Five years of active returns taken to a tracking error and an information ratio, gross and net, with the mistakes that move the answer more than the manager does.
Tracking error is the standard deviation of the active returns, the portfolio return minus the benchmark return period by period; the information ratio is the mean active return divided by that tracking error. On the Marchwood mandate's five years the gross active returns average 0.3712 points with a sample tracking error of 0.6601, an information ratio of 0.5624. Net of fees the mean falls to 0.0833 and the ratio to 0.1262.
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Both figures appear on every active manager's factsheet and in most mandate reviews. They are simple to compute and easy to compute slightly wrong, and the two usual slips move the answer by more than the manager's edge.
Marchwood is the illustrative multi-asset pension mandate in the book's companion workbooks, measured against a fixed-weight five-segment composite. Five annual observations, in per cent:
| Year | Portfolio, gross | Benchmark | Active, gross | Portfolio, net | Active, net |
|---|---|---|---|---|---|
| Year 1 | 10.840 | 9.890 | 0.950 | 10.5431 | 0.6531 |
| Year 2 | −5.932 | −5.755 | −0.177 | −6.1938 | −0.4388 |
| Year 3 | 12.549 | 11.340 | 1.209 | 12.2431 | 0.9031 |
| Year 4 | 5.883 | 6.065 | −0.182 | 5.5923 | −0.4727 |
| Year 5 | 3.151 | 3.095 | 0.056 | 2.8667 | −0.2283 |
| Mean | 0.3712 | 0.0833 |
Step 1, active returns. at = Rp,t − Rb,t. Year 1: 10.840 − 9.890 = 0.950.
Step 2, mean. (0.950 − 0.177 + 1.209 − 0.182 + 0.056) ÷ 5 = 0.3712.
Step 3, deviations squared. 0.5788² = 0.3350, (−0.5482)² = 0.3005, 0.8378² = 0.7019, (−0.5532)² = 0.3060, (−0.3152)² = 0.0994. Sum 1.7428.
Step 4, tracking error. Divide by n − 1 = 4: variance 0.4357. Square root: TE = 0.6601.
Step 5, information ratio. IR = 0.3712 ÷ 0.6601 = 0.5624.
In Excel, active returns in D2:D6: =AVERAGE(D2:D6)/STDEV.S(D2:D6).
With monthly data the same steps apply, and the results are annualised: tracking error by multiplying by the square root of 12 (3.4641), so a monthly 0.19 becomes 0.6582 a year; mean active return by 12; the information ratio therefore by the square root of 12.
Everything above is ex post: it measures the active risk the manager actually ran. A risk system will also quote an ex ante tracking error, a forecast built from current holdings and a covariance model. The two answer different questions, the first about the record and the second about the portfolio today, and a mandate review should say which one a limit refers to before comparing either with the information ratio.
Gross of fees the manager added 0.3712 points a year for 0.6601 of active risk. Net of the mandate's fee the mean active return is 0.0833. There are two defensible net ratios: 0.0833 over the gross tracking error gives 0.1262, and over the tracking error of the net active series, 0.6472, gives 0.1287. The fee removed about four fifths of the ratio.
An information ratio is also a t-statistic in disguise: multiply it by the square root of the number of observations. Gross, 0.5624 × √5 = 1.2575, short of the 2 that would suggest skill rather than luck. To reach 2 at this ratio takes (2 ÷ 0.5624)² = 12.65 years of data; at the net 0.1262, about 251.29 years. Five years of a 0.5 information ratio cannot prove anything, which is a reason to state the ratio with its observation count.
Hold the mean gross active return at 0.3712 and vary the risk taken to earn it.
| Tracking error | Information ratio | Years for a t-statistic of 2 |
|---|---|---|
| 0.25 | 1.4848 | 1.8 |
| 0.50 | 0.7424 | 7.3 |
| 0.6601 | 0.5624 | 12.6 |
| 1.00 | 0.3712 | 29.0 |
| 2.00 | 0.1856 | 116.1 |
The same outperformance is worth much more when it arrives steadily. A manager who earns 0.3712 a year with a tracking error of 2.00 has a ratio of 0.1856 and would need more than a century of data to distinguish that from noise.
Population instead of sample standard deviation. Dividing by 5 rather than 4 gives a tracking error of 0.5904 and an information ratio of 0.6287, flattering the manager by 12 per cent. With five observations the choice is not cosmetic. Use STDEV.S.
Difference of volatilities. Tracking error is not the portfolio's volatility minus the benchmark's. Here the gross portfolio volatility is 7.3204 and the benchmark's 6.7918; the difference, 0.5286, has nothing to do with active risk. Two portfolios can have identical volatility and a large tracking error if they hold different things.
Mixing gross and net. Subtracting an annualised compound fee drag from an arithmetic gross mean produces a net ratio that is neither one thing nor the other. Compute the net active return year by year and average that, as above.
Reading it beside the Sharpe ratio without thinking. On these figures the manager beat the benchmark even net of fees and yet has the lower Sharpe ratio, 0.2944 on net returns against 0.3043, using a mean cash return of 2.86. The portfolio carried more equity than the benchmark, so it took more total risk than its outperformance paid for. The information ratio measures active skill; the Sharpe ratio measures the whole portfolio. Both can be right at once.
Compute active returns period by period, take their sample standard deviation, and divide the mean by it. Report the number of observations beside the ratio, and show gross and net separately. The free attribution and risk workbook for this case builds every figure here from the five years of segment data. For the Brinson split of the same active return into allocation and selection, see how to calculate Brinson allocation and selection effects.
Above 0.5 is commonly treated as good and above 1.0 as exceptional, but the number of observations matters as much as the level. A ratio of 0.5624 over five years is a t-statistic of only 1.2575; reaching 2 at that ratio takes 12.65 years. Net of fees the same manager's ratio here is 0.1262.
STDEV.S, the sample standard deviation, because the observed periods are a sample of the manager's possible results. On five annual active returns it gives 0.6601; STDEV.P gives 0.5904, raising the information ratio from 0.5624 to 0.6287. The smaller the sample, the larger the flattery from the wrong function.
Multiply it by the square root of 12, which is 3.4641, assuming the monthly active returns are independent. A monthly tracking error of 0.19 points becomes 0.6582 a year. The mean active return is multiplied by 12, so a monthly information ratio is annualised by the same square root of 12.
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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