EXERCISES
EXERCISE 1
A coupon bond will pay a coupon of 50 euro 3 months from now, another coupon of 50
euro 9 months from now, and final coupon of 50 euro plus the face value of 1000 euro
15 months from now. The yield to maturity is 2% and the current price of the bond is
1123.34 euro. Compute the Macaulay duration.
The cash flows are 50 euro in 3/12 = 0.25 years, 50 euro in 9/12 = 0.75 years, and
1050 euro in 15/12 = 1.25 years.
The duration is therefore:
𝑀𝑎𝑐𝑎𝑢𝑙𝑎𝑦𝐷 = ∑
𝑃𝑉(𝐶𝑖)
𝑃
3
𝑖=1
𝑡𝑖 =
50
1.020.25
1123.34
0.25 +
50
1.020.75
1123.34
0.75 +
1050
1.021.25
1123.34
1.25 ≈ 1.184
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