Measuring performance

RiskAtoms Group runs three equity funds, and each one can truthfully claim to be the best performer of the last five years. Defensive wins on the Sharpe ratio, Focus on Treynor's ratio and Jensen's alpha, Dynamic on the information ratio. Which claim matters depends on what the fund is for.

A raw return says little on its own. A fund that borrowed to double its market exposure will beat a cautious fund in a rising market without any skill at all. Every performance measure therefore divides or adjusts the return by some measure of risk. The measures differ in which risk they charge for, and that choice should follow from how the fund will be used.

Two ideas carry over from earlier lessons. The Sharpe ratioExcess return over the risk-free rate divided by total volatility: reward per unit of all the risk taken. from L2.1.03: Risk and return charges for total volatility. Beta and the security market line from L3.1.09: CAPM, beta and the security market line say that only market risk earns a premium, because specific risk can be diversified away.

Five measures, three kinds of risk.

Treynor's ratioExcess return divided by beta: reward per unit of market (systematic) risk. and Jensen's alphaThe intercept of a regression of the fund's excess returns on the market's: return above what CAPM predicts for the fund's beta. both charge only for beta. The information ratioAverage active return divided by tracking error: benchmark-beating return per unit of benchmark-relative risk. compares the fund with its benchmarkThe index a fund is measured against, such as a global equity index.: the active returnThe fund's return minus its benchmark's return. divided by the tracking errorThe standard deviation of active returns: how far, typically, the fund strays from its benchmark.. M²Modigliani's M²: the return the fund would have earned if geared up or down to the market's volatility. turns the Sharpe ratio into a percentage return.

The league table. Here are all the measures for the three funds. The best fund on each is marked.

Why they disagree.

• Focus picks about 25 shares, so it carries a lot of specific risk: its residual volatility is 15.72% a year. The Sharpe ratio charges for that risk and pushes Focus down. Treynor and Jensen ignore it, because in a large portfolio it would diversify away, so they put Focus first with an alpha of 3.00%.
• Dynamic gears up the index to a beta of 1.45. In five years when the market beat cash by 6.00% a year, that alone added 2.70% a year over the index. The information ratio counts that as active return, so Dynamic tops it. Yet its alpha is 0.00%: an investor could copy it by borrowing to buy the index.
• Defensive has the lowest raw return but also the lowest volatility, and the best Sharpe ratio. Against a full-market benchmark its beta of 0.65 makes it lag in rising markets, so its information ratio is negative (−0.119).

Skill or luck? Five years is a short track record. Jensen's alpha is a regression estimate, so it has a standard error (M-D03: Linear regression). For Focus the t statistic of alpha is only 0.42; for Defensive, 0.51. Neither is near the 5% critical value of 2.00. Roughly, the t statistic grows with the square root of time: $t \approx (\alpha/\sigma_\varepsilon)\sqrt{T}$, with $T$ in years. At Defensive's ratio of alpha to specific risk, about 76 years of data would be needed to reach $t = 2$; for Focus, about 110. Performance measures describe the past; proving skill takes far longer.

Write $\bar R_p$ for the fund's mean return, $\sigma_p$ for its volatility, $\beta_p$ for its beta against the market $m$, and $R_b$ for the benchmark (here the market itself). All figures are a year.

Jensen's alpha from a regression. In practice alpha and beta are estimated together, by least squares on monthly excess returns (M-D03: Linear regression). This is the security market line of L3.1.09: CAPM, beta and the security market line with an intercept added.

With $x_t = R_{m,t} - r_f$ and $y_t = R_{p,t} - r_f$, the least-squares estimates and the standard error of alpha are:

The regression is run on monthly data; alpha is then multiplied by 12 to give an annual figure. The residual standard deviation $s$ uses $n - 2$ because two numbers (alpha and beta) were estimated.

How the measures are linked. Three short derivations show why they can disagree.

1. Jensen and Treynor rank by the same thing, scaled by beta. Take $\beta_p$ out of the alpha formula:
   $$\begin{aligned} \alpha_p &= (\bar R_p - r_f) - \beta_p(\bar R_m - r_f) \\ &= \beta_p\left[\frac{\bar R_p - r_f}{\beta_p} - (\bar R_m - r_f)\right] \\ &= \beta_p\,(T_p - T_m) \end{aligned}$$
   A fund beats the market on Treynor's ratio exactly when its alpha is positive (for positive beta). For Focus: $1.05 \times (8.86\% - 6.00\%) = 3.00\%$.
2. The information ratio mixes alpha with a beta bet. Subtract the benchmark's return from the regression:
   $$\begin{aligned} \overline{R_p - R_m} &= \alpha_p + \beta_p(\bar R_m - r_f) + r_f - \bar R_m \\ &= \alpha_p + (\beta_p - 1)(\bar R_m - r_f) \\[4pt] TE_p^2 &= (\beta_p - 1)^2\sigma_m^2 + \sigma_\varepsilon^2 \end{aligned}$$
   A beta above 1 earns active return whenever the market beats cash. Dynamic's active return is $0.00\% + (1.45 - 1) \times 6.00\% = 2.70\%$, almost all of it beta.
3. M² ranks exactly as Sharpe does, because $M^2_p - r_f = S_p\,\sigma_m$ just multiplies every Sharpe ratio by the same $\sigma_m$.

Start by picking what the fund is for, and see which measure it selects and who wins. Then switch the chart to “Return vs beta” and see alpha as the height above the security market line. Finally raise the risk-free rate to 6%: low-beta Defensive gains alpha, because the line it is judged against becomes flatter.

The regression uses monthly excess returns over the 60 months. Its summary statistics are: market excess mean $\bar x = 0.5000\%$, Focus excess mean $\bar y = 0.7750\%$, $S_{xx} = 0.119475$ and $S_{xy} = 0.125449$. Try each step before revealing it.

Performance figures are marketing, so rules and standards govern how they are shown. The Global Investment Performance Standards (GIPS), published by the CFA Institute, are a voluntary industry standard for presenting performance fairly: for example, firms must not show only their best funds. In the UK, the Financial Conduct Authority requires authorised fund managers to assess, at least once a year, whether each fund's charges are justified by the value it delivers, considering among other things how it performed against its objectives after charges, and to publish the conclusions. Insurers that invest policyholders' money face their own investment rules under Solvency II, covered in M4.6: Solvency II and insurance regulation.

• Sharpe divides excess return by total volatility; M² restates it as the return at the market's volatility.
• Treynor divides by beta, and Jensen's alpha subtracts the CAPM return: both charge only for market risk.
• The information ratio divides return above the benchmark by tracking error; a beta above 1 can inflate it.
• Use Sharpe or M² for a whole portfolio, Treynor or Jensen for one fund in a diversified portfolio, and the information ratio for an active mandate.
• Alpha is an estimate with a standard error. A few years of data rarely prove skill.

Sharpe ratio

Excess return over the risk-free rate divided by total volatility: reward per unit of all the risk taken.

Treynor ratio

Excess return divided by beta: reward per unit of market (systematic) risk.

Jensen's alpha

The intercept of a regression of the fund's excess returns on the market's: return above what CAPM predicts for the fund's beta.

Benchmark

The index a fund is measured against, such as a global equity index.

Active return

The fund's return minus its benchmark's return.

Tracking error

The standard deviation of active returns: how far, typically, the fund strays from its benchmark.

Information ratio

Average active return divided by tracking error: benchmark-beating return per unit of benchmark-relative risk.

Appraisal ratio

Jensen's alpha divided by the volatility of the regression residuals: alpha per unit of specific risk. Some books call this the information ratio.

M²

Modigliani's M²: the return the fund would have earned if geared up or down to the market's volatility.