Drawdown witnesses and recovery clocks

What you will learn

Derive maximum drawdown from a running-peak recursion, certify an ordered peak and trough, and construct equal-terminal-return paths with different depths while distinguishing recovery gains and censored durations.

Prerequisites: Linked returns and external cash flows · Borrowing, equity and collateral repair

Declare the path and observation clock

A final return compresses a capital history into two endpoints. That is useful for performance measurement, but it cannot tell a learner how far the account fell below a previously attained level. Drawdown keeps the ordering information. Imagine a fictional unlevered capital index observed at equally spaced ticks numbered zero through five. Its initial value is 100 chips. Every interval return applies multiplicatively to the value just before it. There are no deposits, withdrawals, financing charges, distributions outside the index, fees or taxes. All observed values are strictly positive. We study one supplied path rather than a random sample, so no probability of future loss is implied.

These assumptions matter before computing any statistic. A deposit can lift account wealth without recovering investment performance. If external flows exist, first build the cash-flow-neutral unit index taught in linked returns, then apply the present calculations to that index. Equally spaced ticks let us count intervals, but do not turn those intervals into calendar days. Our drawdown is measured only at observations. An unobserved intraperiod crash can make a continuously monitored drawdown larger. A short report should therefore name its series, window, flow treatment and sampling interval alongside its depth.

Carry a running peak forward

Write Wt for observed wealth and Ht=max(W0,...,Wt) for its running peak. Start with H0=W0, including the initial capital as a possible peak. Update by Ht=max(Hprevious,Wt). Define the nonnegative drawdown depth dt=1−Wt/Ht. At a new or repeated high, dt is zero. Below that high it is positive. Some reports display underwater returns as negative numbers; our depth convention displays their positive magnitudes. Maximum drawdown is D=max(dt) across the declared observation window. Its denominator changes whenever a new high is reached.

There is an equivalent pair certificate: maximize 1−Wj/Wi over ordered indices i≤j. For a fixed trough candidate j, the expression grows when Wi grows, since Wj is positive. Therefore the best preceding denominator is exactly Hj. This proves that the running-peak scan and the ordered-pair search agree. It also explains why an unrestricted global maximum and global minimum are insufficient: their chronological order may be reversed. Our tie convention selects the earliest trough attaining D and the earliest preceding occurrence of that trough's running peak. Ties change reported dates even when depth is unchanged.

Derive a complete peak and trough witness

Apply returns +50%, −20%, −25%, +25%, then 0%. The wealth path is 100, 150, 120, 90, 112.5, 112.5. Check each multiplication: 100×1.5=150; 150×0.8=120; 120×0.75=90; 90×1.25=112.5; multiplying by one leaves 112.5. Running peaks are 100,150,150,150,150,150. The corresponding depths are zero, zero, 20%, 40%, 25%,25%. Thus tick one and tick three certify maximum drawdown 40%. The worst single interval lost 25%, while the full path finished 12.5% above inception. Neither number answers the peak-to-trough question.

Independently enumerate the candidate troughs instead of trusting the scan. At ticks zero and one there is no decline. At tick two the largest earlier wealth is 150, so the worst ordered ratio is 120/150. At tick three it is 90/150. At ticks four and five it is 112.5/150. Their smallest ratio is 0.6, proving the largest decline is 1−0.6=0.4. Using the initial hundred as denominator would produce 10% at tick three and miss the fact that the account first earned fifty chips before losing sixty. Reporting the sixty-chip loss divided by trough wealth would instead measure a recovery gain.

Separate recovery magnitude from recovery time

For a selected peak H and positive trough T, a future cumulative gain g recovers the old level exactly when T(1+g)=H. Thus g=H/T−1. Since T=H(1−D), substitution yields g=D/(1−D). With H=150 and T=90, g=150/90−1=2/3, or about 66.666667%. Independently, ninety times five thirds equals 150. A 40% gain gives only 126, leaving the account underwater. This gain is a required cumulative return from the trough, not a prediction, annual rate or recommended exposure. As D approaches one, its denominator approaches zero and the requirement becomes unbounded. At zero capital the positive-wealth formula no longer applies.

Define recovery tick as the first observation strictly after the selected trough whose wealth is at least the selected old peak. Equality is enough. Peak-to-recovery duration is recovery tick minus peak tick; trough-to-recovery duration is recovery tick minus trough tick. Those are different clocks. In the example, tick five wealth 112.5 still misses 150, so recovery is open at the observation boundary. We know the episode remains unrecovered for four intervals after its peak, but do not know its eventual total duration. Replacing the final zero return with +50% gives 168.75 at tick five. Recovery then occurs at tick five, with four peak-to-recovery intervals and two trough-to-recovery intervals.

Change order while preserving terminal performance

Use the workshop to swap adjacent return tiles. Keep the same four numbers, but construct +50%, −25%, +25%, −20%. With a zero final return the path becomes 100,150,112.5,140.625,112.5,112.5. Its running peak is again 150 after tick one. Its worst depth is now 25%, first attained at tick two and repeated at ticks four and five. The tie rule therefore selects peak one and trough two. Recovery from that selected trough requires 150/112.5−1=1/3, or 33.333333%. Select those rows and enter a certificate before saving it. A numerically correct depth without the correct ordered pair does not complete this construction.

Both orderings end at 112.5 because multiplication of their positive gross return factors commutes: 1.5×0.8×0.75×1.25=1.125. Their collection of interval returns, arithmetic mean and empirical interval variance are also identical. Maximum drawdown differs because the set of eligible earlier peaks changes with order. Save both witnesses to make this distinction visible, then switch the final return to +50% and locate the first recovery tick on each path. Edited paths clear answers, while saved certificates retain their original returns. A smaller realized drawdown on one constructed ordering proves a path distinction, not that a policy has lower prospective risk. Horizon extension cannot reduce the recorded maximum, but an earlier open recovery can become observed later.

Hands-on: Construct a drawdown witness

Reorder identical returns, certify an ordered maximum-loss pair and distinguish recovered from open episodes.

Common mistakes

• Using a global high that occurs after the selected trough.
• Calling a 25% single-period loss the 40% cumulative drawdown.
• Dividing missing wealth by trough wealth when reporting drawdown depth.
• Treating an unobserved recovery as zero duration or as proof recovery never occurs.
• Applying drawdown to raw account balances with external deposits instead of a flow-neutral index.

Keep these ideas

• Running peaks certify the maximum over all chronologically eligible peak-to-trough pairs.
• Depth uses peak wealth; recovery gain uses trough wealth and equals D/(1−D).
• Name both observation spacing and duration start; mark unfinished recoveries open.
• Permuting identical returns preserves terminal wealth but can change drawdown.

Further reading

• MSCI IndexMetrics, June 2020, page 51: Maximum Drawdown ↗

  Read Appendix page 51 definition and ordered peak-to-trough formula. Supports the maximum-drawdown definition only. We declare a nonnegative depth convention, discrete observation grid and our own recovery and tie rules. Original tile construction, proof and diagnostics; no source dataset or quotation reused. Checked 2026-10-25.

• CME Group: Controlling Risk, Making Up for Trading Losses ↗

  Read HTML Making Up for Trading Losses and Table 4. Supports the changing denominator in percentage recovery. Our algebra and multi-tick examples are original. Embedded video was not watched or reviewed; no trading rule or empirical guarantee is adopted. Checked 2026-10-25.