Deflated Sharpe ratio: formula, worked example and how many backtests are too many

A researcher tests 200 versions of a monthly trend-following rule on ten years of data, changing the lookback window, the entry threshold and the position sizing. The best version shows an annualized Sharpe ratio of 1.20. Its monthly returns are negatively skewed, with a skewness of −0.8 and a kurtosis of 6. Taken on its own, that record has a 99.9% probability of coming from a strategy whose true Sharpe ratio is above zero. Once the other 199 versions are counted, the probability falls to 73%.

The second figure is the deflated Sharpe ratio.

The measure starts from the ordinary Sharpe ratio, which our Sharpe ratio guide computes step by step on the S&P 500. Every Sharpe ratio in the formulas below is measured at the frequency of the data, monthly or daily, and annualized only for display.

Why the best of many backtests has an inflated Sharpe ratio

A Sharpe ratio measured on historical data is an estimate with a margin of error. When every strategy in a set has a true Sharpe ratio of zero, their measured ratios still scatter around zero, some above and some below. Picking the highest one picks the strategy that drew the most favorable noise.

The expected size of that maximum has a closed-form approximation. For N independent trials whose measured Sharpe ratios are normally distributed with a standard deviation σ, the expected best ratio is:

Applied to strategies with no skill at all, tested over two, five and ten years of data:

Forty-five trials on five years of data produce an expected best annualized Sharpe ratio of 1.0 from strategies that will earn nothing out of sample. Bailey, Jonathan Borwein, López de Prado and Qiji Jim Zhu turned that relationship into a minimum backtest length in the Notices of the American Mathematical Society in 2014: the number of years of data needed to keep the expected best Sharpe ratio of skill-less configurations below a chosen level. On two years of data, seven independent configurations are enough to expect a best Sharpe ratio of about 1.0.

The expected maximum grows with the square root of the logarithm of N. Going from 10 to 100 trials adds about one standard deviation to the expected best result. Going from 1,000 to 10,000 adds 0.6.

The same arithmetic applies outside backtests. A consultant who screens 500 hedge funds and presents the one with the best three-year Sharpe ratio is reporting a maximum, and so is a database sorted by Sharpe ratio. Selection bias also enters through unreported experiments, journals that publish only positive results, hedge fund indices that drop funds after they close, and managers who start reporting a track record once it looks good, the survivorship and backfill biases that inflate hedge fund index returns.

From Lo's standard errors to the probabilistic Sharpe ratio

The deflated Sharpe ratio corrects two separate sources of inflation. Selection is the first. The second is the margin of error on a single Sharpe ratio, which depends on the length of the record and on the shape of the return distribution.

Andrew Lo's 2002 article "The Statistics of Sharpe Ratios", in the Financial Analysts Journal, gave the standard error of a Sharpe ratio measured on T independent, normally distributed returns: the square root of (1 + SR²/2) ÷ T. Our Sharpe ratio article applies it to the S&P 500 and covers Lo's adjustment for serially correlated returns. Elmar Mertens extended the formula the same year to skewed and fat-tailed returns:

In the opening example, the best variant's monthly Sharpe ratio is 0.346. Lo's formula gives a standard error of 0.094 over 120 months. With a skewness of −0.8 and a kurtosis of 6, the standard error rises to 0.110, about 16% wider. Most of the increase comes from the skewness term.

The probabilistic Sharpe ratio and the minimum track record length

Bailey and López de Prado turned that standard error into a probability in "The Sharpe Ratio Efficient Frontier", published in the Journal of Risk in 2012. The probabilistic Sharpe ratio (PSR) is the probability that the true Sharpe ratio exceeds a benchmark SR*, given the measured ratio, the number of observations and the third and fourth moments of the returns:

With SR* set to zero, the best of the 200 variants scores a PSR of 0.9992. The same paper defines the minimum track record length, the number of observations needed for the PSR to reach a chosen confidence level:

MinTRL = 1 + [1 − γ₃SR + (γ₄ − 1)/4 × SR²] × [Z⁻¹(0.95) ÷ (SR − SR*)]²

For an annualized Sharpe ratio of 2 to be judged above 1 at 95% confidence, normal monthly returns need a 3.24-year record. With the skewness of −0.72 and kurtosis of 5.78 of the HFR aggregate hedge fund index, the requirement grows to 4.99 years. Over January 2000 to May 2011, only 9 of the HFR style indices had an annualized Sharpe ratio above 0.5 at 95% confidence by this measure, among them merger arbitrage, equity market neutral, macro and distressed securities.

A PSR of 0.9992 describes one strategy tested once. The researcher in the example tested 200.

The deflated Sharpe ratio formula

The deflated Sharpe ratio is a probabilistic Sharpe ratio whose benchmark is SR₀, the Sharpe ratio expected from the best of N skill-less trials. If the strategy has no skill, a search of N trials is expected to turn up a ratio of about SR₀, so the measured ratio has to beat that level.

The inputs, and where each one comes from:

SR, the selected strategy's Sharpe ratio

Measured at the frequency of the data and left unannualized. A monthly ratio of 0.346 corresponds to an annualized 1.20.

T, the number of return observations

120 for ten years of monthly returns, about 1,250 for five years of daily returns.

γ₃ and γ₄, skewness and kurtosis

Computed on the selected strategy's returns. The formula expects raw kurtosis, which is 3 for a normal distribution.

N, the number of independent trials

Every configuration examined before the selection, reduced to the number of effectively independent ones when the trials are correlated.

V[SR_n], the variance of the trials' Sharpe ratios

The cross-sectional variance of the Sharpe ratios of all the trials, at the same frequency as SR. It comes from the trials themselves, so every trial has to be recorded.

A plain Sharpe ratio rests on two estimates, the mean and the standard deviation of returns. The deflated version adds five: skewness, kurtosis, sample length, the number of trials and the dispersion of their results. With a single trial there is no search to correct for, and the DSR reduces to the PSR measured against a benchmark of zero.

Worked example: 200 variants of a trend rule

The opening example, computed in full. The inputs:

• 200 variants tested, treated as independent for now
• annualized Sharpe ratios across the 200 variants with a standard deviation of 0.35
• best variant: annualized Sharpe ratio of 1.20 over 120 months, skewness of −0.8, kurtosis of 6

1. Convert to monthly figures. The best Sharpe ratio becomes 1.20 ÷ √12 = 0.3464. The standard deviation of the trials' ratios becomes 0.35 ÷ √12 = 0.1010.
2. Expected maximum of 200 standard normal draws. Z⁻¹(1 − 1/200) = 2.576 and Z⁻¹(1 − 1/(200e)) = 2.904. Weighted by 0.4228 and 0.5772, they give 2.766.
3. Deflated benchmark. SR₀ = 0.1010 × 2.766 = 0.2794 a month, or 0.97 annualized. Two hundred variants with no skill would be expected to produce a best annualized Sharpe ratio of about 0.97.
4. Non-normality term. 1 − (−0.8 × 0.3464) + (6 − 1)/4 × 0.3464² = 1 + 0.2771 + 0.1500 = 1.4271. Its square root is 1.1946.
5. Test statistic. (0.3464 − 0.2794) × √119 ÷ 1.1946 = 0.0670 × 10.909 ÷ 1.1946 = 0.612.
6. Deflated Sharpe ratio. Z(0.612) = 0.73.

A DSR of 0.73 is a 73% probability that the true Sharpe ratio is above zero once the search is accounted for, short of the 0.95 threshold. The same track record with a different number of trials behind it:

The record clears 0.95 with up to 11 independent trials behind it. With 200 trials, the best variant would need an annualized Sharpe ratio of about 1.64 to reach 0.95, or, keeping its Sharpe ratio of 1.20 and the same return distribution, a track record of about 72 years.

In the last row the order of the two columns flips. Once the measured ratio falls below SR₀, the wider margin of error from negative skewness pulls the probability back toward 0.5.

Bailey and López de Prado's Treasury auction example

The 2014 paper illustrates the measure with a strategist who trades the US Treasury auction cycle, selling off-the-run bonds a few days before an auction and buying the new issue a few days after it. After backtesting combinations of pre-auction and post-auction windows, tenors, holding periods and stop-losses, the best configuration shows an annualized Sharpe ratio of 2.5 on five years of daily data. The investor approached for funding asks for four numbers:

• 100 independent trials
• a variance of 0.5 across the trials' annualized Sharpe ratios
• 1,250 daily observations
• a skewness of −3 and a kurtosis of 10

The deflated benchmark comes out at 0.113 a day, 1.79 annualized, and the DSR at 0.9004, below 0.95. The investor declines. Had the strategist stopped after 46 trials, the DSR would have been 0.9505. With normal returns and the same Sharpe ratio, up to 88 trials would have passed.

How to count the number of independent trials

Published backtests rarely report N. A grid of 200 parameter combinations seldom contains 200 independent strategies either: a 50-day and a 55-day moving average produce nearly the same trades. Counting all 200 overstates SR₀, and counting only the final specification sets N to 1.

The 2014 paper offers an interpolation based on the average correlation ρ between the M trials actually run:

N ≈ ρ + (1 − ρ) × M

With ρ = 0 every trial counts, and with ρ = 1 the whole grid is a single trial. For the 200 variants of the example:

• average correlation of 0.3: N ≈ 140, DSR of 0.76
• average correlation of 0.6: N ≈ 81, DSR of 0.82
• average correlation of 0.9: N ≈ 21, DSR of 0.92

Even at an average correlation of 0.9, the example stays below 0.95.

Average correlation is a rough measure. When the number of trials exceeds the number of observations, as it does whenever 1,000 configurations are run on 120 months of returns, the correlation matrix cannot be estimated reliably. The paper suggests reducing its dimension first or using information-theoretic measures of redundancy. Principal component analysis and clustering of the trials' return series do the same job: they group trials that behave alike and count the groups.

A trial covers more than the final grid search. Variables tried and dropped, universes tested, alternative cost assumptions and earlier versions of the idea all belong in M. A research log that records every backtest, failures included, is what makes N and V[SR_n] computable.

The paper also proposes a stopping rule taken from the secretary problem in probability theory. Sample about 37% (1/e) of the configurations that theory justifies, at random. Then keep testing one at a time and stop at the first configuration that beats every one seen so far.

Errors that change the deflated Sharpe ratio

• Annualized Sharpe ratios in the formula. The formula expects Sharpe ratios at the frequency of T. Running the example with the annualized figures, 1.20 and a trial standard deviation of 0.35, against T = 120 months gives a DSR of 0.90 instead of 0.73.
• Excess kurtosis where raw kurtosis is expected. scipy.stats.kurtosis and pandas' kurt both return excess kurtosis by default, 0 for a normal distribution. Entering 3 in place of 6 moves the example's DSR from 0.730 to 0.736. The gap grows with the per-period Sharpe ratio, because the kurtosis term is multiplied by SR².
• Total trials in place of independent trials. Counting every correlated variant overstates the deflated benchmark. Counting only the final model, as a backtest that does not disclose its search implicitly does, removes the correction altogether.
• An assumed variance of trial Sharpe ratios. V[SR_n] has to come from the recorded trials. A conventional value such as 1 has no connection to the search that was run, and without the Sharpe ratios of the trials the DSR cannot be computed.
• Smoothed or autocorrelated returns. The formula assumes returns that are independent over time. Monthly returns from illiquid holdings are often positively autocorrelated, which understates volatility and overstates the Sharpe ratio before any deflation, as our Sharpe ratio article shows for hedge funds holding mortgage-backed securities. The serial correlation has to be dealt with first.
• The standard error evaluated at a different point. The 2014 formula uses the selected strategy's own Sharpe ratio in the denominator. Some implementations, such as the ML4T diagnostic library, evaluate it at SR₀, which gives 0.738 in the example instead of 0.730.

How much backtested Sharpe ratios shrink out of sample

Across 215 commercially promoted alternative beta strategies in five asset classes, the median Sharpe ratio fell 73% between the backtest and live trading. The Sharpe ratios of the most complex strategies fell by over 30 percentage points more than those of the simplest.

On the Quantopian platform, 888 trading algorithms with at least six months of out-of-sample results showed almost no link between backtest and live performance. Backtest metrics such as the Sharpe ratio explained less than 2.5% of the variation in out-of-sample performance (an R² below 0.025). The more backtests an author had run on a strategy, the wider the gap between its backtest and its live results.

Published academic factors lose part of their returns after publication too, and the count of more than 300 proposed factors led Campbell Harvey, Yan Liu and Heqing Zhu to raise the t-statistic hurdle for a new one to 3.0, as the factor zoo section of our factor investing article explains. Because a t-statistic equals the annualized Sharpe ratio times the square root of the number of years, a t-statistic of 3.0 on ten years of data corresponds to an annualized Sharpe ratio of 0.95, close to the 0.97 deflated benchmark of the 200-variant example.

Haircut Sharpe ratios, PBO and other corrections for multiple testing

The 2014 paper presents the DSR as complementary to other corrections and suggests computing it against more than one threshold.

Harvey and Liu's haircut Sharpe ratio

Campbell Harvey and Yan Liu's "Backtesting", published in the Journal of Portfolio Management in 2015, converts a Sharpe ratio into a t-statistic, adjusts its p-value for the number of tests with the Bonferroni, Holm or Benjamini-Hochberg-Yekutieli method, and converts the result back into a lower Sharpe ratio. The cut depends on the starting level. With 50 tests, an earnings-to-price strategy with a Sharpe ratio of 0.43 lost almost half of it, and a betting-against-beta strategy at 0.78 lost 7.9%. Across their scenarios, ratios below 0.4 almost always lost more than 50% and ratios above 1.0 lost at most 25%, which led them to reject the common practice of halving every backtested Sharpe ratio. The haircut produces a smaller Sharpe ratio, where the DSR produces a probability.

The probability of backtest overfitting

The probability of backtest overfitting (PBO), from Bailey, Borwein, López de Prado and Zhu and published in the Journal of Computational Finance in 2017, is estimated by combinatorially symmetric cross-validation. The return history of all trials is split into blocks, every combination of half the blocks serves once as the in-sample period, and PBO is the share of combinations in which the configuration that ranked best in sample ranks below the median out of sample. It makes no assumption about the distribution of returns and works with any performance statistic, but it needs the full return series of every trial.

Bayesian false discovery rates

A 2025 paper by López de Prado, Alexander Lipton and Vincent Zoonekynd for ADIA Lab, "How to Use the Sharpe Ratio", revisits the PSR, the minimum track record length and the DSR and adds Bayesian false discovery rate methods. In its Monte Carlo experiments, the corrected measures outperform standard t-tests and the common multiple-testing corrections.

Computing the deflated Sharpe ratio in Python

The function below uses only Python's standard library and reproduces the worked example. It takes Sharpe ratios at the frequency of the returns, raw kurtosis, and at least two trials.

from math import e, sqrt
from statistics import NormalDist

def deflated_sharpe(sr, t, skew, kurt, n_trials, var_trials):
    z = NormalDist()
    gamma = 0.5772156649
    emax = ((1 - gamma) * z.inv_cdf(1 - 1 / n_trials)
            + gamma * z.inv_cdf(1 - 1 / (n_trials * e)))
    sr0 = sqrt(var_trials) * emax
    se = sqrt(1 - skew * sr + (kurt - 1) / 4 * sr ** 2)
    return z.cdf((sr - sr0) * sqrt(t - 1) / se)

# 200 trials, 120 months, best annualized Sharpe ratio 1.20
sr = 1.20 / sqrt(12)
var_trials = (0.35 / sqrt(12)) ** 2
print(deflated_sharpe(sr, 120, -0.8, 6.0, 200, var_trials))  # 0.7296

From raw data, each trial's Sharpe ratio is its mean excess return divided by the standard deviation of its excess returns, var_trials is the variance of those ratios across trials, and the skewness and kurtosis come from the selected strategy's returns, with scipy.stats.kurtosis(returns, fisher=False) giving raw kurtosis. For a single strategy tested once, the PSR with a benchmark of zero replaces the function, because the expected-maximum formula needs N of at least 2.

Limits of the deflated Sharpe ratio

The expected-maximum formula assumes the trials' Sharpe ratios are normally distributed around a common mean, which fits a family of variants of one idea better than a mix of unrelated strategies. It is also an approximation for large N. In the paper's simulations it overstates the simulated maximum by less than 0.05 standard deviations when there are fewer than 50 trials, and by about 0.006 at 1,000 trials.

Skewness and kurtosis are noisy on short samples, and a single extreme observation can move the kurtosis of a whole series, as our article on fat tails shows with the October 1987 crash in S&P 500 daily returns. When the moments are uncertain, Bailey and López de Prado recommend entering a lower bound for skewness and an upper bound for kurtosis in place of the point estimates.

The DSR measures statistical evidence and nothing else. Transaction costs, capacity limits and a market change that removes the effect after the test period all sit outside it, so a strategy with a DSR above 0.95 still has to be tested out of sample and after costs. The correction also covers only the trials that were recorded and disclosed.

Deflated Sharpe ratio questions

What is a good deflated Sharpe ratio?

A deflated Sharpe ratio of 0.95 or more is the usual threshold: it means a 95% probability that the true Sharpe ratio is above zero after accounting for the number of strategies tried. A value of 0.5 means the selected strategy did exactly as well as the best of the same number of skill-less trials would be expected to do. In the worked example, a 1.20 annualized Sharpe ratio picked from 200 backtests on ten years of monthly data scores 0.73.

What is the difference between the probabilistic and the deflated Sharpe ratio?

Both give the probability that a strategy's true Sharpe ratio exceeds a benchmark, after correcting for track record length, skewness and kurtosis. In the probabilistic Sharpe ratio the user chooses the benchmark, often zero. In the deflated Sharpe ratio the benchmark is the Sharpe ratio expected from the best of N skill-less trials, so it rises with the number of strategies tested. With a single trial the two are the same.

Who created the deflated Sharpe ratio?

David H. Bailey and Marcos López de Prado, in "The Deflated Sharpe Ratio: Correcting for Selection Bias, Backtest Overfitting and Non-Normality", published in the Journal of Portfolio Management in 2014 (volume 40, issue 5, pages 94 to 107). A free copy of the paper is hosted on David Bailey's website, davidhbailey.com.

Does a deflated Sharpe ratio of 0.42 mean the strategy kept 42% of its Sharpe ratio?

No. The deflated Sharpe ratio is a probability between 0 and 1. A value of 0.42 means a 42% probability that the true Sharpe ratio is above zero once the search is accounted for. Any value below 0.5 means the selected strategy's Sharpe ratio is lower than the one expected from the best of the same number of skill-less trials.

Can the deflated Sharpe ratio be used to choose a fund?

The arithmetic applies to funds as well as backtests. When an investor or consultant screens many managers in the same strategy class and keeps the one with the highest Sharpe ratio, N is the number of managers screened. The calculation measures statistical evidence only. It says nothing about fees, capacity or whether the strategy will keep working, and this article is general information rather than investment advice.