’s co-
variance with the entire portfolio; and each of these assets’ portfolio covariances
is, in turn, a weighted average of the asset’s covariance with all components of the
portfolio.

Example 19.3
We compute the portfolio expected excess return, the assets’ covariances with the
portfolio return, and the portfolio variance when the risky assets’ weights are 0.50
and 0.40 (implyingx 0 = 0.10):

E( ̃rj−r) cov( ̃rj, ̃r 1 ) cov( ̃rj, ̃r 2 )

1 0.200 0.16 0.05

2 0.122 0.05 0.09

E( ̃rp−r) = 0. 50 × 0 .200 + 0. 40 × 0 .122 = 0. 1488

cov( ̃r 1 ,r ̃p) = 0. 50 × 0 .160 + 0. 40 × 0 .050 = 0. 1000

cov( ̃r 2 ,r ̃p) = 0. 50 × 0 .050 + 0. 40 × 0 .090 = 0. 0610

⇒cov( ̃rp,r ̃p) = 0. 50 × 0 .100 + 0. 40 × 0 .061 = 0. 0744

How do these numbers change when the portfolio weights are being tweaked?
First look at a two-risky-assets example and see how mean and variance are affected
by a small change in the weight of asset 1 (implicitly matched by a small offsetting
change in the weight for the risk-free bond, asset zero):

E( ̃rp−r) = x 1 E( ̃r 1 −r) +x 2 E( ̃r 2 −r);

⇒

∂E( ̃rp−r)

∂x 1

= E( ̃r 1 −r), (19.16)

(^5) We use the fact that, inside a variance, risk-free returns added or subtracted play no role:
var(r+
P
xj( ̃rj−r)) = var(
P
xj ̃rj).