# Autocorrelation-adjusted Sharpe calculator

How does positive autocorrelation change a Sharpe ratio scaled over twelve periods?

With a per-period Sharpe of 0.1, square-root scaling over 12 periods gives 0.346. If adjacent returns have a lag-one correlation of 0.2 under an AR(1) model, Lo's aggregation formula lowers the twelve-period figure to 0.288.

Annualize a periodic Sharpe ratio with a stated autocorrelation and horizon using Lo's aggregation identity. Inspect AR(1) dependence over a horizon.

## Aggregation under serial dependence

The model assumes a stationary AR(1) return process whose lag-k correlation is rho to the power k. The periodic mean and variance are treated as known estimates.

```
SR(q) = q × SR(1) / sqrt(q + 2 sum[k=1..q−1] (q−k) rho^k)
```

Positive serial correlation raises aggregated variance and lowers the horizon Sharpe relative to square-root-of-time scaling. Negative correlation reverses that comparison.

## Twelve-period sensitivity check

For a per-period Sharpe of 0.1, rho = 0, and q = 12, the result is 0.346410. Setting rho = 0.2 reduces it to 0.287885 because adjacent returns move together.

The ordinary square-root rule applies when all nonzero-lag covariances vanish. The AR(1) coefficient exposes the dependence assumption hidden by treating monthly returns as independent observations. A positive coefficient matters most when the horizon contains many adjacent observations. A negative coefficient can raise the converted ratio, although a strongly negative value should prompt a check of the sampling and pricing process. Enter the periodic Sharpe directly: do not enter an annualized figure, or the horizon effect will be applied twice. The output is a ratio for a chosen aggregation horizon; it does not measure whether the estimated mean differs from zero. For inference, the sample size, higher moments and uncertainty of the autocorrelation estimate must also be considered. As a sensitivity check, compare zero, modest positive and modest negative rho while keeping the same periodic Sharpe.

## What this conversion leaves out

The output converts a Sharpe ratio across horizons; it supplies no confidence interval for the estimate. An AR(1) model may miss long memory, volatility clustering and changing correlation.

## Lo's derivation

Lo, A. W. (2002), The Statistics of Sharpe Ratios, Financial Analysts Journal 58(4), 36–52. The page uses his time-aggregation identity with AR(1) lag correlations. [Original or institutional source](https://doi.org/10.2469/faj.v58.n4.2453).

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```
<iframe id="stochastly-acsharpe" src="https://stochastly.com/embed/autocorrelation-adjusted-sharpe-calculator" title="Autocorrelation-adjusted Sharpe calculator by Stochastly" width="100%" height="700" style="border:0;max-width:720px" loading="lazy" sandbox="allow-scripts allow-same-origin allow-popups allow-popups-to-escape-sandbox" allow="clipboard-write"></iframe>
<p style="font-size:13px"><a href="https://stochastly.com/tools/autocorrelation-adjusted-sharpe-calculator">Autocorrelation-adjusted Sharpe calculator</a> by Stochastly</p>
<script>
(function () {
var allowed = ["https://stochastly.com"];
window.addEventListener("message", function (e) {
var f = document.getElementById("stochastly-acsharpe");
if (!f || allowed.indexOf(e.origin) < 0 || e.source !== f.contentWindow) return;
var d = e.data;
if (!d || d.type !== "stochastly-acsharpe-height" || typeof d.height !== "number") return;
if (!(d.height >= 200 && d.height <= 4000)) return;
f.style.height = Math.ceil(d.height) + "px";
});
})();
</script>
```

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```
Stochastly (2026). Autocorrelation-adjusted Sharpe calculator [web calculator]. https://stochastly.com/tools/autocorrelation-adjusted-sharpe-calculator
```

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The method behind the autocorrelation-adjusted Sharpe calculator comes from the paper below: cite that paper for the method, and the reference above for the autocorrelation-adjusted Sharpe calculator itself.

Lo, A. W. (2002). The Statistics of Sharpe Ratios. Financial Analysts Journal 58(4), 36-52.

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