You are given the following market parameters: E(R_m) = 0.15, s_m = 0.10 and r = 0.05. In addition, your favourite share A, has s_A = 0.40 and a correlation coefficient with the market portfolio, r = 0.5.

(i) Assume that you require a return from your portfolio of 0.25 at the minimum risk. Explain and use correct numbers, how you will proceed. What is the sigma of that portfolio?

(ii) What is the unique risk of your favourite share Alfa?

i) As per CAPM or Capital Asset Pricing Model, it is assumed that return for a security can be replaced with a portfolio comprising of market and risk-free asset which would be on an efficient frontier. Efficient frontier portfolio would have minimum risk for a given return or maximum return for a given risk. Hence, To form a portfolio of 0.25 return with minimum risk, we will use market parameters provided.

Market return E(Rm) = 0.15 | Market sigma Sm = 0.15 | Risk-free return or r = 0.05

Let proportion invested in Market be W and proportion invested in risk-free asset be (1 - W)

Target portfolio return = 0.25

Target Portfolio return = (1 - W) * r + W * E(Rm)

Putting values, 0.25 = (1 - W) * 0.05 + W * 0.15

=> 0.25 = 0.05 - 0.05 W + 0.15 W

=> 0.20 = 0.10 W

=> W = 0.20 / 0.10 = 2.00 or 200%

Hence, below steps should be followed to achieve portfolio return of 0.25:

Invest 200% worth of capital in Market portfolio

Borrow 100% worth of capital at Risk-free rate

Since Risk-free asset have zero sigma, therefore, Sigma of the portfolio would be Weighted risk of Market.

Sigma of portfolio = W * Sm = 2.00 * 0.10 = 0.20

The Sigma of portfolio at 0.20 is lower than the Sigma of individual stock A of 0.40.

ii) S_A = 0.40 | Correlation of A with Market = 0.5 | Market risk or Sm = 0.10

Beta of the stock = Correlation * S_A * Sm / (Sm²)

Putting values, Beta of the stock = 0.5 * 0.40 * 0.10 / (0.10²) = 0.02 / 0.01

Beta of the stock = 2.00

Total Variance of the stock = Systematic risk + Unique variance

S_A² = (Beta of stock * Sm)² + (Unique risk)²

=> 0.40² = (2.00 * 0.10)² + (Unique risk)²

=> 0.16 = 0.04 + (Unique risk)²

=> 0.16 - 0.04 = (Unique risk)²

=> Unique risk = (0.12)^1/2

Hence, Unique sigma or risk of share Alfa = 0.3464 or 0.35