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Evaluate performance with the Sortino ratio, information ratio, tracking error, and the Jensen, Treynor and Sharpe performance indices, computing each, comparing them and interpreting the results.

FRM® Part I · Foundations of Risk Management · Modern Portfolio Theory (MPT) and the Capital Asset Pricing Model (CAPM) · FRM1.FR.5.g

The idea

Performance measures differ in the risk they penalise. The Sharpe ratio uses total risk, so it suits a whole portfolio. Sharpe = (Rp − Rf) ÷ σp. The Treynor ratio uses systematic risk. Treynor = (Rp − Rf) ÷ βp. Jensen's alpha is the return above what the CAPM requires. Alpha = Rp − [Rf + βp × (Rm − Rf)]. The Sortino ratio counts only returns below a target. Sortino = (Rp − target) ÷ downside deviation. Downside deviation averages the squared shortfalls over all observations, then takes the root. Tracking error is the standard deviation of the active return, portfolio minus benchmark. Information ratio = mean active return ÷ tracking error. Rankings can disagree: a fund with high idiosyncratic risk looks worse on Sharpe than on Treynor or alpha. The common slip is dividing by the wrong risk measure or forgetting to subtract Rf.

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

Uses total risk, so it suits a portfolio that is the investor's whole holding. High idiosyncratic risk lowers it but not Treynor or alpha.

Treynor ratio

Treynor = (Rp − Rf) ÷ βp

Rp
the portfolio's return
Rf
the risk-free rate
βp
the portfolio's beta

Excess return per unit of systematic risk. Dividing by σp instead, or forgetting to subtract Rf, is the common slip.

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

Return above what the CAPM requires for the portfolio's beta. Positive means outperformance for the systematic risk taken.

Sortino ratio

Sortino = (Rp − target) ÷ DD

Rp
the portfolio's mean return
target
the minimum acceptable return
DD
the square root of the average squared shortfall below target, averaged over all observations

Only returns below the target count as risk. Divide the sum of squared shortfalls by all observations, not just the ones below target.

Information ratio

IR = mean active return ÷ tracking error

mean active return
the average of portfolio return minus benchmark return
tracking error
the standard deviation of that active return

Measures active skill against a benchmark, not against the risk-free rate.

A practice question

From the question bank, with the reason each wrong option is wrong.

Jensen's alpha for a portfolio is best described as:

  1. The portfolio's excess return over the risk-free rate divided by its betaWhy not: Describes the Treynor ratio, which scales rather than subtracts.
  2. The portfolio's return minus the return the CAPM requires for its beta - correct
  3. The portfolio's return minus the benchmark return divided by tracking errorWhy not: Describes the information ratio.
  4. The standard deviation of the portfolio's return minus the benchmark returnWhy not: Describes tracking error, a risk measure rather than a return measure.

Jensen's alpha = Rp - [Rf + beta_p x (Rm - Rf)]. Positive alpha means the portfolio earned more than its systematic risk required.

A harder one

A portfolio returned 11%, with a beta of 1.2 and a volatility of 18%. The risk-free rate was 3%. Its Treynor ratio is closest to:

  1. 9.17%Why not: Divides the total return by beta instead of the excess return.
  2. 9.60%Why not: Multiplies the excess return by beta rather than dividing.
  3. 6.67% - correct
  4. 0.444Why not: Divides by the 18% volatility, which gives the Sharpe ratio rather than the Treynor ratio.

Treynor = (Rp - Rf) / beta = (11% - 3%) / 1.2 = 6.67%. It measures excess return per unit of systematic risk.

The hardest tier

The risk-free rate is 2% and the expected market return is 8%. Fund X returned 12% with a volatility of 14% and a beta of 1.5. Fund Y returned 10% with a volatility of 12% and a beta of 1.0. Which ranking is correct?

  1. X ranks higher on the Sharpe ratio; Y ranks higher on both the Treynor ratio and Jensen's alpha - correct
  2. X ranks higher on all three measures because it has the higher returnWhy not: Ranks on raw return; Treynor and Jensen penalise X's beta of 1.5.
  3. Y ranks higher on all three measures because it has the lower betaWhy not: Applies the beta-based ranking to Sharpe too, which divides by total volatility and favours X.
  4. X ranks higher on Treynor and Jensen; Y ranks higher on SharpeWhy not: Reverses the pattern: it assumes beta-based measures favour the higher-beta fund.

Sharpe: X = 10 / 14 = 0.714, Y = 8 / 12 = 0.667. Treynor: X = 10 / 1.5 = 6.67%, Y = 8 / 1.0 = 8.00%. Alpha: X = 12 - (2 + 1.5 x 6) = 1.0%, Y = 10 - (2 + 6) = 2.0%. Sharpe uses total risk and the rest use beta.

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