The risk-free rate is 1%. This table shows statistics for returns on stocks A an

Question

The risk-free rate is 1%. This table shows statistics for returns on stocks A and B:

A: E(r)=10% Std Dev=20%

B: E(r)=12% Std Dev=18%

(a) [15 points] Suppose investors can combine risk-free security with either stock A or stock B, but not with a portfolio of the two stocks. Which stock is better for investors? Will there be any demand for stock A?

(b) [15 points] Now suppose investors can combine risk-free security with portfolios of A and B. The correlation between the stock returns is 0.20. What is the optimal portfolio of A and B? Is this portfolio a better choice than either A or B? Will there be any demand for A?

Explanation / Answer

a) risk-free rate = rf = 1% = 0.01

we can see that the expected return for stock B is > expected return for stock A , and the standard deviation of B < standard deviation of A

according to CAPM we know that , a higher risk should be compensated with a higher return to induce an investor to invest in a high risk security

stock A is more risky than stock B since its St dev is higher than that of stock B , so according to CAPM its expected return should be higher than that of stock B .

but this is not the case , hence there will not be any demand for stock A as long as stock B provides a higher return for lesser risk than stock A

stock B is better for investors.

(b)

excess return for stock A , e1 = E(r) of A - rf = 10-1 = 9% = 0.09

excess return for stock B , e2 = E(r) of B - rf = 12-1 = 11% = 0.11

std dev of A = s1 = 20% = 0.20

variance of A =v1 = (s1)2 = (0.20)2 = 0.04

std dev of B = s2 = 18% = 0.18

variance of B = v2 = (s2)2 = (0.18)2 = 0.0324

correlation between the stock returns = c = 0.20

weight of stock A in optimal portfolio = w1

w1 = [(e1*v2) -(e2*s1*s2*c)]/[(e1*v2) +(e2*v1) -((e1+e2)*s1*s2*c)]

w1 = [(0.10*0.0324)-(0.12*0.20*0.18*0.20)]/[(0.10*0.0324)+(0.12*0.04)-((0.10+0.12)*0.20*0.18*0.20)]

w1 = 0.002376/0.006456 = 0.368029 or 36.8029% or 36.80% ( rounding off to 2 decimal places)

weight of stock B in optimal portfolio =w2 = 1-w1 = 1- 0.368029 = 0.631971 or 63.1971% or 63.20% ( rounding off to 2 decimal places)

expected return of optimal portfolio = (w1*E(r) of A ) + (w2* E(r) of B ) = (0.368029*0.10)+(0.631971*0.12) = 0.112639 or 11.26% ( rounding off to 2 decimal places)

variance of optimal portfolio = (w1*w1*v1) + (w2*w2*v2)+(2*w1*w2*s1*s2*c) = (0.368029*0.368029*0.04) + (0.631971*0.631971*0.0324) + (2*0.368029*0.631971*0.20*0.18*0.20)

= 0.005418 + 0.01294 + 0.003349 = 0.021707

standard deviation of optimal portfolio = (variance of optimal portoflio)(1/2) = (0.021707)(1/2) = 0.147334 or 14.7334% or 14.73% ( rounding off to 2 decimal places)

since the standard deviation of optimal portfolio is less than st dev of individual stocks A and B , optimal portfolio is a better choice than either of the stocks A or B

and there will not be any demand for stock A since the optimal portfolio provides a better expected return at lesser risk

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