Maximum Drawdown Calculator: Working Peak to Trough

A maximum drawdown pass takes a series of portfolio values in time order, carries the highest value seen so far, and reports the largest percentage fall from any such high to a later low inside the window entered. On the Anantara Multi-Asset Portfolio's stated year the worst fall was 9.7 per cent, and returning to that earlier high needs a subsequent rise of 10.74 per cent.

The box above runs the pass and lays out every step of it. Each step is worked below, so that when a number comes back it is clear which two readings produced it and which divisions were run on them. A drawdown's meaning, and why it is not the same animal as volatility, is covered under drawdown measures.

Everything worked here comes from one place. The Anantara Multi-Asset Portfolio is an invented discretionary mandate of Rs 5,00,00,00,000/- run for an invented charitable endowment against a composite benchmark of 60 per cent a broad equity index and 40 per cent a broad bond index, neither named anywhere. For one stated twelve month period its worst fall from a high point to a later low point was 9.7 per cent and the benchmark's was 8.1 per cent. The two depths of 9.7 and 8.1 per cent are the only measured drawdown facts on record, and every other number below is derived from them by arithmetic anybody can check.

What does this calculator compute?

One number, and three labels that travel with it. The number is the deepest percentage fall the value seriesThe list of what the portfolio was worth at each point in time, in order, with each reading struck on the same basis as the ones around it. contains, measured from the highest reading seen up to that moment down to the lowest reading that follows it before a new high is set. The three labels are the window the readings cover, the sampling frequencyHow often the portfolio was valued for this series: daily, weekly, monthly or quarterly. The frequency fixes how much of the path the series can possibly contain. they were struck at, and the basis on which the values were adjusted.

A depth without its window and its frequency is not a smaller answer, it is an uninterpretable one. Think of a shopkeeper who says the worst day of the year took a third of the day's takings. The claim only lands once it is clear whether he counted every day or looked at the till on four Sundays. The percentage is identical in both tellings. The fact behind it is not.

The rise needed to get back to the earlier high comes out alongside it, because it is arithmetic on the same two readings and because it is what a committee asks for the moment the depth is read out. Nothing comes out about how long the fall lasted or how often falls of that size happen. Duration and frequency of falls are not recorded for this portfolio, and neither can be conjured from a single path.

What has to go in before it can compute anything?

Three things, and only the first of them looks like data. The first is the value series itself, in time order, with each reading struck on a comparable basis. The second is the measurement windowThe stretch of time the readings cover, from the first reading to the last. Two different windows over the same portfolio give two different drawdown figures, and neither is wrong., the stretch of time the first and last readings bracket. The third is the sampling frequency, how often a reading was taken inside that stretch.

The window and the frequency are inputs and not labels: change either one and the output changes, which is why the box takes them as fields that can be driven rather than as captions. Set the first reading to six, or take every third reading, and watch the headline figure move while the readings themselves sit untouched. Publishing a depth without its window and its frequency is the commonest defect in the way drawdowns are reported. The defect is one of presentation rather than of arithmetic, and that is precisely why it survives review after review.

How does the running peak actually work, reading by reading?

The readings are taken in order, carrying one extra number along the way. The extra number is the running peakThe highest value reached at any point up to and including the current reading. The running peak is updated only when a reading beats it, so it never falls.: the highest value seen so far. At each reading, if the reading is higher, the running peak becomes the reading and the fall at that point is zero. If it is lower, one less the reading divided by the running peak gives the fall from the best point reached down to the current reading. The largest such fall seen is carried forward, and at the end of the series that kept figure is the answer. The table inside the box runs those four columns down the readings entered.

The running peak only ever rises. A single pass does the whole working, and a later high never erases an earlier fall. The rule is not a convenience of the method, it is the point of it. A measure that let a good month nine wipe out a bad month four would be a measure of where the portfolio ended up, and that is what the return figure is for.

The thirteen readings the box opens on are stated in index valueA value series rescaled so that the numbers are easy to divide, with the units dropped. The percentage falls it produces are identical to those the rupee series would produce. form rather than in rupees. The readings are scaled so that the opening reading of 95.00 and the closing reading of 108.49 reproduce the recorded 14.2 per cent return for the stated year. The high of 100.00 and the low of 90.30 reproduce the recorded 9.7 per cent worst fall. The shape between those anchor points is illustrative and claims nothing about any particular month.

Notice what happens at month eight. The reading of 101.60 beats the old high of 100.00, so the peak steps up and every fall after that point is measured from the new, higher line. The 9.70 per cent recorded at month four is untouched. The figure is called a maximum for exactly that reason: it is the largest member of a set of falls that were each computed against the best point available at the time.

Why is the fall measured on the peak rather than on the starting value?

Because the peak is the value the holder actually had and then did not have. The starting value is what the holder had before anything happened, and measuring against it answers a different question: how am I doing against where I began. The return figure already answers that question, and answering it twice adds nothing.

Work both divisions on the same series and the difference is immediate. The troughThe lowest reading that follows a running peak before a new high is set. The trough is found by the same pass that finds the peak, not searched for separately. reading of 90.30 against the high of 100.00 is a fall of 9.70 per cent. The same 90.30 against the opening value of 95.00 is a fall of 4.95 per cent, because 90.30 divided by 95.00 is 0.9505. Nearly half the fall vanishes, and only the denominator changed.

Measured against the starting value instead of the peak, a portfolio that rose strongly and then fell hard would report almost no fall at all, which is precisely the case the measure exists to catch. Picture a shopkeeper who opens the year with Rs 10,00,000/- of stock, builds it to Rs 14,00,000/- through a good season, then is left holding Rs 11,00,000/-. Against the opening he is up. Against the best the shop ever held, close to a quarter of it is gone, and that is the number he has to rebuild from.

How much does it take to get back from a given fall?

The derivation runs to two lines. Suppose the peak was worth one unit and the fall was d. The remainder is one less d. Returning to one unit means adding d to what is left, and the gain required is that addition expressed on the base that remains. Dividing d by one less d is the whole recovery gainThe rise from the low reading that would bring the portfolio back to the earlier high. The recovery gain is arithmetic on two readings and says nothing about how long such a rise would take. identity.

The fall is computed on the peak. The gain is computed on the smaller base the fall left behind, so the gain needed to recover always exceeds the fall itself. At shallow falls the two are nearly the same and the difference looks like a rounding quibble. At deep falls they part company completely.

Dividing 0.10 by 0.90 gives 0.1111, so a fall of 10 per cent needs 11.11 per cent back. A fall of 20 needs 25.00, a fall of 30 needs 42.86, and a fall of 50 needs 100.00. Halving a portfolio requires doubling what is left of it.

What does the working look like on the recorded fall?

State the window before the number, every time. Over one stated twelve month period the Anantara Multi-Asset Portfolio's worst fall from a high point to a later low point was 9.7 per cent. In index form the high inside that window is 100.00 and the low that follows it is 90.30. The high and the low are an arithmetic restatement of the locked depthThe size of the fall, given as a percentage of the peak. Depth is one of several things a drawdown could be described by, and the only one recorded for this portfolio. so that the working can be seen, and not two extra facts about the portfolio.

Step one, the fall: one less 90.30 divided by 100.00 is 0.097, or 9.70 per cent, the recorded depth exactly. Step two, the recovery: from 90.30 back to 100.00 is a rise of 9.70 on a base of 90.30. The division gives 0.10742, or 10.74 per cent to two places. The identity behind step two is 0.097 divided by 0.903, and it gives the same figure, as it must.

Now run the composite benchmark over the identical window. Its recorded worst fall for the same stated year was 8.1 per cent, so the gain it would need is 0.081 divided by 0.919. The answer is 8.81 per cent. The depths of 9.7 and 8.1 per cent are 1.6 points apart. The recoveries they demand, 10.74 and 8.81 per cent, are 1.93 points apart. Required gain accelerates with depth, and the wider of the two gaps is the proof of it.

What does the calculator get wrong before any arithmetic does?

Two things, and neither of them produces an error message. Silence is what makes them dangerous rather than merely annoying.

The first is the wrong window. If the readings begin after the low point, or end before it, the trough is not in the data and the pass cannot find what it was never given. Set the first reading to six in the box above and watch it happen: the deepest fall the pass can now report is 3.53 per cent, from the high of 104.90 down to 101.20. The 3.53 per cent is arithmetically flawless and useless as a description of the year.

The second is the wrong frequency. A trough that opened and closed between two sampled readings is invisible, and no care in the arithmetic recovers it. Set the box to take every third reading and five are left: 95.00, 96.50, 97.10, 104.90 and 108.49. Every one is higher than the one before, so the running peak never has anything to fall from and the maximum fall reads 0.00 per cent. The high of 100.00 and the low of 90.30 both sat between quarter ends and neither was ever looked at.

The second series the box opens with is the harder version of the same failure. The second series grinds down over five readings rather than dropping in two: 95.00, 96.20, 97.30, 98.40, 100.00, 98.10, 96.40, 94.20, 92.00, 90.30, 96.00, 102.00 and 108.49. The slower path opens and closes where the first series does, returns the same 14.2 per cent for the year, spends the same six readings below its running peak, and reports the same maximum fall of 9.70 per cent. Two quite different shapes, one figure. Now take every third reading of each: the first reports 0.00 per cent and the second reports 8.23 per cent, from a high of 98.40 down to 90.30. Neither path moved. Only the sampling did.

Neither the wrong window nor the wrong frequency produces a warning of any kind, and that silence is exactly why both survive in published figures for years at a time.

What happens when money moves in or out of the portfolio?

The third failure corrupts the series before the arithmetic is even reached. A drawdown compares each reading with an earlier one, and that comparison only means something if the two readings differ because of what the holdings did. If money came in or went out between them, dividing one by the other measures a mixture of performance and cash flow.

Worked on this portfolio's size, the high inside the window corresponds to Rs 5,00,00,00,000/-, and a like for like low at a depth of 9.7 per cent is Rs 4,51,50,00,000/-. Now suppose the endowment transfers Rs 25,00,00,000/- of new money in on exactly that day. The custodian's statement for the day reads Rs 4,76,50,00,000/-. A pass run over the statement values reports one less 4,76,50,00,000 over 5,00,00,00,000, or 4.70 per cent. Entered in the box above, that amount drops the headline from 9.70 to 4.70. The measure has recorded a deposit as performance.

A drawdown has to be computed on a value series adjusted for money in and money out, or a large inflow at a low point reads as a recovery that never happened. The contribution adjustmentRestating the value series so each reading reflects only what the holdings did, with money paid in and money taken out stripped out first. is not a refinement to apply if time allows. Without it the output is not a less precise drawdown. The output is a different quantity with no name.

The household picture is exact. Someone watching their savings drop through a hard month, who then receives a bonus on the worst day, sees the balance recover on the screen while nothing about the hard month improved. The screen is adding two different things together, and so is an unadjusted drawdown.

What goes in, and where does each number come from?

Every field in the box at the top is copied out of a document, and the note under it names the document and the line rather than saying what the figure means. The value series comes from the custodian's valuation statements, one total value line per valuation date. The window comes from the first and last of those dates, not from the reporting period on the cover sheet. The frequency comes from the custodian's valuation calendar. The adjustment basis comes from the cash flow schedule for the same window, a list of every contribution and withdrawal by date.

Reading the fields is a sourcing exercise rather than an interpretive one, and if a field cannot be sourced the honest response is to leave the calculator alone rather than to estimate the input. An estimated reading does not produce a slightly less reliable depth. It produces a depth for a portfolio nobody holds.

A date column would invite the box to report a duration, and no duration is recorded for this portfolio. The box therefore takes no date column. The second series it sets beside the first is not a ranking. The only honest comparison is against a drawdown computed over the same window, at the same frequency, on a series adjusted the same way, and the box requires all three to be matched.

What can one depth figure not support?

One path gives one depth. The depth carries no duration, no recovery time and no information about how often a fall of that size occurs, and the three absences are not gaps that a longer run of the same calculation would fill. Each absence is a different question requiring different data.

The honest output of this tool is a number accompanied by a window, a frequency, an adjustment basis and an explicit list of what is not in it. For the Anantara Multi-Asset Portfolio that list is short: the date the high was set, the date the low was reached, and the date the earlier high was regained. None of the three dates is recorded, and no arithmetic on a value series can recover a date the series was never given.

There is a discipline in printing the absences rather than omitting them. A reader handed 9.7 per cent with nothing else fills the gaps with assumptions nobody can predict. A reader handed 9.7 per cent alongside three lines marked unavailable knows exactly how far the figure reaches.

Who actually reaches for this number, and what do they do with it?

Rukmini Deshpande chairs the investment committee of the endowment that holds the Anantara Multi-Asset Portfolio, and Faiz Ahmad Ansari runs the mandate. When the quarterly pack lands, the depth figure is read as a liquidity question: if the endowment had needed to draw money during the worst stretch of the window, what would it have been drawing against. A committee that plans a distribution schedule against the peak value of a portfolio is planning against a number that was true on one day.

A lending officer runs the same arithmetic on collateral, and runs it more often than a portfolio manager does. Securities pledged against a facility are valued at the market. Her question is what the worst observed fall in that collateral was over a stated window at a stated frequency, and the answer is what sizes the margin she asks for. She will also ask whether the series was daily or monthly, and a manager who cannot answer has not given her a usable figure.

An analyst comparing two mandates uses it as a screening question rather than a ranking one. Two depths are only comparable when the window, the frequency and the adjustment basis all match. The analyst's first job is to establish that they do, not to note the smaller number. Where the three cannot be established, the professional answer is that no comparison is available, which is a finding and not a failure to produce one.

A household does a rough version of this without naming it. Anyone who has watched a single salary carry a home through a bad quarter has felt the question directly: not what the average month looked like, but how far below the best month the worst one sat. The arithmetic here is that instinct, written down and given a window.

The Anantara Multi-Asset Portfolio, the charitable endowment that holds it, the composite benchmark it is measured against, Rukmini Deshpande and Faiz Ahmad Ansari are invented.
Educational material. Not advice on any investment, tax, budget or market position.