714 CHAPTER 19. SETTING THE COST OF INTERNATIONAL CAPITAL
Figure 19.2:The risk-return bound with just risky assets6 E( ̃rp)sd( ̃-rp)and the same (half)line. This gives us the total picture: all feasible combinations
withx≥0 are on a halfline from (sd( ̃rp),E( ̃rp)) = (0,r 0 ) through (sd( ̃rs),E( ̃rs)).
The slope of that halfline is called theSharpe Ratio:
∀x≥0 :
E( ̃rp−r 0 )
sd( ̃rp)=
E( ̃rs−r 0 )
sd( ̃rs)
=s’s Sharpe Ratio. (19.9)Now look at a second simple case, where the portfolio consists of two imperfectly
correlated risky assets, subscripted 1 and 2. Now we have
r ̃p=x 1 r ̃ 1 + (1−x 1 ) ̃r 2 ; (19.10)⇒
E( ̃rp) = E( ̃r 1 ) +x 1 [E( ̃r 1 )−E( ̃r 2 )],sd( ̃rp) =√
x^21 var( ̃r 1 ) + 2x 1 (1−x 1 )cov( ̃r 1 ,r ̃ 2 ) + (1−x 1 )^2 var( ̃r 2 )(19.11)
From the first implication we conclude thatx 1 and expected return are still two
sides of the same thermometer. The sd function looks messier. But we immediately
see that variance is quadratic inx 1 and, therefore, in E( ̃rp) too. This means a
rotated “U”-shape-like graph (or a rounded V, if you want) opening towards the
right. Warping the risk axis by taking squareroots does not fundamentally change
the shape of the relation, as you can check using a spreadsheet. We end up with a
feasible set like in Figure 19.2. Basic textbooks will tell you that, if there are more
than two risky assets, the feasible combinations in a (std, E) space graph is still
similar.
The last step is to look atN risky assets and a risk-free one. We return to
Figure 19.1 except that the risky part of the portfolio,s, must be chosen from a
feasible set shaped like in Figure 19.2. A risk-averse mean-variance investor wants