The Factor Model: Return Split by Common Drivers

A factor model explains a return by exposures shared with other holdings rather than by anything specific to one of them. The Anantara portfolio's beta of 1.08 against its composite benchmark is a single factor loading, and over the stated year that one factor explains 90.6 per cent of the portfolio's variation. The remaining 9.4 per cent is what one factor cannot see.

The 90.6 per cent explained share can be checked without fitting anything. The record already carries three dispersion figures for one stated twelve month period, and the share falls straight out of them.

The running example is the Anantara Multi-Asset Portfolio, an invented Rs 500 crore mandate run by Faiz Ahmad Ansari for a 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. Its benchmark is a composite of 60 per cent a broad equity index and 40 per cent a broad bond index, both unnamed.

Over the stated year the portfolio returned 14.2 per cent and the composite benchmark 12.6 per cent, both before the cost of running the mandate, against a risk-free rateThe return available over the same period without taking on the risk being discussed. of 6.5 per cent. Portfolio volatility was 11.8 per cent, benchmark volatility 10.4 per cent, the betaThe slope of a portfolio's movement against a benchmark's, so 1.08 means it moved about eight per cent more than the benchmark did. against the benchmark 1.08 and the tracking errorThe standard deviation of the difference between a portfolio's return and its benchmark's, over a stated period. 3.7 per cent. The four dispersion figures are not four free measurements: given any three of them, arithmetic fixes the fourth.

What is a factor model actually for?

A factor modelAn arrangement that relates a return to one or more common influences, so the part moving with those influences separates from the part that does not. answers one question: how much of what happened to this portfolio also happened to everything else like it. Not why, and not whether anybody did well. Just how much was shared.

Twelve shops line one road with different trades, and takings fall by a third in one month. Interviewing twelve shopkeepers yields twelve true, specific stories. Or the road turns out to have been dug up for four weeks. A factor model is the arithmetic version of noticing the road: it measures the movement common to everybody standing on it and leaves the twelve stories alone.

A factor model divides and does not explain: it establishes that a third of the fall was road and the rest was not, and says nothing about why the tailor did better than the stationer.

The one factor version relates a portfolio's movement to a single common influence, usually a stated benchmark, and calls everything else the leftover. The split into a market part and a residual is attributed to William Sharpe, and the later expansion to several factors at once, the three factor model, to Eugene Fama and Kenneth French in their 1993 work. One common influence and one leftover is the whole of the arithmetic below.

What does a factor loading of 1.08 actually say?

A factor loadingHow much of a common influence a portfolio carries: a loading of 1.08 means it moved 1.08 units for every one unit the factor moved. is how much of the factor the portfolio carries. In the single index modelThe simplest factor model, using one common influence and treating everything else as a leftover with no structure of its own. it is the familiar beta, and for the Anantara portfolio it is 1.08 against the composite benchmark for the stated year.

The loading says one thing: on average over that year the portfolio moved about eight per cent further than the benchmark in whatever direction the benchmark went. The loading is symmetric, so the same 1.08 that flatters the portfolio in a rising period punishes it in a falling one, and nothing in the number indicates which kind of period comes next.

A loading also says three things it does not say. A loading is an average over the window it was measured in, and a different window gives a different number. So the 1.08 does not say the extra exposure was intended, that it was rewarded, or that it was stable. Each is a judgement a person makes, using information the model never touched.

The record locks one loading, 1.08, for the whole stated twelve month period and carries none for any part of it. Four equal stretches of three months can be constructed as denominators, but the numerators are not there, and the honest thing to draw in their place is the words NOT SUPPLIED.

How much of the variation does one factor explain?

A beta is not a bare slope. A beta is a correlationA measure of how closely two series move together, running from minus 1.00 through zero to plus 1.00 and never outside. scaled by the ratio of the two dispersions, so writing it the other way round recovers the correlation from figures already in the record.

The correlation is the beta multiplied by the benchmark volatility and divided by the portfolio volatility. For the Anantara portfolio the sum is 1.08 times 10.4, or 11.232, divided by 11.8. The answer is 0.9519. Squaring it gives the explained variationThe share of one series' up and down movement that moves in step with another series. It is the square of the correlation between them.: 0.9060, or 90.6 per cent. The 90.6 per cent is a headline statistic about the Anantara portfolio derived from three numbers sitting in plain sight, in two multiplications and a division.

The same split works out in squared percentages, the units the arithmetic actually uses. The portfolio's total variation is 11.8 squared, or 139.24. The part attributable to the single factor is the beta squared multiplied by the benchmark's variation: 1.1664 times 108.16, or 126.16. The leftover is 139.24 less 126.16, or 13.08. Dividing 126.16 by 139.24 gives 0.906 again, from the other direction.

One tenth of how the Anantara portfolio moved about over the stated year was not shared with its benchmark, and every conversation about individual holdings is a conversation about that one tenth.

What is the residual, and what is it not?

The residualWhatever is left of a series after the modelled part has been removed. It is defined by subtraction and carries no explanation of its own. is the 9.4 per cent: the share of the portfolio's variation over the stated year that did not move with the single factor. The residual is defined by subtraction, so it is what is left rather than a thing measured.

Three readings of that number are wrong, and each one costs money in a real committee. The residual is not skill, it is not alpha in the return sense, and it is not error.

The residual is not skill. Nothing in the arithmetic looked at a decision. The same share would come out if every position had been chosen by drawing names out of a bowl, and a share of variation has no sign, so in an invented year where the leftover pulled the portfolio down it would still be 9.4 per cent.

The residual is not alpha in the return sense either. A residual share of variation is measured in squared percentages and reported as a share; alpha is a return quantity in percentage points. The measure named after Michael C. Jensen is the return one, carried under Jensen's alpha in the risk monitoring and performance evaluation sequence.

And it is not error. A single factor was never going to account for everything, and a residual of zero would be the strange result, not this one.

Two numbers near 9.4 sit in the same record. The residual share of variation is 9.4 per cent; the worst drawdown over the stated year was 9.7 per cent. The residual share and the drawdown share a first digit and nothing else: one is a share of dispersion with no units of return in it, the other a fall in value measured from the highest point to the lowest inside a stated window. The window has to be named every time a drawdown is quoted.

Why can a loading not take any value at all?

A correlation cannot exceed 1.00, because two things cannot move together more than perfectly. Since the beta equals the correlation multiplied by the ratio of the two volatilities, the beta is capped at that ratio.

For the Anantara portfolio the ratio is 11.8 divided by 10.4, or 1.1346. No beta above 1.1346 is compatible with those two volatilities, so a report showing one has a reporting fault rather than an interesting result. The recorded 1.08 sits comfortably underneath.

The relationship between the loading and the explained share is not a straight line. Because the share is the square of the correlation, moving the loading from 0.60 to 0.70 buys much less explanation than moving it from 1.00 to 1.08: at 0.60 the factor explains 28.0 per cent, and at the ceiling it explains the whole of it.

Is residual volatility the same thing as tracking error?

No, and they sit close enough in this record that the mistake is easy. Residual volatilityThe dispersion of what is left after the modelled part of a series has been subtracted, expressed in the same units as the original series. removes 1.08 times the benchmark. Tracking error removes the benchmark once.

Both come from the same three inputs. Residual volatility is the square root of 139.24 less 126.16. The square root of 13.08 is 3.62 per cent. Tracking error is the square root of 139.24 plus 108.16 less twice 1.08 times 108.16. The square root of 13.7744 is 3.71 per cent, matching the record's stated 3.7 per cent. The two statistics are 0.09 percentage points apart in this record, and that closeness is a fact about these particular numbers rather than a rule.

Removing 1.00 times the benchmark and removing it once are the same act, so at a loading of exactly 1.00 the two would be the same number. The gap opens as the loading moves away from 1.00 in either direction. A loading of 1.08 has not moved far, so only 0.09 points sit between them.

How Factor Investing Fits Into Portfolio Construction, and what does it leave untouched?

A factor approach chooses holdings by their loadings rather than one name at a time. Instead of asking whether a business is a good one, it asks which names carry more of a stated characteristic. A factor approach is therefore a rule for filling a sleeve, not a different sleeve and not a different portfolio, and every constraint written before the rule still applies to every order it produces.

Much is fixed before a factor rule gets near the market. The Anantara portfolio is Rs 500 crore, its equity sleeve Rs 300 crore at 60.0 per cent, set by the allocation work before any name was considered. The mandate caps any single holding at Rs 25 crore, 5 per cent of the portfolio, and the equity band runs from 50 to 70 per cent. A factor rule inherits all of that.

Suppose a factor rule wants Rs 32 crore of the sleeve in one name. Rs 32 crore is 6.4 per cent of the Rs 500 crore portfolio, outside a cap written at 5 per cent, so the order is cut to Rs 25 crore and the remaining Rs 7 crore goes elsewhere. The rule ran into a constraint written down before it existed.

The largest holding in the sleeve is Rs 23 crore: 4.6 per cent of the Rs 500 crore portfolio and 7.67 per cent of the Rs 300 crore sleeve. Neither figure is wrong and they answer different questions, so the base has to be named in the same sentence as the number every time.

The same law caps what a factor rule can do to concentration. The Rs 300 crore sleeve is held across 28 names, and the ten largest are Rs 155 crore between them, 51.7 per cent of the sleeve and 31.0 per cent of the Rs 500 crore portfolio. Ten of twenty eight holdings can never be less than ten twenty eighths of the sleeve, a floor of 35.71 per cent. Any reported top ten below that floor is an arithmetic impossibility rather than a well spread portfolio.

A factor approach leaves the size of the sleeve, the cap, the equity band and the settlement plumbing exactly as they were. A factor approach does reach the cost of trading, set out under trading costs. A rule that turns the sleeve over faster pays that cost more often, and every return figure above is struck before it is taken.

What does adding a second factor require?

Three things, and the record for the Anantara portfolio holds one of them. A first loading, measured; a second loading measured over the same twelve month period, against a stated construction of the second factor; and a statement of how far the two overlap. A factor is not a natural object, so somebody has to say how the second one was built.

The record for the Anantara portfolio carries one loading and no second series, so no second factor is fitted. Inventing a series so the arithmetic looks complete would produce a number with nothing behind it.

The overlap requirement is the one people underestimate. A household asked why its month was expensive might answer that it was the wedding season and also that relatives visited. The relatives visited for the wedding, so the two answers are not independent explanations, and adding them up double counts one event. Two factors that move together do the same to a return.

Where does the loading show up in the return itself?

Variation is how a portfolio moved about. The return is where it finished, and the two connect like this.

Over the stated year the composite benchmark returned 12.6 per cent against a risk-free rate of 6.5 per cent, so it returned 6.1 points above that rate. A portfolio carrying a loading of 1.08 on that benchmark would be expected to return 6.5 plus 1.08 times 6.1: 6.5 plus 6.588, or 13.088 per cent. The Anantara portfolio returned 14.2 per cent gross of the cost of running the mandate. Subtract: 14.2 less 13.088 is 1.112 percentage points.

Now split the headline. The gross excess return over the benchmark was 1.6 points, being 14.2 less 12.6. Of that, 13.088 less 12.6, or 0.488 points gross, is simply the reward for carrying more of the factor than the benchmark did. The leftover of 1.112 points gross is everything else. Check the addition: 0.488 plus 1.112 is exactly 1.600. So of a gross excess of 1.6 points, roughly three tenths came from carrying more market exposure and roughly seven tenths did not, and a headline gap that is not split like this has said nothing about which is which.

In rupees, the leftover the record names as 1.11 points is Rs 5,55,00,000/- of the Rs 500 crore portfolio. The subtraction shown in full gives 1.112 points, or Rs 5,56,00,000/-. The Rs 1,00,000/- between them is where the rounding was taken, and that convention is settled under the information ratio in the risk monitoring and performance evaluation sequence.

Rs 5,55,00,000/- is 1.85 per cent of Rs 300 crore, so if that whole leftover had arisen inside the Rs 300 crore equity sleeve it would be 1.85 points of the sleeve. The word IF is doing real work: the record does not split the leftover between the sleeves. Anything achieved inside the equity sleeve reaches the Rs 500 crore portfolio at 0.60 of its size. A sleeve figure and a portfolio figure are never interchangeable.

Which decomposition is this, and which one is it not?

The same 1.6 point gross gap can be pulled apart two ways, and the two splits are not competing estimates of one quantity. The beta decomposition, the one run here, gives 0.488 points of exposure and 1.112 points of leftover. The attribution decomposition belongs to the risk monitoring and performance evaluation sequence, and asks where the gap came from across the sleeves: an allocation effect of plus 0.35 points and a selection effect of plus 1.25 points, adding to the same 1.6. Neither is the true split, and a sentence that pairs the allocation effect with the residual, or the selection effect with the exposure part, has invented a quantity that nobody measured.

What happens to all of this after the cost of running the mandate?

Every return figure above is gross of the cost of running the mandate, and the record carries that cost. The management fee was Rs 6.25 crore and the performance fee was Rs 3.15 crore, or Rs 9.40 crore in total. The two fees together come to 1.88 per cent of the Rs 500 crore portfolio. Subtract that from the gross 14.2 per cent and the net return is 12.32 per cent, against a benchmark of 12.6 per cent.

So the gross excess of plus 1.6 points for the stated year becomes a net shortfall of minus 0.28 points for the very same year, and the two figures describe opposite outcomes for one portfolio. None of the factor arithmetic changes. The leftover is simply worth less to the holder once the cost of producing it has been paid.

One number in that fee build collides with another. Rs 6.25 crore is the management fee for the stated year, and also what the selection effect of 1.25 points comes to on the Rs 500 crore portfolio. A sentence that lets the reader guess is worse than no sentence, so wherever Rs 6.25 crore appears, name which of the two it is.

What does a factor model refuse to say about a holding?

Everything. The model reads a series of portfolio returns and a series of factor returns and compares their movements. No business, no balance sheet, no management, no valuation ever entered the arithmetic.

The model cannot say whether one particular holding is worth holding, and the fact that it cannot is a property of the tool rather than a shortcoming in it. A weighing scale cannot say whether the ingredient is fresh either.

Whether a business is a good one is settled in a different part of this platform, where company analysis is taught. Portfolio construction deals with what happens after a view exists: how a chosen holding enters a constrained portfolio, what it displaces, what it costs, and how the result is later split apart.

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

Faiz Ahmad Ansari brings the record to Rukmini Deshpande's investment committee. The temptation is to open the holdings list and start discussing positions. Opening the list feels like real work. The discipline is to spend four minutes first on three lines nobody has to look anything up for.

Line one is the explained share and the residual. Line two is the gross split of 0.488 and 1.112 points, and the net shortfall of 0.28 points once the 1.88 per cent cost is taken. Line three names the residual as the part that moved differently. The three lines take four minutes to state and they change what the remaining fifty minutes are worth arguing about.

The same shape appears far from Rs 500 crore. A household with a bank deposit, a monthly equity plan and a house in the town where its only salary is earned can ask how much of what happened to its savings last year happened to everybody in that town. The arithmetic is heavier at Rs 500 crore and the question is identical, and answering it first stops a conversation about details standing in for one about the whole.

A committee holding the split can reach three more numbers from the same record: return over volatility, net of the 6.5 per cent risk-free rate, is 7.7 over 11.8 for the portfolio, or 0.653, against 6.1 over 10.4 for the benchmark, or 0.587, and the information ratio is 1.6 over 3.7, or 0.43. Each of the three carries the risk-free rate and the benchmark it was struck against. Without both, none of them can be compared with anything.

An analyst reading somebody else's report uses the ceiling as an integrity check: divide the larger volatility by the smaller and see whether the quoted beta could be produced by them. A lender whose collateral is a portfolio uses the explained share the same way. A high share says the collateral moves with the market it is pledged against.