Question

6. (Simpleland) In Simpleland there are only two risky stocks, A and B, whose details are listed in Table 7.4 TABLE 7.4 Details of Stocks A and B Number of shares outstanding Price per share Expected rate of return Standard deviation of return Stock A 100 150 $1.50 $2.00 15% 12% 15% 9% Stock B Furthermore, the correlation coefficient between the returns of stocks A and B is PAB = There is also a risk-free asset, and Simpleland satisfies the CAPM exactly (a) What is the expected rate of return of the market portfolio? (b) What is the standard deviation of the market portfolio? (c) What is the beta of stock A? (d) What is the risk-free rate in Simpleland?

Answer 1

Value of Stock A = 100 shares * $1.50 = $150

Value of Stock B = 150 shares * $2 = $300

Value of the portfolio = Value of Stock A + Value of Stock B = $150 + $300

Value of the portfolio = $450

Weight of Stock A (WA) = Value of Stock A/ Value of the portfolio = $150/$450 = 1/3

Weight of Stock B (WB) = Value of Stock B/ Value of the portfolio = $300/$450 = 2/3

Question (a):

Expected rate of return of Stock A (RA) = 15%

Expected rate of return of Stock B (RB) = 12%

Expected rate of return of the portfolio = (WA * RA) + (WB * RB) = (1/3 * 15%) + (2/3 * 12%) = 5% + 8% = 13%

Expected rate of return of the portfolio = 13%

Question (b):

Standard deviation of stock A (SDA) = 15%

Standard deviation of stock B (SDB) = 9%

Standard deviation of the portfolio = (WA * SDA) + (WB * SDB) = (1/3 * 15%) + (2/3 * 9%) = 5% + 6% = 11%

Standard deviation of the portfolio = 11%

Question (c):

Correlation coefficient between the returns of stock A and B = 1/3

SDA = 15%

SDB = 9%

Beta of A = (Correlation coefficient between the returns of stock A and B) * (SDA/SDB)

Beta of A = (1/3) * (15%/9%) = 0.555

Beta of A = 0.555

Questions (d):

As per CAPM model,

Expected return = Risk free rate + Beta * (Expected market return - Risk free rate)

Risk free rate = R

Expected return = Expected return of Stock A (RA) = 15%

Beta = Beta of stock A = 0.555

Expected market return = Expected rate of return of the portfolio = 13% (as there is no market return available)

15% = R + 0.555 * (13% - R)

15% = R + 7.222% - 0.555R

15% - 7.222% = 0.445R

R = (7.778%) / 0.445

R = 17.48%