Alpha is a leftover, and leftovers depend on the subtraction

The return splits into two parts, and only one is the manager's

Beta is the share of a portfolio's movement that came from the market moving. Alpha is what remains once you subtract it. One is a measurement; the other is a leftover — and a leftover depends entirely on what was taken away first.

Start with the case that makes the difference visible. Two funds finished a year up 30%. The index both of them name as their benchmark rose 20%. Read that as a 10-point achievement each and you have already made the error this article is about, because the two did not take the same amount of the index's movement to get there.

Suppose the first amplified the index by 1.4 times and the second by 0.9. The market part of the first fund's year is 1.4 × 20%, or 28%, leaving two points unexplained. The market part of the second is 0.9 × 20%, or 18%, leaving twelve. Same return, same benchmark, six times the leftover — and the fund that looks better on the naive comparison is the one that looks worse here.

The multiplier is beta. The leftover is alpha. The figures above are chosen to make the arithmetic easy and describe no actual scheme; every number in this article is an illustration with its inputs stated, not a record of what happened.

Beta is a slope, and a slope needs a fit to mean anything

Beta comes from fitting a straight line. Plot the fund's return in each period against the index's return in the same period, draw the line that sits closest to those points, and the steepness of that line is beta. A beta of 1.4 says that in a period when the index moved one point, this fund tended to move 1.4.

Two things follow that are routinely skipped. The first is that beta is symmetric: the same 1.4 that turns a 20% rise into a 28% gain turns a 20% fall into a 28% loss. Nothing in the number says which of those the period contained.

The second matters more. A line can be drawn through any cloud of points, however scattered, and the slope will come out to something. The statistic that says whether the line describes the data at all is the R-squared — the share of the fund's variation that the index's movement accounts for. A beta of 0.9 with a low R-squared is a slope through a cloud, and the alpha derived from it is mostly the scatter, not the manager.

One more mechanical wrinkle, because it explains figures that otherwise look wrong. Beta computed from daily returns of a portfolio holding thinly traded names comes out lower than beta computed from monthly returns of the same portfolio, because prices that update with a lag record only part of each day's move. The holdings did not change. The measurement frequency did.

Why a high beta looks like skill in a rising market

Take the first fund from the opening — beta 1.4, up 30% in a year the index rose 20% — and lay the two readings side by side. The naive comparison hands the manager 10 percentage points. Subtracting the market part first hands over 2. The eight points in between were not earned; they were rented from the index and they arrive with a debt attached, because the same 1.4 is waiting on the way down.

Call it borrowed beta: return that came from taking more of the market's movement, presented as return that came from choosing better. It is not a trick anyone has to play deliberately. A fund that runs concentrated positions, holds less cash than its peers or leans towards more volatile parts of the market will carry a beta above one as a consequence of its mandate, and in a rising market that mandate alone produces numbers that read as skill.

The reason this survives so well is that a run of rising years produces a run of flattering figures on every measure that does not adjust for beta — which is most of the figures a saver actually encounters. The correction arrives only in the fall, and by then the record being compared covers a different period. Concentrated and thematic mandates make the same point in a sharper form.

The reverse case deserves equal weight, because it is the one that gets a good manager fired. Against that same index up 20%, a fund with a beta of 0.9 takes 18% from the market, so anything short of two points of genuine alpha still leaves it behind the index — and behind the index is the entirety of what a one-year comparison table shows. Historical figures describe what happened, not what will happen; past performance does not indicate future results.

Change the benchmark and the alpha changes

Alpha is measured against something. Change that something and the number moves, with nothing in the portfolio touched. This is not a technicality at the edges of the calculation — it is the calculation.

Hold the fund fixed at 30% for the year and vary only what it is compared against. Each row below states its own inputs, so the arithmetic can be redone in a line: the market part is beta times the benchmark's return, and the alpha is 30% less that.

Three defensible calculations, three different answers, one unchanged set of holdings. The figures are illustrative and the dividend contribution in the third row is an assumption of the illustration rather than any index's actual payout — but the direction of each effect is mechanical and does not depend on the numbers chosen.

The third row is worth sitting with, because it is invisible in a comparison table. An index quoted as a price series records only the change in the constituents' prices; the dividends those companies paid are simply absent from it. A fund receives those dividends, and comparing the fund against the price series therefore credits the manager with the market's own dividend stream. Whether Indian schemes must be measured against a total-return series is a regulatory question this article does not answer — the arithmetic gap is the same wherever a price series is used.

The second row is the more common problem in practice. A fund whose mandate has drifted towards mid-sized companies, while its stated benchmark remains a broad large-company index, is being measured against something it no longer resembles. Its beta to that benchmark will be high and its R-squared low, which is the pair of symptoms that says the comparison has stopped working. The benchmark named in the scheme's own documents is the starting point, not the conclusion.

Alpha is whatever the model could not explain

The benchmark is only half the choice. The other half is the model, and it moves the answer just as far.

The simplest model has one explanatory input: the market. It says a portfolio's expected return is a baseline return — what a short government instrument would have paid, since the money could have sat there instead — plus beta times the market's return above that baseline. Put the earlier fund through it with a baseline of 6%, a figure chosen to keep the multiplication easy and not a claim about any actual rate: 6 + 1.4 × (20 − 6) gives 25.6%, so the alpha is 4.4 points rather than the 2 that came out when the baseline was ignored. The baseline is an input like any other, and inputs move outputs.

Now add explanatory inputs. Models in wide use include the tilt towards smaller companies, towards cheaply valued ones, towards recent winners, towards steadier and more profitable businesses. Each is a pattern a rule can reproduce — screen, rank, hold — without anyone forming a judgement about a particular company. Every one of them that gets added to the model absorbs part of what the previous model called alpha, because the return it explains is no longer unexplained.

Being a residual has a second consequence, and it is the one that decides how much weight a published alpha can carry. Everything the model failed to account for lands in it, in one undifferentiated pile: an ill-fitting benchmark, cash held between decisions, a factor nobody included, the expense ratio if the figure was computed before costs, and ordinary luck. The costs alone are worth checking, since an expense ratio is charged every year on the whole corpus — an alpha quoted gross of it describes a return no investor received.

None of which makes alpha useless. It makes it conditional. An alpha figure without its benchmark, its model, its baseline and its period attached is not a small figure or a large one; it is an incomplete sentence.

Why the estimate is less certain than the number of observations suggests

Suppose the benchmark and the model are both settled, so the alpha means one definite thing. It is still an estimate from a sample, and the sample is smaller than it looks.

Take ten years of monthly returns — 120 months — and compute a rolling 3-year alpha, stepping forward one month at a time. That yields 120 − 36 + 1, or 85 windows, which sounds like a substantial record. But each window moves only one month past the one before it — the second drops the first month and picks up a thirty-seventh, so it shares 35 of its 36 observations with the first. The count of genuinely independent windows is the length of the series divided by the length of the window, or 120 ÷ 36. Roughly three, not eighty-five.

Confidence in an estimate grows with the square root of the number of independent observations, so the gap between three and eighty-five is not a rounding matter. A significance test run on the 85 overlapping windows will report a certainty the data cannot support, and it will do so without any error being made in the arithmetic — the arithmetic assumes an independence the windows do not have. The same mechanism, applied to returns rather than to alpha, is set out in rolling returns.

A second source of uncertainty sits underneath even a clean, non-overlapping sample. Alpha is where the fitted line crosses the vertical axis — the fund's return in a period when the index did nothing — and that crossing point, the line's intercept, carries an error bar whose width depends on how far the individual points scatter around the line. The wider that scatter relative to the alpha being measured, the longer the record needed to tell a small alpha apart from zero, and monthly equity returns scatter widely against an alpha of a couple of points a year. How long a record, for Indian data specifically, is an empirical question this article does not answer.

The practical form of all this is a habit rather than a formula: treat a published alpha as one draw from a distribution, not as a measurement. Ask how wide the distribution was before asking where this draw sat in it.

What the two numbers throw away

Every risk measure compresses a distribution into one number, and the useful question about any of them is what got discarded in the compression. Four things, here.

• Direction. One slope is fitted across rising and falling periods alike, so a fund can show a beta of 0.9 overall while behaving like 1.2 in the months the index fell. Up-capture and down-capture, described in the risk measures article, split exactly that number in two.
• Shape. Alpha is an average of residuals, and an average survives one extraordinary observation comfortably. A decade's alpha can rest on a single quarter, in which case the median month was negative while the mean was positive. Dropping the best month and refitting is a cheap test of whether that is what happened.
• Depth. Neither number says how far the portfolio fell from a peak on the way, or how long it took to climb back — its drawdown. Two funds with identical alpha and beta can have had completely different worst moments, and the worst moment is what decides whether an investor was still holding at the end.
• The investor. Both are computed on the scheme's own price series. What a particular person earned depends on when their money went in and came out, which is a different calculation entirely — the distinction between a scheme's return and an investor's.

The trade-off in using either measure is worth stating plainly, since it is the price of the convenience. Compressing a decade into two numbers is what makes funds comparable at all; it is also what removes the ordering, the asymmetry and the depth of the falls. The comparability and the loss are the same operation, and you cannot keep one without the other.

Reading an alpha figure without being fooled by it

Given that the number is conditional, the useful skill is knowing which conditions to ask for. Seven questions, and a published figure that cannot answer them is not evidence of anything.

1. Against which benchmark? And does that benchmark still resemble what the fund holds today, rather than what it held when the mandate was written?
2. Price or total return? A price series excludes the constituents' dividends, and the manager collects them.
3. What period, at what frequency? Daily, weekly and monthly returns produce different betas for the same portfolio, and a period without a fall has not tested anything.
4. What is the R-squared? A low fit means the beta is unstable and the alpha built on it inherits the instability.
5. Gross or net of costs? An alpha quoted before the expense ratio and one quoted after it differ by the whole fee, every year.
6. Were the windows overlapping? If so, discount the apparent significance rather than the figure.
7. Does the sign survive? Drop the best month, refit, and look again. If the alpha disappears, it was a description of one event.

There is a quieter conclusion in that list. If a fund's alpha vanishes when the benchmark is made to match its actual holdings, and vanishes again when a size or value tilt is added to the model, then what the fund offered was an exposure rather than an edge — and an exposure is precisely the thing a rule can deliver, which is what index funds and exchange-traded funds are built to do. A fee buying selection and a fee buying a rule are not the same purchase. That is a comparison of what is being paid for, not a suggestion about what to own; the answer depends on facts about a particular investor that an article does not have.

Beta has a second job that has nothing to do with judging a manager, and it is arguably the more useful one: it describes how a holding is likely to sit alongside everything else owned. That is the province of asset allocation, where the question is not whether the number is high but whether the portfolio already contains four things carrying the same one.

Testing the sensitivity rather than arguing about it

Every claim above is arithmetic, but the size of each effect is not — how far an alpha actually moves when the benchmark or the window changes is something to ask of a price history rather than debate.

FNOTrader's Mutual Funds app runs on the full AMFI NAV history — around 34 million rows of net asset value, the per-unit price of a scheme — and reports rolling-return distributions across every start date available, alongside comparison against a chosen benchmark and maximum drawdown. Running the same scheme against two different benchmarks, and the same benchmark over two different windows, is the direct way to see how much of a published figure was the choice rather than the fund. The stock backtesting universe covers roughly 2,390 stocks and 17 NSE sector and size indices, so the same exercise can be run on an index rather than a scheme.

FNOTrader is not a SEBI-registered investment adviser or research analyst, and nothing here is a recommendation to buy, hold or avoid any scheme or security. What the figures in this article are is a method for asking better questions of the ones you are shown.

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