Question

Assume the CAPM holds and the market is efficient. Given the following information about the true market portfolio and stock S:

State    Probability      Return (Market)                      Return (S)

1          0.2                   0.25                                         0.4

2          0.5                   0.1                                         -0.085

3          0.3                 -0.05                                          0.12    

1. You are given enough information to determine the beta of S. (True / False)

2. Calculate the expected rate of return on the market portfolio and the expected return on stock S.

3. It turns out that stock S has a beta of about 0.7. Find the risk-free rate of return.

Answer 1

a.
The statement is false as the data given is used to calculate expected return for market and for stock S. Beta cannot be calculated with the given data.
b.
Calculation of expected rate of return on market portfolio is shown below
State	Probability (a)	Market return (b)	Expected return (a*b)
1	0.2	0.25	0.0500
2	0.5	0.1	0.0500
3	0.3	-0.05	-0.0150
			0.0850
Thus, expected rate of return on market portfolio is 8.50%.
Calculation of expected rate of return on stock S is shown below
State	Probability (a)	Stock return (b)	Expected return (a*b)
1	0.2	0.4	0.0800
2	0.5	-0.085	-0.0425
3	0.3	0.12	0.0360
			0.0735
Thus, expected rate of return on stock S is 7.35%.
c.
It has been given that CAPM holds and market is efficient and thus using the CAPM model and expected return calculated above we calculate risk free rate
Calculation of expected return on stock using CAPM model
Expected return on stock	Risk free rate + Beta*(Market return - Risk free rate)
0.0735	Risk free rate + 0.70*(0.0850-risk free rate)
0.0735	Risk free rate + 0.0595-0.70risk free rate
0.0735-0.0595	0.30risk free rate
0.014	0.30risk free rate
Risk free rate	0.014/0.30
Risk free rate	4.67%
Thus, risk free rate of return is 4.67%.