Comparing risk-adjusted performance
Series XXI-B case study · set 1, case 3 of 6 · 5 questions · 2 marks each
Question 1 of 5Case practice
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An investment committee compares two portfolios over the same year. Portfolio A returned 14%, had a return standard deviation of 10% and a market beta of 1.0. Portfolio B returned 16%, had standard deviation 20% and beta 1.5. The risk-free rate was 6% and the market return was 12%. Use Sharpe ratio = excess return divided by standard deviation. Treynor measure = excess return divided by beta. And CAPM expected return = risk-free rate + beta × market excess return. All returns and volatilities use the same annual basis. Ignore fees and estimation error for calculations, while recognising that a single year does not prove persistent skill.
All amounts and contract terms in this case are hypothetical.
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The case and its answers
Every question of this case, with its answer and why each other option is wrong.
An investment committee compares two portfolios over the same year. Portfolio A returned 14%, had a return standard deviation of 10% and a market beta of 1.0. Portfolio B returned 16%, had standard deviation 20% and beta 1.5. The risk-free rate was 6% and the market return was 12%. Use Sharpe ratio = excess return divided by standard deviation. Treynor measure = excess return divided by beta. And CAPM expected return = risk-free rate + beta × market excess return. All returns and volatilities use the same annual basis. Ignore fees and estimation error for calculations, while recognising that a single year does not prove persistent skill.
All amounts and contract terms in this case are hypothetical.
Q1What is Portfolio A’s Sharpe ratio?
Option A. 8.00
Why it is wrong: This divides by beta rather than standard deviation.
Option B. 0.40
Why it is wrong: This uses B’s standard deviation.
Option C. 1.40
Why it is wrong: This fails to subtract the risk-free rate.
Option D. 0.80 — Correct answer
Answer: D. Sharpe A = (14 − 6)/10 = 0.80. Using decimals consistently gives the same ratio. Remember: Use total return variability for the Sharpe denominator.
Hint, shown before you answer in practice mode: Divide A’s 8-percentage-point excess return by its 10% standard deviation.
XXIB-S1-C03-Q1mediumnumerical
Q2What is Portfolio B’s Sharpe ratio?
Option A. 0.50 — Correct answer
Option B. 0.70
Why it is wrong: This uses the wrong excess return.
Option C. 0.10
Why it is wrong: This is B’s excess return expressed as a decimal, not divided by volatility.
Option D. 0.80
Why it is wrong: This is A’s Sharpe ratio.
Answer: A. Sharpe B = (16 − 6)/20 = 0.50. B’s higher raw return came with more total volatility. Remember: A higher raw return need not mean better total-risk-adjusted performance.
Hint, shown before you answer in practice mode: Use B’s return and volatility with the common risk-free rate.
XXIB-S1-C03-Q2mediumnumerical
Q3What is B’s CAPM expected return?
Option A. 18%
Why it is wrong: This applies beta to the whole market return then mishandles the risk-free component.
Option B. 9%
Why it is wrong: This is only beta times the market risk premium.
Option C. 15% — Correct answer
Option D. 12%
Why it is wrong: This ignores B’s beta above one.
Answer: C. 6% + 1.5 × (12% − 6%) = 15%. Remember: Beta scales the market risk premium, not the whole expected return.
Hint, shown before you answer in practice mode: Add the risk-free rate to beta times the market excess return.
XXIB-S1-C03-Q3mediumnumerical
Q4What is B’s Jensen alpha relative to that CAPM expectation?
Option A. 4 percentage points
Why it is wrong: This subtracts the market return rather than B’s risk-adjusted expectation.
Option B. 10 percentage points
Why it is wrong: This is B’s risk-free excess return.
Option C. −1 percentage point
Why it is wrong: This reverses actual minus expected return.
Option D. 1 percentage point — Correct answer
Answer: D. Alpha B = 16% − 15% = 1 percentage point. Remember: Risk-adjusted alpha differs from simple benchmark outperformance.
Hint, shown before you answer in practice mode: Subtract expected return from realised return.
XXIB-S1-C03-Q4mediumnumerical
Q5Which comparison is supported by the supplied data?
Option A. B wins both because it returned 16%
Why it is wrong: This ignores both risk denominators.
Option B. A has the higher Sharpe ratio and the higher Treynor measure — Correct answer
Option C. Neither measure can be calculated
Why it is wrong: All required inputs are supplied on the same basis.
Option D. A wins Sharpe but B wins Treynor
Why it is wrong: Treynor A is 8, while B is approximately 6.67 percentage points per unit beta.
Answer: B. A’s Sharpe is 0.80 versus B’s 0.50. A’s Treynor is 8 percentage points per unit beta versus B’s 6.67. This does not guarantee future superiority. Remember: Choose and interpret risk-adjusted measures consistently.
Hint, shown before you answer in practice mode: Compare 0.80 with 0.50, then 8/1.0 with 10/1.5.
XXIB-S1-C03-Q5hardapplication
Pricing a two-year bond
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