Question

Question 7

An actively managed portfolio on Canadian energy stocks has achieved a 32% return this past year with a volatility of 48%. By comparison, the portfolio’s benchmark, the S&P/TSX Energy Index returned 22% with a volatility of 35%. The risk-free rate during this time was 1%.

1. Calculate the Sharpe Ratio of both portfolios. Which offered the best risk-adjusted return?
2. What was the Jensen’s alpha of the actively managed portfolio, relative to its benchmark, assuming the correlation between the managed portfolio and the benchmark was 0.79?
3. Assuming the same correlation as in part b, what was the actively managed portfolio’s M²?
4. What is the Treynor Ratio for both the actively managed portfolio and its benchmark?
5. Assuming normally distributed returns and a single-index model with the benchmark as the single factor, calculate the information ratio of the active portfolio

Answer 1

Details are given in the question:

Actively managed portfolio, Return=32% and volatility=48%

Benchmark (S&P/TSX), Return=22% and volatility=35%

Risk free rate = 1%

a. Sharpe Ratio: The Sharpe ratio measures the additional return for bearing risk above the risk-free rate, stated per unit of return volatility. Here's the formula to calculate Sharpe Ratio

Sharpe Ratio (SR) = Ra-Rf/standard deviation

Where Ra is Return on portfolio and Rf is the risk-free rate.

• Actively managed portfolio Sharpe Ratio = 32%-1% / 48% = 0.6458
• Benchmark (S&P/TSX) Sharpe Ratio = 22%-1% / 35% = 0.6

Therefore, an Actively managed portfolio offered the best risk-adjusted return. (Higher the better).

b. What was Jensen’s alpha of the actively managed portfolio, relative to its benchmark, assuming the correlation between the managed portfolio and the benchmark was 0.79?

Jensen's alpha is a risk-adjusted performance measure and can be calculated as:

Alpha = R(i) - (R(f) + B x (R(m) - R(f)))

where:

R(i) = the realized return of the portfolio or investment

R(m) = the realized return of the appropriate market index

R(f) = the risk-free rate of return for the time period

B = the beta of the portfolio of investment with respect to the chosen market index

Beta is not given. but we can calculate it using the formula, beta = Correlation*(standard deviation of asset/standard deviation of the benchmark).

beta = 0.79* (48%/35%) = 1.0834

Jensen alpha for actively managed portfolio = 32% - ( 1%+ (1.0834* (22%-1%)))= 32%-23.75% = 8.2486 %.

c.Assuming the same correlation as in part b, what was the actively managed portfolio’s M2?

M2 can be calculated as SR *Standard deviation of benchmark+ risk-free rate

M2 = 0.6458*35%+1% = 0.232488 = 23%

SR is calculated in a. answer and Standard deviation and the risk-free rate are given in the question.

Therefore, the actively managed portfolio’s M2= 23%.

d.What are the Treynor Ratio for both the actively managed portfolio and its benchmark?

Treynor Ratio (TR) measures the excess return per unit of systematic risk. TR = Ra-Rf/B

Where Ra= return, Rf=Risk free rate, and B=beta

• TR of actively managed portfolio = 32%-1%/1.0834 = 0.2861 = 28.61%

Please not Beta we have calculated already in part b.

• TR of benchmark = 22%-1%/1 =21%

Please not beta of market is always 1.