Beta is a slope and alpha is what is left after it. A fund can beat its index every year and still show no alpha at all.
Beta is the slope of the fund's excess returns over cash regressed on the index's excess returns; Jensen's alpha is the fund's average excess return minus beta times the index's. On twelve illustrative quarters a fund beat its index by 1.57 per cent a year, but with a beta of 1.27 its Jensen's alpha is minus 0.04 per cent a quarter, about minus 0.17 a year. All of the outperformance was extra market exposure in a rising market.
Worked in full in Asset Management by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Jensen's alpha comes from the capital asset pricing model: a fund with market exposure beta should earn the cash rate plus beta times the market's excess return. Whatever it earns beyond that is alpha. Both terms are estimated from the same excess returns, so subtract cash from every period first.
β = Cov(Rp − Rf, Rm − Rf) / Var(Rm − Rf)
α = mean(Rp − Rf) − β × mean(Rm − Rf)
Excel: =SLOPE(FundExcess,IndexExcess) and =INTERCEPT(FundExcess,IndexExcess)
Three years of quarterly returns for a fictional equity fund and its benchmark index, with cash at an illustrative 0.75 per cent a quarter, 3.00 a year. Returns are in per cent.
| Quarter | Index | Fund | Index less cash | Fund less cash |
|---|---|---|---|---|
| Q1 | 4.0 | 5.2 | 3.25 | 4.45 |
| Q2 | -2.5 | -3.1 | -3.25 | -3.85 |
| Q3 | 6.0 | 6.9 | 5.25 | 6.15 |
| Q4 | 1.5 | 2.4 | 0.75 | 1.65 |
| Q5 | -4.0 | -5.9 | -4.75 | -6.65 |
| Q6 | 5.5 | 6.8 | 4.75 | 6.05 |
| Q7 | 3.0 | 3.3 | 2.25 | 2.55 |
| Q8 | 2.0 | 2.6 | 1.25 | 1.85 |
| Q9 | -1.0 | -1.4 | -1.75 | -2.15 |
| Q10 | 7.0 | 8.6 | 6.25 | 7.85 |
| Q11 | 2.5 | 2.3 | 1.75 | 1.55 |
| Q12 | 4.5 | 5.5 | 3.75 | 4.75 |
| Mean | 2.3750 | 2.7667 | 1.6250 | 2.0167 |
The fund beat the index in eight of twelve quarters and by 0.3917 a quarter on average, 1.57 a year on the simple times-four convention. On that number alone the manager looks skilled.
Step 1, beta. The sample covariance of the two excess-return columns is 14.8091 and the sample variance of the index column is 11.6875. Beta is the ratio, 1.2671. The fund moves 1.27 per cent for every 1 per cent the index moves over cash: it lost more in all three falling quarters, by 0.97 on average, and gained more in the rising ones, by 0.84 on average.
Step 2, the return that exposure should earn. The index earned 1.6250 a quarter over cash, so a fund with a beta of 1.2671 should have earned 1.2671 × 1.6250 = 2.0590 over cash with no skill at all.
Step 3, alpha.
α = 2.0167 − 2.0590 = −0.0424 per quarter
Annualised ≈ −0.0424 × 4 = −0.17 per cent a year
Times four is the usual approximation for quarterly data; compounding the quarters changes little at this size. The regression explains 99.1 per cent of the fund's variance: the fund behaves like a geared position in its own index, not like a stock picker.
Alpha is an estimate with a standard error. The residuals, what is left after the market line, have a standard deviation of 0.4380 a quarter, and the standard error of alpha is 0.1411. The t-statistic is minus 0.30, nowhere near the 2 usually asked for. Beta is measured much more precisely: its standard error is 0.0386, so 1.27 is clearly above 1. The honest reading is that the fund's beta is high and its alpha is indistinguishable from zero. To detect a quarterly alpha of 0.0424 against that residual noise at a t-statistic of 2 would take about 428 quarters, or 107 years.
Alpha depends entirely on the beta you charge. The same fund, scored against different assumed betas:
| Beta used | Required excess, quarter | Alpha, quarter | Alpha, year |
|---|---|---|---|
| 1.00 (simple outperformance) | 1.6250 | 0.39 | 1.57 |
| 1.10 | 1.7875 | 0.23 | 0.92 |
| 1.20 | 1.9500 | 0.07 | 0.27 |
| 1.27 (estimated) | 2.0590 | -0.04 | -0.17 |
| 1.30 | 2.1125 | -0.10 | -0.38 |
Each 0.10 of beta moves annual alpha by about 0.65 point, because the index returned 1.6250 a quarter over cash. In a rising market a high-beta fund always looks like it outperforms; in a falling one the same fund looks like it underperforms. Neither is skill. A ratio that rewards return per unit of total risk instead is worked in how to calculate the Sharpe ratio of a fund. Measuring the active return against its own volatility, with no beta adjustment, is the information ratio in how to calculate tracking error and the information ratio.
The tracking error, both information ratios and both Sharpe ratios for a five-year mandate, with the attribution beside them, are in the free workbook for this case.
Outperformance is fund return minus index return, which assumes a beta of 1. Jensen's alpha deducts the return the fund's actual market exposure should have earned. In the worked case the fund beat its index by 0.39 per cent a quarter, 1.57 a year, but a beta of 1.27 required 2.06 per quarter over cash against 2.02 delivered, so alpha is minus 0.04 a quarter.
Put the fund's and the index's excess returns over cash in two columns and use =SLOPE(fund_excess, index_excess), or =COVARIANCE.S(fund_excess, index_excess)/VAR.S(index_excess), which is the same thing. On the twelve worked quarters the covariance is 14.8091 and the variance 11.6875, so beta is 1.27. =INTERCEPT on the same ranges gives Jensen's alpha per period.
Far more than most track records hold. The t-statistic is alpha over its standard error, which shrinks only with the square root of the number of periods. In the worked case a quarterly alpha of 0.0424 against a residual standard deviation of 0.4380 would need about 428 quarters, or 107 years, to reach a t-statistic of 2.
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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