Alpha: The Return a Benchmark and Beta Cannot Explain

Alpha is what a portfolio returned less what a portfolio of its exposure was expected to return. At a beta of 1.08 against its composite benchmark and a risk-free rate of 6.5 per cent, the Anantara Multi-Asset Portfolio was expected to return 13.088 per cent for its one stated year. The portfolio returned 14.2, leaving alpha at plus 1.112 percentage points.

The word does a great deal of work in ordinary conversation, and almost none of it is the arithmetic. People say a manager delivered alpha when they mean the portfolio finished ahead of something. The measure is stricter than the word, and the strictness is the only reason anybody computes it. The looser meaning is where the trouble starts.

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 year returned 14.2 per cent against a composite benchmark that returned 12.6 per cent, with a risk-free rate of 6.5 per cent and a beta against that benchmark of 1.08. All four belong to a single twelve month period, and every other number below is worked out from those four rather than carried in from anywhere.

What is alpha, and why is it not just beating the benchmark?

Alpha is a leftover. The calculation begins with what a portfolio actually returned, sets against that a constructed figure for what a portfolio carrying its exposure was expected to return, and alpha is the difference between the two. Nothing in the definition mentions the benchmark's own return on its own, and nothing in it says anything about skill. The whole of alpha is a subtraction with a constructed number on one side of it.

Here is the everyday shape. A cousin runs a tea stall outside a college gate and takes Rs 4,000/- on a Tuesday. Whether that is good cannot be said until the ordinary take is known for a stall of that size, in that spot, on a college day. Suppose the ordinary Tuesday take for a stall in that position is Rs 3,600/-. The Rs 400/- above it is the only part of the day that the position does not already account for. If the stall were twice the size, the ordinary take would be higher and the leftover would be smaller on the same Rs 4,000/-. The leftover moves with the yardstick, and the yardstick is built rather than observed.

Outperformance against a benchmark and alpha are different quantities, and they are equal only in the single case where the betaHow much a portfolio moves for a given move in its benchmark. A beta of 1.08 means the portfolio has historically moved about 8 per cent more than the benchmark did, in both directions. is exactly 1.00. The Anantara portfolio's gross return finished 1.6 percentage points ahead of its composite benchmark for the stated year. Its alpha for that same year is plus 1.112 points. Both figures are correct, neither is a rounding of the other, and the gap between them is not an error to be reconciled away. The gap is what carrying a beta of 1.08 rather than 1.00 did.

Treating the two words as synonyms is by some distance the commonest error attached to the term, and it is a comfortable error to make because in casual reporting the two arrive in the same sentence. The difference between them shows up in the numbers.

What expectation is the residual measured against?

Since alpha is a leftover, everything depends on what it is left over from. The expectation is built in three moves and each move has a job.

Move one is the risk-free rateThe return available on an instrument treated as carrying no repayment uncertainty over the period being measured. The rate is the starting point every other return is measured from.. A portfolio carrying no exposure to the benchmark at all would still expect to earn something, and that something is the rate. For the stated year that is 6.5 per cent. Move two is the benchmark's own excess returnA return measured against something else rather than against zero. A return less the risk-free rate names the part that carrying exposure is being paid for. over that rate, which is 12.6 less 6.5, or 6.1 percentage points. The 6.1 points are what carrying the benchmark's own exposure, no more and no less, was expected to add. The portfolio does not carry the benchmark's exposure; it carries 1.08 of it. Move three therefore scales that 6.1 by the beta. So 1.08 times 6.1 is 6.588 points, and the expectation is 6.5 plus 6.588, or 13.088 per cent.

Notice the order of operations. The usual slip lives there. The beta scales the benchmark's excess over the rate, 6.1 points, and never the benchmark's return itself of 12.6. Scaling 12.6 by 1.08 gives 13.608 per cent, and the answer is wrong. Scaling the whole return charges the portfolio 1.08 times a risk-free rate it never carried 1.08 of. The rate is a floor everyone stands on once, and only the part above it is exposure being scaled.

The expectation is a construction and not an observation, so everything downstream of it inherits whichever inputs went into it. Nobody watched a portfolio return 13.088 per cent. The 13.088 was assembled out of one rate, one benchmark return and one estimate, and if any of the three had been different the leftover would have been different without a single holding behaving differently.

Alpha vs Beta: which part of the return did anybody need a manager for?

The distinction between the two gives either one whatever force it has, so the pair is worth naming rather than absorbing. Both are settled quantities before the split begins. Beta is fixed in the risk and return sequence as the portfolio's sensitivity to the benchmark. Alpha is the leftover once that sensitivity has been paid for.

Take beta first, on its own terms. Beta is market exposureThe part of a portfolio's behaviour that follows from moving with a market rather than from anything specific to what is held. More exposure means larger moves in both directions. expressed as a quantity. Beta is obtainable. Anybody who wants a beta of 1.08 against a composite benchmark can construct it by holding that benchmark's composition and adjusting how much of it they hold, and no judgement about any individual holding is required to do it. Obtainability is the whole force of the idea.

Now take alpha on its own terms. Alpha is not a thing at all, so alpha is not obtainable in that sense. Alpha is the subtraction itself. Alpha is defined as whatever the beta and the benchmark failed to explain, so no arrangement of holdings produces alpha the way an arrangement produces a beta of 1.08. A return that came from beta is a return nobody needed a manager to obtain, and separating the pair is what makes that difference visible.

Defining alpha by what the beta and the benchmark failed to explain makes it a definition rather than an achievement. The arithmetic yields a leftover and stops. Whether the leftover repeats, whether it came from anything anyone did on purpose, and whether it is large enough to mean anything are three separate questions, and performance appraisal already works out how long a record the last of them needs. Alpha names the size of the leftover and establishes nothing about its cause.

How is the excess return separated into exposure and residual?

The arithmetic of the split runs exactly in both directions. One figure goes in, the 1.6 points of gross excess by which the portfolio finished ahead, and two figures come out that add back to it exactly.

The exposure part is the easier half. The portfolio carried a beta of 1.08 where the benchmark by definition carries 1.00, so it carried an extra 0.08 of the benchmark's exposure. Each whole unit of that exposure was expected to add the benchmark's 6.1 point excess over the risk-free rate. So the extra 0.08 was expected to add 0.08 times 6.1, or 0.488 percentage points. The exposure part of the year's outperformance is arithmetic, and it follows from the beta and the benchmark alone.

The residualWhatever is left after a model has taken out everything it can account for. A residual is defined by subtraction, so it collects anything the model did not include. is the other half, and it is the alpha: 1.6 less 0.488, which is 1.112 percentage points. Add the two back: 0.488 plus 1.112 is 1.600 exactly. Both parts must be computed, and reporting only the 1.6 is precisely the omission that hides the exposure half.

As shares of the year, 0.488 over 1.6 is 30.5 per cent and 1.112 over 1.6 is 69.5 per cent. So roughly three tenths of the outperformance was carrying more of the market than the benchmark did, and the reconciliation is not a bookkeeping formality. The reconciliation is the check that the split was honest. A split whose parts do not add back to the total has quietly dropped or double counted something.

What does the whole working look like end to end?

Before any figure appears, the split being run has to be named. The exposure question asks how much of the year's excess was simply carrying more market. Attribution by allocation and selection asks a different thing entirely, namely where the excess came from, and the two splits must be kept apart.

Now the inputs, each with its job. The Anantara Multi-Asset Portfolio returned 14.2 per cent over the stated twelve months. The composite benchmark returned 12.6 per cent over the same twelve months. The risk-free rate over that period was 6.5 per cent, and it appears twice: once to define the benchmark's excess and once as the floor the expectation is built up from. The beta against that benchmark was 1.08.

Points are easy to shrug at. Put money against them. On Rs 500 crore the whole 1.600 points of outperformance is Rs 8,00,00,000/-. The exposure part, 0.488 points, is Rs 2,44,00,000/- of that gross figure, and the residual of 1.112 points is Rs 5,56,00,000/-. A holder told only that the portfolio beat its benchmark by Rs 8,00,00,000/- has been told a true thing that hides the fact that Rs 2,44,00,000/- of it followed from the beta by arithmetic alone.

Four things must be printed beside any alpha figure for it to be checkable at all: the window it belongs to, the benchmark it was struck against, the risk-free rate used, and the beta together with the window and frequency that beta was estimated over. The fourth of those is not a formality, and moving the beta estimate on its own shows why.

How far does alpha move when only the beta estimate moves?

Hold the 14.2, the 12.6 and the 6.5 completely still and move only the beta. Because the expectation is the risk-free rate plus beta times 6.1, every 0.01 added to the beta adds 0.061 points to the expectation and therefore takes 0.061 points away from the leftover. The relationship is a straight line, and its slope is the benchmark's excess.

Walk it. At a beta of 0.90 the expectation is 6.5 plus 5.49, or 11.99, so alpha is plus 2.21. At 1.00 the expectation is exactly the benchmark's 12.6 and alpha is plus 1.60, the one case where alpha and gross outperformance coincide. At the recorded 1.08 alpha is plus 1.112. At 1.20 the expectation is 6.5 plus 7.32, or 13.82, so alpha is plus 0.38. At 1.30 the expectation is 14.43, above the realised return, so alpha is minus 0.23.

There is a beta at which alpha is exactly zero, and it does not have to be hunted for. Setting the realised return equal to the expectation: 14.2 equals 6.5 plus beta times 6.1. Rearranged, beta is the portfolio's own excess over the risk-free rate divided by the benchmark's, or 7.7 over 6.1, giving 1.2623. Above that estimate the portfolio finished ahead of its benchmark and behind its own expectation at the same time, and both statements stay true together.

Across beta estimates from 0.90 to 1.30, a spread no wider than the disagreement between two reasonable estimation windows, this portfolio's alpha runs from plus 2.21 points to minus 0.23 points, and its own returns never changed by a single basis point. That is a travel of 2.44 percentage points, which on Rs 500 crore is Rs 12,20,00,000/-, produced entirely by the choice of an input.

Alpha is therefore not a measurement of a portfolio, it is a measurement of a portfolio against an estimate, and that estimate came out of a regression run over a window somebody chose, at a frequency somebody chose. Change either choice and the estimate moves, and the leftover moves with it.

What does a residual inherit from the choices behind it?

A leftover carries whatever was decided upstream of it, and here there are four such decisions. The list is what gets printed beside the figure, so the four are worth naming one by one.

First, the estimation windowThe stretch of history a statistic was calculated over, together with how often observations were taken inside it. Two people using different windows on the same portfolio get different answers, and neither is a mistake. and the frequency inside it. A beta estimated on monthly observations over twelve months is a different estimate from one on weekly observations over three years, and the second is not a better version of the first. Second, the benchmark, including any structural mismatch with the mandate: the Anantara mandate holds 10 per cent cash at its policy weight and its composite benchmark contains no cash at all, so a slice of the portfolio is being measured against something it does not resemble. Third, the risk-free rate used, appearing twice in the working and therefore moving the answer twice. Fourth, the beta itself, already shown above to swing the leftover by more than two points on its own.

Every one of those four is a decision somebody made, and the alpha carries all of them while displaying none of them. That is not a criticism of the measure. Carrying decisions invisibly is what a residual does. A leftover cannot show what it was left over from unless somebody writes it down beside it.

Why must this split never be mixed with the other one?

One split of the Anantara portfolio's 1.6 points has terms of plus 0.488 and plus 1.112. Attribution by allocation and selection runs a different split of the same 1.6 points, and its terms are plus 0.35 of allocation effect and plus 1.25 of selection effect. Both sets total 1.600. Neither is a better estimate of the other.

The two splits are separate instruments because they answer separate questions. The exposure split asks how much of the excess followed from carrying more market than the benchmark carried. The other split asks where the excess came from, in terms of which decisions produced it. The bases differ, the arithmetic differs, and the fact that both totals land on 1.600 is a property of both being complete decompositions, not evidence that they are two views of one quantity.

The two never appear in the same sentence, neither is the true split, and the residual of 1.112 is not a check on the selection effect of 1.250 however close the two happen to look. That closeness is the trap. A reader who notices 1.11 and 1.25 sitting near each other feels the pull to treat one as corroborating the other, and there is nothing there to corroborate. The two were never computed from the same working. Change the beta estimate and the 1.112 moves while the 1.250 does not.

What does a positive alpha not establish?

Two things, and both matter more than the sign of the number.

A positive alpha does not establish skill. One window is one draw, and performance appraisal works out how long a record that question actually needs before a single year's figure separates from noise. A positive residual in one stated twelve month period is one observation of one leftover.

A positive alpha also does not establish that the residual came from anything the manager did, and the sharper point follows straight from the definition. Because alpha is what the model failed to explain, everything the model left out lands in it automatically. Exposures nobody estimated land there. Timing inside the window lands there. Costs that the chosen return basis hides land there. A structural mismatch between the mandate and the benchmark lands there. And plain chance in a single draw lands there too.

Alpha names the leftover, and naming a leftover is not explaining it. The residual against a market model carries Michael C. Jensen's name, which is why the term is often written as Jensen's alpha. The original derivation sits in the academic record.

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

Three habits, and none of them requires building a model.

An investment committee like Rukmini Deshpande's does not ask whether the alpha was good. The committee asks for the four inputs, then asks what the leftover would have been at two other defensible beta estimates. If the answer is that the leftover crosses zero inside the range of defensible estimates, the committee has learned something the single figure would never have told it, and it has learned it in about four minutes.

An analyst comparing two managers does the same thing in reverse. Two alphas struck against different benchmarks, over different windows, at different frequencies, are not comparable, and putting them in one column does not make them so. The honest output is often that the comparison cannot be run, and the sensitivityHow much an answer moves when one of its inputs is changed by a small amount. A result with high sensitivity to an input is really a statement about that input as well. arithmetic above is how that is shown rather than asserted.

A household does a version of this without any of the machinery. Someone reports that their savings did better than the market last year. The useful question is not by how much; it is how much more of the market they were carrying while they did it. If they held a more aggressive mix than the thing they are comparing themselves against, a part of the difference follows from the mix and required no cleverness at all. Splitting a result into the part that came from carrying more and the part that did not is the same habit at every size, and it is almost never done at any of them.