PORTFOLIO THEORY – EXERCISES 19/03/2024
Dr. Andrea Rigamonti
EXERCISE 1
In a perfectly competitive market where the interest rate r remains
constant over time, security A pays 1200 euro after two years and it
costs 1000 euro.
Security B pays 1100 euro after one year. If there is a 20% tax on
financial profits, how much does the investor who buys B earn after
one year?
The interest rate is:
1000 =
1200
(1 + 𝑟)2
(1 + 𝑟)2
=
1200
1000
1 + 𝑟 = 1.21 2⁄
𝑟 = 1.21 2⁄
− 1 = 0.0954 = 9.54%
Since the interest rate is 9.54% and we are in a perfect market, the
price of B has to be:
𝑃𝐵 =
1100
1 + r
=
1100
1 + 0.0954
= 1004.199
The investor earned 1100 – 1004.199 = 95.801 euro, but has to pay
a 20% tax on it, so after taxes he actually earns:
95.801 − 95.801 ∗ 0.2 = 76.6408
EXERCISE 2
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 zerocoupon
bonds, which must be equal to the price of the coupon bond,
is:
97 ∗ 2 + 95 ∗ 2 + 90 ∗ 12 = 1464
EXERCISE 3
The log-return of the four assets included in an equally weighted
portfolio is:
𝑟1 = 0.1, 𝑟2 = −0.06, 𝑟3 = 0.07, 𝑟4 = 0.05
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(−0.06) − 1 = −0.0582
𝑅3 = exp( 𝑟3) − 1 = exp(0.07) − 1 = 0.0725
𝑅4 = exp( 𝑟4) − 1 = exp(0.05) − 1 = 0.0513
We can now compute the return of the portfolio:
𝑅 𝑝 = 𝑤1 𝑅1 + 𝑤2 𝑅2 + 𝑤3 𝑅3 + 𝑤4 𝑅4 =
= 0.25 ∗ 0.1052 + 0.25 ∗ (−0.0582) + 0.25 ∗ 0.0725 + 0.25
∗ 0.0513 = 0.0427
EXERCISE 4
Suppose that the conditions for the CAPM are fully respected. The
market portfolio expected return is 5% and the risk-free rate is 1%. We
create a portfolio in which 30% of the wealth is placed in the security A
whose beta is equal to that of the market, 50% in the security B whose
beta is twice that of the market, and the remaining 20% is invested in
the risk-free asset. What is the expected return of the portfolio?
The beta of the market portfolio is equal to 1 by definition. Therefore
the beta of A is also equal to 1, and the beta of B is equal to 2. Since
the CAPM holds, the expected return of A is equal to that of the market
portfolio (because it has the same beta), while that of B is:
𝐸[ 𝑅 𝐵] = 𝑅𝑓 + 𝛽 𝐵 ∗ (𝐸[ 𝑅 𝑀] − 𝑅𝑓) = 0.01 + 2 ∗ (0.05 − 0.01)
= 0.01 + 0.08 = 0.09
Hence, the expected return of the portfolio is:
𝐸[ 𝑅 𝑃] = 0.3 ∗ 0.05 + 0.5 ∗ 0.09 + 0.2 ∗ 0.01 = 0.062 = 6.2%
EXERCISE 5
Consider the following series of unadjusted monthly closing prices (in
euro) of a stock that undergoes the corporate events indicated next to
the price.
January: 7
February: 9
March: 5 -- 2 for 1 stock split
April: 4.8
What is the monthly mean return for an investor that buys these shares
at the end of January (i.e., for 7 euro per share) and keeps them until
the end of April (i.e., until they are worth 4.8 euro per share)?
First we need to adjust for the stock split:
January: 7/2 = 3.5
February: 9/2 = 4.5
March: 5
April: 4.8
We can now compute the returns:
𝑅1 =
4.5 − 3.5
3.5
≈ 0.2857
𝑅2 =
5 − 4.5
4.5
≈ 0.1111
𝑅3 =
4.8 − 5
5
≈ −0.04
The mean return is:
𝑅̅ =
0.2857 + 0.1111 − 0.04
3
≈ 0.1189