The Mathematics of Financial Modelingand Investment Management

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)