19.2. THE SINGLE-COUNTRYCAPM 717
and
var( ̃rp) = x^21 var( ̃r 1 ) + 2x 1 x 2 cov( ̃r 1 , ̃r 2 ) +x^22 var( ̃r 2 );⇒∂var( ̃rp)
∂x 1
= 2x 1 var( ̃r 1 ) + 2x 2 cov( ̃r 1 ,r ̃ 2 ),
= 2[x 1 cov( ̃r 1 ,r ̃ 1 ) +x 2 cov( ̃r 1 ,r ̃ 2 )],
= 2[cov( ̃r 1 ,x 1 r ̃ 1 ) + cov( ̃r 1 ,x 2 r ̃ 2 )],
= 2cov( ̃r 1 ,x 1 r ̃ 1 +x 2 r ̃ 2 ),
= 2cov( ̃r 1 ,r ̃p). (19.17)Similarly,∂E( ̃∂xrp 2 −r^0 )= E( ̃r 2 −r) and ∂var( ̃∂x 2 rp)= 2cov( ̃r 2 , ̃rp).
DoItYourself problem 19.3
Recompute the expected excess return and variance when, in the previous exam-
ple,x 1 is increased from 0.50 to 0.51. Check how the scaled change in the mean,
∆E/∆x 1 , is exactly the first asset’s own expected excess return. Likewise, check
how the scaled change in the variance, ∆var/∆x 1 , is about twice the first asset’s
own covariance with the portfolio return.^6
DoItYourself problem 19.4
Consider a portfolio with, initially,x 1 = 0.5 andx 2 = 0 so that var( ̃rp) = var(0.5 ̃r 1 ) =
0. 52 var( ̃r 1 ). Then increase the second weight to 0.001. Write out the change in the
variance, and check whether it is far from 2cov( ̃r 2 , ̃rp)× 0 .001.
19.2.4 Efficient Portfolios: A Review
Recall that a portfolio is efficient if it has the highest expected return among all con-
ceivable portfolios with the same variance of return. We just reviewed the probably
familiar result that any efficient portfolio is a combination of two building blocks:
the risk-free asset, and the tangency portfolio of risky assets (Figure 19.3). But
what is perhaps less obvious is how the tangency portfolio must be constructed and
what this implies for the risk-return relation. Let us consider this.
It is easily shown that, if a portfolio is to be efficient, then for each and every
asset the marginal risk-return ratio—the ratio of any asset’s marginal “good” (its
contribution to the portfolio’s expected excess return) to the asset’s marginal “bad”
(^6) For the variance, the scaled difference is not perfectly the same as the partial derivative because
the function is quadratic in the weights, not linear. (For non-linear functions, obviously, ∆y/∆x 6 =
dx/dy.) But note how the scaled change in fact equals the average of the original and the revised
covariances (0.1000 whenx 1 = 0.50, and 0.1016 whenx 1 = 0.51). In the limit, the two covariances
are so close that they become indistinguishible from their average.