Drawdown and path statistics

Maximum drawdown

The largest peak-to-trough decline in the equity curve over the sample, expressed as a fraction of the peak. It is computed on the compounded path, which means it depends on the ordering of returns and not only on their distribution.

Field	Value
Formula	E_t = prod_{s<=t} (1 + r_s); P_t = max_{s<=t} E_s; D_t = 1 - E_t/P_t; MaxDD = max_t D_t
Worked	Base series: the peak is 1.216586 at month 9. The trough is 1.124165 at month 21. MaxDD = 1 - 1.124165/1.216586 = 0.075967 = 7.5967 percent
Path dependence	The same 24 returns in a different order give a different maximum drawdown. Every distributional statistic in the performance section - mean, volatility, skewness, Sharpe, Sortino, Omega - is unchanged by reordering. Maximum drawdown is not
Recovery arithmetic	Recovering a drawdown of d requires a gain of d/(1-d). Here 0.07596737/0.92403263 = 8.2213 percent, not 7.5967 percent
Sampling frequency	Computed on monthly closes. The same path sampled daily would show a drawdown at least as large and almost always larger, because intra-month troughs are invisible to monthly data

• Maximum drawdown is non-decreasing in the sample length by construction. A longer track record cannot have a smaller maximum drawdown than a subset of itself, so any comparison between records of different lengths is biased toward the shorter one.
• The statistic uses exactly two observations out of T - one peak and one trough. Its sampling variability is correspondingly enormous and it has no standard error in general use, so it cannot support a confidence statement.
• The frequency at which the equity curve is sampled is part of the definition. A monthly maximum drawdown and a daily maximum drawdown on the same strategy are different numbers, and a report that does not state the frequency is not reproducible.
• Maximum drawdown is not subadditive across time or across sub-portfolios. The maximum drawdown of a combination is not bounded by any combination of the components' maximum drawdowns, in either direction, which is why it cannot be used in a risk-budgeting framework the way variance can.

Average drawdown, and the divisor question again

The mean of the drawdown series. As with the Sortino denominator, the divisor is either the full sample length - treating a month at a new high as a zero drawdown - or the count of in-drawdown months. The first is a property of the path; the second is a conditional statistic.

Field	Value
Formula	AvgDD over all periods = (1/T) sum_t D_t. AvgDD over in-drawdown periods = (1/n_dd) sum over the n_dd periods with D_t > 0. Sometimes called the pain index in the divisor-T form
Worked, divisor T	Base series: sum of the 24 drawdown values = 0.481879, divided by 24 = 0.020078 = 2.0078 percent
Worked, divisor n_dd	16 months with a nonzero drawdown. 0.481875/16 = 0.030117 = 3.0117 percent, 50 percent higher
Relation	The ratio of the two is exactly T/n_dd = 24/16 = 1.500000, so a strategy that spends less time under water reports a proportionally higher conditional average drawdown
Comparison with the maximum	Average drawdown 2.0078 percent against maximum 7.5967 percent, a ratio of 0.264

• The divisor-T form is the useful one because it uses every observation and is comparable across strategies with different time-under-water. The conditional form penalises a strategy for spending less time in drawdown, which is the opposite of the intended direction.
• Average drawdown is far more stable than maximum drawdown - it averages T observations rather than selecting two - and it is correspondingly less reported, because it is smaller and less dramatic.
• The average drawdown says nothing about episode structure. One long shallow episode and many short deep ones can produce the same average, and they are not the same experience for a holder facing redemption.

Ulcer index

The root mean square of the drawdown series. It penalises deep drawdowns more than shallow ones, unlike the average, and it uses every observation, unlike the maximum, which makes it the most statistically stable of the drawdown family.

Field	Value
Formula	UI = sqrt( (1/T) sum_t D_t^2 ), with D_t the drawdown at t expressed in percent or as a fraction, stated consistently
Worked	Base series: sum of squared drawdowns = 0.021343 in fractional units. UI = sqrt(0.021343/24) = sqrt(0.00088930) = 0.029821 = 2.9821 percent
With the conditional divisor	sqrt(0.021343/16) = 3.6523 percent. As with average drawdown, the divisor must be stated
Ordering	UI = 2.9821 percent sits between the average drawdown of 2.0078 percent and the maximum of 7.5967 percent, which is where a root-mean-square always sits
Martin ratio	(g_ann - rf_ann) / UI = (0.093278 - 0.024266)/0.029821 = 2.314199. Also called the Ulcer Performance Index

• The Ulcer index is the drawdown analogue of a standard deviation and should be preferred over maximum drawdown wherever a single risk number is needed, precisely because it averages rather than selects. It is still path-dependent and still sensitive to sampling frequency.
• Units matter. The Ulcer index computed on drawdowns expressed in percent is 100 times the value computed on fractions, and both conventions appear in published implementations. The Martin ratio changes by a factor of 100 accordingly.
• Unlike maximum drawdown, the Ulcer index does not increase mechanically with sample length; adding a long stretch of new highs lowers it. That makes it comparable across records of different lengths in a way maximum drawdown is not.

Source: Martin and McCann 1989

Drawdown duration and recovery time

Duration is the number of periods from the prior peak to the recovery of that peak; time to trough and time from trough to recovery are its two components. They are separate statistics from depth and are frequently the binding constraint on a real allocation.

Field	Value
Formula	For an episode beginning after peak at t0 and ending at t1 when E_t1 >= E_t0: duration = t1 - t0; time to trough = argmin(E) - t0; recovery time = t1 - argmin(E)
Worked, worst episode	Base series: peak at month 9 (equity 1.216586), trough at month 21 (1.124165), sample ends at month 24 with equity 1.195257, still 1.7532 percent below the peak. Time to trough 12 months, elapsed since trough 3 months, total duration at least 15 months
Censoring	The episode has not recovered, so its duration is right-censored: 15 months is a lower bound, not the duration. Averaging censored and uncensored episode lengths together understates the mean duration
First episode	Month 6 only: depth 0.8000 percent, duration 1 month, recovered immediately at month 7
Asymmetry	Time to trough 12 months against a recovery that has taken more than 3 and is unfinished. Depth and duration are separate axes; a shallow, long drawdown can be harder to hold than a deep, fast one

• The final drawdown episode of any sample is almost always unrecovered and therefore censored. Reporting a mean or maximum drawdown duration without saying how the open episode was handled is the single most common error in path statistics.
• Recovery time is bounded below by depth divided by the mean return: at 0.766667 percent per month, an 7.5967 percent drawdown needs at least 9.9 months of average performance to recover even with no further losses. That arithmetic is a useful sanity check on any claimed recovery.
• Duration is measured in periods, so it inherits the sampling frequency. An episode that is 15 months at monthly resolution is roughly 315 trading days, and neither number is more correct than the other.

Time under water

The fraction of the sample spent below a prior peak. It is the simplest path statistic, it has no divisor ambiguity, and it captures the thing a holder actually experiences, which is duration rather than depth.

Field	Value
Formula	TUW = (1/T) * count of periods with D_t > 0. The complement, time at new highs, is 1 - TUW
Worked	Base series: 16 of 24 months have a drawdown above zero. TUW = 16/24 = 0.666667 = 66.67 percent. Time at new highs = 8/24 = 33.33 percent
Consistency check	The 8 months at new highs are months 1, 2, 3, 4, 5, 7, 8 and 9. Every other month sits below the running peak
Independence from depth	Time under water and maximum drawdown are separate. A strategy can be under water 90 percent of the time with a 2 percent maximum drawdown, or 10 percent of the time with a 40 percent one

• For a positive-drift random walk, time under water is high by default. A strategy at new highs in only a third of its months is not unusual and is not evidence of a problem; the base series makes 9.3 percent a year while spending two thirds of its life below a prior peak.
• Because it counts periods rather than averaging magnitudes, time under water is the only statistic in this section that is invariant to the size of the returns. Scaling every return by 2 leaves it unchanged and changes every other drawdown statistic.
• It is still sampling-frequency dependent, and in the opposite direction from depth statistics: sampling more finely finds more moments below the running peak, so daily time under water exceeds monthly time under water on the same path.

Expected maximum drawdown of a random walk

A path with no drift and no skill still produces drawdowns, and their expected maximum has a closed form in the zero-drift case. Comparing an observed maximum drawdown with this number is how a reader distinguishes a risk characteristic from an artefact of path length and volatility.

Field	Value
Formula	Zero drift: E[MaxDD] = sqrt(pi/2) * sigma * sqrt(Ty) = 1.253314 * sigma * sqrt(Ty), with sigma the annualised volatility and Ty the horizon in years. Nonzero drift mu > 0: E[MaxDD] = (2*sigma^2/mu) * Qp(mu^2*Ty/(2*sigma^2)), where Qp has no elementary closed form
Worked, the null	Base series annualised volatility 7.2063 percent, horizon 2 years. E[MaxDD] = 1.253314 * 0.072063 * 1.414214 = 12.7728 percent
Comparison	Observed maximum drawdown 7.5967 percent, divided by the driftless expectation of 12.7728 percent = 0.594759. The realised drawdown is 41 percent SMALLER than a no-skill path of the same volatility would be expected to produce over the same horizon
Scaling	E[MaxDD] grows as sqrt(Ty): 9.0317 percent at 1 year, 12.7728 at 2, 20.1956 at 5, 28.5608 at 10, 40.3911 at 20, all at 7.2063 percent volatility
Inverting for volatility	An observed maximum drawdown of d over Ty years implies a volatility of at least d/(1.253314*sqrt(Ty)) under the driftless null. Here 0.075967/(1.253314*1.414214) = 4.2860 percent

• The drift-free case is the useful null because it is the case that assumes nothing. With positive drift the expected maximum drawdown is smaller, so a driftless benchmark is conservative: a drawdown that already looks small against the driftless expectation looks smaller still against the correct one.
• The nonzero-drift result requires the function Qp, which is given as a series and a numerical approximation in the source paper and has no closed form. Any quoted E[MaxDD] for a drifting process that does not name its approximation should be treated as unverified.
• This is an EXPECTED maximum drawdown, not a bound. The distribution of the maximum drawdown of a random walk has a long right tail, so a single observed drawdown above the expectation is not evidence of anything at all.
• The result assumes continuous monitoring. A monthly-sampled path realises a smaller maximum drawdown than the continuous path it was sampled from, so comparing a monthly-observed drawdown against the continuous-time expectation is biased toward finding the strategy well-behaved.

Source: Magdon-Ismail, Atiya, Pratap and Abu-Mostafa 2004, On the maximum drawdown of a Brownian motion

Why maximum drawdown is a biased estimator of risk

Maximum drawdown estimates nothing well. It is an extreme order statistic of a path, so its expectation depends on sample length and monitoring frequency as much as on the underlying risk, and it is systematically understated in short samples and by coarse sampling.

Field	Value
Formula	E[MaxDD] scales as sigma*sqrt(Ty) under the driftless null, so the ratio MaxDD/sigma is not comparable across samples of different length. The scale-free quantity is MaxDD/(sigma*sqrt(Ty))
Worked, length bias	At 7.2063 percent volatility, the expected maximum drawdown is 9.0317 percent over 1 year and 28.5608 percent over 10. A ten-year record showing three times the drawdown of a one-year record shows the same underlying risk
Frequency bias	Monthly sampling of the base series gives 7.5967 percent. The same path sampled daily can only be worse, never better, since a coarser sample is a subset of the peaks and troughs the finer one sees
Scale-free version	Base series: 0.075967/(0.072063*1.414214) = 0.745420 in units of sigma*sqrt(Ty), against a driftless expectation of 1.253314
Selection bias	A strategy chosen from many candidates because it had the smallest maximum drawdown has an understated maximum drawdown for exactly the reason a selected Sharpe ratio is overstated

• The honest use of maximum drawdown is as a scale-free ratio against the driftless expectation for the same volatility and horizon. In raw percent it is not a risk measure; it is a fact about one path.
• Because maximum drawdown is non-decreasing in T, it can never revise downward. A strategy's reported maximum drawdown will only ever grow, which means the number in a marketing document is a lower bound on the number a future holder will see.
• There is no accepted standard error for maximum drawdown and no simple bootstrap gives one, because resampling returns destroys the path ordering the statistic depends on. Block bootstraps preserve some of it and give wide intervals.
• None of this makes drawdown useless. It is the binding operational constraint for a leveraged book and for any mandate with a stop. It is simply not an estimator of the underlying risk, and should not be used as one.

Sampling frequency changes every drawdown statistic

Drawdown statistics are computed on a discretely sampled equity curve, and the sample is a subset of the true path. Coarser sampling misses interior peaks and troughs, so it understates depth and can either overstate or understate duration.

Field	Value
Formula	For sampling frequencies f1 finer than f2 on the same path: MaxDD(f1) >= MaxDD(f2) always, since the coarse sample's peak-trough pairs are a subset of the fine sample's. TUW has no such ordering
Worked, direction of the depth bias	Strictly monotone: refining the sampling can only find a deeper drawdown. The monthly maximum drawdown of 7.5967 percent on the base series is a lower bound on the daily figure; the same holds for every individual drawdown D_t, since the running peak at t can only rise when more points are observed
Direction, time under water	Not monotone. Refining can find brief excursions to new highs inside a month, which shortens time under water, and brief dips, which lengthen it
Practical size	Under the driftless null, the expected maximum drawdown of a continuously monitored path is 1.253314*sigma*sqrt(Ty). Monthly monitoring of the same process realises less than this by an amount that grows with the volatility per sampling interval
Reporting rule	A drawdown figure requires three declarations: the sampling frequency, the window, and whether it is computed on total or excess returns

• Comparing a hedge fund's monthly maximum drawdown with an index's daily maximum drawdown is a systematic comparison in the fund's favour, of unknown but nonzero size. The fix is to compute both at the coarser frequency.
• A fund reporting monthly and holding illiquid marks understates its drawdown twice: once for the sampling frequency, once for the smoothing that shows up as positive autocorrelation. The two effects compound.
• Intraday drawdown on a daily-marked strategy is invisible in the same way and matters for the same reason: it is what a margin desk sees.