116 Mathematics for Finance
cw′+(1−c)w′′for anyc∈Rand only of such portfolios.
Proof
By Proposition 5.10 the minimum variance line consists of portfolios whose
weights are given by a certain linear function of the expected returnμVon the
portfolio,w=aμV+b.Ifw′andw′′are the weights of two different portfolios
on the minimum variance line, thenw′=aμV′+bandw′′=aμV′′+bfor some
μV′=μV′′. Because numbers of the formcμV′+(1−c)μV′′forc∈Rexhaust
the whole real line, it follows that portfolios with weightscw′+(1−c)w′′for
c∈Rexhaust the whole minimum variance line.
This proposition is important. It means that the minimum variance line has
the same shape as the set of portfolios constructed from two securities, studied
in great detail in Section 5.2. It also means that the shape of the attainable
set on theσ, μplane (the Markowitz bullet), which we have seen so far for
portfolios constructed from two or three securities, will in fact be the same for
any number of securities.
Once the shape of the minimum variance line is understood, distinguishing
the efficient frontier is easy, also in the case ofnsecurities. This is illustrated
in Figure 5.10. The efficient frontier consists of all portfolios on the minimum
variance line whose expected return is greater than or equal to the expected
return on the minimum variance portfolio.
Figure 5.10 Efficient frontier constructed from several securitiesThe next proposition provides a property of the efficient frontier which will
prove useful in the Capital Asset Pricing Model.
Proposition 5.12
The weightswof any portfolio belonging to the efficient frontier (except for