Up Capture and Down Capture: Rising and Falling Markets

Up capture is the portfolio's compounded return across the sub-periods when the benchmark rose, divided by the benchmark's own compounded return across those same sub-periods. Down capture runs the identical arithmetic over the sub-periods when the benchmark fell. Both need the series split before anything is divided, so a single twelve month return produces neither.

A reader usually arrives with a good question. How far a portfolio moves with its benchmark on average is already established; what that average cannot hold is whether the portfolio gave back less than its share on the bad days. The average and the bad-day behaviour are two different questions. The second one needs a different measurement, built from inputs the first one never required, and most of the trouble on this subject comes from people reaching for a number they already have instead of admitting they lack the one they need.

The running example throughout is the Anantara Multi-Asset Portfolio, an invented discretionary mandate of Rs 500 crore run by Faiz Ahmad Ansari for an invented charitable endowment whose investment committee is chaired by Rukmini Deshpande. Its policy weights are equity 60.0 per cent at Rs 300 crore, fixed income 30.0 per cent at Rs 150 crore and cash 10.0 per cent at Rs 50 crore. Over one stated twelve month period the portfolio returned 14.2 per cent against 12.6 per cent for its composite benchmark, with a betaA single number for how far a holding or a portfolio has tended to move when the thing it is measured against moves by one unit. The number is one slope fitted across every period at once. of 1.08, volatilities of 11.8 and 10.4 per cent, a tracking error of 3.7 per cent and a risk-free rate of 6.5 per cent. Every one of those is a whole year figure or a whole path figure. A capture ratio can be built from one material only, a series of sub-period returns, and not one of them is that.

What are up capture and down capture, and what are they for?

Start with what the beta already does. The beta fits one slope across every period in the window at once, rising periods and falling periods thrown into the same regression, and hands back a single number. The slope is genuinely useful and it is genuinely one number. One slope was never built with a place to put a difference between the two directions, so the beta cannot describe a portfolio that behaves differently on the way down from the way it behaves on the way up.

A capture pairThe two figures, up capture and down capture, reported together. The reading that matters is how far apart the two figures are, so neither one means much alone. asks the same question twice instead of once. The capture pair asks how much of the benchmark's rise the portfolio picked up across the periods when the benchmark rose, and separately how much of the benchmark's fall the portfolio took across the periods when it fell. Two questions, two answers, reported together. A portfolio that moves less on the way down than it does on the way up cannot be described by one number at all, and the capture pair exists precisely to hold that possibility open.

The household version is easier to feel than the finance version. A small tailoring shop next to a wedding hall shows it. In a good month, when weddings are booked solid, the shop's takings rise almost as fast as the hall's bookings. In a bad month, when the hall goes quiet, the shop still has school uniforms and alterations, so its takings fall by much less. Asking the shopkeeper for one number describing how the shop tracks the hall produces an average that flatters neither the good months nor the bad ones. Asking twice, once about good months and once about bad ones, describes how the shop actually earns.

Which side's sign decides the split?

Before any division happens at all there is a sorting step, and it is the step people skip. The series of period returns is split into two piles: the sub-periodsThe individual stretches the measurement window is cut into, usually months or quarters. Each one carries a return for the portfolio and a return for the benchmark. in which the benchmark's return was positive, and the sub-periods in which it was negative. Every sub-period goes into exactly one pile and none is discarded. Only then does any arithmetic begin.

The classificationThe sorting step that decides which pile each sub-period belongs to. The sign of one series decides it, and never the sign of the other. is run on the benchmark's sign and never on the portfolio's, because the question being asked is how the portfolio behaved when the market did something. Reversed, it produces nonsense. Classifying by the portfolio's own sign makes the up pile, by construction, the periods in which the portfolio rose, so the numerator is positive in every one of them. The question becomes how the portfolio did in the periods when the portfolio did well, and that answer is always flattering and always empty.

How is each ratio actually computed?

With the two piles in hand the arithmetic is short. Take the rising pile. Compound the portfolio's returns across those sub-periods to get one cumulative returnWhat a series of period returns comes to when they are chained together by multiplication rather than added up. Each period's gain is earned on the result of the one before. for the portfolio. Compounding the benchmark's returns across the same sub-periods gives one cumulative return for the benchmark. The first divided by the second, expressed as a percentage, is up capture. The same procedure run over the falling pile gives down capture.

Both figures are ratios of compounded returns, not averages of period by period ratios, and the two routes give genuinely different answers on identical data. This is not a rounding quibble. Averaging the period ratios lets a single small denominator dominate the result: a sub-period in which the benchmark rose 0.1 per cent and the portfolio rose 0.4 per cent contributes a ratio of 400 to the average, even though almost nothing happened in either series. A period in which nothing happened contributes almost nothing to either cumulative figure, so compounding first refuses the distortion.

Why can one annual return never produce either figure?

The honest centre of the subject is worth stating flatly. A twelve month return is already the compounded result of every sub-period inside it, rising and falling stretches multiplied together into a single number. Compounding is a one way street. A list of sub-period returns produces the annual figure. An unlimited number of different lists produce exactly the same annual result, so the annual figure can never recover the list.

Look at what that means concretely. The Anantara Multi-Asset Portfolio returned 14.2 per cent over its stated twelve months. Three sub-period paths are drawn below, all of them invented, and all three compound to 14.2 per cent. One contains no falling sub-period at all, so its down capture would not exist. One contains a single fall. One contains two. No beta, no volatility and no tracking error recovers the parts from the whole, so the only route to a capture pair is the underlying period series itself.

What does the pair say that a single beta does not?

Now put the two side by side properly. A beta of 1.08 says that across every period in the window taken together, the portfolio moved about 8 per cent further than the benchmark for a given benchmark move. The beta is an average of behaviour, and averages are silent about the shape of what they average.

Take the simplifying case, and this assumption belongs in the same sentence as the conclusion it produces: suppose the rising and falling sub-periods are equal in number and equal in weight. The single slope is describing that average, so the two capture figures must average to about 108. A pair at 108 and 108 satisfies it. So does 120 and 96. So does 130 and 86. So does 96 and 120. The same width points the other way there, describing a portfolio that lagged the rise and took more of the fall. The beta fixes the centre of the pair and says nothing whatever about its width, and the width is the only thing a capture pair adds to what the beta had already given.

The width gives the reading rule for any capture pair. The beta had already given the level, so the level of either half is not read on its own. The difference between the halves is what is read. A pair reported without its beta beside it invites exactly the wrong reading: an up capture of 130 looks impressive until the beta turns out to have been 1.30 all along and the portfolio simply carried more of everything, in both directions, all year.

Does the length of the period change the answer?

The answer changes completely, and the period lengthHow finely the measurement window is cut before the sorting step, most often into months or into quarters. The cut is a choice made by whoever runs the calculation. is almost never stated beside the figure. The sorting step works on whatever unit it is given. Given months it sorts months. Given quarters it sorts quarters. A quarter that finished higher goes into the rising pile in one piece, taking any falling month inside it along with it, and that month never appears in the down figure at all.

Work one quarter through. A benchmark rises 3.00 per cent, falls 2.00 per cent, then rises 2.50 per cent. Chained together that quarter came to plus 3.46 per cent. Classified by month there are two rising sub-periods and one falling sub-period, and the fall contributes to the down figure. Classified by quarter there is one rising sub-period and no falling sub-period whatever, and the fall has simply disappeared from the measurement. A monthly capture pair and a quarterly capture pair drawn from the identical path are two different measurements, so laying one beside the other is not a comparison at all.

What does the invented record actually carry about a fall?

Something real, and it is not a capture figure. Inside the stated twelve months the Anantara Multi-Asset Portfolio fell 9.7 per cent from its highest point to its lowest before recovering, against 8.1 per cent for the composite benchmark over that same window. Both are peak to troughMeasured from the highest point a value reached down to the lowest point it reached afterwards, before any recovery. A drawdown is one distance along one path, and a different window gives a different distance. distances, and both carry that window with them wherever they are quoted, because a different window produces a different depth.

A drawdown is a single distance along a single path inside one window. A down capture ratio is a compounded ratio across every falling sub-period in the series. The pair of depths is not a down capture figure and must never be presented as one. The two share nothing except that both involve falls. One is about the worst single stretch; the other is about the total of all the falling stretches, however scattered they were. Reading a ratio of drawdowns as a capture ratio is the single most common way a capture figure gets fabricated, and it is worth recognising on sight.

Two quantities can be pulled out of one constructed series to show how far apart they really sit. The path below is the eight invented benchmark sub-periods drawn earlier, chained into an index that starts at 100.000.

What does the beta account for in that recorded fall?

Since the two depths are in the record and the beta is too, there is one calculation worth running, provided it is labelled honestly at every step. At a beta of 1.08, the exposure the mandate was already carrying would on its own have turned the benchmark's 8.1 per cent fall into 1.08 times 8.1, or 8.75 per cent. The portfolio actually fell 9.7 per cent. The gap leaves 0.95 points of depth beyond what the exposure alone accounts for.

Say plainly what those two numbers are before anyone quotes them. The two figures compare two single peak to trough distances inside one stated twelve month window. They are not a down capture ratio, they are not evidence about behaviour across falling periods in general, and 1.20 is not 120 per cent of anything a capture ratio measures. The 0.95 points is also small enough to be worth naming as small: most of the depth of the recorded fall is simply the exposure the mandate had already chosen, which is a decision taken in the policy weights long before any fall arrived.

What would have to arrive before a pair could be produced here?

Four things, and none of them is exotic. The portfolio's return in every sub-period of the window. The benchmark's return in every one of those same sub-periods. The period length the sorting was run at, stated rather than assumed. And the window the series covers, with its start and its end written down. Supply those four and the pair falls out in a few lines of arithmetic. Withhold any one of them and no honest figure exists.

Reporting that no capture figure can be produced is itself a result, and substituting the nearest available ratio is not a weaker version of the same answer but a different quantity wearing a borrowed name. This is worth insisting on because the alternative is so tempting. There is a ratio sitting right there in the record, it involves falls, it comes out at 1.20, and nobody reading a published sheet can tell what arithmetic produced it. The custodian and the manager hold the series that would settle it; the invented record does not.

How does anybody use this in a room, on a Tuesday?

An analyst handed a factsheet with a capture pair on it does three things in about ninety seconds. The first is to look for the period length beside the figures. If the sheet does not say whether the sorting was monthly or quarterly, the pair cannot be compared with any other pair and the analyst writes that down rather than arguing with it. The second is to find the beta and subtract it mentally from the level of the pair. The beta already gave the level, and only the width is new. The third is to ask where the working is: a pair with no stated route, no stated window and no stated unit is a claim rather than a measurement.

An investment committee like Rukmini Deshpande's uses the same three questions in a different order. The committee is not usually deciding whether a number is impressive; it is deciding whether a number is a number. So the standing instruction to a manager such as Faiz Ahmad Ansari is simple: where the series exists, publish the pair with its unit and its window; where the series is absent, publish nothing and say why. The committee that accepts a fabricated pair once will be shown fabricated pairs forever.

A household does the same work with a bank passbook and a pen. One list records the months in which income rose and the months in which it fell. Against those two piles goes what the savings did. The two piles show something one annual figure cannot: whether the household holds on in the thin months or only accumulates in the fat ones. Two piles, two answers, and the same discipline the analyst applies, at a size that can be checked by hand.

What can a capture pair never establish?

Three things, and each of them gets asked of it anyway. A capture pair cannot establish that the behaviour will recur. A pair describes one set of sub-periods inside one window; it is a record of what happened, not a property of the portfolio. Change the window and the pair changes with it. The change is a fact about the measurement rather than a fact about the manager.

A capture pair cannot establish that anybody did anything. A soft down figure can be produced entirely by what the portfolio held before the fall began, with no decision taken during the fall at all. A mandate with a large cash position and a short duration on its fixed income sleeve will show a soft down figure in most falls without anyone lifting a finger. Attributing the shape of the pair to skill during the fall requires evidence the pair does not contain.

And a capture pair carries no originator's name, so it cannot borrow authority from one. Several measures a reader meets in this part of the subject are attached to the people who set them out: Michael C. Jensen, William F. Sharpe, Jack L. Treynor, Gary P. Brinson for the decomposition of a result, and K. J. Martijn Cremers and Antti Petajisto for the measure of how different a portfolio is from its benchmark. No single originator stands behind the capture pair, so the working is set out here with no name attached rather than with a guessed one, and a reader who meets it credited to somebody should ask to see the original text.