A return series in, the full statistic set out, with both annualisation rules and both divisor conventions wherever a statistic has two. Every input is encoded in the URL. Everything computes in the browser; the page makes no network request.
Two annualisation rules are shown because they are not equivalent. Multiplying a per-period Sharpe ratio by the square root of the periods per year is correct only when returns have no serial correlation. Where they do, the Lo (2002) factor eta(P) replaces sqrt(P), and with positive autocorrelation it is smaller, so the naive figure overstates the annual ratio.
An episode begins the first period the equity index sits below its prior peak and ends when it regains that peak. An episode still open at the end of the sample is censored: its duration is a lower bound, not a measurement.
The URL carries the whole return series, so a pasted series survives the round trip. Long series make long URLs; that is the trade for a page with no server.
| Parameter | Meaning | Example |
|---|---|---|
returns | Comma-separated periodic returns in percent, oldest first. At least four values for the moment estimators to mean anything. | 1.60,4.10,2.30,3.40 |
periods | Periods per year used for annualisation. 12 monthly, 4 quarterly, 252 trading-day, 365 calendar-day. | 12 |
rf | Risk-free rate per period in percent, held constant. Defaults to 0. | 0.20 |
benchmark | Comma-separated benchmark returns in percent, same length and order as returns. Enables beta, alpha, information ratio, Treynor, M-squared and the capture ratios. | 4.00,3.90,0.70,3.00 |
mar | Minimum acceptable return per period in percent, used as the Sortino target and the Omega threshold. Defaults to the value of rf. | 0 |
Example, the published base series with its benchmark:
https://quants.wiki/calc/?returns=1.60,4.10,2.30,3.40,1.50,-0.80,3.20,2.70,1.90,-3.60,-0.50,0.60,2.10,1.10,-1.20,0.90,0.40,-2.50,-1.80,-0.30,-2.90,2.00,1.40,2.80&benchmark=4.00,3.90,0.70,3.00,0.80,-0.40,2.30,3.20,1.40,-3.90,1.10,-1.30,2.80,3.10,0.30,2.30,-1.40,-2.90,-3.00,-1.40,-3.50,0.40,1.00,1.90&periods=12&rf=0.20
The arithmetic mean is the simple average of the periodic returns and annualises as P times the mean. The geometric mean is the constant per-period rate that reproduces the terminal wealth, so it is the Tth root of the product of one plus each return, minus one, and it annualises by compounding: (1+g) raised to the power P, minus one. The geometric mean is always at or below the arithmetic mean, and the gap widens with variance. That gap is the volatility drag, and it is the reason a series can have a positive average return and still lose money. Volatility here is the sample standard deviation with T − 1 in the denominator, annualised by multiplying by the square root of P. Skewness and kurtosis use the plain moment estimators with divisor T, and the kurtosis reported is the full fourth standardised moment, not excess kurtosis; the normal value is 3, and the excess figure is shown alongside so neither can be mistaken for the other.
Sharpe (1966) introduced the measure under the name reward-to-variability ratio, defined on the excess return of a fund over a riskless rate. Sharpe (1994) generalised it: the ratio is defined on a differential return, the return of the fund minus the return of a benchmark, and the denominator is the standard deviation of that differential return rather than of the fund's total returns. The two coincide exactly when the benchmark is a constant riskless rate, which is the case computed here, and they diverge as soon as the benchmark is itself risky. That is a real difference, not a notational one, and a Sharpe ratio quoted without saying which construction was used is ambiguous. Sharpe's own 1994 text also notes that he uses the population form of the standard deviation, with T in the denominator rather than T − 1, while remarking that either is workable when the sample length is the same across the funds being compared. This page uses T − 1, which is the convention in the academic performance-measurement literature, and reports the T-divisor value beside it so the difference is visible rather than hidden.
Sources retrieved: William F. Sharpe, “The Sharpe Ratio”, Journal of Portfolio Management, Fall 1994, full text on the author's own page at Stanford, including endnote 1 on the choice of divisor and the note that the ratio times the square root of T equals the t-statistic of the mean differential return. Sharpe (1966), “Mutual Fund Performance”, Journal of Business, is cited within that text as the origin of the measure.
Multiplying a per-period Sharpe ratio by the square root of P assumes the periodic returns are serially uncorrelated. Lo (2002) derives the correct scaling factor in the general stationary case. Writing eta(q) for the factor that converts a one-period ratio to a q-period ratio, eta(q) equals q divided by the square root of q plus twice the sum over k from 1 to q − 1 of (q − k) times the k-th autocorrelation. Under an AR(1) structure the k-th autocorrelation is the first raised to the k-th power, which is the plug-in this page uses. When autocorrelation is zero the factor reduces exactly to the square root of q. When it is positive the factor is smaller, so the naive annualisation overstates the annual ratio; when it is negative the factor is larger. Lo's own abstract states the size of the effect in practice: an annual hedge fund Sharpe ratio can be overstated by as much as 65 percent by the square-root rule when monthly returns are serially correlated, and the ranking of funds can change once the correlation is accounted for.
Source retrieved: Andrew W. Lo, “The Statistics of Sharpe Ratios”, Financial Analysts Journal 58(4), 2002, pages 36–52, doi 10.2469/faj.v58.n4.2453; also listed on the author's page at MIT. The article itself is paywalled: what was retrieved is the abstract and the bibliographic record, which state the result but not the algebra. The eta(q) expression implemented here is the AR(1) plug-in form; it has not been read off the published page images, and that limitation is stated rather than glossed.
Under iid normal returns the standard error of the estimated per-period Sharpe ratio is the square root of (1 plus half the squared ratio) divided by T. That is the figure used for the confidence interval here, and it is what makes short samples uninformative: at T = 24 the standard error is large enough that most reported ratios are indistinguishable from zero. Relaxing normality, the widely used correction adds a skewness term and a kurtosis term, giving a numerator of 1 minus skewness times the ratio plus (kurtosis minus 1) over 4 times the squared ratio, with kurtosis the full fourth moment. This page reports two non-normal standard errors, because the published corpus on this site and the canonical form do not agree: the corpus value corresponds to a numerator using (kurtosis minus 3) over 4 rather than (kurtosis minus 1) over 4, which is the same expression with the half-squared-ratio term omitted. Both are shown and both are labelled. Neither is presented as the other.
Sources retrieved: the peer-reviewed comparison of thirteen performance measures by Martin Eling and Frank Schuhmacher, “Does the choice of performance measure influence the evaluation of hedge funds?”, Journal of Banking & Finance, 2007, which states the Sharpe, Sortino, Omega, Calmar, Sterling, Jensen and Treynor formulas explicitly and uses the T − 1 divisor. The non-normal correction is attributed in the literature to a 2002 research note by Elmar Mertens, “Comments on variance of the IID estimator in Lo (2002)”; that note itself was not retrieved, only bibliographic references to it in indexed academic work, and it is named here on that basis alone.
The ratio of a sample mean to a sample standard deviation is biased upward in small samples, because the sample standard deviation is biased downward. Miller and Gehr (1978) give the exact factor. Dividing the estimate by the square root of (T − 1) over 2, times the ratio of the gamma function at (T − 2)/2 to the gamma function at (T − 1)/2, removes it. The correction does not change the ranking of series measured over identical lengths; it changes the level, and at T = 24 it is a shade over three percent.
The Sortino ratio replaces the standard deviation with downside deviation: the root mean square of the shortfalls below a minimum acceptable return, with every period at or above the target contributing zero. The load-bearing detail is the divisor. Downside deviation is the square root of the second-order lower partial moment, and the lower partial moment divides by the total number of periods T, not by the count of periods below the target. Dividing by the below-target count turns the measure into an ordinary conditional standard deviation and destroys the property that matters: a strategy that rarely breaches the target should earn a smaller denominator for it. This page reports the T-divisor value as the Sortino ratio and the below-count value beside it, labelled, because the second is common in implementations and is roughly double the first on the published base series.
Sources: Sortino and van der Meer (1991), “Downside Risk”, Journal of Portfolio Management 17, pages 27–31; refined in Sortino and Price (1994), “Performance Measurement in a Downside Risk Framework”, Journal of Investing 3(3), pages 59–64, doi 10.3905/joi.3.3.59 — publisher record retrieved, the articles themselves paywalled. The divisor-T convention was verified against the explicit lower partial moment definition in Eling and Schuhmacher (2007) linked above, which defines LPM of order n as one over T times the sum of the shortfalls raised to the power n, and the Sortino ratio as excess return over the square root of the second-order LPM.
Omega at a threshold is the sum of the amounts by which returns exceed the threshold divided by the sum of the amounts by which they fall short of it. It uses the entire distribution and no moment estimates at all, which is its appeal, and it is scale-free in a way that makes it unusable as a signed quantity: it is 1 when gains and shortfalls balance, unbounded above, and it cannot go below zero. Eling and Schuhmacher give the algebraically equivalent representation as excess return over the first-order lower partial moment, plus one; the two forms return the same number.
The equity index is the running product of one plus each return starting from 1. The running peak is the highest value so far. Drawdown at each period is one minus the ratio of equity to that peak. Maximum drawdown is the largest of those. Two averages are reported because both are in use: the mean over all periods, which includes the zeros at new highs, and the mean over in-drawdown periods only. Time under water is the count of periods with a drawdown above zero. The gain required to recover from the maximum is d over (1 − d), which is always larger than d.
The Ulcer index is the root mean square of the percentage drawdowns. Its author's published algorithm accumulates squared drawdowns only in the periods below the prior peak, but divides by the total number of periods, so periods at a new high contribute zero to the numerator and one to the denominator. That is the primary figure here. The in-drawdown-only variant is shown beside it and is not the author's definition. The Martin ratio, also called the Ulcer Performance Index, is the excess return over the Ulcer index, in the author's own words total return minus risk-free return, divided by the Ulcer index.
Source retrieved: Peter G. Martin, “Ulcer Index: An Alternative Approach to the Measurement of Investment Risk & Risk-Adjusted Performance”, the author's own page, retrieved in full including the pseudo-code that divides the sum of squares by NumOfPeriods and the definition UPI = (Total return − Risk-free return) / UI. Martin dates the measure to 1987 and its first published description to The Investor's Guide to Fidelity Funds (1989).
Calmar divides a return by the maximum drawdown. Which return, and over what window, is where the definitions part company. Young's own 1991 definition, quoted verbatim in the reference below, is the average annual rate of return for the last 36 months divided by the maximum drawdown for the last 36 months, recomputed monthly. The peer-reviewed statement in Eling and Schuhmacher uses the excess return, mean return minus the risk-free rate, over the maximum drawdown. The corpus published on this site uses the geometric annualised return with no risk-free deduction, over the whole sample rather than a 36-month window. This page computes the corpus convention as the headline figure and shows the excess-return variant beside it, and states that Young's 36-month window is not applied because the window length is not an input.
Sterling is the same idea with the denominator softened. The older managed-futures convention divides the annual return by the average of the annual maximum drawdowns plus ten percentage points, an offset whose only justification is that it stops the ratio exploding when drawdowns are small. Kestner's later definition, as stated in Eling and Schuhmacher, drops the offset and takes the mean of the N largest drawdowns. Both offset and no-offset figures are reported, using the average of the per-year maximum drawdowns as the drawdown input, and they differ by nearly a factor of three on the base series, which is the point of showing both.
Sources retrieved: the Calmar ratio reference page, which quotes Terry W. Young's definition verbatim and cites “Calmar Ratio: A Smoother Tool”, Futures, 1 October 1991, and records that the name is an acronym of California Managed Accounts Reports. The 1991 magazine article itself is not available online and was not retrieved; it is cited here as the origin, on the strength of the verbatim quotation, and not as a document read. Sterling and Calmar formulas as used in the academic literature: Eling and Schuhmacher (2007), equations 6 and 7, linked above, attributing Sterling to Kestner (1996) and Calmar to Young (1991).
Beta is the sample covariance of the series with the benchmark divided by the benchmark variance. Correlation and R-squared follow from it. Jensen's alpha is the excess return of the series minus beta times the excess return of the benchmark, exactly as stated in Eling and Schuhmacher's equation 12, annualised by multiplying by P. Its t-statistic is the alpha divided by the ordinary least squares standard error of a regression intercept, which uses the residual standard deviation on T − 2 degrees of freedom and the leverage term for the mean of the regressor. Tracking error is the standard deviation of the arithmetic active return, series minus benchmark. The information ratio is the mean active return over the tracking error. The appraisal ratio is alpha over the residual standard deviation of the regression, which is a different denominator from the tracking error and a different number. Treynor is the annual excess return over beta. M-squared restates the Sharpe ratio as a return by levering the series to the benchmark's volatility. Up and down capture are the mean return of the series in periods when the benchmark rose or fell, divided by the mean benchmark return in those same periods; they are ratios of means, not means of ratios, and the two are not the same.
A reported drawdown means nothing on its own, because a process with no edge whatsoever produces drawdowns, and the deeper the volatility the deeper they get. Magdon-Ismail, Atiya, Pratap and Abu-Mostafa (2004) analyse the maximum drawdown of a Brownian motion with drift, giving an infinite series for its distribution and, in the zero-drift case, a closed-form expression for its expected value; their abstract records that the limiting behaviour in the horizon is square-root for zero drift, logarithmic for positive drift and linear for negative drift. The zero-drift expected maximum drawdown scales as the square root of pi over two, about 1.253314, times the volatility times the square root of the horizon in years. This page reports that quantity at the volatility and horizon of the series entered, and the ratio of the observed maximum drawdown to it. A ratio below one means the observed drawdown is shallower than a driftless random walk of the same volatility would be expected to produce, which is not evidence of skill; it is the absence of evidence of unusual risk.
Source retrieved: Magdon-Ismail, Atiya, Pratap and Abu-Mostafa, “On the Maximum Drawdown of a Brownian Motion”, Journal of Applied Probability 41(1), 2004, pages 147–161, doi 10.1239/jap/1077134674 — the institutional record and full abstract were retrieved; the article is paywalled. The abstract confirms the square-root scaling and the existence of an analytic zero-drift expectation. The specific constant, the square root of pi over two, was not read from the paper and is used here as published in the corpus this page belongs to.
The corpus this calculator belongs to publishes a constructed 24-month return series and a benchmark, and derives every statistic on the site from them. Those are the default inputs, so the page loads reproducing the published numbers. The figures below are the published values, for comparison against the live table above.
| Statistic | Published monthly | Published annualised |
|---|---|---|
| Arithmetic mean return | 0.766667 % | 9.2000 % |
| Geometric mean return | 0.745939 % | 9.3278 % |
| Terminal wealth from 1.00 | — | 1.19525673 |
| Standard deviation | 2.080273 % | 7.2063 % |
| Skewness / kurtosis | −0.508966 / 2.368633 | — |
| Lag-1 autocorrelation | 0.297578 | eta(12) = 2.621035 |
| Sharpe ratio | 0.272400 | 0.943622 at sqrt(12) |
| Sharpe ratio, Lo-corrected | 0.272400 | 0.713970 |
| Sharpe ratio, bias-corrected | 0.263403 | 0.912456 |
| Standard error, iid normal | 0.207876 | 0.720104 |
| Standard error, non-normal | 0.216692 | 0.750643 |
| Sortino ratio, MAR 0.20 % | 0.444302 | 1.539108 |
| Sortino, below-count divisor | 0.256518 | 0.888604 |
| Omega at 0 / at 0.20 % | 2.352941 / 1.894737 | — |
| Maximum drawdown | — | 7.5967 % |
| Average drawdown, all / in-drawdown | — | 2.0078 % / 3.0117 % |
| Ulcer index, all / in-drawdown | — | 2.9821 % / 3.6523 % |
| Calmar / Martin | — | 1.227869 / 2.314199 |
| Sterling, 10-point offset / none | — | 0.594269 / 1.637532 |
| Gain required to recover | — | 8.2213 % |
| Year 1 / year 2 maximum drawdown | — | 4.0820 % / 7.3105 % |
| E[MaxDD], driftless, 2 years | — | 12.7728 %, ratio 0.594759 |
| Beta / correlation / R-squared | 0.750000 / 0.855946 | 0.732644 |
| Jensen's alpha | 0.266667 % | 3.2000 %, t = 1.170625 |
| Tracking error | 1.228526 % | 4.2557 % |
| Information ratio / appraisal ratio | 0.135664 / 0.242466 | 0.469954 / 0.839927 |
| Treynor / M-squared | — | 0.090667 / 10.1606 % |
| Up / down capture | — | 0.909938 / 0.612360 |