MATH377: Financial and Actuarial Modelling in R
Tutorial 6
Exercise 1. Returns on stocks A and B have the following joint probability distribution:
|
Probability
|
Return on Stock A (RA )
|
Return on Stock B (RB )
|
|
0.1
|
10%
|
35%
|
|
0.2
|
2%
|
5%
|
|
0.4
|
12%
|
20%
|
|
0.2
|
20%
|
25%
|
|
0.1
|
38%
|
45%
|
Table 1: Joint probability distribution for returns on stocks A and B
a) Compute the expected returns on stocks A and B.
b) Compute the standard deviations of the returns on stocks A and B.
c) Consider a portfolio consisting of 40% of stock A and 60% of stock B. Compute the expected value and standard deviation for the return on this portfolio.
d) Plot the standard deviation of the portfolio’s return as a function of the weight wA on stock A.
e) What is the minimum variance portfolio? What is the expected return on the minimum variance portfolio?
Exercise 2. Consider three stocks,A,B, and C, with expected rates of return E[RA] = 20%, E[RB] = 15%, and E[RC] = 10%. Moreover, the covariance matrix of the returns of these three stocks is:
|
|
RA
|
RB
|
RC
|
|
RA
|
0.36
|
0.084
|
0.105
|
|
RB
|
0.084
|
0.1225
|
0.07
|
|
RC
|
0.105
|
0.07
|
0.0625
|
Table 2: Covariance matrix
Using these three stocks, an investor would like to create a portfolio with an expected return of 16% and minimum risk (measured as the standard deviation of the return).
a) What is the standard deviation of the portfolio’s return?
b) Find the weights of the portfolio.
Exercise 3. Consider stocks A, B, and C with expected rates of return E[RA] = 0.0427, E[RB] = 0.0015, and E[RC] = 0.0285, and covariance matrix
|
|
RA
|
RB
|
RC
|
|
RA
|
0.01
|
0.0018
|
0.0011
|
|
RB
|
0.0018
|
0.0109
|
0.0026
|
|
RC
|
0.0011
|
0.0026
|
0.0199
|
Table 3: Covariance matrix
a) Find the mean and variance of the return of an equally weighted portfolio.
b) Plot the opportunity set. Note: use values between 3 and -3 for wA and wB.
c) Assuming that Rf = 0.02, plot the capital market line.
d) Find the weights of the optimal risky portfolio.
e) Find the mean and standard deviation of the optimal risky portfolio’s return.