PORTFOLIO THEORY – EXERCISES 3
Dr. Andrea Rigamonti
EXERCISE 1
An investor with risk-aversion 𝛾 = 4 invests in a portfolio of one risk-free asset and two risky assets whose
excess returns have mean vector, covariance matrix and inverse covariance matrix equal to:
µ = [
0.006
0.004
] 𝚺 = [
0.003 0.002
0.002 0.0015
] 𝚺−𝟏
≈ [
3000 −4000
−4000 6000
]
Compute the portfolio weights that maximize mean-variance utility, and the corresponding utility.
The weights for the risky assets are:
𝒘 𝑼 =
1
𝛾
𝚺−𝟏
𝝁 =
1
4
[
3000 −4000
−4000 6000
] [
0.006
0.004
] = [
750 −1000
−1000 1500
] [
0.006
0.004
]
= [
750 ∗ 0.006 + (−1000) ∗ 0.004
−1000 ∗ 0.006 + 1500 ∗ 0.004
] = [
0.5
0
]
The weight for the risk-free asset is: 1 − 0.5 = 0.5
The formula we used maximizes the utility given by 𝑈(𝒘) = 𝒘′𝝁 −
𝛾
2
𝒘′
Ʃ𝒘. Therefore, the utility is:
𝑈(𝒘) = 𝒘′𝝁 −
𝛾
2
𝒘′
Ʃ𝒘 = [0.5 0] [
0.006
0.004
] −
4
2
[0.5 0] [
0.003 0.002
0.002 0.0015
] [
0.5
0
]
= 0.5 ∗ 0.006 + 0 ∗ 0.004 − 2 ∗ [0.5 ∗ 0.003 + 0 ∗ 0.002 0.5 ∗ 0.002 + 0 ∗ 0.0015] [
0.5
0
]
= 0.003 − 2 ∗ [0.0015 0.001] [
0.5
0
] = 0.003 − 2 ∗ 0.0015 ∗ 0.5 = 0.003 − 0.0015 = 0.0015
Equivalently, we can use the formula for the utility of an optimal mean-variance portfolio, which is only
valid for a portfolio with optimal weights:
𝑈(𝒘 𝑼) =
1
2𝛾
𝝁′
Ʃ−𝟏
𝝁 =
1
2 ∗ 4
[0.006 0.004] [
3000 −4000
−4000 6000
] [
0.006
0.004
]
=
1
8
[0.006 ∗ 3000 + 0.004 ∗ (−4000) 0.006 ∗ (−4000) + 0.004 ∗ 6000] [
0.006
0.004
]
=
1
8
[2 0] [
0.006
0.004
] =
1
8
∗ 2 ∗ 0.006 = 0.0015
EXERCISE 2
An investor with risk-aversion 𝛾 = 2 invests in a portfolio of one risk-free asset and two risky assets whose
excess returns have mean vector, covariance matrix and inverse covariance matrix equal to:
µ = [
0.005
0.004
] 𝚺 = [
0.003 0.002
0.002 0.0015
] 𝚺−𝟏
≈ [
3000 −4000
−4000 6000
]
Compute the portfolio weights that maximize mean-variance utility given a full-investment constraint.
The weights are given by:
𝒘 𝑼∗ =
Ʃ−𝟏
𝛾
(𝝁 +
𝛾 − 𝝁′
Ʃ−𝟏
𝟏
𝟏′Ʃ−𝟏 𝟏
𝟏)
=
1
2
[
3000 −4000
−4000 6000
] {[
0.005
0.004
] +
2 − [0.005 0.004] [
3000 −4000
−4000 6000
] [
1
1
]
[1 1] [
3000 −4000
−4000 6000
] [
1
1
]
[
1
1
]}
= [
1500 −2000
−2000 3000
] {[
0.005
0.004
] +
2 − [0.005 0.004] [
3000 ∗ 1 − 4000 ∗ 1
−4000 ∗ 1 + 6000 ∗ 1
]
[1 1] [
3000 ∗ 1 − 4000 ∗ 1
−4000 ∗ 1 + 6000 ∗ 1
]
[
1
1
]}
= [
1500 −2000
−2000 3000
] {[
0.005
0.004
] +
2 − [0.005 0.004] [
−1000
2000
]
[1 1] [
−1000
2000
]
[
1
1
]}
= [
1500 −2000
−2000 3000
] {[
0.005
0.004
] +
2 − [0.005 ∗ (−1000) + 0.004 ∗ 2000]
−1000 + 2000
[
1
1
]}
= [
1500 −2000
−2000 3000
] {[
0.005
0.004
] +
−1
1000
[
1
1
]} = [
1500 −2000
−2000 3000
] {[
0.005
0.004
] + [
−0.001
−0.001
]}
= [
1500 −2000
−2000 3000
] [
0.004
0.003
] = [
1500 ∗ 0.004 − 2000 ∗ 0.003
−2000 ∗ 0.004 + 3000 ∗ 0.003
] = [
0
1
]
EXERCISE 3
Given a risk-free asset and two risky assets whose excess returns have mean vector, covariance matrix and
inverse covariance matrix equal to
µ = [
0.01
0.008
] 𝚺 = [
0.004 0.002
0.002 0.003
] 𝚺−𝟏
= [
375 −250
−250 500
]
compute the weights for the optimal mean-variance portfolio with target return 𝑅𝑒 = 0.01.
The weights for the risky assets are:
𝒘 𝒎𝒗 =
𝑅𝑒
𝝁′𝚺−𝟏 𝝁
𝚺−𝟏
𝝁 =
0.01
[0.01 0.008] [
375 −250
−250 500
] [
0.01
0.008
]
[
375 −250
−250 500
] [
0.01
0.008
]
=
0.01
[0.01 ∗ 375 + 0.008 ∗ (−250) 0.01 ∗ (−250) + 0.008 ∗ 500] [
0.01
0.008
]
[
375 ∗ 0.01 + (−250) ∗ 0.008
−250 ∗ 0.01 + 500 ∗ 0.008
]
=
0.01
[1.75 1.5] [
0.01
0.008
]
[
1.75
1.5
] =
0.01
0.0175 + 0.012
[
1.75
1.5
] =
0.01
0.0295
[
1.75
1.5
] ≈ [
0.59
0.51
]
The weight for the risk-free asset is 1 − (0.59 + 0.51) = −0.1
EXERCISE 4
Given a risk-free asset and two risky assets whose excess returns have mean vector, covariance matrix and
inverse covariance matrix equal to
µ = [
0.01
0.008
] 𝚺 = [
0.004 0.002
0.002 0.003
] 𝚺−𝟏
= [
375 −250
−250 500
]
compute the weights for the optimal mean-variance portfolio with target return 𝑅𝑒 = 0.01 given a fullinvestment
constraint.
The weights are:
𝒘 𝒎𝒗∗ = Ʃ−𝟏
[
𝐶𝑅𝑒 − 𝐵
𝐴𝐶 − 𝐵2
𝝁 +
𝐴 − 𝐵𝑅𝑒
𝐴𝐶 − 𝐵2
𝟏]
where 𝐴 = 𝝁′Ʃ−𝟏
𝝁, 𝐵 = 𝟏′
Ʃ−𝟏
𝝁 and 𝐶 = 𝟏′
Ʃ−𝟏
𝟏.
First we compute the three scalars:
𝐴 = 𝝁′Ʃ−𝟏
𝝁 = [0.01 0.008] [
375 −250
−250 500
] [
0.01
0.008
] =
= [0.01 ∗ 375 + 0.008 ∗ (−250) 0.01 ∗ (−250) + 0.008 ∗ 500] [
0.01
0.008
] =
= [1.75 1.5] [
0.01
0.008
] = 1.75 ∗ 0.01 + 1.5 ∗ 0.008 = 0.0295
𝐵 = 𝟏′
Ʃ−𝟏
𝝁 = [1 1] [
375 −250
−250 500
] [
0.01
0.008
] =
= [1 ∗ 375 + 1 ∗ (−250) 1 ∗ (−250) + 1 ∗ 500] [
0.01
0.008
] =
= [125 250] [
0.01
0.008
] = 125 ∗ 0.01 + 250 ∗ 0.008 = 3.25
𝐶 = 𝟏′
Ʃ−𝟏
𝟏 = [1 1] [
375 −250
−250 500
] [
1
1
] =
= [1 ∗ 375 + 1 ∗ (−250) 1 ∗ (−250) + 1 ∗ 500] [
1
1
] =
= [125 250] [
1
1
] = 125 ∗ 1 + 250 ∗ 1 = 375
Now we can compute the weights:
𝒘 𝒎𝒗∗ = Ʃ−𝟏
[
𝐶𝑅𝑒 − 𝐵
𝐴𝐶 − 𝐵2
𝝁 +
𝐴 − 𝐵𝑅𝑒
𝐴𝐶 − 𝐵2
𝟏]
= [
375 −250
−250 500
] {
375 ∗ 0.01 − 3.25
0.0295 ∗ 375 − 10.5625
[
0.01
0.008
] +
0.0295 − 3.25 ∗ 0.01
0.0295 ∗ 375 − 10.5625
[
1
1
]}
= [
375 −250
−250 500
] {
0.5
0.5
[
0.01
0.008
] +
−0.003
0.5
[
1
1
]}
= [
375 −250
−250 500
] {[
0.01
0.008
] + [
−0.006
−0.006
]} = [
375 −250
−250 500
] [
0.004
0.002
]
= [
375 ∗ 0.004 − 250 ∗ 0.002
−250 ∗ 0.004 + 500 ∗ 0.002
] = [
1
0
]
EXERCISE 5
Given a risk-free asset and two risky assets whose excess returns have mean vector, covariance matrix and
inverse covariance matrix equal to
µ = [
0.01
0.008
] 𝚺 = [
0.004 0.002
0.002 0.003
] 𝚺−𝟏
= [
375 −250
−250 500
]
compute the tangency portfolio.
The weights are given by:
𝒘 𝒕𝒂𝒏 =
Ʃ−𝟏
𝝁
𝟏′Ʃ−𝟏 𝝁
=
[
375 −250
−250 500
] [
0.01
0.008
]
[1 1] [
375 −250
−250 500
] [
0.01
0.008
]
=
[
375 ∗ 0.01 − 250 ∗ 0.008
−250 ∗ 0.01 + 500 ∗ 0.008
]
[1 1] [
375 ∗ 0.01 − 250 ∗ 0.008
−250 ∗ 0.01 + 500 ∗ 0.008
]
=
[
1.75
1.5
]
[1 1] [
1.75
1.5
]
=
[
1.75
1.5
]
1 ∗ 1.75 + 1 ∗ 1.5
=
1
3.25
[
1.75
1.5
] ≈ [
0.54
0.46
]
EXERCISE 6
Given two risky assets with mean return, covariance matrix and inverse covariance matrix equal to
µ = [
0.007
0.004
] 𝚺 = [
0.004 0.002
0.002 0.003
] 𝚺−𝟏
= [
375 −250
−250 500
]
compute the weights for the minimum variance portfolio.
The weights are:
𝒘 𝒗 =
1
𝟏′𝚺−𝟏 𝟏
𝚺−𝟏
𝟏 =
1
[1 1] [
375 −250
−250 500
] [
1
1
]
[
375 −250
−250 500
] [
1
1
]
=
1
[1 ∗ 375 + 1 ∗ (−250) 1 ∗ (−250) + 1 ∗ 500] [
1
1
]
[
375 ∗ 1 + (−250) ∗ 1
−250 ∗ 1 + 500 ∗ 1
]
=
1
[125 250] [
1
1
]
[
125
250
] =
1
125 + 250
[
125
250
] =
1
375
[
125
250
] ≈ [
0.33
0.67
]
EXERCISE 7
A mean-variance utility investor with risk-aversion coefficient 𝛾 = 5 can invest in two risky assets and one
risk-free asset. The mean and covariance matrix of the excess returns of the two risky assets are:
µ = [
0.01
0.008
] Ʃ = [
0.005 0.002
0.002 0.003
]
Compute the weights for the 1/N portfolio that optimally allocates the wealth between the risky assets
and the risk-free asset.
We compute the weights for the risky assets:
𝒘 𝟏/𝑵 =
1
𝛾
𝟏′𝝁
𝟏′Ʃ𝟏
𝟏 =
1
5
[1 1] [
0.01
0.008
]
[1 1] [
0.005 0.002
0.002 0.003
] [
1
1
]
[
1
1
]
=
1
5
1 ∗ 0.01 + 1 ∗ 0.008
[1 ∗ 0.005 + 1 ∗ 0.002 1 ∗ 0.002 + 1 ∗ 0.003] [
1
1
]
[
1
1
]
=
1
5
0.018
[0.007 0.005] [
1
1
]
[
1
1
] =
1
5
0.018
0.007 ∗ 1 + 0.005 ∗ 1
[
1
1
]
=
1
5
0.018
0.007 ∗ 1 + 0.005 ∗ 1
[
1
1
] =
1
5
0.018
0.012
[
1
1
] =
0.018
0.06
[
1
1
] = 0.3 [
1
1
] = [
0.3
0.3
]
The two risky assets get a weight of 0.3 each, so the weight of the risk-free asset is:
1 − (0.3 + 0.3) = 0.4