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.
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.
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.
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.
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.
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.
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.