PORTFOLIO THEORY – EXERCISES 07/05/2024
Dr. Andrea Rigamonti
EXERCISE 1
The vector of weights and the covariance matrix of a portfolio with two
assets are:
𝒘 = [
0.6
0.4
] 𝜮 = [
0.002 0.001
0.001 0.003
]
Compute, using matrix form, the variance of the portfolio.
𝑉𝑎𝑟( 𝑅 𝑃) = [0.6 0.4] [
0.002 0.001
0.001 0.003
] [
0.6
0.4
] =
[0.6 ∗ 0.002 + 0.4 ∗ 0.001 0.6 ∗ 0.001 + 0.4 ∗ 0.003] [
0.6
0.4
] =
[0.0016 0.0018] [
0.6
0.4
] = 0.0016 ∗ 0.6 + 0.0018 ∗ 0.4 = 0.00168
EXERCISE 2
An investor with risk-aversion 𝛾 = 4 invests in a portfolio of one risk-free
asset and two risky assets with mean excess return, covariance matrix and
inverse covariance matrix equal to:
µ = (
0.006
0.004
) ∑ = [
0.003 0.002
0.002 0.0015
] ∑−𝟏
≈ [
3000 −4000
−4000 6000
]
Compute the portfolio weights.
The weights for the risky assets are:
𝒘 𝒎𝒗 =
1
𝛾
∑−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
EXERCISE 3
Given two risky assets with mean return, covariance matrix and inverse
covariance matrix equal to
µ = (
0.007
0.004
) ∑ = [
0.004 0.002
0.002 0.003
] ∑−𝟏
= [
375 −250
−250 500
]
compute the weights for the minimum variance portfolio.
The weights for the risky assets are:
𝑤𝑣 =
1
𝟏′∑−𝟏 𝟏
∑−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 4
Write the first order conditions for the problem of computing portfolio
weights that minimize the variance given a target return of 1% and a riskfree
rate of 0.1%:
min
𝒘
𝒘′
∑𝒘
subject to:
𝒘′
𝝁 + (1 − 𝒘′𝟏)0.001 = 0.01
First we write the Lagrangian function:
𝐿( 𝒘, 𝜆) = 𝒘′
∑𝒘 + 𝜆[0.01 − 𝒘′
𝝁 − (1 − 𝒘′
𝟏)0.001]
= 𝒘′
∑𝒘 + 𝜆[0.009 − 𝒘′
𝝁 + 𝒘′
𝟎. 𝟎𝟎𝟏]
The first order conditions are the two partial derivatives of the Lagrangian
with respect to 𝒘 and 𝜆 set equal to zero:
𝜕𝐿
𝜕𝒘
= 2∑𝒘 − 𝜆𝝁 + 𝜆𝟎. 𝟎𝟎𝟏 = 𝟎
𝜕𝐿
𝜕𝜆
= 0.009 − 𝒘′
𝝁 − 𝒘′
𝟎. 𝟎𝟎𝟏 = 0
EXERCISE 5
Given a vector of theoretically optimal weights
𝒘 = [
−0.2
0.4
0.5
0.3
]
compute the weights 𝒘∗
of the shrinkage portfolio obtained as
𝒘∗
= 𝛿𝒘 𝑵𝑨𝑰𝑽𝑬 + (1 − 𝛿)𝒘
with shrinkage parameter 𝛿 = 0.4
The weights of the shrinkage portfolio are:
𝒘∗
= 0.4 [
0.25
0.25
0.25
0.25
] + 0.6 [
−0.2
0.4
0.5
0.3
] = [
0.1
0.1
0.1
0.1
] + [
−0.12
0.24
0.3
0.18
] = [
−0.02
0.34
0.4
0.28
]
EXERCISE 6
Given the following series of returns:
0.05 , -0.01 , 0.02 , 0.03 , 0 , 0.005
Compute the downside deviation for an investor that sets the benchmark
B=0.01
We compute the semivariance:
𝜎 𝐵
2
=
1
6
[(−0.01 − 0.01)2
+ (0 − 0.01)2
+ (0.005 − 0.01)2]
=
1
6
0.000525 = 0.0000875
The downside deviation is simply the square root of the semivariance:
𝜎 𝐵 = √0.0000875 = 0.009354143
EXERCISE 7
Consider the set of weights
𝑤𝑡−1 = [
0.3
0.7
] 𝑤𝑡 = [
0.4
0.6
]
Compute the turnover taking into account the effect of the realized
returns 𝑅𝑡 = [
0.1
0.2
]
We have to use the formula
𝑇𝑂𝑡 = ∑|𝑤𝑖,𝑡 − 𝑤𝑖,𝑡−1
+
|
𝑁
𝑖=1
To apply it correctly we first need to compute 𝑤𝑖,𝑡−1
+
.
The first asset experienced a +10% returns, therefore we have 0.3 +
0.3 × 0.1 = 0.33
The second asset experienced a +20% returns, therefore we have 0.7 +
0.7 × 0.2 = 0.84
The weights now sum to 1.17. We need to normalize them so that they
sum up to 1 again:
𝑤𝑖,𝑡−1
+
=
[
0.33
1.17
0.84
1.17]
≈ [
0.28
0.72
]
Now we can apply the formula and compute the turnover:
𝑇𝑂𝑡 = |0.4 − 0.28| + |0.6 − 0.72| = 0.12 + 0.12 = 0.24
This means we need to trade 14% of our wealth in order to update the
weights.