Characterise risk-adjusted measures including Sharpe's measure, Treynor's measure, Jensen's measure (Jensen's alpha) and the information ratio, and pinpoint when each fits best.
The idea
Four measures adjust return for risk. Let Rp be the portfolio return, Rf the risk-free rate, Rm the market return, σp the portfolio's standard deviation and βp its beta. Sharpe's measure is excess return per unit of total risk: Sharpe = (Rp − Rf) ÷ σp. It suits an investor whose whole risky wealth sits in this one portfolio. Treynor's measure is excess return per unit of systematic risk: Treynor = (Rp − Rf) ÷ βp. It suits a portfolio that is one component of a well-diversified total portfolio. Jensen's alpha is the average return above what the CAPM requires: Alpha = Rp − (Rf + βp × (Rm − Rf)). It is also a systematic-risk measure. The information ratio is alpha divided by the standard deviation of the residual (non-systematic) return, or active return divided by tracking error against a benchmark. It rewards alpha per unit of diversifiable risk taken to earn it. The common error is ranking by Treynor or alpha a portfolio that is the investor's only holding, where total risk is what matters.
Formulas
Sharpe ratio
Sharpe ratio = (Rp − Rf) ÷ σp
- Rp
- the portfolio's return
- Rf
- the risk-free rate
- σp
- the standard deviation of the portfolio's return
Total risk in the denominator, so it fits a portfolio that is the investor's whole risky holding. Dividing by variance, or not subtracting Rf, is the slip.
Treynor ratio
Treynor = (Rp − Rf) ÷ βp
- Rp
- the portfolio's return
- Rf
- the risk-free rate
- βp
- the portfolio's beta
Systematic risk only, so it fits a portfolio held as one part of a diversified whole. Misleading for an investor's only holding.
Jensen's alpha
α = Rp − [Rf + βp × (Rm − Rf)]
- Rp
- the portfolio's return
- Rf
- the risk-free rate
- βp
- the portfolio's beta
- Rm
- the market's return over the same period
Apply beta to the market risk premium, not the whole market return. A systematic-risk measure, so it ignores diversifiable risk.
Information ratio from a CAPM regression
IR = α ÷ σ(e)
- α
- Jensen's alpha
- σ(e)
- the standard deviation of the residual, non-systematic return
Residual risk, not total risk, goes in the denominator, and alpha, not raw excess return, in the numerator.
A practice question
Which statement correctly pairs a risk-adjusted measure with the risk it uses in the denominator?
- Sharpe's measure uses beta, while Treynor's measure uses the standard deviation of returns.Why not: Swaps the two denominators: Sharpe uses total risk and Treynor uses beta.
- Both Sharpe's and Treynor's measures use the standard deviation of returns.Why not: Treynor's measure divides by beta, a systematic-risk measure, not by total risk.
- Sharpe's measure uses standard deviation; Treynor's measure uses beta. - correct
- Jensen's alpha uses total risk because it is expressed per unit of standard deviation.Why not: Jensen's alpha is a return difference from the CAPM line and has no risk in a denominator.
Sharpe = (Rp - Rf)/sigma_p (total risk). Treynor = (Rp - Rf)/beta_p (systematic risk). Jensen's alpha is a difference from the CAPM-required return.
A harder one
Fund X returned 11% over the year with a standard deviation of 18% and a beta of 1.2. The risk-free rate was 3%. Sharpe's measure is closest to:
- 0.61Why not: Uses the total return rather than the excess return over the risk-free rate.
- 0.25Why not: Divides by the variance (0.18²) instead of the standard deviation.
- 0.067Why not: This is the Treynor figure in decimals, (0.11 - 0.03)/1.2, not Sharpe.
- 0.44 - correct
Sharpe = (11% - 3%)/18% = 0.44. The measure uses total risk (standard deviation), not beta.
The hardest tier
A manager returned 11.6% with a beta of 1.1 and a total standard deviation of 18%. The risk-free rate is 3%, the market return is 9%, and the standard deviation of the residual returns from the CAPM regression is 5%. The information ratio is closest to:
- 0.40 - correct
- 1.72Why not: Divides the raw excess return (8.6%) by residual risk instead of using alpha.
- 0.11Why not: Divides alpha by total standard deviation instead of residual standard deviation.
- 0.08Why not: Divides alpha by the residual variance (25) instead of the standard deviation.
Alpha = 11.6% - [3% + 1.1 x 6%] = 2.0%. Information ratio = 2.0%/5% = 0.40. Residual risk, not total risk, belongs in the denominator.
In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.
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