PORTFOLIO THEORY – EXAM 14/6/2024
Dr. Andrea Rigamonti
EXERCISE 1
A three-year 1000-euro coupon bond pays a 200-euro coupon each year.
Knowing that a bond that pays 100 euro after one-year costs 97 euro, one that pays 100 euro after two years
costs 95 euro, and one that pays 100 euro after three years costs 90 euro, what is the price at which the
coupon bond should trade in a market in which the law of one price holds?
Due to the law of one price, equal cash flows should have the same price. The coupon bond pays 200 euro
after one year, 200 euro after two years and 1200 (the 1000 euro plus the 200 coupon) after three years.
Therefore, we can replicate its cash flows with two one-year zero-coupon bonds, two two-year zero-coupon
bonds and twelve three-year zero-coupon bonds. The price of this portfolio of zero-coupon bonds, which
must be equal to the price of the coupon bond, is:
97 ∗ 2 + 95 ∗ 2 + 90 ∗ 12 = 1464
EXERCISE 2
The security S pays 990 euro after one year and it costs 900 euro.
If there is a 20% tax on financial profits and the inflation rate is 2% per year, what is the real return paid by
security S after taxes?
After one year we earn 990 − 900 = 90 euro.
On this we have to pay a 20% tax, so we actually earn 90 − 90 ∗ 0.2 = 72 euro
The nominal return net of taxes is therefore:
𝑅 =
972 − 900
900
= 8%
We now need to account for the inflation. The real return is:
𝑅 𝑟 =
1 + 𝑅
1 + 𝑖
− 1 =
1 + 0.08
1 + 0.02
− 1 ≈ 0.0588
EXERCISE 3
Consider a portfolio of three assets. The vector of weights and log-returns are:
𝒘 = [
0.6
0.5
−0.1
] 𝒓 = [
0.1
−0.02
0.04
]
What is the return of the portfolio?
Log-returns are not asset additive. We first need to convert them to simple returns:
𝑅1 = exp(𝑟1) − 1 = exp(0.1) − 1 ≈ 0.1052
𝑅2 = exp(𝑟2) − 1 = exp(−0.02) − 1 ≈ −0.0198
𝑅3 = exp(𝑟3) − 1 = exp(0.04) − 1 ≈ 0.0408
We can now compute the return of the portfolio:
𝑅 𝑝 = 𝑤1 𝑅1 + 𝑤2 𝑅2 + 𝑤3 𝑅3 = 0.6 × 0.1052 + 0.5 × (−0.0198) − 0.1 × 0.0408
= 0.06312 − 0.0099 − 0.00408 = 0.04914 ≈ 4.9%
EXERCISE 4
Given a risk-free asset and two risky assets with mean return, covariance matrix and inverse covariance matrix
equal to
µ = (
0.01
0.008
) ∑ = [
0.004 0.002
0.002 0.003
] ∑−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:
𝑤𝑣 =
𝑅𝑒
𝝁′∑−𝟏 𝝁
∑−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
0.012
)
[
1.75
1.5
] =
0.01
0.0175 + 0.018
[
1.75
1.5
] =
0.01
0.0355
[
1.75
1.5
] ≈ [
0.49
0.42
]
The weight for the risk-free asset is 1 − (0.49 + 0.42) = 0.09
EXERCISE 5
Consider the set of weights
𝑤𝑡−1 = [
0.2
0.8
] 𝑤𝑡 = [
−0.1
1.1
]
Compute the turnover taking into account the effect of the realized returns 𝑅𝑡 = [
0.05
0.1
]
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 +5% returns, therefore we have 0.2 + 0.2 × 0.05 = 0.21
The second asset experienced a +10% returns, therefore we have 0.8 + 0.8 × 0.1 = 0.88
The weights now sum to 1.09. We need to normalize them so that they sum up to 1 again:
𝑤𝑖,𝑡−1
+
=
[
0.21
1.09
0.88
1.09]
=≈ [
0.19
0.81
]
Now we can apply the formula and compute the turnover:
𝑇𝑂𝑡 = |−0.1 − 0.19| + |1.1 − 0.81| = 0.29 + 0.29 = 0.58
This means we need to trade 58% of our wealth in order to update the weights.
EXERCISE 6
1. What are the four properties of a coherent risk measure?
2. Assuming no change in the risk and in the interest rate, how does the price of a zero-coupon bond
change over time on the secondary market?
3. The effective annual rate of an investment is 10%. What is the monthly rate?
4. What is the efficient portfolio in a situation in which all the assumptions of the CAPM perfectly hold?
5. What is the bid-ask spread?
1. Monotonicity, translation invariance, positive homogeneity, subadditivity.
2. The price gradually rises until it reaches the face value.
3. The monthly rate is (1 + 0.1)1/12
− 1 ≈ 0.008 = 0.8%
4. The efficient portfolio is equal to the market portfolio.
5. It is the difference between the price at which you can buy a security on the market and the price at
which you can sell it.