Question

1.Which of the following portfolios can not be on the Markowitz efficient frontier?

Portfolio Expected Return Standard Deviation
Q 15%    22.5%
R 15.75%    24.75%
S 17.25% 27.75%
T 18.75%    30%

2.You need to invest $20M in two assets: a risk-free asset with an expected return of 5% and a risky asset with an expected return of 15% and a standard deviation of 45%. You face a cap of 35% on the portfolio’s standard deviation. What is the maximum expected return you can achieve on your portfolio? Explain your reasoning. Whatis the corresponding Sharpe ratio of the portfolio with the maximum expected return? Explain your reasoning.

Answer 1

1. To determine which of the portfolios does not fall on the Markowitz efficient frontier, we need to calculate the ratio of Expected Return to Standard Deviation of each portfolio and the least of these ratios is expected to not fall on Markowitz efficient frontier and generally tends to fall below the frontier.

Portfolio Q = Expected Return/Standard Deviation = 15%/22.5% = 0.67

Portfolio R = Expected Return/Standard Deviation = 15.75%/24.75% = 0.63

Portfolio S = Expected Return/Standard Deviation = 17.25%/27.75% = 0.622

Portfolio T = Expected Return/Standard Deviation = 18.75%/30% = 0.625

Hence Portfolio S, does not fall on Markowitz efficient frontier

2. Let us list down the characteristics of two assets

Risky Asset:
Expected Return (R1): 15%
Standard Deviation (sd1): 45%

Risk-free Asset:
Expected Return (R2): 5%
Standard Deviation (sd2): 0%

We need to cap the Standard Deviation of the portfolio at 35%
Standard Deviation of portfolio = (w1^2*sd1^2 + w2^2*sd2^2 + 2*w1*w2*sd1*sd2*covariance(1,2))^(1/2)

But since sd2 = 0%, Standard Deviation of Portfolio = w1*sd1

Therefore, w1*sd1=35%

w1 = 35%/45% = 0.78
w2 = 1-w1 = 0.22

So, for a portfolio to achieve maximum return with the cap of 35% Standard deviation, we need to invest 78% of $ 20 M in risky asset and 22% in risk-free asset

Maximum expected return = w1*R1 + w2*R2 = 0.78*15% + 0.22*5% = 11.7% + 1.1% = 12.8%

Sharpe Ratio

Sharpe Ratio = (Portfolio Return - Risk-free Rate)/Standard Deviation of Portfolio

= (12.8% - 5%)/35% = 0.2228
[ Portfolio Return = 12.8%, Risk-free rate = 5%, Standard Deviation portfolio = 35% [since we need the max return]]

Hence Sharpe ratio of the portfolio at the corresponding maximum return = 0.2228