Smoothed monthly returns make volatility look lower than it is, and the square-root-of-12 rule turns that into a Sharpe ratio the fund never earned. Lo's correction puts it back.
Positive autocorrelation in monthly returns makes the usual square-root-of-12 rule overstate a hedge fund's annual Sharpe ratio, because it assumes good and bad months are independent when they actually cluster. On an illustrative fund earning 0.50 per cent a month over cash with 1.60 per cent monthly volatility and a first-order autocorrelation of 0.35, the reported Sharpe ratio is 1.08; Lo's correction gives 0.78, so the reported figure overstates it by 39 per cent.
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The problem is common in strategies that hold assets without a daily price: credit, convertible arbitrage, asset-backed securities, anything marked with a model or a broker quote. Marks lag, losses get recognised over several months instead of one, and the monthly series becomes smoother than the economics underneath it. Monthly volatility looks low. Annualising it with the square root of 12 then carries the error into the annual figure, and the annual figure is the one that goes into the allocation memo.
| Input | Value |
|---|---|
| Mean monthly excess return | 0.50% |
| Standard deviation of monthly returns | 1.60% |
| First-order autocorrelation, ρ1 | 0.35 |
| Higher lags | ρk = 0.35k |
| Periods per year, q | 12 |
The first-order autocorrelation is measured from the fund's own track record: in Excel, with monthly returns in B2:B61, =CORREL(B3:B61,B2:B60). Assuming that higher lags decay geometrically (0.35, then 0.122, 0.043, 0.015 and so on) is the simplest model of a smoothed series. Lo's paper uses the sample autocorrelation at each lag instead, and the variant below shows what changes if only the first lag is counted.
Monthly Sharpe ratio = 0.50 ÷ 1.60 = 0.3125
Reported annual Sharpe = 0.3125 × √12 = 0.3125 × 3.464 = 1.08
Lo factor η(q) = q ÷ √[ q + 2 × Σk=1..q−1 (q − k) × ρk ]
Weighted sum = 11 × 0.35 + 10 × 0.122 + 9 × 0.043 + … = 5.63
Variance multiple = 12 + 2 × 5.63 = 23.27, against 12 with no autocorrelation
η(12) = 12 ÷ √23.27 = 2.488
Adjusted annual Sharpe = 0.3125 × 2.488 = 0.78
Excel, with lags in D2:D12 and autocorrelations in E2:E12: =12/SQRT(12+2*SUMPRODUCT(12-D2:D12,E2:E12))*Monthly_Sharpe
The factor 2.488 replaces 3.464, so the adjusted ratio is 71.8 per cent of the reported one. The mechanism is visible in the variance multiple. With independent months, the variance of a year's return is 12 times the monthly variance. With autocorrelation of 0.35 it is 23.27 times, because each month's error is partly repeated in the months that follow. Annual volatility is 1.60 × √23.27 = 7.72 per cent, not the 5.54 per cent the simple rule gives. The return is untouched: 6.0 per cent a year over cash either way. Dividing 6.0 by 7.72 gives the same 0.78.
The fund's true risk-adjusted performance is a Sharpe ratio of about 0.78. The reported 1.08 is 39 per cent too high, or put the other way, the correction removes 28 per cent of the reported figure. Nothing in the fund's returns was misstated month by month; the error comes entirely from annualising a series that is not independent.
| ρ1 | Variance multiple | Lo factor η(12) | Adjusted Sharpe | Annual volatility | Reported overstates by |
|---|---|---|---|---|---|
| 0.00 | 12.00 | 3.464 | 1.08 | 5.54% | 0% |
| 0.10 | 14.42 | 3.160 | 0.99 | 6.08% | 10% |
| 0.20 | 17.38 | 2.879 | 0.90 | 6.67% | 20% |
| 0.35 | 23.27 | 2.488 | 0.78 | 7.72% | 39% |
| 0.50 | 32.00 | 2.121 | 0.66 | 9.05% | 63% |
The overstatement grows faster than the autocorrelation. At 0.50, a level not unusual for funds holding illiquid credit, the reported Sharpe ratio is 63 per cent above the true one and annual volatility is 9.05 per cent rather than 5.54.
If you count only the first lag and set the rest to zero, the variance multiple is 12 + 2 × 11 × 0.35 = 12 + 7.70 = 19.70, the factor is 2.704 and the adjusted Sharpe ratio 0.84, with annual volatility of 7.10 per cent. That is a smaller correction than the geometric version, but still a large one. Which to use depends on what the data show: if the measured second- and third-lag autocorrelations are clearly positive, the first-lag-only version understates the problem.
Put the illustrative fund beside a liquid long-short manager earning 0.45 per cent a month over cash with the same 1.60 per cent volatility and no autocorrelation. Reported, the smoothed fund wins: 1.08 against 0.97, a lead of 0.11. Adjusted, it loses: 0.78 against 0.97, a gap of 0.20 the other way. The liquid manager earns less, 5.4 per cent a year over cash against 6.0, but its volatility is real. A screen that sorts managers on the reported annual Sharpe ratio systematically favours the strategies whose prices are the least reliable.
The common mistake is to annualise every monthly series with the square root of 12 because the formula is standard, and then compare the results across strategies with very different liquidity. The square-root rule is a special case of Lo's formula, the case where every autocorrelation is zero. Measure the autocorrelation first; if it is near zero, the shortcut is fine, and if it is not, the shortcut is the largest error in the comparison.
The same smoothing that inflates a hedge fund's Sharpe ratio is what makes private equity look uncorrelated with listed markets. The article on unsmoothing private equity reverses it in the return series itself, and the free workbook for this case measures the first-order autocorrelation from a reported series. For the plain version of the ratio, see how to calculate the Sharpe ratio of a fund.
Before comparing annual Sharpe ratios, compute the first-order autocorrelation of each fund's monthly returns. Replace the square root of 12 with Lo's factor: at an autocorrelation of 0.35 it is 2.488, and a reported 1.08 becomes 0.78. A fund whose ranking depends on low measured volatility in hard-to-price assets should be compared on the adjusted figure, or not compared at all. This is arithmetic on an illustrative series, not a view on any manager.
Andrew Lo's 2002 correction replaces the square root of 12 with a factor that allows for autocorrelation: 12 divided by the square root of 12 plus twice the sum of (12 minus k) times the lag-k autocorrelation. With no autocorrelation the factor is 3.464 and nothing changes; with an illustrative first-order autocorrelation of 0.35 decaying geometrically it is 2.488, which cuts a reported 1.08 Sharpe ratio to 0.78.
Liquid long-short equity and managed futures funds usually show first-order autocorrelation near zero. Persistently positive values point to stale or model-based marks on hard-to-price holdings, or to deliberate smoothing. In the illustrative sensitivity, even 0.20 overstates the annual Sharpe ratio by 20 per cent, and 0.50 overstates it by 63 per cent, so anything above about 0.10 deserves a question to the manager about pricing.
No. Smoothing moves returns between months but leaves the long-run average roughly where it was: the illustrative fund still earns 6.0 per cent a year over cash. What changes is the measured risk. Its annual volatility is 7.72 per cent once the correlation between months is counted, against 5.54 per cent from the square-root-of-12 rule.
This article is one calculation from Alternative Investments. 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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