Common mistake 1 with Performance & drawdown

Comparing Sharpe ratios computed at different frequencies or with different risk-free assumptions.

Common mistake 2 with Performance & drawdown

Reading a short, calm track record's Sharpe as durable - the drawdown that has not happened yet is missing from the sample.

Common mistake 3 with Performance & drawdown

Annualising the Sharpe of a series with strong positive autocorrelation; the √P scaling overstates it.

Common mistake 4 with Performance & drawdown

Comparing a Sharpe ratio against a benchmark's without noting that the benchmark is not investable. Index returns carry no fees, no tracking error and no cash drag.

EconStatsCraft › Performance & drawdown

Performance & drawdown

Turns a return series into the numbers an investment committee asks for - CAGR, annualised volatility, Sharpe, Sortino, the maximum drawdown and how long it lasted - with an optional benchmark for tracking error and capture ratios.

What it needs

Return series
Benchmark (optional)

The intuition

A return of 12% means nothing on its own. Twelve percent earned smoothly is a different product from twelve percent earned through a 40% drawdown. Every performance ratio is the same shape - reward divided by some definition of risk - and they differ only in what they are willing to call risk. Sharpe counts all variation; Sortino counts only downside; the information ratio counts only deviation from the benchmark.

When to use it

Evaluating a strategy or fund, comparing managers on a like-for-like basis, or stress-testing whether a track record is actually holdable - the maximum drawdown is the number that answers that.

Where the estimator comes from

Sharpe is S=(\bar{R}-R_f)/\sigma using total standard deviation. Sortino replaces the denominator with downside deviation, \sqrt{\frac{1}{T}\sum_t \min(R_t-\tau,0)^2} against a target \tau, on the argument that upside variation is not a risk anyone wishes to be compensated for. The information ratio is (\bar{R}-\bar{R}_b)/\sigma(R-R_b), where the denominator is tracking error - the standard deviation of the ACTIVE return, not of the fund. Annualising multiplies the numerator by k and the denominator by \sqrt{k}, so the ratio itself scales by \sqrt{k}: a monthly Sharpe of 0.25 is 0.25\sqrt{12}=0.87 a year.

The model

\text{Sharpe} = \frac{\bar{r}-r_f}{\sigma}\sqrt{P},\qquad \text{Sortino} = \frac{\bar{r}-r_f}{\sigma_{\text{down}}}\sqrt{P},\qquad \text{MDD} = \min_t\left(\frac{W_t}{\max_{s\le t}W_s}-1\right)

What it assumes

The return series is complete and at a regular frequency; gaps distort the compounding and the drawdown path.
Annualising by √P assumes returns are serially uncorrelated - smoothed or illiquid returns (private assets, some hedge funds) violate this and flatter the Sharpe ratio.
The risk-free rate used for excess returns matches the return frequency.
The track record is the whole record. Performance statistics computed on surviving funds overstate the average by the amount the closed ones lost, and a single fund's history has the same problem if it began reporting after a good year.

Statistical properties

The square-root-of-time annualisation assumes independent returns. Positive autocorrelation, which is common in funds holding illiquid assets, inflates the annualised Sharpe ratio substantially (Lo, 2002).
Sharpe ratios are themselves estimates with large standard errors. Roughly, the standard error is about the square root of (1 + S squared/2) divided by the square root of T. Three years of monthly data gives a standard error near 0.33.
Sharpe rewards smoothness, so strategies that sell insurance - writing options, carry trades, credit - look excellent right up until they do not. Their returns are negatively skewed and the ratio does not see skewness at all.
Sortino exceeds Sharpe whenever returns are positively skewed and falls below it when negatively skewed. Comparing the two is a free skewness diagnostic.
The information ratio is the right measure for a fund with a mandate to beat an index; the Sharpe ratio is the right one for a standalone allocation. They answer different questions and are not comparable.
Maximum drawdown is highly sensitive to the sample window and is not a statistic with a usable sampling distribution. Report it, but do not rank funds on it.

How to read the output

Sharpe uses total volatility; Sortino only penalises downside deviation, so a strategy with big upside surprises scores better on Sortino than on Sharpe.
Maximum drawdown is the worst peak-to-trough loss an investor would have sat through. A high Sharpe with a 60% drawdown is not a strategy most people can actually hold.
Calmar (CAGR ÷ |max drawdown|) is the return-per-unit-of-pain number that allocators to trend and macro strategies watch.
Down capture below 1 with up capture near 1 is the asymmetry every active manager claims; check whether it actually holds in the data.
Maximum drawdown is path-dependent, so it depends on the start and end dates in a way the volatility does not - two funds with identical returns in a different order have different drawdowns.
Up and down capture separate where the performance came from. Beating the benchmark through smaller losses is a different business from beating it through larger gains, and the two need different explanations.

A worked reading

A 180-month record: total return 295.94%, a CAGR of 9.61% on annualised volatility of 19.30%, Sharpe 0.571 and Sortino 0.635. Sortino only 11% above Sharpe says the downside deviation is close to the total volatility - this record is roughly symmetric, not the positively skewed kind where Sortino runs far ahead. Against the benchmark, tracking error is 10.57% for an information ratio of 0.330, with up-capture 1.007 and down-capture 0.817: it keeps pace in rising months and loses less in falling ones, which is where the active return comes from. The number that deserves the most attention is the maximum drawdown of -34.17%, running 13 periods from peak to trough and taking a further 35 to recover. A Calmar ratio of 0.281 puts that in perspective - nearly four years of average growth to work off the worst decline. Ratios computed on 180 observations still carry a standard error on the Sharpe of roughly 0.077, so 0.571 is about 0.571 ± 0.15 at two standard errors: comfortably positive, and nowhere near precisely measured.

Symptom → diagnosis → what to do

If you see this	It means	Do this
Sharpe above 2 for a long-only fund	Too short a sample, or an illiquid mark-to-model	Check the return autocorrelation
Sortino far above Sharpe	Positive skew, or a target set too low	State the target explicitly
Sortino below Sharpe	Negative skew: losses are larger than gains	The Sharpe ratio is flattering this fund
Information ratio high, Sharpe low	Beats the benchmark but the benchmark did badly	Both statements are true; report both
Returns strongly autocorrelated	Stale pricing or smoothing	Annualised figures are inflated. Apply the Lo correction
Ratio changes sign on a small window shift	The mean is not distinguishable from zero	The sample is too short for any of this

Common mistakes

Comparing Sharpe ratios computed at different frequencies or with different risk-free assumptions.
Reading a short, calm track record's Sharpe as durable - the drawdown that has not happened yet is missing from the sample.
Annualising the Sharpe of a series with strong positive autocorrelation; the √P scaling overstates it.
Comparing a Sharpe ratio against a benchmark's without noting that the benchmark is not investable. Index returns carry no fees, no tracking error and no cash drag.

Where to read this properly

Sharpe (1966), "Mutual Fund Performance", Journal of Business 39(1).
Sortino and Price (1994), "Performance Measurement in a Downside Risk Framework", Journal of Investing 3(3).
Lo (2002), "The Statistics of Sharpe Ratios", Financial Analysts Journal 58(4) - the autocorrelation correction.
Grinold and Kahn, Active Portfolio Management (2000), ch. 5, on the information ratio.

Related Finance & risk

Black–Scholes option pricing · Bond price, duration & convexity · Merton distance to default · EMI / home loan planner · CAPM / beta estimation · Multi-factor model · Portfolio optimization · Value at Risk & Expected Shortfall · Time value of money & NPV/IRR · Equity valuation (DDM, FCF, residual income) · Cost of capital (WACC) · Binomial option pricing · Option strategies (payoff & profit) · Forwards, futures & swaps · Credit risk & Basel capital · Extreme value theory (tail risk)

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Performance & drawdown

What it needs

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

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