- Portfolio Management 99
To avoid clutter, we introduce the following notation for the expectation
and variance of a portfolio and its components:
μV=E(KV),σV=√
Var(KV),
μ 1 =E(K 1 ),σ 1 =√
Var(K 1 ),
μ 2 =E(K 2 ),σ 2 =√
Var(K 2 ).We shall also use the correlation coefficient
ρ 12 =Cov(K 1 ,K 2 )
σ 1 σ 2. (5.6)
Formulae (5.4) and (5.5) can be written as
μV=w 1 μ 1 +w 2 μ 2 , (5.7)
σ^2 V=w^21 σ^21 +w^22 σ^22 +2w 1 w 2 ρ 12 σ 1 σ 2. (5.8)Remark 5.3
For risky securities the returnsK 1 andK 2 arealwaysassumedtobenon-
constant random variables. Because of thisσ 1 ,σ 2 >0andρ 12 is well defined,
since the denominatorσ 1 σ 2 in (5.6) is non-zero.
Example 5.6
We use the following data:
Scenario Probability ReturnK 1 ReturnK 2
ω 1 (recession) 0. 4 −10% 20%
ω 2 (stagnation) 0. 2 0% 20%
ω 3 (boom) 0. 4 20% 10%We want to compare the risk of a portfolio such thatw 1 = 40% andw 2 =
60% with the risks of its components as measured by the variance. Direct
computations give
σ^21 ∼= 0. 0184 ,σ^22 ∼= 0. 0024 ,ρ 12 ∼=− 0. 96309.By (5.8)
σ^2 V∼=(0.4)^2 × 0 .0184 + (0.6)^2 × 0. 0024
+2× 0. 4 × 0. 6 ×(− 0 .96309)×√
0. 0184 ×
√
0. 0024
∼= 0. 000736.
Observe that the varianceσ^2 Vis smaller thanσ 12 andσ 22.