720 CHAPTER 19. SETTING THE COST OF INTERNATIONAL CAPITAL
a relation that comes in good stead to derive theCAPMin the next subsection.
Using a variety of proxies for the market portfolio and a variety of methodologies,
[19.22] has been used to estimate theusaverage risk aversion. The estimates vary,
but the consensus in long-term tests is thatλexceeds unity. Also this result will
come in handy later.
19.2.5 The Market Portfolio as the Benchmark
Let us now go from an individual investor’s portfolio to the market portfolio—
defined as the aggregate asset holdings of all investors in a particular group. The
group typically considered in the standardCAPMis composed of all investors in
the economy. What exactly “the” economy corresponds to in practice—a country?
a region?—is left vague, but, crucially, this set of investors is assumed to have
homogeneous opportunities, that is, equal access to the same list of assets, and
homogeneous expectations, that is, equal perceptions about the return characteristics
of the assets.
The effect of these homogeneity assumptions is that all of the investors agree
about the composition of the tangency portfolio. If each investor holds the risk-free
asset plus the same tangency portfolio, then also the aggregate portfolio must be
a combination of the risk-free asset plus that very same tangency portfolio. But
any such combination is efficient. Therefore, for the market portfolio (denoted
by subscriptm), the efficiency condition Equation [19.1] must hold, withλmnow
defined as the market’s risk aversion (which can be shown to be a kind of weighted
average of the individuals’ risk aversions):
E( ̃rj−r)
cov( ̃rj, ̃rm)
=λm,for all risky assetsj=1, ... N. (19.23)Although Equation [19.23] is not yet written in the standardCAPMform, this equa-
tion already is an embryonic capital asset pricing model because it tells us what
the expected excess return should be as a function of the asset’s covariance risk in
the market portfolio. To implement the model, we need to know the relative risk
aversion for the average investor. But we just found a way to infer this: just use
[19.22] to identify the market’s relative risk aversion. This leads us straight to the
CAPM:
E( ̃rj−r) =λmcov( ̃rj, ̃rm) =
E( ̃rm−r)
var( ̃rm)cov( ̃rj, ̃rm)= βj,mE( ̃rm−r), (19.24)In Equation [19.24],βj,m= cov( ̃rj, ̃rm)/var( ̃rm) is the asset’s rescaled covariance
risk, or the asset’s beta. The advantage of rescaling the covariance risk is thatβj,mis
also the slope coefficient from the so-calledmarket model, the regression of the return
from asset j, on the return from the market portfolio, ̃rj =αj,m+βj,m ̃rm+j,m.