International Finance: Putting Theory Into Practice

19.2. THE SINGLE-COUNTRYCAPM 715

Figure 19.3:Efficient Portfolios & the Tangency Portfolio

6 x 6 E( ̃rp)

sd( ̃-rp)

0 r 0 

t

...

...

...

...

...

...

...

...

...

...

..

sd( ̃rt)

rt)1 ...................E( ̃ r

to be leftward/upward in the graph: high return, low risk. Soswill be chosen
from the left-upper risky-asset bound. Among all such portfolios, the best one is
the portfolio that rotates the halfline from (sd = 0,E =r 0 ) as far upward/leftward
as is feasible—the one with the highest Sharpe Ratio. It follows that the optimal
choice is thetangency portfolio, the one where the halfline from (sd = 0,E =r 0 )
just touches the V-curve that bounds the risky-assets risk-return set. All portfolios
on this halfline are efficient. They all are combinations of the risk-free asset and the
tangency portfolio, subscriptedt.

We now want to take a peek at the analytical solution and its implication. To
understand how the tangency portfolio can be found we need to understand first
how a small change in one of the portfolio weights affects the expected return and
the variance of the portfolio return.

19.2.3 How Portfolio Choice Affects Mean and Variance of the

Portfolio Return

We want to understand what happens if investors choose portfolios on the basis of
the mean and variance of the portfolio return. To figure out how these people think,
we need to understand how portfolio choice affects the mean and variance of the
total return. The link is, of course, Equation [19.7]: ̃rp=r+

∑N

j=1xj( ̃rj−r). From
this it follows that

E( ̃rp) = r+

∑N

j=1

xjE( ̃rj−r), (19.12)

var( ̃rp) =

∑N

j=1

xj

∑N

k=1

xkcov( ̃rj, ̃rk). (19.13)