- Portfolio Management 123
for some numbersγ>0andμ. The beta factor of the portfolio with weights
wV can, therefore, be written as
βV=Cov(KV,KM)
σ^2 M=
wMCwTV
wMCwTM=
γ(m−μu)wTV
γ(m−μu)wTM=
μV−μ
μM−μ.
To findμconsider the risk-free security, with returnrFand beta factorβF=0.
SubstitutingβF andrF forβV andμV in the above equation, we find that
μ=rF. We have proved the following remarkable property.
Theorem 5.13
The expected returnμV on a portfolio (or an individual security) is a linear
function of the beta coefficientβVof the portfolio,
μV=rF+(μM−rF)βV. (5.19)The expected return plotted against the beta coefficient of any portfolio or
individual security will form a straight line on theβ, μplane, called thesecurity
market line. This is shown in Figure 5.13, in which the security market line is
plotted next to the capital market line for comparison. A number of different
portfolios and individual securities are indicated by dots in both graphs.
Figure 5.13 Capital market line and security market lineSimilarly as in formula (5.18) for the capital market line, the term (μM−
rF)βVin (5.19) is therisk premium, interpreted as compensation for exposure
to systematic risk. However, (5.18) applies only to portfolios on the capital
market line, whereas (5.19) is much more general: It applies toallportfolios
and individual securities.
Exercise 5.19
Show that the characteristic lines of all securities intersect at a common
point in the CAPM. What are the coordinates of this point?