17-Capital Asset Pricing Model Page 514 Wednesday, February 4, 2004 1:10 PM
514 The Mathematics of Financial Modeling and Investment Management
cally, we can show that the appropriate risk that investors should be com-
pensated for accepting is not the variance of an asset’s return but some
other quantity. In order to do this, let’s take a closer look at risk.
We can do this by looking at the variance of the portfolio.
The proof is as follows. The variance of a portfolio consisting of N
assets is equal to
N N
var(Rp) = ∑ ∑ wiwjcov(Ri, Rj)
i = 1 j = 1
If we substitute M (market portfolio) for p and denote by wiM and wjM,
the proportion invested in asset i and j in the market portfolio, then the
above equation can be rewritten as
N N
var(RM) = ∑ ∑ wiMwjMcov(Ri, Rj)
i = 1 j = 1
It can be demonstrated that the above equation can be expressed as follows:
var(RM)
N N
= w 1 M ∑ wjMcov(R 1 , Rj) + w 2 M ∑ wjMcov(R 2 , Rj )
j = 1 j = 1
N
+ ... + wNM ∑ wNMcov(RN, Rj)
j = 1
The covariance of asset i with the market portfolio, cov(Ri, RM), is
expressed as follows:
N
cov(Ri, RM) = ∑ wjMcov(Rj, Rj)
j = 1
Substituting the right-hand side of the left-hand side of the equation
into the prior equation, gives
var(RM)
= w1M cov(R 1 , RM) + w2M cov(R 2 , RM) +... + wNM cov(RN, RM)