Active Share: How Far the Portfolio Sits From Its Benchmark

Active share measures how far a portfolio's weights sit from its benchmark's weights, as half the sum of the absolute differences across every position. Run across the Anantara Multi-Asset Portfolio's three asset buckets against a benchmark holding 60 per cent equity and 40 per cent bonds, the figure is 10 per cent. Run across holdings it would be a different number entirely.

Active share hides a small amount of arithmetic and one large trap, and the trap matters far more than the arithmetic does. The arithmetic takes five steps and a school child can run it. The trap is that the same portfolio, compared with the same benchmark, using exactly the same five steps, produces several different figures depending on how finely the two lists are cut before the working starts. Every one of those figures is correct. Only one of them is the one somebody means.

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 stated shape is 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. Equity, fixed income and cash at those levels are policy weightsThe share of a portfolio the holder has decided each asset class should carry. A policy weight is decided in advance, and the actual weights drift away from it as prices move., so they describe the shape the holder chose rather than the shape the market left behind on any particular day. The benchmark is a composite holding 60 per cent of a broad equity index and 40 per cent of a broad bond index, and it holds nothing in cash.

What does active share actually measure?

Active share compares two lists of weights and reports how much of the portfolio is invested differently from the benchmark. The idea stops there. The measure does not ask what the portfolio returned, what the benchmark returned, how volatile either was, or whether the person running it was any good. The question is about position, not about outcome.

A household grocery order set beside a neighbour's makes the same point. The two lists can be compared on the morning both are placed, before anybody has eaten anything, and it is possible to say precisely how much of one basket differs from the other. There is no need to wait and see who enjoyed dinner more. Which basket was the better one is a completely different question, answered by a completely different kind of evidence, and it takes a month rather than a minute.

Because it needs nothing but two lists of weights, active share can be computed on the day a portfolio is built, before a single return exists, and that is the property that separates it from every other measure in this part of the subject. A tracking errorThe dispersion of the difference between a portfolio's returns and its benchmark's returns over a stated period. Tracking error is computed from a history of returns, so it cannot exist before that history does. figure needs a run of return differences before it can exist at all. So does an information ratio, so does an alpha, so does a maximum drawdown. Active share needs a spreadsheet and an afternoon.

How is the figure worked, step by step?

Five steps, and only one of them ever surprises anybody. First, line up every position that appears in either list. The step matters more than it sounds. Second, take the portfolio weight less the benchmark weight for each line, giving a weight differenceThe portfolio's weight in one position less the benchmark's weight in the same position. The difference is positive where the portfolio holds more and negative where it holds less. that can be positive or negative. Third, take the absolute differenceThe size of a difference with its sign thrown away, and minus 10.0 and plus 10.0 both count as 10.0. The absolute difference answers how far apart, not which way. on each line. Throwing the sign away makes being short and being long count the same. Fourth, add the absolute differences. Fifth, halve the total.

The first step carries a rule people skip. Absence is a position, so a holding that appears in one list and not in the other still enters the sum, with the missing side counted as a weight of zero. The Anantara portfolio holds 10.0 per cent in cash and the composite benchmark holds none, and that line contributes as much to the working as any line where both sides hold something. Comparing only the positions the two lists have in common would measure a different thing entirely, and it would flatter every portfolio that had wandered somewhere the benchmark has never been.

Here is the working laid out on the Anantara portfolio's three lines. The last column is the whole measure. Read it first.

Twenty points of absolute difference. Half of twenty is ten. So the active share of the Anantara Multi-Asset Portfolio, computed across its three asset bucketsA broad container such as equity, fixed income or cash, used when weights are compared at the coarsest level rather than holding by holding., is 10.0 per cent. Notice also that the plain difference column sums to zero. The zero is not a coincidence, and it explains why the sum is halved.

Why is the sum halved rather than reported as it stands?

Because both lists add to the same total, and that single fact does all the work. The portfolio's weights sum to 100.0 points. The benchmark's weights sum to 100.0 points. So if the portfolio holds ten points more of something than the benchmark does, those ten points had to be taken from somewhere else in the same portfolio. There is no other place they could have come from.

On the Anantara lines it is visible in one glance. The portfolio holds 10.0 points of cash that the benchmark does not hold at all, and it holds 10.0 points less in fixed income than the benchmark does. The two lines are not two separate departures. One departure is being seen from both ends: money left the bond sleeve and arrived in cash. The raw sum of absolute differences counts that single move twice, once where the money went and once where it came from, so halving it is not a tidying convention but the step that converts a double count back into the share of the portfolio genuinely invested differently.

The result can be checked without any of the arithmetic. A check that agrees is a good sign that the halving is honest. Exactly Rs 50 crore of the Rs 500 crore portfolio sits in a bucket the benchmark does not hold at all. Rs 50 crore is 10.0 per cent of Rs 500 crore. The working produced the same 10.0 per cent, arrived at by subtracting weights rather than by counting money.

What do the two ends of the scale actually mean?

Zero means the two lists are identical, position by position and weight by weight. Nothing in the portfolio departs from the benchmark, so nothing about the portfolio's composition can produce a result different from the benchmark's. One hundred per cent means the opposite extreme: no position is shared at all, so every rupee sits somewhere the benchmark does not go.

Both ends are fixed, so the measure sits on a bounded scaleA measure that cannot go below a floor or above a ceiling, however extreme the underlying situation. The floor here is zero and the ceiling is one hundred per cent.. A bounded scale makes the figure comparable straight across portfolios of wildly different sizes, and that is unusual among the measures sitting near it. A Rs 5 crore portfolio and a Rs 5,000 crore portfolio both report on the same nought to one hundred ruler. A rupee figure offers no such ruler. A dispersion figure has no ceiling to press against, and so it offers none either. The convenience is genuine, and it is also why the measure gets quoted so casually. A number that always lands between two familiar posts feels safer than it is.

How does a weight distance differ from a return dispersion?

Active share and tracking error are the pairing that gets muddled most often, so both sides are worth setting down before they are compared. Active share is a distance between two lists of weights, computed from composition alone, available immediately, and bounded at nought and one hundred. Tracking error is the dispersion of a series of return differences over a stated period, computed from performance, unavailable until that performance exists, and with no ceiling. The two are measured in different units, from different inputs, at different times.

The two measures can move independently, and both awkward combinations genuinely occur. A portfolio can hold names the benchmark does not hold at all while those names happen to move almost in step with the benchmark, giving a large weight distance and a small return dispersion at the same time. Picture two neighbouring tea stalls. One buys its leaves from a different wholesaler, different sacks, different labels, nothing in common on the shelf, and yet both stalls have a good week when the weather turns and a poor week when it does not. Their inventories share nothing; their fortunes share almost everything.

The reverse happens too. A portfolio can hold nearly the same names as its benchmark in slightly different sizes, giving a small weight distance, and still show a wide dispersion of return differences if those small tilts sit in the positions that swing hardest. Neither measure is a substitute for the other, and a report carrying only one of them has answered only one of the two questions.

Why does the level the comparison is run at change the answer?

Because the five steps never specify how finely to cut the two lists before lining them up, and every cut is a legitimate application of the same working. The Anantara portfolio compared bucket by bucket gives three lines and an answer of 10.0 per cent. The same portfolio compared sector by sector would give as many lines as there are sectors either list touches, and a different answer. Compared name by name it would give a line for each of the 28 equity holdings, a line for every name the benchmark holds that the portfolio does not, and a different answer again.

All three are correct, none of them is more correct than the others, and a figure quoted without its level of aggregationHow finely the two lists of weights were cut before being compared: by broad asset bucket, by sector, or by individual holding. The same portfolio gives a different answer at each level. stated beside it cannot be read at all. The direction of the drift is predictable, which makes it worse rather than better: coarser cuts hide differences inside the buckets, so a bucket level figure is normally the smallest one the portfolio can honestly produce. If nobody fixes the level before the number is computed, the level will end up being chosen by whoever wants a particular answer, and they will not have to falsify anything to get it.

What does the working give on the Anantara portfolio?

The level has to be declared before the working is run, and fixing it first is the whole discipline. The record for this mandate carries the Anantara Multi-Asset Portfolio's three policy weights and the composite benchmark's two, and nothing finer. So the figure below is computed across three asset buckets, and it is not the number people usually mean by this measure.

The three lines are equity 60.0 against 60.0, fixed income 30.0 against 40.0, and cash 10.0 against 0.0. The absolute differences are 0.0, 10.0 and 10.0, summing to 20.0 points, and half of 20.0 points is 10.0 per cent. Read as money, exactly Rs 50 crore of the Rs 500 crore portfolio sits in a bucket the composite benchmark does not hold at all, and that Rs 50,00,00,000/- is the same 10.0 per cent arrived at by counting rupees instead of subtracting weights.

Now the honest part. The usual computation is run at holding levelComparing the two lists name by name rather than by broad bucket, so each individual holding contributes its own line to the working., and running it that way here would need the benchmark's weight in each of the 28 equity names the portfolio holds, plus the weight of every name the benchmark holds that the portfolio does not. None of that is in this record. The holding level figure therefore cannot be computed from what is available, and naming the missing input is the output.

Saying which way the figure would move is fair. Leaving that direction sounding like a measurement is not. A 28 name sleeve set against a broad index holding a great many more names must differ on a great many lines, so a holding level figure would be substantially higher than 10.0 per cent. Substantially higher is a direction. A direction is not a number, it cannot be put in a table, and it must never be written down beside real figures as though it were one.

How few names does a 5 per cent cap actually permit?

The Anantara mandate carries a concentration capA stated limit on how large any single holding may be, written against a named base. A cap restrains the biggest position and says nothing about how many positions there are. of 5 per cent of the portfolio on any single holding, and equity is 60.0 per cent of the portfolio. Five per cent of Rs 500 crore is Rs 25 crore. Twelve holdings at Rs 25 crore each is Rs 300 crore, the entire equity sleeve. So the cap, obeyed to the letter, permits an equity sleeve of exactly twelve names.

The sleeve in fact holds 28 names. The largest is 4.6 per cent of the portfolio, or Rs 23 crore, sitting just inside the cap. The top ten come to Rs 155 crore, or 31.0 per cent of the Rs 500 crore portfolio and 51.7 per cent of the Rs 300 crore sleeve. The other eighteen names hold Rs 145 crore between them, an average of Rs 8.06 crore each. A ceiling on any one position imposes no floor at all on how many positions there are, so every one of those figures came from decisions taken inside the cap rather than from the cap itself. A ceiling and a count are two entirely different constraints, and they get read as one constantly.

Notice how the base moves the story. The trap is the level of aggregation trap wearing a different costume. The largest holding is 4.6 per cent of the portfolio and 7.7 per cent of the equity sleeve. Both figures are right. The two answer different questions, and the cap is written against the portfolio, so 4.6 per cent is the figure that decides whether the mandate is being observed.

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

An investment committee like Rukmini Deshpande's uses the figure as a consistency check rather than as a verdict. The mandate describes a portfolio run differently from its benchmark. The figure says how differently, in a unit anybody in the room can hold in their head, and it says so before the first quarter of performance arrives. If the described intention and the measured distance point in opposite directions, the conversation that follows is about the description or about the portfolio, and it can happen a year earlier than a return based conversation could.

A lender or an analyst reading somebody else's report uses it the other way round, as a question generator. Three questions, in order. At what level was it computed. On what date were the two lists struck. Both drift between rebalancings. And which benchmark, described in full. A distance measured against the wrong reference is precisely measured nonsense. None of those three questions needs the report to be produced again, and any report that cannot answer all three has supplied a number that cannot be used.

A household runs the same check with a pen and no arithmetic at all. One list records where the savings actually sit, a second records the default arrangement the bank or employer would have put them in, and the lines where the two differ are marked. The marked fraction is the same measure. The fraction shows how much of the position is a choice made rather than a choice accepted, and it says nothing whatever about whether the choice was a good one.

What can this measure never tell the reader?

There are no returns anywhere inside the measure, so it cannot say whether the differences were good ones. Two departures of identical size can behave in completely unrelated ways depending on what sits on either side of them, so the measure cannot say how much risk a departure carries. Skill is a claim about outcomes and active share is a statement about position, so the measure cannot say whether the portfolio was run with skill.

A large figure is a description of how much departure there is, and on its own it is never an argument for or against anything, in either direction. The temptation runs both ways and both are errors: reading a large figure as evidence of conviction, and reading a small one as evidence of caution. A portfolio can be far from its benchmark for excellent reasons, for poor reasons, or for no reason anybody wrote down. The measure cannot separate the three and does not try.

The measure is K. J. Martijn Cremers' and Antti Petajisto's, and it is worth naming them because the attribution is part of the term rather than an ornament attached to it. Their original work sits at ideas.repec.org.