108156.pdf

104 Mathematics for Finance

Figure 5.2 The minimum ofσ^2 Vas a function ofs

The line on theσ, μplane defined by the parametric equations (5.11) and
(5.12) represents all possible portfolios with givenσ 1 ,σ 2 >0and− 1 ≤ρ 12 ≤1.
The parameterscan be any real number whenever there are no restrictions on
short selling. If short selling is not allowed, then 0≤s≤1 and we only obtain a
segment of the line. Assincreases from 0 to 1, the corresponding point (σV,μV)
travels along the line in the direction from (σ 1 ,μ 1 )to(σ 2 ,μ 2 ). Figure 5.3 shows
two typical examples of such lines, withρ 12 close to but greater than−1 (left)
and withρ 12 close to but smaller than 1 (right). Portfolios without short selling
are indicated by the bold line segments.

Figure 5.3 Typical portfolio lines with− 1 <ρ 12 < 1

Figure 5.4 illustrates the following corollary.

Corollary 5.6

Suppose thatσ 1 ≤σ 2. The following three cases are possible:

1) If− 1 ≤ρ 12 <σσ^12 , then there is a portfolio without short selling such that

σV<σ 1 (lines 4 and 5 in Figure 5.4);

2) Ifρ 12 =σσ^12 ,thenσV≥σ 1 for each portfolio (line 3 in Figure 5.4);