Performance Attribution: Where the Return Came From

Performance attribution takes the gap between what a portfolio returned and what its benchmark returned, then splits that gap into the decisions behind it. Over the Anantara Multi-Asset Portfolio's one stated year, 14.2 per cent against 12.6 per cent leaves 1.6 percentage points, and the recorded split is an allocation effect of plus 0.35 points and a selection effect of plus 1.25 points.

A portfolio returned 1.6 percentage points more than its benchmark. Most people read that as one fact. The gap is not one fact. A gap is a total, and a total is what remains when somebody has already added up several different things and passed on only the sum. Attribution is the working that puts the pieces back on the table.

The running example throughout is the Anantara Multi-Asset Portfolio, an invented discretionary mandate of Rs 500 crore run by Faiz Ahmad Ansari for a charitable endowment whose investment committee is chaired by Rukmini Deshpande. Its policy weightsThe share of a portfolio the holder decided in advance that each asset class should carry. It is a decision written down, not a measurement of what is actually held today. are equity 60.0 per cent, fixed income 30.0 and cash 10.0, drawn below in rupees. Its benchmark is an unnamed composite of 60 per cent a broad equity index and 40 per cent a broad bond index, and it holds no cash at all.

The portfolio's 14.2 per cent against the benchmark's 12.6 per cent is the excess returnThe portfolio's return less the benchmark's return over the same window. It is a difference in percentage points, not a return in its own right., plus 1.6 percentage points, or Rs 8,00,00,000/- on Rs 500 crore. A reported 1.6 points can only be believed or doubted. A split into plus 0.35 and plus 1.25 can be checked term by term against the weights and returns that produced it. The difference in what can be argued with is the whole reason attribution exists.

What does an attribution add that a performance report does not?

How Performance Attribution Explains Portfolio Returns

A performance report states what happened. Every word of it is true and none of it is an explanation. Nobody reading those two returns can say which decision earned the gap, what any decision cost, or whether the same decisions are still in place.

An attribution answers a different question. Out of everything the manager actually decided, what moved the result, and by how much? Answering it means going back to the level at which decisions were taken. A multi-asset mandate takes them at two levels. The first is how much of each thing to hold. The second is what to hold inside each thing. The result can be split along the same two lines the decisions were taken along, and attribution is nothing more exotic than that correspondence made arithmetic.

Think of a household that runs a monthly food budget and ends the year 8 per cent above the neighbouring household of the same size. Either the proportions changed, more of the expensive categories and less of the cheap ones, or the basket did, with the same proportions bought at pricier shops. The 8 per cent on its own cannot tell them which. Split it, and it can.

The vocabulary follows straight from that. The proportions decision becomes the allocation effect. The inside-the-basket decision becomes the selection effect.

What is the allocation effect actually measuring?

The allocation effect is the part of the excess return that came from holding a different amount of a bucketOne of the groups the portfolio and its benchmark are both broken into for the working, such as an asset class, a sector or a region. The list of groups is chosen before anything is computed. than the benchmark held. Nothing more, and in particular nothing about what was chosen inside it.

The allocation effect stays blind to what was picked, through a trick in the pricing. The allocation effect multiplies the weight difference by the bucket's benchmark return, not the portfolio's own, asking what would have happened if the portfolio's weights had been held alongside the benchmark's own holdings inside each bucket. A weight decision priced at benchmark returns credits nothing at all to what was picked inside the bucket, and that blindness is the whole point of the term.

For the Anantara mandate the weight differences fall out at once: zero on equity, minus 10 points on fixed income where the portfolio's 30 meets the benchmark's 40, and plus 10 points on cash where the portfolio's 10 meets a benchmark holding none. Because both sets of weights add to 100 per cent, the three weight differences sum to zero, and they must. The zero total is a property of every attribution ever run rather than a coincidence of this mandate, and it does real work twice further down.

And what is the selection effect measuring?

The selection effect is the part that came from what was held inside a bucket doing better or worse than that bucket's own benchmark. If the equity sleeve returned more than the broad equity index over the same window, that difference is selection, whatever the weight was.

The selection effect is priced with the mirror-image trick. The selection effect multiplies the return difference inside the bucket by the benchmark's weight, not the portfolio's own, asking what would have happened if the benchmark's proportions had been held alongside the portfolio's own holdings inside each one. A holding decision priced at benchmark weights credits nothing at all to how much of the bucket was held.

The symmetry of those two mirror-image tricks is the design, and it is also the reason a third term has to exist.

Are these two terms simply halves of one thing?

Allocation Effect vs Selection Effect

No, and treating them as halves is the first serious error available here. Allocation and selection are answers to two different questions asked of the same year. A portfolio can score well on one and badly on the other, and the two scores do not trade against each other in any fixed way.

The clean way to see it is to build four notional portfolios out of the same ingredients, swapping in one set of weights and one set of returns at a time: the benchmark itself, one where only the proportions changed, one where only the holdings changed, and the real portfolio. The four notional portfolios are numbered I to IV in the grid below. The allocation effect is II less I, the selection effect is III less I, and the total excess return is IV less I.

Now count. II less I, plus III less I, does not reach IV less I. The remainder is the part of the excess return that exists only because both changes happened at once. The third piece is the interaction effect, and it arises precisely when a bucket was both held above its benchmark weight and filled with holdings that beat that bucket's benchmark, so the good picking got applied to more money than the benchmark had there.

Interaction is real, and small and awkward enough that many working papers do not report it separately. The common alternative folds interaction into selection, by pricing the selection term at the portfolio's own weights rather than the benchmark's, and the arithmetic still closes. Folding interaction into selection changes the selection figure while leaving the total untouched, so two honest attributions of the same year can print different selection numbers and both reconcile perfectly. So a selection effect is only comparable with another when both forms are stated.

How is the working actually run, step by step?

How to Run Brinson Performance Attribution

The partition of an excess return into an allocation effect and a selection effect is Gary P. Brinson's, with L. Randolph Hood and Gilbert L. Beebower as co-authors on the work that made it standard practice. The original papers are indexed at ideas.repec.org.

The procedure has a fixed order, drawn as a ladder below, and no step may be skipped. The bucket list is fixed once; here that is equity, fixed income and cash, the level at which the mandate's allocation decisions were actually taken. The window is one stated period, the same for every bucket. Then exactly four numbers per bucket go in: the portfolio weight, the benchmark weight, the portfolio return and the benchmark return. The three terms come out bucket by bucket, each is summed across the buckets, and the ladder closes on a check.

The closing check is not a formality. The check is the entire point of running a procedure rather than making an assertion: if the three terms do not reconcileTo add up to the figure the working began with, exactly, with no unexplained remainder left over. A working that does not reconcile has not finished. to the excess return the working began with, then some bucket weight, some bucket return or some window is wrong, and any explanation resting on those terms explains a portfolio nobody holds.

People skip the check because the terms look plausible on their own. The terms should not be trusted alone: an allocation effect of plus 0.35 points has no meaning until it sits beside a selection effect and an interaction term that together close the gap. Reconciliation is what converts three plausible numbers into one verified statement.

Does the convention chosen change the answer?

There are two common forms of the allocation term, and the difference between them gets stated loosely almost everywhere. One multiplies the weight difference by the bucket's benchmark return as it stands. The other first subtracts the benchmark's own total return from that bucket return, then multiplies. Overweighting a bucket now scores positively only if the bucket beat the benchmark overall, and that is closer to how the decision felt when it was taken.

Bucket by bucket the two do differ, by that bucket's weight difference multiplied by the benchmark's total return. Against a benchmark that returned 12.6 per cent over the stated year, the fixed income bucket's allocation contribution is 1.26 points higher under the second form, being 0.10 times 12.6; the cash bucket's is 1.26 points lower, and equity, whose weight difference is zero, shifts by nothing.

But the two totals are identical, always, and the reason is the property noted earlier: the weight differences sum to zero, so the benchmark's total return gets multiplied by zero and cancels out of the sum entirely. The convention changes which bucket gets credited, not how much allocation there was in total. Which of those two levels is meant has to be stated, or the reader is taught something false.

Quote a total allocation effect and the convention does not need to travel with it. Quote a bucket-level contribution and the convention has to travel with it. Under the second form 1.26 points of credit sits on a different line of the table, on a choice nobody wrote down.

Which weights does the working actually need?

The choice of weights does more damage than the convention question. Attribution needs the average actual weightsThe weights the portfolio really carried on average across the window being measured, rather than the weights it was supposed to carry. They are computed from the holdings over time. the portfolio carried across the window, not the policy weights, and the two are not the same thing.

The reason is driftThe slow movement of a portfolio's actual weights away from the weights the holder chose, caused by the parts moving at different speeds rather than by anyone deciding anything.. The Anantara mandate's policy weights are 60, 30 and 10, but prices move at different speeds, so the equity share wanders above and below 60 between rebalancings, along the shape drawn below. Some of the year's allocation difference against the benchmark was therefore never decided at all. It happened.

Now watch the trap. Suppose the average actual weights are not to hand, so somebody substitutes the policy weights instead. The policy weights always are to hand. Every weight difference now shrinks to the deliberate part, and the allocation term shrinks with it. Because the total is pinned to the excess return by construction, everything that drift produced does not disappear. The drift silently moves out of allocation and into selection, the total still reconciles perfectly to 1.60, and nothing anywhere in the working looks wrong.

The error is a dangerous one because the reconciliation check, the very thing that makes the procedure trustworthy, still passes. The check catches arithmetic faults, not a wrong input that happens to be internally consistent. The only defence is to state which weights were used, in writing, beside the result.

How else can the same active return be cut?

Brinson and Factor Attribution Compared

Brinson attribution partitions by parts of the portfolio. The working takes a bucket list and four numbers per bucket, and hands back allocation, selection and interaction. Brinson attribution is entirely a story about the structure of what is held.

Factor attribution partitions by shared behaviours instead, regressing the active returnThe portfolio's return less the benchmark's return, measured over the same window. Another name for the excess return when the comparison is deliberate rather than incidental. on a set of common exposures, reporting a contribution for each and leaving a residualThe part of a result that a model does not account for. It is what is left after every term the model does carry has been subtracted. at the end. The shared behaviours cut clean across the buckets: two holdings in different asset classes can share one exposure, and two holdings in the same bucket can share none.

Brinson asks which parts of the portfolio produced the excess and factor attribution asks which shared behaviours produced it, so both can be correctly computed on the same year and still tell stories that do not line up, with neither thereby wrong. An overweight in one bucket can be an allocation decision to Brinson and an exposure to a common behaviour to a factor working, both descriptions accurate at once.

A factor working needs an estimated exposure for every common behaviour it reports, and the Anantara mandate's record carries none, so no factor contribution can be computed for that portfolio.

Why does one excess return have more than one honest split?

A split is defined by the question it answers and by the base it is computed on. Change the question and the same number splits differently, and both splits can be entirely correct.

The Anantara portfolio's 1.6 points has two splits recorded against it. The first asks which decisions produced the excess: plus 0.35 of allocation and plus 1.25 of selection. The second asks how much of the excess was simply carrying more market than the benchmark, and at a betaA measure of how much a portfolio tends to move when the benchmark moves. Above 1.00 means it has tended to move more, below 1.00 less. of 1.08, against a benchmark 6.1 points above the 6.5 per cent risk-free rate, it lands on plus 0.488 points of extra market exposure and plus 1.112 points of residual, which is Michael C. Jensen's alpha and is not taught here. Its four steps are drawn below, and the calculator on portfolio risk and attribution drives them live from the same figures.

Both splits add to 1.60, neither is an estimate of the other, and no term from one may ever appear beside a term from the other in the same sentence.

And now the trap those four figures set. The four figures sit close: selection at 1.25 and the residual at 1.112 differ by 0.138 points, and allocation at 0.35 and the exposure part at 0.488 differ by 0.138 points as well. The two differences are one single gap of 0.138 appearing twice. When two pairs both sum to 1.60, whatever one term gains the other must lose, so the gap is forced by arithmetic rather than found in the data. The agreement is a property of subtraction, not corroboration of anything.

What does the Anantara portfolio's own year look like through the working?

Start with the totals, the only things the whole exercise has to land back on. The mandate's record gives a gross excess return of plus 1.6 percentage points over the stated twelve month period, split into an allocation effect of plus 0.35 points and a selection effect of plus 1.25 points.

The first thing to do with any split is add it. 0.35 plus 1.25 is 1.60, the gross excess return, so the split reconciles. Then express the parts as shares of the whole: 0.35 divided by 1.6 is 21.9 per cent and 1.25 divided by 1.6 is 78.1 per cent, and those two add to 100.0 per cent. On Rs 500 crore, allocation is worth Rs 1,75,00,000/- and selection is worth Rs 6,25,00,000/-, and together they make the Rs 8,00,00,000/- the gap represents.

Now the derivation that makes the 0.35 mean something. The equity weight matches the benchmark's exactly, so equity contributes nothing whatever to allocation, by construction rather than by luck. The entire allocation effect must therefore come out of the 10 point fixed income underweight and the 10 point cash overweight, and there is nowhere else for it to be.

The two remaining terms collapse neatly. Minus 0.10 multiplied by the fixed income bucket's benchmark return, plus 0.10 multiplied by the cash bucket's benchmark return, is 0.10 times the difference between the two. Set that equal to the recorded plus 0.35 points and the difference has to be 3.5 percentage points: the cash bucket outran the fixed income bucket by 3.5 points over the stated year. Because the weight differences sum to zero, the 3.5 point result is the same under either allocation convention.

The 3.5 is produced by a definition rather than observed anywhere, and it rests on three choices made before the arithmetic started. The bucket list is these three asset classes and nothing finer. The weights are the policy weights standing in for average actual weights the mandate's record does not carry, and substituting them is exactly the trap set out above. And the cash bucket's benchmark return is a convention. The benchmark holds no cash at all, so it defines no return for that bucket. Move any one of those three and the implied 3.5 points moves, so quoting 3.5 without its three conditions manufactures a fact out of a definition.

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

An investment committee like Rukmini Deshpande's does not read an attribution to find out whether the year was good. The committee reads one to find out whether the manager is doing what the mandate hired him to do, and attribution is one of very few tools that can reach that.

Suppose the mandate was written on the understanding that the manager's contribution would come from picking holdings inside the asset classes, with the asset class weights held close to policy. A year at plus 1.6 points with 78.1 per cent of it in selection fits that mandate. The same 1.6 points with allocation carrying almost all of it would not, and the committee would have a live question: the return was fine, but it came from a lever nobody agreed in advance would be the main one.

An analyst looking from outside asks a narrower thing: is the split stable enough to be worth reading? One year is one draw, so the honest use is to log the shape and wait, not to conclude. A lender or a trustee reading the same attribution mostly wants to know whether the terms reconcile and which weights were used. Those two checks are what separate a working from a story.

The household version is the same shape at a different size: at the end of a year of saving, a household can ask how much of the difference against a plain benchmark came from keeping more in cash than the benchmark assumed, and how much came from which particular deposits and schemes were picked, and neither question can be answered from the total alone. A total conceals a structure, and the structure is where the questions live.

What can an attribution never settle?

Three things. First, it cannot say whether any effect will recur: one window is one draw, and a split computed over twelve months describes those twelve months and nothing beyond them.

Second, it cannot say whether a positive selection effect came from judgement. A bucket that did well for reasons nobody anticipated produces exactly the same figure as one chosen with insight. The difference is not in the numbers. Telling a result apart from luck is a separate working, covered separately.

Third, it cannot say whether the result was worth what it cost: turnover, fees and taxes sit outside every term in the working, so a gross excess return says nothing about what the holder ended up with. Attribution explains a result and settles nothing at all about repetition, skill or worth, and the moment it is asked to do any of those three it stops being arithmetic and starts being a story with numbers attached.