100 Mathematics for Finance
Example 5.7
Consider another portfolio with weightsw 1 = 80% andw 2 = 20%, all other
things being the same as in Example 5.6. Then
σ^2 V∼=(0.8)^2 × 0 .0184 + (0.2)^2 × 0. 0024
+2× 0. 8 × 0. 2 ×(− 0 .96309)×√
0. 0184 ×
√
0. 0024
∼= 0. 009824 ,
which is betweenσ 12 andσ^22.
Proposition 5.3
The varianceσ^2 V of a portfolio cannot exceed the greater of the variancesσ^21
andσ^22 of the components,
σV^2 ≤max{σ^21 ,σ^22 },if short sales are not allowed.
Proof
Let us assume thatσ 12 ≤σ^22. If short sales are not allowed, thenw 1 ,w 2 ≥0and
w 1 σ 1 +w 2 σ 2 ≤(w 1 +w 2 )σ 2 =σ 2.Since the correlation coefficient satisfies− 1 ≤ρ 12 ≤1, it follows that
σ^2 V=w 12 σ^21 +w^22 σ 22 +2w 1 w 2 ρ 12 σ 1 σ 2
≤w 12 σ^21 +w^22 σ 22 +2w 1 w 2 σ 1 σ 2
=(w 1 σ 1 +w 2 σ 2 )^2 ≤σ^22.Ifσ 12 ≥σ 22 , the proof is analogous.
Example 5.8
Now consider a portfolio with weightsw 1 =−50% andw 2 = 150% (allowing
short sales of security 1), all the other data being the same as in Example 5.6.
The variance of this portfolio is
σV^2 =∼(− 0 .5)^2 × 0 .0184 + (1.5)^2 × 0. 0024
+2×(− 0 .5)× 1. 5 ×(− 0 .96309)×√
0. 0184 ×
√
0. 0024
∼= 0. 0196 ,
which is greater than bothσ^21 andσ^22.