Excess return per unit of beta: a useful ranking for a fund inside a diversified portfolio, and a misleading one for a fund that is the whole portfolio.
The Treynor ratio is a fund's average return over cash divided by its beta: excess return per unit of market risk. On twelve illustrative quarters an index earns 1.6250 per cent a quarter over cash, so its Treynor ratio is 1.6250, or 6.50 per cent a year. A high-beta fund that beat it by 1.57 a year scores only 6.37, and a low-beta fund that lagged it scores 9.15, which is the right ranking only if the fund sits inside an already diversified portfolio.
Worked in full in Asset Management by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Jack Treynor's ratio uses the same ingredients as Jensen's alpha: the excess return over the risk-free rate and the beta from a regression of fund excess returns on index excess returns. Where alpha subtracts what the beta should have earned, Treynor divides by the beta.
Treynor = mean(Rp − Rf) / βp
βp = Cov(Rp − Rf, Rm − Rf) / Var(Rm − Rf)
Excel: =AVERAGE(FundExcess)/SLOPE(FundExcess,IndexExcess)
The index always has a beta of 1, so its Treynor ratio is simply its own excess return. That is the bar: a fund with a higher Treynor ratio earned more per unit of market exposure than the market itself.
Three years of quarterly returns, in per cent, for a fictional index and two fictional funds, with cash at an illustrative 0.75 per cent a quarter, 3.00 a year. Fund A is the geared equity fund worked in how to calculate a fund's alpha and beta. Fund B is a defensive fund with a lot of stock-specific risk.
| Quarter | Index | Fund A | Fund B |
|---|---|---|---|
| Q1 | 4.0 | 5.2 | 6.5 |
| Q2 | -2.5 | -3.1 | -2.0 |
| Q3 | 6.0 | 6.9 | 2.4 |
| Q4 | 1.5 | 2.4 | 4.2 |
| Q5 | -4.0 | -5.9 | -4.6 |
| Q6 | 5.5 | 6.8 | 1.8 |
| Q7 | 3.0 | 3.3 | 5.6 |
| Q8 | 2.0 | 2.6 | -1.2 |
| Q9 | -1.0 | -1.4 | 2.3 |
| Q10 | 7.0 | 8.6 | 3.4 |
| Q11 | 2.5 | 2.3 | 5.9 |
| Q12 | 4.5 | 5.5 | 0.5 |
| Mean | 2.3750 | 2.7667 | 2.0667 |
Step 1, subtract cash. Take 0.75 from every quarter. The mean excess returns are 1.6250 for the index, 2.0167 for Fund A and 1.3167 for Fund B.
Step 2, estimate beta. The sample variance of the index excess column is 11.6875. Fund A's covariance with it is 14.8091, so its beta is 1.2671. Fund B's covariance is 6.7273, so its beta is 0.5756.
Step 3, divide.
Index: 1.6250 / 1.00 = 1.6250 a quarter, 6.50 a year
Fund A: 2.0167 / 1.2671 = 1.5916 a quarter, 6.37 a year
Fund B: 1.3167 / 0.5756 = 2.2875 a quarter, 9.15 a year
To annualise quarterly data, multiply the Treynor ratio by four: the numerator is a return that scales with time, the beta does not. The Sharpe ratio is different, because its denominator is a volatility, and it is annualised by the square root of four.
| Excess, year | Beta | R² | Treynor, year | Sharpe, year | |
|---|---|---|---|---|---|
| Index | 6.50 | 1.00 | 100.0% | 6.50 | 0.95 |
| Fund A | 8.07 | 1.27 | 99.1% | 6.37 | 0.93 |
| Fund B | 5.27 | 0.58 | 33.4% | 9.15 | 0.77 |
Fund A earned the most and is last on Treynor, slightly below the index. Its extra return is all leverage to the market, and the gap between its Treynor ratio and the index's, minus 0.0334 a quarter, is exactly its Jensen's alpha of minus 0.0424 divided by its beta.
Fund B earned the least and is first on Treynor by a wide margin, 1.41 times the index. Yet its Sharpe ratio is 0.77 against 0.95, 0.81 of the index's. Both are correct. The difference is the 66.6 per cent of Fund B's variance that the index does not explain. Treynor ignores it; Sharpe charges for it.
Use Treynor when the fund is one of many in a diversified portfolio, where stock-specific risk washes out and only beta is left. Use Sharpe when the fund is the investor's whole risky exposure. A client who owns only Fund B carries all of its specific risk and is badly served by a 9.15 Treynor ratio.
Treynor is only as good as its beta, and Fund B's beta is poorly measured: with an R² of 33.4 per cent its standard error is 0.2570, against 0.0386 for Fund A. Move it by one or two standard errors and the ratio swings across the index.
| Beta used | Beta | Treynor, quarter | Treynor, year |
|---|---|---|---|
| Estimate less 2 s.e. | 0.06 | 21.37 | 85.50 |
| Estimate less 1 s.e. | 0.32 | 4.13 | 16.53 |
| Estimate | 0.58 | 2.29 | 9.15 |
| Estimate plus 1 s.e. | 0.83 | 1.58 | 6.33 |
| Estimate plus 2 s.e. | 1.09 | 1.21 | 4.83 |
One standard error up and Fund B falls below the index's 6.50. As beta approaches zero the ratio explodes, and with a negative beta it changes sign and stops meaning anything. For Fund A the problem is small: with a beta between 1.20 and 1.30 its Treynor ratio stays between 6.72 and 6.21.
The Sharpe ratio on the same logic is worked in how to calculate the Sharpe ratio of a fund. Tracking error, both information ratios and both Sharpe ratios on a five-year mandate are in the free workbook for this case.
There is no absolute level: compare the fund's Treynor ratio with the market's, which is simply the index's excess return over cash because its beta is 1. In the illustrative case the index earns 6.50 per cent a year over cash, so a fund scoring above 6.50 earned more per unit of market risk. Fund B scores 9.15, but only with a beta whose standard error is 0.2570.
Both divide excess return over cash by a risk measure. Sharpe divides by total volatility, Treynor by beta, so Treynor ignores stock-specific risk. On the worked data a fund with an R-squared of 33.4 per cent ranks first on Treynor, 9.15 a year against 6.50 for the index, and last on Sharpe, 0.77 against 0.95. Use Treynor for a component of a diversified portfolio, Sharpe for a stand-alone holding.
Put the fund and index excess returns over cash in two columns and use =AVERAGE(FundExcess)/SLOPE(FundExcess,IndexExcess). On twelve illustrative quarters Fund A averages 2.0167 over cash with a beta of 1.2671, a quarterly Treynor ratio of 1.5916. Multiply by four for an annual figure, 6.37.
This article is one calculation from Asset Management. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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