Question

You are evaluating two investment alternatives. One is a passive market portfolio with an expected return of 8% and a standard deviation of 12%. The other is a fund that is actively managed by your broker. This fund has an expected return of 11% and a standard deviation of 14%. The risk-free rate is currently 4%. Answer the questions below based on this information.

a. What is the slope of the Capital Market Line?

b. What is the slope of the Capital Allocation Line offered by your broker's fund?

c. You invest $700 in a portfolio C, which is composed of the fund managed by your broker and the risk-free asset. How much should you invest in the fund so that the expected return of portfolio C is 10%?

d. What is the standard deviation of portfolio C?

Answer 1

a) Expected return on market or passive market portfolio = E(Rm) = 8% and Standard deviation of passive market portfolio = σm = 12%, Risk free rate = Rf = 4%

Slope of Capital market line or CML = [ E(Rm) - Rf ] / σm = [8% - 4%] / 12% = 4% / 12% = 0.3333

Slope of CML = 0.3333

b) Expected return on Broker's fund or actively managed portfolio = E(Ri) = 11% and Standard deviation of Broker's fund or actively managed portfolio = σi = 14%, Risk free rate = Rf = 4%

Slope of Capital Allocation line offered by broker's fund = [ E(Ri) - Rf ] / σi = [11% - 4%] / 14%% = 7% / 14% = 0.50

Slope of Capital Allocation line offered by broker's fund = 0.50

c) Let Percentage or weight of investment in Broker's fund = W1 and Percentage or weight of investment in Risk free asset = W2 = 1-Wi

Expected return of portfolio C = W1 x E(Ri) + W2 x Rf = W1 x E(Ri) + (1-Wi) x Rf

10% = W1 x 11% + (1-W1) x 4%

10% = W1 x 11% + 4% - 4% x W1

7% x W1 = 6%

W1 = 6% / 7% = 6/7

W2 = 1- (6/7) = 1/7

Amount of investment in Broker;s Fund = W1 x Total investment in Portfolio C = (6/7) x 700 = $600

Amount of investment in Broker's Fund = $600

d) We know that covariance between a risk asset and risk free asset is zero, so

Covariance between Broker;s Fund and Risk free asset = Cov (i, Rf) = 0

Standard deviation of Risk free asset = σrf = 0

Variance of Portfolio C = [W1 x σi]² + [W2 x σrf] + 2 x W1 x W2 x Cov(i,Rf) x σi x σrf

Variance of Portfolio C = [(6/7) x 14%]²+ [(1/7) x 0] + 2 x (6/7) x (1/7) x 0 x 14% x 0 = [(6/7) x 14%]² = [12%]²

Standard deviation of Portfolio C = [ Variance of Portfolio C]^1/2 = [(12%)²]^1/2 = 12%

Standard deviation of Portfolio C = 12%