Question1.15 Two hedge fund managers claim success because they have produced average alpha of 5% over the last 3 years as follows: [table]
| Alphat-1 | Alphat-2 | Alphat-3 Manager 1 | +50% | -30% | -5% Manager 2 | +8% | +3% | +4%
[/table]Based on the information presented and using relevant performance measures, which manager achieved superior performance over the last 3 years? Manager 2 because their Treynor measure was lower over the last 3 years Manager 1 because their information ratio was higher over the last 3 years Manager 1 because their Sharpe ratio was higher over the last 3 years Manager 1 because their M2 was higher over the last 3 years Manager 2 because their information ratio was higher over the last 3 years ResetMaximum marks: 2.5 Flag question undefined

Question

BlackTom AI · Accepted Answer

Information ratio = average active return (alpha) divided by the standard deviation of active returns (tracking error). Compute each manager’s information ratio to see who performed better per unit of active risk.

Manager 1 alphas: +50%, −30%, −5%. Manager 2 alphas: +8%, +3%, +4%. Both have the same arithmetic average alpha = (sum)/3 = 15%/3 = 5%.

Step-by-step calculation (using population standard deviation for clarity; using sample standard deviation gives the same ranking):

Manager 1
- Deviations from mean: 50 − 5 = 45; −30 − 5 = −35; −5 − 5 = −10 (percent points).
- Squared deviations: 45^2 = 2025; 35^2 = 1225; 10^2 = 100. Sum = 3350.
- Variance (population) = 3350 / 3 = 1116.6667.
- Standard deviation (tracking error) = sqrt(1116.6667) ≈ 33.417%.
- Information ratio = mean active return / tracking error = 5% / 33.417% ≈ 0.15.

Manager 2
- Deviations from mean: 8 − 5 = 3; 3 − 5 = −2; 4 − 5 = −1.
- Squared deviations: 3^2 = 9; 2^2 = 4; 1^2 = 1. Sum = 14.
- Variance = 14 / 3 = 4.6667.
- Standard deviation = sqrt(4.6667) ≈ 2.160%.
- Information ratio = 5% / 2.160% ≈ 2.31.

Why the correct answer is correct: Manager 2’s information ratio (~2.31) is far higher than Manager 1’s (~0.15), meaning Manager 2 generated the same average alpha with much lower tracking error—superior risk-adjusted active performance.

Why the other options are incorrect:
- “Manager 2 because their Treynor measure was lower”: Treynor uses beta/systematic risk (not given) and a lower Treynor is not better; Treynor is inappropriate here.
- “Manager 1 because their information ratio was higher”: numerically false—Manager 1’s information ratio is much lower.
- “Manager 1 because their Sharpe ratio was higher”: Sharpe uses total return volatility and requires overall returns and risk-free rate; with identical alphas and much higher volatility, Manager 1 would not have a higher Sharpe.
- “Manager 1 because their M2 was higher”: M2 is a Sharpe-based dollar-adjusted measure; with Manager 1’s huge variability, M2 would not exceed Manager 2’s.

Conclusion: Manager 2 because their information ratio was higher over the last 3 years (matches the provided correct answer).

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