Both formulas, why only the geometric one compounds across periods and survives currency translation, and how far apart they drift.
Arithmetic excess return is the fund's return minus the benchmark's; geometric excess return is the ratio of their growth less one, (1 + fund) / (1 + benchmark) − 1. Over an illustrative five years in which a fund grew 49.12 per cent and its benchmark 34.64, the arithmetic excess is 14.48 points but the geometric excess is 10.76 per cent: the investor ended with 10.76 per cent more wealth than the index would have given, not 14.48. Only the geometric figure compounds across periods and survives a change of currency.
Worked in full in Asset Management by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Both measures are correct answers to different questions. The arithmetic one says how many percentage points separate two returns. The geometric one says how much richer the fund made the client relative to the benchmark, which is the question a client, a consultant and a performance fee are usually asking. The difference is small in a single quiet year and large over a long or volatile period.
| Year | Fund | Benchmark | Arithmetic excess | Geometric excess |
|---|---|---|---|---|
| 2021 | 12.0 | 9.0 | 3.00 | 2.75 |
| 2022 | −6.0 | −8.0 | 2.00 | 2.17 |
| 2023 | 15.0 | 12.0 | 3.00 | 2.68 |
| 2024 | 9.0 | 8.0 | 1.00 | 0.93 |
| 2025 | 13.0 | 11.0 | 2.00 | 1.80 |
Arithmetic excess = Rfund − Rbenchmark
Geometric excess = (1 + Rfund) / (1 + Rbenchmark) − 1
Multi-period geometric excess = Π(1 + geometric excesst) − 1
In Excel, with fund and benchmark returns as decimals in B and C: =B2-C2 and =(1+B2)/(1+C2)-1; for the whole period =PRODUCT(1+B2:B6)/PRODUCT(1+C2:C6)-1.
One year. In 2021 the fund returned 12.0 per cent against 9.0. Arithmetic excess: 3.00 points. Geometric: 1.12 / 1.09 = 1.0275, so 2.75 per cent. Every unit of currency in the benchmark grew to 1.09; in the fund it grew to 1.12, which is 2.75 per cent more, not 3.
A falling year. In 2022 the fund lost 6.0 per cent and the benchmark 8.0. The ratio is 0.94 / 0.92 = 1.0217: a geometric excess of 2.17 per cent, larger than the arithmetic 2.00, because losing less is worth more relative to a smaller base.
Five years. The fund's growth factor is 1.4912 and the benchmark's 1.3464. Arithmetic cumulative excess: 49.12 − 34.64 = 14.48 points. Geometric: 1.4912 / 1.3464 = 1.1076, so 10.76 per cent.
| Measure | Excess | Reconciles with the cumulative returns? |
|---|---|---|
| Sum of annual arithmetic excess | 11.00 | No |
| Annual arithmetic excess compounded | 11.48 | No |
| Cumulative arithmetic excess | 14.48 | Yes, as a difference of points |
| Annual geometric excess compounded | 10.76 | Yes, equals the cumulative geometric excess |
The geometric figure is the only one that chains. Compounding the five annual geometric excesses gives exactly the 10.76 per cent obtained from the cumulative returns; neither summing nor compounding the annual arithmetic excesses reaches 14.48. Annualised, the fund returned 8.32 per cent and the benchmark 6.13: an arithmetic excess of 2.19 points a year, a geometric one of 2.06.
This is why multi-period attribution is awkward in arithmetic form. Brinson effects computed arithmetically each year do not add up to the cumulative excess without a smoothing algorithm; computed geometrically, they compound to it.
Take an investor whose own currency weakened 10 per cent against the fund's over the period. In their currency the fund returned 64.04 per cent and the benchmark 48.11. The arithmetic excess becomes 15.93 points, up from 14.48, although the manager did nothing different. The geometric excess is unchanged at 10.76 per cent, because the currency factor appears in both the numerator and the denominator and cancels.
| Benchmark return | Fund return | Arithmetic excess | Geometric excess |
|---|---|---|---|
| −10.0% | −7.0% | 3.00 | 3.33 |
| 2.0% | 5.0% | 3.00 | 2.94 |
| 6.0% | 9.0% | 3.00 | 2.83 |
| 20.0% | 23.0% | 3.00 | 2.50 |
Three points of outperformance in a market up 20 per cent is a smaller achievement, relative to the client's wealth, than three points in a market down 10. The geometric measure says so; the arithmetic one treats them as identical.
The common mistake is to report a cumulative arithmetic excess and describe it as the extra wealth created. Saying the fund beat the index by 14.48 per cent over five years invites the reader to multiply it by the amount invested; the correct extra wealth relative to the index is 10.76 per cent of what the index would have delivered. The second mistake is mixing the two: annual figures reported arithmetically and a cumulative figure computed geometrically, so that the annual numbers visibly fail to add up to the total and the reader suspects an error where there is only a convention.
The same distinction runs through performance attribution, worked in how to calculate Brinson allocation and selection effects, and through the information ratio in how to calculate tracking error and the information ratio. The book's four returns for one mandate, with the benchmark beside them, are in the free workbook for this case on the Asset Management companion page.
Whichever the clause names, so the clause should name one. Geometric excess suits multi-period reporting because it compounds and is currency-neutral; arithmetic is simpler to read in a single year. Over the illustrative five years the two are 10.76 per cent and 14.48 points, so a fee on outperformance drafted without a formula leaves a gap of several points for the two sides to argue over.
Because it is measured relative to the benchmark's ending wealth, which is smaller after a fall. A fund down 7.0 per cent against a benchmark down 10.0 per cent has an arithmetic excess of 3.00 points but a geometric excess of 3.33 per cent: 0.93 is 3.33 per cent more than 0.90. In a market up 20 per cent, the same three points are worth 2.50 per cent.
The arithmetic measure, yes; the geometric, no. Translating the illustrative record into a currency that weakened 10 per cent lifts the fund to 64.04 per cent and the benchmark to 48.11, so arithmetic excess rises from 14.48 to 15.93 points, while geometric excess stays at 10.76 per cent because the currency factor cancels.
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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