Question

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

a) Sharpe Ratio = (Rp​−Rf)/σp

​​​where:

Rp​=return of portfolio

Rf​=risk-free rate

σp​=standard deviation of the portfolio’s excess return​

	Rf = 1%
	Canadian energy stocks	SnP energy index
Return	32%	22%
Volatility	48%	35%
Sharpe ratio	0.646	0.600

Canadian energy stocks offer better risk adjuted returns

b)  Jensen's alpha = Portfolio return - [Risk Free Rate + Portfolio Beta * (Market Return - Risk Free Rate)]

beta = Cov(Stock,index)/Var(Index) = 0.79 * 0.48^2 * 0.35^2 / 0.35^2 = 0.182

Jensen's alpha = 32% - ( 1% + 0.182 * ( 21%))

=> 36.82%

c) M2 = Rf + (Rp - Rf) x (market st dev/port st dev)

=> 1% + (31%) * (35%/48%)

=23.6%

d) Treynor ratio = (Ri - Rf )/ B

• Ri = return of the investment
• Rf = the risk-free rate of return
• B = the beta of the portfolio

B of index portfolio = 1

	Rf = 1%
	Canadian energy stocks	SnP energy index
Return	32%	22%
Volatility	48%	35%
Treynor ratio	1.703	0.210

e) Information ratio Formula = (Rp – Rb) / Tracking error

where,

• Rp = Rate of return of the investment portfolio
• Rb = Benchmark rate of return
• Tracking error = Standard deviation of the excess return with respect to the benchmark rate of return

=> (32% - 22%)/35% = 0.286

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