106 Mathematics for Finance
Using Theorem 5.5, we compute
s 0 ∼=− 1. 1663 ,smin=0.It follows that in the portfolio with minimum risk the weights of securities
should bew 1 ∼= 2 .1663 andw 2 ∼=− 1 .1663 if short selling is allowed. Without
short sellingw 1 =1andw 2 =0.
Exercise 5.10
Compute the weights in the portfolio with minimum risk for the data in
Example 5.6. Does this portfolio involve short selling?We conclude this section with a brief discussion of portfolios in which one of
the securities is risk-free. The variance of the risky security (a stock) is positive,
whereas that of the risk-free component (a bond) is zero.
Proposition 5.7
The standard deviationσV of a portfolio consisting of a risky security with
expected returnμ 1 and standard deviationσ 1 >0, and a risk-free security
with returnrF and standard deviation zero depends on the weightw 1 of the
risky security as follows:
σV=|w 1 |σ 1.
Proof
Letσ 1 >0andσ 2 = 0. Then (5.7) reduces toσV^2 =w^21 σ^21 , and the formula for
σV follows by taking the square root.
Figure 5.5 Portfolio line for one risky and one risk-free security