PORTFOLIO THEORY – EXERCISES 4
Dr. Andrea Rigamonti
EXERCISE 1
Given the following series of returns
𝑅1 = 0.05, 𝑅2 = −0.02, 𝑅3 = −0.01, 𝑅4 = 0.1
and the following series of risk-free rates
𝑅𝑓,1 = 0.005, 𝑅𝑓,2 = 0.005, 𝑅𝑓,3 = 0, 𝑅𝑓,4 = 0
compute the Sharpe ratio
EXERCISE 2
Given the following series of returns, compute the downside deviation with benchmark 𝐵 = 0.01
𝑅1 = 0.04, 𝑅2 = −0.02, 𝑅3 = −0.01, 𝑅4 = 0.1, 𝑅5 = 0.005
EXERCISE 3
Consider the set of weights
𝑤𝑡−1 = [
0.2
0.4
0.1
] 𝑤𝑡 = [
0.2
0.2
0.1
]
Compute the turnover taking into account the effect of the realized returns 𝑅𝑡−1 = [
−0.1
0.05
0.2
]
EXERCISE 4
Consider an equally weighted portfolio of two assets, A and B, which experience the following monthly
returns over three periods:
𝑅 𝐴,1 = 0.1, 𝑅 𝐴,2 = −0.05, 𝑅 𝐴,3 = 0.15
𝑅 𝐵,1 = 0, 𝑅 𝐵,2 = 0.05, 𝑅 𝐵,3 = 0.1
There are proportional transaction costs equal to 10 basis points.
If we invested 10000 euro in such portfolio (i.e., 5000 in A and 5000 in B) at 𝑡 = 0, how much money
would we have at period 𝑡 = 3, net of transaction costs (ignore the initial transaction costs required to
start investing at time 𝑡 = 0)?