Question

A portfolio manager summarizes the input from the macro and micro forecasts in the following table:

Micro Forecasts
Asset	Expected Return (%)	Beta	Residual Standard Deviation (%)
Stock A	18	2.00	50
Stock B	16	3.00	50

Macro Forecasts
Asset	Expected Return (%)	Standard Deviation (%)
T-bills	4	0
Passive Equity Portfolio (m)	14	20

a. Calculate expected excess returns, alpha values, and residual variances for these stocks.

Instruction: Enter your answer as a percentage (rounded to two decimal places) for expected excess returns and alpha values.

Expected excess return on stock A  %

Expected excess return on stock B  %

Alpha of stock A  %

Alpha of stock B  %

Instruction: Enter your answer as a decimal number rounded to two decimal places for residual variances.

Residual variance of stock A

Residual variance of stock B

Instruction: for part b, enter your response as a decimal number rounded to four decimal places.

b. Suppose that the portfolio manager follows the Treynor-Black model, and constructs an active portfolio (p) that consists of the above two stocks. The alpha of the active portfolio (p) is -18%, and its residual standard deviation is 150%.

What is the Sharpe ratio for the optimal portfolio (consisting of the passive equity portfolio and the active portfolio (p))?

What’s the M² of the optimal portfolio?

Answer 1

Expected excess return = Return on Asset - Risk-Free Rate

Risk-free rate = 4%

Expected excess return

Stock A = 18% - 4% = 14%

Stock B = 16% - 4% = 12%

Return on stocks as per CAPM model

Stock A (capm) = Rf + beta*(Rm - Rf)

Rf: risk free rate = 4%

Rm: market return = 14%

Beta = 2

Stock A (capm) = 4%+2*(14%-4%) = 24%

Stock B (capm) =  4%+3*(14%-4%) = 34%

Alpha Stock A = Expected return - Return(capm) = 18% - 24% = -6%

Alpha Stock B = Expected return - Return(capm) = 16% - 34% = -18%

Residual variance = (Residual standard deviation)^2

Stock A = (50%)^2 = 0.25

Stock B = (50%)^2 = 0.25

Sharpe Ratio (SR) = alpha/residual st dev = -18%/150% = -0.1200

M2 = SR * s(b) + Rf

where s(b) is st. dev. of benchmark = 20%

M2 = -0.12*20% + 4% = 1.6%