122 Mathematics for Finance
The beta factor is an indicator of expected changes in the return on a
particular portfolio or individual security in response to the behaviour of the
market as a whole. SinceμV =βVμM+αV, the return on a security with
a positive beta factor tends to increase as the return on the market portfolio
increases, while the return on a security with a negative beta factor tends to
increase if the return on the market portfolio goes down.
In what follows we discuss another interpretation of the beta factor. The
riskσ^2 V=Var(KV) of a security or portfolio can be written as
σV^2 =Var(εV)+βV^2 σ^2 M.This formula is easy to verify upon substituting the expressionεV =KV−
(αV+βVKM) for the residual random variable. The first term Var(εV) is called
theresidual varianceordiversifiable risk. It vanishes for the market portfolio,
Var(εM) = 0. This part of risk can ‘diversified away’ by investing in the market
portfolio. The second termβ^2 VσM^2 is called thesystematicorundiversifiable
risk. The market portfolio involves only this kind of risk. The beta factorβV
can be regarded as a measure of systematic risk associated with a security or
portfolio.
This interpretation of the beta factor is of crucial importance. In the CAPM
systematic risk, measured byβV, will be linked to the expected returnμV
and hence to the pricing of individual securities and portfolios: The higher
the systematic risk, the higher the return required by investors as a premium
for exposure to this kind of risk. However, diversifiable risk will attract no
additional premium, having no effect onμV. This is because diversifiable risk
can be eliminated by spreading an investment in a portfolio of many securities
and, in particular, by investing in the market portfolio. The next section is
devoted to establishing the link betweenβV andμV.
Exercise 5.18
Show that the beta factorβVof a portfolio consisting ofnsecurities with
weightsw 1 ,...,wnis given byβV=w 1 β 1 +···+wnβn,whereβ 1 ,...,βn
are the beta factors of the securities.5.4.3 Security Market Line ...............................
Consider an arbitrary portfolio with weightswV. The weights in the market
portfolio will be denoted bywM. The market portfolio belongs to the efficient
frontier of the attainable set of portfolios consisting of risky securities. Thus,
by Proposition 5.12
γwMC=m−μu