PORTFOLIO THEORY – EXERCISES 1
EXERCISE 1
The security S pays 990 euro after eight months 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 annual return paid
by security S after taxes?
After eight months 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
= 0.08 = 8%
The annual nominal return is:
𝑅12 = (1 + 𝑅)
12
8 – 1 = (1 + 0.08)
3
2 – 1 ≈ 0.122
We now need to account for the inflation. The real return is:
𝑅 𝑟 =
1 + 𝑅12
1 + 𝑖
− 1 =
1 + 0.122
1 + 0.02
− 1 = 0.1 = 10%
EXERCISE 2
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 3
The returns of a security over four periods are:
𝑅𝑡=1 = 0.2, 𝑅𝑡=2 = −0.1, 𝑅𝑡=3 = 0.08, 𝑅𝑡=4 = 0.04
If we invested 1000 euro in this asset at t=0, how much is our investment worth at t=4?
The value of the investment is:
𝑉4 = 𝑉0 + 𝑉0 [∏(1 + 𝑅𝑡)
4
𝑡=1
− 1] = 1000 + 1000[(1 + 0.2)(1 − 0.1)(1 + 0.08)(1 + 0.04) − 1]
≈ 1000 + 1000 ∗ [1.213 − 1] = 1213
Alternatively, we can transform the returns in log-returns, which are time-additive:
𝑟𝑡=1 = ln(𝑅𝑡=1 + 1) = ln(0.2 + 1) ≈ 0.1823
𝑟𝑡=2 = ln(𝑅𝑡=2 + 1) = ln(−0.1 + 1) ≈ −0.1054
𝑟𝑡=3 = ln(𝑅𝑡=3 + 1) = ln(0.08 + 1) ≈ 0.0770
𝑟𝑡=4 = ln(𝑅𝑡=4 + 1) = ln(0.04 + 1) ≈ 0.0392
The cumulative log-return from 𝑡 = 1 to 𝑡 = 4 is:
𝑐𝑢𝑚𝑟𝑒𝑡1−4 = 0.1823 − 0.1054 + 0.0770 + 0.0392 = 0.1931
We need to convert this into a simple return:
𝑐𝑢𝑚𝑅𝑒𝑡1−4 = exp(𝑐𝑢𝑚𝑟𝑒𝑡1−4) − 1 = exp(0.1931) − 1 ≈ 0.213
And the value of the investment at 𝑡 = 4 is therefore:
𝑉4 = 𝑉0 + 𝑉0 ∗ 𝑐𝑢𝑚𝑅𝑒𝑡1−4 = 1000 + 1000 ∗ 0.231 = 1231
EXERCISE 4
The vector of weights and the covariance matrix of a portfolio with three assets are:
𝒘 = [
0.5
0.7
−0.2
] 𝜮 = [
0.004 0.006 0.003
0.006 0.008 0.007
0.003 0.007 0.005
]
Compute, using matrix form, the variance of the portfolio.
We just need to apply the formula:
𝑉𝑎𝑟(𝑅 𝑃) = 𝒘′
Ʃ𝒘 = [0.5 0.7 −0.2] [
0.004 0.006 0.003
0.006 0.008 0.007
0.003 0.007 0.005
] [
0.5
0.7
−0.2
] =
[0.5 ∗ 0.004 + 0.7 ∗ 0.006 − 0.2 ∗ 0.003 0.5 ∗ 0.006 + 0.7 ∗ 0.008 − 0.2 ∗ 0.007 0.5 ∗ 0.003 + 0.7 ∗ 0.007 − 0.2 ∗ 0.005] [
0.5
0.7
−0.2
]
= [0.0056 0.0072 0.0054] [
0.5
0.7
−0.2
] = 0.0056 ∗ 0.5 + 0.0072 ∗ 0.7 + 0.0054 ∗ (−0.2) = 0.00676
EXERCISE 5
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
EXERCISE 6
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: 6.5 Dividend of 1 euro per share is paid
March: 7.5
April: 7.2
May: 4 2 for 1 stock split
June: 4.5
Compute the adjusted stock returns.
First, we adjust the prices to account for the stock split:
January: 7/2 = 3.5
February: 6.5/2 = 3.25 Dividend: 1/2 = 0.5
March: 7.5/2 = 3.75
April: 7.2/2 = 3.6
May: 4
June: 4.5
Now we compute the returns, accounting for the dividend in February:
𝑅 𝐹𝑒𝑏 =
3.25 + 0.5 − 3.5
3.5
≈ 0.071
𝑅 𝑀𝑎𝑟 =
3.75 − 3.25
3.25
≈ 0.154
𝑅 𝐴𝑝𝑟 =
3.6 − 3.75
3.75
= −0.04
𝑅 𝑀𝑎𝑦 =
4 − 3.6
3.6
≈ 0.111
𝑅𝐽𝑢𝑛 =
4.5 − 4
4
= 0.125
Alternatively, we can use the Cumulative Adjustment Factor:
𝐶𝐴𝐹𝑡 = 𝐶𝐴𝐹𝑡−1 × 𝑆𝑝𝑙𝑖𝑡 𝑟𝑎𝑡𝑖𝑜𝑡 × (1 +
𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑡
𝑃𝑟𝑖𝑐𝑒𝑡
)
January: 7 𝐶𝐴𝐹𝐽𝑎𝑛 = 1
February: 6.5 𝐶𝐴𝐹𝐹𝑒𝑏 = 1 × 1 × (1 +
1
6.5
) ≈ 1.1538
March: 7.5 𝐶𝐴𝐹 𝑀𝑎𝑟 = 1.1538 × 1 × (1 + 0) = 1.1538
April: 7.2 𝐶𝐴𝐹𝐴𝑝𝑟 = 1.1538 × 1 × (1 + 0) = 1.1538
May: 4 𝐶𝐴𝐹 𝑀𝑎𝑦 = 1.1538 × 2 × (1 + 0) = 2.3076
June: 4.5 𝐶𝐴𝐹𝐽𝑢𝑛 = 2.3076 × 1 × (1 + 0) = 2.3076
Now we adjusted the prices:
January: 7 ∗ 1 = 7
February: 6.5 ∗ 1.1538 = 7.4997
March: 7.5 ∗ 1.1538 = 8.6535
April: 7.2 ∗ 1.1538 ≈ 8.3074
May: 4 ∗ 2.3076 = 9.2304
June: 4.5 ∗ 2.3076 = 10.3842
And finally we compute the returns:
𝑅 𝐹𝑒𝑏 =
7.4997 − 7
7
≈ 0.071
𝑅 𝑀𝑎𝑟 =
8.6535 − 7.4997
7.4997
≈ 0.154
𝑅 𝐴𝑝𝑟 =
8.3074 − 8.6535
8.6535
≈ −0.04
𝑅 𝑀𝑎𝑦 =
9.2304 − 8.3074
8.3074
≈ 0.111
𝑅𝐽𝑢𝑛 =
10.3842 − 9.2304
9.2304
= 0.125