108156.pdf

5. Portfolio Management 105

3) If σσ^12 <ρ 12 ≤1, then there is a portfolio with short selling such that

σV<σ 1 , but for each portfolio without short sellingσV≥σ 1 (lines 1 and 2

in Figure 5.4).

Figure 5.4 Portfolio lines for various values ofρ 12

Proof

1. If− 1 ≤ρ 12 <σσ 21 ,thenσ 1 σ+^1 σ 2 >s 0 >0. But σ 1 σ+^1 σ 2 <1, so 0<s 0 <1,
   which means that the portfolio with minimum variance, which corresponds to
   the parameters 0 , involves no short selling and satisfiesσV<σ 1.


2. Ifρ 12 =σσ^12 ,thens 0 = 0. As a result,σV≥σ 1 for every portfolio because
   σ 12 is the minimum variance.


3. Finally, ifσσ^12 <ρ 12 ≤1, thens 0 <0. In this case the portfolio with
   minimum variance that corresponds tos 0 involves short selling of security 1
   and satisfiesσV<σ 1 .Fors≥s 0 the varianceσVis an increasing function ofs,
   which means thatσV>σ 1 for every portfolio without short selling.

The above corollary is important because it shows when it is possible to
construct a portfolio with risk lower than that of any of its components. In
case 1) this is possible without short selling. In case 3) this is also possible, but
only if short selling is allowed. In case 2) it is impossible to construct such a
portfolio.

Example 5.9

Suppose that

σ^21 =0. 0041 ,σ^22 =0. 0121 ,ρ 12 =0. 9796.

Clearly,σ 1 <σ 2 andσσ^12 <ρ 12 <1, so this is case 3) in Corollary 5.6. Our task
will be to find the portfolio with minimum risk with and without short selling.