EXERCISES
EXERCISE 1
A corporation paid a dividend of 10 euro per share in 2023 and a dividend of 10.1 euro
in 2024. We expect the dividend to keep growing at the same rate forever. The annual
cost of equity is 5% and the annual cost of debt is 2%. What should the price of the
shares be according to the dividend-discount model?
The annual growth rate of the dividend from 2023 to 2024 was 10.1/10 − 1 = 0.01.
We know that it will stay like this forever, so the growth rate will always be 1%. We
have therefore a growing perpetuity.
The cost of debt is irrelevant in the dividend-discount model. The cost of equity is 5%.
Therefore the price of the share should be:
𝑃0 =
𝐷𝑖𝑣2025
𝑅 𝐸 − 𝑔
=
𝐷𝑖𝑣2024 + 𝑔𝐷𝑖𝑣2024
𝑅 𝐸 − 𝑔
=
10.1 + 0.01 ∗ 10.1
0.05 − 0.01
=
10.201
0.04
= 255.025
EXERCISE 2
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: 18
February: 18.6
March: 5.6 3 for 1 split
April: 6.4
May: 3.4 2 for 1 stock split
June: 3.7
Compute the adjusted stock returns.
The prices before the first stock split must be adjusted for both splits. We can do this
by dividing those prices by a cumulative adjustment factor given by the product of the
two splits’ ratios. The prices in March and April only need to be corrected for the May
split, so we can just divide them by 2.
January: 18/(3 ∗ 2) = 3
February: 18.6/(3 ∗ 2) = 3.1
March: 5.6/2 = 2.8
April: 6.4/2 = 3.2
May: 3.4
June: 3.7
Now we compute the returns:
𝑅 𝐹𝑒𝑏 =
3.1 − 3
3
≈ 0.033
𝑅 𝑀𝑎𝑟 =
2.8 − 3.1
3.1
≈ −0.097
𝑅 𝐴𝑝𝑟 =
3.2 − 2.8
2.8
≈ 0.143
𝑅 𝑀𝑎𝑦 =
3.4 − 3.2
3.2
= 0.0625
𝑅𝐽𝑢𝑛 =
3.7 − 3.4
3.4
≈ 0.0.88
EXERCISE 3
Given the following series of returns
0.048 , 0.02 , -0.01 , 0.1
and the following series of risk-free rates
0.005 , 0.005 , 0 , 0
Compute the Sharpe ratio of the investment.
The Sharpe ratio is given by:
𝑆𝑅 =
𝐸[𝑅 𝑝 − 𝑅𝑓]
𝑆𝐷(𝑅 𝑝 − 𝑅𝑓)
First we compute the excess return:
0.048 − 0.005 = 0.043
0.02 − 0.005 = 0.015
−0.01 − 0 = −0.01
0.1 − 0 = 0.1
Then we compute the mean and the standard deviation of the excess returns:
𝑅 𝑒𝑥𝑐
̅̅̅̅̅̅ =
0.043 + 0.015 − 0.01 + 0.1
4
= 0.037
𝜎2
=
1
𝑇 − 1
∑(𝑅 𝑒𝑥𝑐,𝑡 − 𝑅 𝑒𝑥𝑐
̅̅̅̅̅̅)
2
4
𝑡=1
=
(0.043 − 0.037)2
+ (0.015 − 0.037)2
+ (−0.01 − 0.037)2
+ (0.1 − 0.037)2
3
≈
0.000036 + 0.000484 + 0.002209 + 0.003969
3
≈ 0.0022
𝜎 = √0.0022 ≈ 0.0469
So the Sharpe ratio is:
𝑆𝑅 =
0.037
0.0469
≈ 0.789
EXERCISE 4
A risky investment is estimated to deliver the following returns.
After 9 months:
• 𝑅 = −0.15 with a 20% probability
• 𝑅 = 0.1 with a 70% probability
• 𝑅 = 0.25 with a 10% probability
After 24 months:
• 𝑅 = −0.2 with a 20% probability
• 𝑅 = 0.15 with a 60% probability
• 𝑅 = 0.3 with a 20% probability
The annual inflation rate is 3%.
What is the real cumulative expected return after 24 months?
First we need to compute the expected return at 9 months and at 24 months.
𝐸[ 𝑅9𝑚] = ∑ 𝑝 𝑅 𝑅
𝑅
= 0.2 ∗ (−0.15) + 0.7 ∗ 0.1 + 0.1 ∗ 0.25 = 0.065
𝐸[ 𝑅24𝑚] = ∑ 𝑝 𝑅 𝑅
𝑅
= 0.2 ∗ (−0.2) + 0.6 ∗ 0.15 + 0.2 ∗ 0.3 = 0.11
The cumulative expected return at 24 months is:
𝑅24𝑚,𝑐𝑢𝑚 = ∏(1 + 𝑅𝑡)
2
𝑡=1
− 1 = (1 + 0.065)(1 + 0.11) − 1 ≈ 0.182
The 24-month inflation rate is:
𝑖24 = (1 + 𝑖12)2
– 1 = (1 + 0.03)2
– 1 = 0.0609
So the real cumulative expected return after 24 months is:
𝑅 𝑟 =
1 + 𝑅
1 + 𝑖
− 1 =
1 + 0.182
1 + 0.0609
− 1 ≈ 0.114