International Finance: Putting Theory Into Practice

19.2. THE SINGLE-COUNTRYCAPM 719

The portfolio in the example will be selected by an investor with relative risk
aversion equal toλ= 2. One way to detect differences in risk aversion among mean-
variance investors is to watch the proportions they invest in the risk-free asset. An
investor with a higher relative risk aversion simply allocates more of his or her wealth
to the risk-free asset, and less to the tangency portfolio of risky assets.

Example 19.5
Suppose that an investor invests half of his or her wealth in the tangency portfolio
identified in the previous example, and the remainder in the risk-free asset. That
is, the weights in portfoliop′arex 0 = 0.5 for the risk-free asset, and (x 1 = 0.2,x 2
= 0.3) for the risky assets. We can easily verify thatp′is still an efficient portfolio
and that this investor has a relative risk aversion equal to 4:

• The risks of the assets in portfoliop′are computed as follows:
  Asset 1: cov( ̃r 1 , x 1 r ̃ 1 +x 2 ̃r 2 ) = 0. 2 × 0 .04 + 0. 3 × 0 .05 = 0. 023 ,
  Asset 2: cov( ̃r 2 , x 1 r ̃ 1 +x 2 ̃r 2 ) = 0. 2 × 0 .05 + 0. 3 × 0 .09 = 0. 037.


• The excess return/risk ratios now both equal 4:
  0. 092
  0. 023

=

0. 148

0. 037

= 4. (19.20)

which implies that the portfolio is also efficient.

Thus, the investor’s relative risk aversion can be inferred from his or her portfolio
choice. Relative to the tangency portfolio chosen by an investor withλ= 2, the
more risk-averse investor withλ= 4 simply reduces the proportion invested in the
risky assets by half. This, as we notice, also halves the (covariance) risks of each
risky asset in the total portfolio. This stands to reason: if the total portfolio risk
falls, assets’ contributions to that total risk must fall too.

There is another, related, way to measure risk aversion: compute the excess-
return-to-variance ratio for the entire portfolio. This ratio produces the same num-
ber as the previous ones since it takes the same linear combination of both numer-
ators and denominators:^8

if

0. 092

0. 023

=

0. 148

0. 037

= 4 then

0. 2 × 0 .092 + 0. 3 × 0. 148

0. 2 × 0 .023 + 0. 3 × 0. 037

= 4. (19.21)

We conclude that, for efficient portfolios, the holder’s relative risk aversion can be
measured by the overall excess-return/risk ratio:

Relative risk aversion =λ=

E( ̃rp−r)

var( ̃rp),

, (19.22)

(^8) The general way to establish this is to write the efficiency condition as E( ̃rj−r) =λcov( ̃rj,r ̃p).
This impliesxjE( ̃rj−r) =λxjcov( ̃rj, ̃rp) and therefore
P
jxjE( ̃rj−r) =λ
P
jxjcov( ̃rj,r ̃p) =
λcov(
P
jxjr ̃j, ̃rp). Thus, E( ̃rp−r) =λcov( ̃rp,r ̃p) =λvar( ̃rp).