118 Mathematics for Finance
5.4 Capital Asset Pricing Model ...............................
In the days when computers where slow it was difficult to use portfolio theory.
For a market withn=1,000 traded securities the covariance matrixCwill have
n^2 =1, 000 ,000 entries. To find the efficient frontier we have to compute the
inverse matrixC−^1 , which is computationally intensive. Accurate estimation
ofCmay pose considerable problems in practice. The Capital Asset Pricing
Model (CAPM) provides a solution that is much more efficient computation-
ally, does not involve an estimate ofC, but offers a deep, even if somewhat
oversimplified, insight into some fundamental economic issues.
Within the CAPM it is assumed that every investor uses the same values of
expected returns, standard deviation and correlations for all securities, making
investment decisions based only on these values. In particular, every investor
will compute the same efficient frontier on which to select his or her portfo-
lio. However, investors may differ in their attitude to risk, selecting different
portfolios on the efficient frontier.
5.4.1 Capital Market Line ................................
Form now on we shall assume that a risk-free security is available in addition
tonrisky securities. The return on the risk-free security will be denoted byrF.
The standard deviation is of course zero for the risk-free security.
Consider a portfolio consisting of the risk-free security and a specified risky
security (possibly a portfolio of risky securities) with expected returnμ 1 and
standard deviationσ 1 >0. By Proposition 5.7 all such portfolios form a broken
line on theσ, μplane consisting of two rectilinear half-lines, see Figure 5.5. By
taking portfolios containing the risk-free security and a security withσ 1 ,μ 1
anywhere in the attainable set represented by the Markowitz bullet on the
σ, μplane, we can construct any portfolio between the two half-lines shown
in Figure 5.11. The efficient frontier of this new set of portfolios, which may
contain the risk-free security, is the upper half-line tangent to the Markowitz
bullet and passing through the point with coordinates 0,rF. According to the
assumptions of the CAPM, every rational investor will select his or her portfolio
on this half-line, called thecapital market line. This argument works as long
as the risk-free returnrFis not too high, so the upper half-line is tangent to
the bullet. (IfrFis too high, then the upper half-line will no longer be tangent
to the bullet.)
The tangency point with coordinatesσM,μMplays a special role. Every
portfolio on the capital market line can be constructed from the risk-free se-
curity and the portfolio with standard deviationσMand expected returnμM.