108156.pdf

102 Mathematics for Finance

Figure 5.1 Typical portfolio lines withρ 12 =−1and1

Suppose thatρ 12 =−1. It follows from the proof of Proposition 5.4 that
σV=|w 1 σ 1 −w 2 σ 2 |. In addition,μV=w 1 μ 1 +w 2 μ 2 by (5.7) andw 1 +w 2 =1
by (5.1). We can chooses=w 2 as a parameter. Then 1−s=w 1 and

σV=|(1−s)σ 1 −sσ 2 |,

μV=(1−s)μ 1 +sμ 2.

These parametric equations describe the line in Figure 5.1 with a broken seg-
ment between (σ 1 ,μ 1 )and(σ 2 ,μ 2 ). Assincreases, the point (σV,μV)moves
along the line in the direction from (σ 1 ,μ 1 )to(σ 2 ,μ 2 ).
Ifρ 12 =1,thenσV =|w 1 σ 1 +w 2 σ 2 |. We chooses=w 2 as a parameter
once again, and obtain the parametric equations

σV=|(1−s)σ 1 +sσ 2 |,

μV=(1−s)μ 1 +sμ 2

of the line in Figure 5.1 with a straight segment between (σ 1 ,μ 1 )and(σ 2 ,μ 2 ).
If no short selling is allowed, then 0≤s≤1 in both cases, which corresponds
to the bold line segments.

Exercise 5.9

Suppose that there are just two scenariosω 1 andω 2 and consider two

risky securities with returnsK 1 andK 2. Show thatK 1 =aK 2 +bfor

some numbersa=0andb, and deduce thatρ 12 =1or−1.

Our next task is to find a portfolio with minimum risk for any givenρ 12
such that− 1 <ρ 12 <1. Again, we takes=w 2 as a parameter. Then (5.7)