284 Mathematics for Finance
5.7First we findE(K 1 ) = 7% andE(K 2 ) = 23%. If the expected return on the
portfolio is to beE(KV) = 20%, then by (5.4) and (5.1) the weights must
satisfy the system of equations
7 w 1 +23w 2 =20,
w 1 +w 2 =1.
The solution isw 1 =18.75% andw 2 =81.25%.
5.8First, we find computeμ 1 =4%andμ 2 = 16% from the data in Example 5.6.
Next, (5.7) and (5.1) give the system of equations
4 w 1 +14w 2 =46,
w 1 +w 2 =1,
for the weightsw 1 andw 2. The solution isw 1 =− 3 .2andw 2 =4. 2 .Finally,
we use (5.7) with the valuesσ 12 ∼= 0 .0184,σ^22 ∼= 0 .0024 andρ 12 ∼=− 0. 96309
computed in Example 5.6 to find the risk of the portfolio:
σ^2 V∼=(− 3 .2)^2 × 0 .0184 + (4.2)^2 × 0. 0024
+2×(− 3 .2)× 4. 2 ×(− 0 .96309)×√
0. 0184 ×√
0. 0024
∼= 0. 40278.5.9The returns on risky securities are non-constant random variables, that is,
K 1 (ω 1 )=K 1 (ω 2 ) andK 2 (ω 1 )=K 2 (ω 2 ). Because of this, the system of equa-
tions
K 1 (ω 1 )=aK 2 (ω 1 )+b,
K 1 (ω 2 )=aK 2 (ω 2 )+b,
must have a solutiona=0andb. It follows thatK 1 =aK 2 +b.
Now, use the properties of covariance and variance to compute
Cov(K 1 ,K 2 )=Cov(aK 2 +b, K 2 )=aCov(K 2 ,K 2 )=aVar(K 2 )=aσ^22 ,
σ^21 =Var(K 1 )=Var(aK 2 +b)=a^2 Var(K 2 )=a^2 σ^22.
It follows thatσ 1 =|a|σ 2 andρ 12 =Cov(K^1 ,K^2 )
σ 1 σ 2= aσ(^22)
|a|σ 22
=± 1.
5.10Using the valuesσ^21 ∼= 0 .0184,σ 22 ∼= 0 .0024 andρ 12 ∼=− 0 .96309 computed in
Example 5.6, we finds 0 from (5.13):
s 0 = σ
(^21) −ρ 12 σ 1 σ 2
σ 12 +σ 22 − 2 ρ 12 σ 1 σ 2
∼= 0. 73809.
This means that the weights in the portfolio with minimum risk arew 1 =
0 .73809 andw 2 =0.26191 and it involves no short selling.
5.11μV=0.06,σV∼= 1 .013.
5.12The weights of the three securities in the minimum variance portfolio are[ w∼=
0 .314 0.148 0. 538
]
,the expected return on the portfolio isμV∼= 0. 173
and the standard deviation isσV∼= 0 .151.