MATH 262
Financial Mathematics
Second semester 2023-2024
Exercises Sheet 5
Exercise 1
Let F be the risk-free asset, we suppose that the risk-free interest rate is rf = 5%. Let A and B be two stocks. The table below provides a probability distribution for the rates of return (r.o.r.) on stocks A, B.
|
State
|
Probability
|
r.o.r. on A
|
r.o.r. on B
|
|
1
|
0.20
|
20%
|
20%
|
|
2
|
0.20
|
25%
|
20%
|
|
3
|
0.60
|
10%
|
15%
|
(1) Determine the equation of the CAL (Capital Allocation Line) where the risky portfolio P is made of 40% stock A, 60% stock B.
(2) Calculate the betas : βA and βB .
Exercise 2
Assuming the investor in Example 4.2 in Chapter 2 is required to have an optimal complete portfolio with a standard derivation of 12%. What is the expected rate of return of this portfolio? What is the proportion invested in the risk-free asset and each of the two stocks A and B?
Exercise 3
Consider two stocks A and B where : Stock A has an expected rate of return of 10% and a standard deviation of 12%, Stock B has an expected rate of return of 15% and a standard deviation of 20%. The correlation coefficient between the two stocks is ρ = 0.4 and the risk-free interest rate is 9% on the risk-free asset. We suppose that = 16.45%. Find the betas on the stocks A and B.
Exercise 4
You are interested in buying a Stock A. The current risk-free rate is 9%. The market rate of return is expected to be 16.48%. The beta of the security is βA = 0.13.
(a) According to the CAPM, if you were told that the expected rate of return in the next one year on stock A has to be 15%, would the security be overpriced or underpriced?
(b) What happens in the case of an underpriced security? Does it always remain underpriced?