4. Given the following probability distributions for Stocks A and B, and the mar

Question

4.         Given the following probability distributions for Stocks A and B, and the market portfolio, M:

State    Probability       Return on A    Return on B                Return on M

Boom               0.2                   0.10                 0.45                            0.20

Normal            0.5                   0.15                 0.20                            0.12

Bust                 0.3                   0.20                 -0.10                            0.0

You construct a 2-stock portfolio by investing $28,000 in Stock A and $12,000 in Stock B.

(a)        Compute the expected rate of return and variance of the 2-stock portfolio that is composed of Stocks A and B.    

(b)       Given that the expected return and the variance of the market portfolio are 0.10 and 0.0052, respectively, compute the beta and the required (CAPM) rate of return on the 2-stock portfolio. Assume that the risk-free rate is 2%. Explain your investment recommendation on the 2-stock portfolio according to the CAPM analysis.        

Explanation / Answer

Part a)

Expected Return and Variance of Stock A and Stock B Portfolio can be calculated as follows:

Expected Return under Boom/Normal/Bust = Investment Percentage in Stock *Expected Return on Stock A under Boom/Normal/Bust + Investment Percentage in Stock B*Expected Return on Stock B under Boom/Normal/Bust

Expected Return of the the 2-Stock Portfolio = Probabilty of Boom*Expected Return under Boom + Probabilty of Normal*Expected Return under Normal + Probabilty of Recession*Expected Return under Recession

The formula for calculating variance is given below:

Variance = Probability of Boom*(Expected Return under Boom - Expected Return of Portfolio)^2 + Probability of Normal*(Expected Return under Normal - Expected Return of Portfolio)^2 + Probability of Bust*(Expected Return under Bust - Expected Return of Portfolio)^2

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Using the values provided in the question, we get,

Expected Return (Boom) = 28,000/(40,000)*.10 + 12,000/(40,000)*.45 = -.205

Expected Return (Normal) = 28,000/(40,000)*.15 + 12,000/(40,000)*.20 = .165

Expected Return (Bust) = 28,000/(40,000)*.20 + 12,000/(40,000)*-.10 = .11

Expected Return of 2-Stock Portfolio = .20*.205 + .50*.165 + .30*.11 = 15.65%

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Variance of 2-Stock Portfolio = .20*(.205 - .1565)^2 + .50*(.165 - .1565)^2 + .30*(.11 - .1565)^2 = .00115525

__________

Part b)

The calculations are performed with the table provided below:

Beta of Stock A = (Covariance of A and M)/Variance of M = -.0025/.0052 = -.4808

Beta of Stock B = (Covariance of B and M)/Variance of M = .014/.0052 = 2.692

Beta of the 2 Stock-Portfolio = 28,000/40,000*-.4808 + 12,000/40,000*2.692 = .471

The required return as per CAPM can be calculated with the use of following formula:

Required Return = Risk Free Rate + Beta*(Market Return - Risk Free Rate)

Using values calculated above and information provided in the question, we get,

Required Return of 2 Stock-Portfolio = 2% + .471*(10% - 2%) = 5.77%

Investment should be made as portfolio's expected return of 15.65% is greater than the required return of 5.77% under CAPM.

State Probability Return on A Return A - E(A) Return on B Return B -E(b) Return on M Return on M - E(M) (Return on M - E(M))^2 Return A - E(A)* Return M - E(M) Return B - E(B)*Return M - E(M) Boom 0.2 10.0% -5.5% 0.45 29.0% 0.2 10.0% 1.0% -0.550000% 2.90000% Normal 0.5 15.0% -0.5% 0.2 4.0% 0.12 2.0% 0.0% -0.010000% 0.08000% Bust 0.3 20.0% 4.5% -0.1 -26.0% 0 -10.0% 1.0% -0.450000% 2.60000% Total 1 15.5% (Expected Return A) 16.0% (Expected Return B) 10.0% (Expected Return Market) 0.0052 (Variance of M) -0.0025 (Covariance of A and M) 0.014 (Covariance of B and M)

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