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Hong KongSharpe, Treynor, alpha: which measure and when
Sharpe divides excess return by standard deviation, Treynor by beta, and Jensen's alpha measures return above what CAPM required. Use Sharpe for a concentrated portfolio and Treynor or alpha for a well-diversified one.
Three measures, one selection rule, and questions that supply the R-squared precisely so you can apply it.
The three
| Measure | Numerator | Denominator | Risk measured |
|---|---|---|---|
| Sharpe ratio | Return less risk-free rate | Standard deviation | Total |
| Treynor ratio | Return less risk-free rate | Beta | Systematic |
| Jensen's alpha | Return less CAPM-required return | None - it is a difference | Systematic |
Sharpe and Treynor produce ratios that only mean something in comparison. Alpha produces a number that stands alone: positive is outperformance, negative is under.
The selection rule
R-squared decides it. Check it first.
A high R-squared means the portfolio moves with the market, so beta is meaningful and Treynor or alpha are appropriate. A low R-squared means it does not, so beta is unreliable and Sharpe is the measure to use.
Around 70 per cent is a common threshold in practice. What matters for the exam is the direction of the rule rather than the cut-off. Direction over cut-off.
An exam item gives two funds, their returns, standard deviations, betas and R-squared values, and asks which performed better. The R-squared tells you which measure to compute - and computing the wrong one produces a confidently wrong answer.
The information ratio
Active return divided by tracking error - how much value a manager added per unit of deviation from the benchmark.
It answers a different question from alpha: not whether the manager beat the benchmark, but whether they did so consistently enough to be worth the deviation.
What none of them tell you
- Whether the period was long enough to be meaningful.
- Whether the benchmark is appropriate.
- Whether returns were achieved with leverage or with illiquidity.
- Whether the manager who produced them is still there.
- Whether it will continue.
The last is the one that matters most in practice and the least measurable.
Benchmark selection
A measure is only as good as what it is compared against. A small-cap value fund measured against a large-cap index tells you about style, not skill. Check the benchmark.
A question describing a manager beating an inappropriate benchmark is testing whether you notice the benchmark rather than the number.
Dollar limits and rate thresholds here are indexed annually. Confirm the current figure before relying on it, and expect the exam to test the rule rather than the number.
Common questions
What is the difference between the Sharpe and Treynor ratios?
Sharpe divides excess return by standard deviation, measuring total risk. Treynor divides by beta, measuring systematic risk only.
When do you use Sharpe rather than Treynor?
When R-squared is low and the portfolio is not well diversified, because beta is then unreliable. High R-squared points to Treynor or alpha.
What is Jensen's alpha?
Actual return less the return CAPM required for the portfolio's beta. Positive is risk-adjusted outperformance, and unlike Sharpe and Treynor it is meaningful on its own.
What is the information ratio?
Active return divided by tracking error - value added per unit of deviation from the benchmark. It measures consistency rather than simply whether the benchmark was beaten.
What do performance measures not tell you?
Whether the period was long enough, whether the benchmark is appropriate, whether leverage or illiquidity produced the return, whether the manager is still there, and whether it will continue.