Mean-Variance Analysis and Its Fragility to Inputs

Mean-variance analysis turns three inputs into one set of portfolio weights: expected returns, expected volatilities and expected correlations. Harry Markowitz set it out in Portfolio Selection in 1952, and what made it matter was showing that a holding counts through its covariance with everything already held rather than through its own volatility. Its fragility is that a small change in one expected return moves the answer a long way.

The framework is settled machinery. How loosely that machinery is bolted to the table is what gets left out, and the arithmetic below puts a figure on it by running one record through the procedure and watching the answer move.

The running example 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 policy weights, set by the mandate rather than derived here, 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.

One point needs flagging before the machinery starts. Volatility is symmetric, settled in the equities layer and not restated here, and the framework treats that symmetric number as the thing an investor dislikes. Whether a symmetric number describes what an investor actually dislikes is one of the three assumptions the framework rests on.

What does mean-variance analysis actually do?

Mean-variance analysisA method that chooses portfolio weights from three forward looking inputs: an expected return and an expected volatility for each asset, and an expected correlation for every pair. is a mapping. Three kinds of input go in at one end, one set of weights comes out at the other, and nothing sits in the middle except arithmetic. A mapping that adds no information cannot be more reliable than the least reliable thing fed into it.

A currency conversion works the same way. The answer is exact to as many decimal places as anybody wants, and every one of them is a property of the rate supplied. If the rate was a guess, the answer is a guess printed to six figures.

Mean-variance analysis reads the three inputs, performs one piece of algebra and stops. Every judgement embedded in its answer was made by whoever set the inputs, and by nobody else. It holds no price history of its own, no view about which asset is cheap, and no memory of the last time it was run.

What are the three inputs, and how many of them are there?

An expected return for each asset, an expected volatility for each asset, and an expected correlation for every pair, all over one stated horizon. Where each comes from is settled under capital market expectations earlier in this sequence. The three kinds do not grow at the same rate, so counting them is worth doing.

Returns and volatilities grow one for one with the number of assets. Correlations grow as the number of assets multiplied by one less than itself, halved, so they rise with the square. Past about six assets the correlations are the majority of everything that has to be set, so most of the input pile is the part that was hardest to set and is least likely to hold.

The mandate's own three asset classes make this concrete: nine inputs, three expected returns, three expected volatilities and three correlations, all set out below. The nine figures are assumptions the holder chose for its own use rather than anybody's published estimates, and a different set of nine produces a different portfolio.

The pair count matters practically. A return assumption can be argued in a committee room: somebody says equity should beat fixed income, somebody else says by how much. Nobody has an opinion about the correlation between the eleventh and the nineteenth holding, so those numbers come from a history of conditions that have already happened.

What was the contribution that made this framework matter?

Before it, risk was thought about one holding at a time and the impressions were added up. Harry Markowitz, in Portfolio Selection in 1952, showed that the question is wrong at the level of grammar: what a holding contributes to the risk of the whole depends on its covarianceHow far two things move together: the correlation between them multiplied by both of their volatilities. with what is already held, not on its own volatility. Every later piece of portfolio arithmetic in this sequence sits on that.

The consequence sounds wrong the first time. A more volatile holding can lower the volatility of the whole, and it falls out of the variance arithmetic settled under portfolio return and risk.

Start from a sleeve holding nothing but fixed income, volatility 5.0 per cent, and add equity at 18.0 per cent, more than three times as jumpy on its own. At the stated correlation of 0.20 the covariance between them is 0.20 times 18.0 times 5.0, or 18.0 in percentage points squared, against a variance of 25.0 for the fixed income leg alone. Because 18.0 is smaller than 25.0, the first slice of equity subtracts more through the covariance term than it adds through its own variance, and the volatility of the whole falls.

The sleeve volatility keeps falling until 2.24 per cent equity, where it reaches 4.9843 per cent against the 5.000 per cent it started at, and after that it climbs. The point is the sign: adding the more volatile thing moved the total in the direction nobody predicts from looking at the thing. The first slice lowers the total whenever the correlation is below the smaller volatility divided by the larger, here 0.2778, and the stated 0.20 sits under it.

The dip is 0.0157 of a percentage point. At the scale above that is under one pixel, so the same green curve is redrawn below with the vertical scale stretched until the dip is visible.

At the mandate's own policy weights the same three inputs give an expected return of 0.60 times 12.0 plus 0.30 times 7.5 plus 0.10 times 6.0, or 10.05 per cent. The variance is built from four terms and only one of them carries the correlation.

If the three parts always moved together, the volatility of the whole would be their weighted average, 0.60 times 18.0 plus 0.30 times 5.0 plus 0.10 times 0.5, or 12.35 per cent. The portfolio as built carries 11.20 per cent. The 1.15 point difference is the diversification, it exists only because the correlation is 0.20 rather than 1.00, and a report calling a portfolio diversified without computing that difference has asserted nothing at all.

What does Mean-Variance Optimization actually return?

Mean-Variance Optimization is the search step. Given the three inputs and one more setting, it hunts across every possible set of weights and returns the one with the lowest variance for a stated expected return. The extra setting is the risk aversion coefficientA single number standing for how much expected return the holder wants in exchange for carrying one more unit of variance., settled earlier in this sequence.

One set of weights comes back. Not a range, not a shortlist, not a probability attached to anything: one point, printed to as many decimal places as the software was asked for. The whole difficulty of the method lives in the gap between that point output and the softness of everything that produced it, and no amount of decimal places narrows that gap at all.

Because the coefficient is a setting rather than a fact, the point moves when the setting moves. Run the procedure on the mandate's two risky classes at its stated 12.0 and 7.5 per cent expected returns, 18.0 and 5.0 per cent volatilities and correlation of 0.20. At a coefficient of 4 it returns 38.2 per cent in equity, at 8 it returns 20.2 and at 16 it returns 11.2, on the same three inputs every time.

No input has been disturbed yet, and three defensible settings of one parameter have already produced three answers, every one outside the mandate's stated band. A procedure returning a single crisp point invites the analyst to read it as an instruction, when it is closer to a reading off a dial the analyst calibrated.

What happens when one expected return moves by half a point?

The stated year's record carries exactly two things with a return and a volatility attached, and those are the two legs: the Anantara portfolio at 14.2 per cent with 11.8 per cent volatility, and the unnamed composite benchmark at 12.6 per cent with 10.4 per cent volatility, over the same period against a risk-free rate of 6.5 per cent, with a beta of 1.08. Treating realised figures as expected figures is a licence: a year that happened is not a view about the year ahead.

The correlation is derived rather than supplied, in the two steps drawn below: beta multiplied by the benchmark variance gives a covariance of 116.8128, and that divided by the product of the two volatilities gives 0.9519. Only the two volatilities and the beta entered.

A correlation of 0.9519 is very high, and it should be. A portfolio and its own benchmark move together almost all the time. The optimiser divides by the variance of the difference between the two legs. The variance of that difference comes to 13.7744 in percentage points squared. The fragility comes from that divisor.

The square root of 13.7744 is 3.7114 per cent. The record's own tracking error for the same period is 3.7 per cent. The divisor of the optimiser and the square of the tracking error are the same quantity written two ways, so the record's four risk figures cannot be set independently: give any three and the fourth follows.

The weight follows in one line: the difference in expected returns divided by the risk aversion coefficient, plus the second leg's variance less the covariance, all over that divisor. Set the coefficient at 8, allow short positions and impose no limits, stated openly because all three change the answer. The numerator comes to 0.00113472 over a divisor of 0.00137744. The answer is 82.4 per cent of the two leg combination held in the Anantara leg.

The numerator is the only thing that moved, by 0.005 divided by 8, or 0.000625. Divided by 0.00137744 that raises the weight 45.4 points to 127.8 per cent. A weight of 127.8 per cent means borrowing to hold more of the first leg, funded by a short position in the second. Half a percentage point of expected return bought 45.4 percentage points of weight, about 91 points of weight for every single point of return, and no volatility, no correlation and no coefficient moved to produce it.

Across a wider stretch the scale becomes plain. Take the expected return on the first leg from 13.2 to 15.2 per cent, a span no honest forward view could rule out. The answer travels from minus 8.4 per cent, a short position in the portfolio leg, to 173.1 per cent, nearly twice the size of the combination on borrowed money. Two points of input bought 181.5 points of travel.

Why is the instability arithmetic rather than a defect of the software?

Because of what is being divided by what. Input sensitivityHow far the answer moves when one input moves by a small amount. here is one divided by the coefficient multiplied by the divisor, and nothing else enters. At a coefficient of 8 and a divisor of 0.00137744 that is 90.7 points of weight per point of return. Every implementation divides the same small difference by the same small number. Change the software, the language or the machine and the number does not move.

When two legs move together almost all the time, the difference between them barely moves, so the variance of that difference is tiny. A correlation of 0.9519 is why the difference variance came to 13.7744 where the legs individually carried 139.24 and 108.16. The optimiser is not confused by two similar assets. Its answer correctly reports that a tiny preference between two near identical things justifies an enormous position.

Equity against fixed income, at the stated 18.0 and 5.0 per cent volatilities and correlation of 0.20, gives a divisor of 324 plus 25 less 36, or 313.0. The result is 22.7 times the divisor of the first run. Same procedure, same coefficient, same size of error, an answer that moves twenty two times less.

Two shops on the same street sell the same rice at almost the same price. Because they are almost identical, a rumour that one is a rupee cheaper sends every buyer there, and a rumour the other way empties it again. The optimiser is that street: the closer two legs are, the more violently a small preference swings the queue.

The high correlation has one more consequence, and it stops a reader expecting a sensible answer at the low risk end. The lowest variance combination of two legs is a mixture of them only when the correlation is below the smaller volatility divided by the larger, and here 0.9519 is above that threshold of 0.8814. So the lowest variance answer holds minus 62.8 per cent of the portfolio leg, a short position in the portfolio itself. Every answer this run produces is a position no committee would sign.

The control below runs that arithmetic, with every input except one pinned in place and shown on screen. The control opens at 14.2 per cent, where the answer is 82.4 per cent of the two leg combination, or Rs 4,12,00,00,000/- on the Rs 500 crore base. The 50 to 70 band is drawn on the scale so the answer can be watched leaving it.

What does the framework assume that is not true?

Three assumptions, each biting in a different place. The inputs are treated as known rather than estimated, so the arithmetic reads 14.2 per cent as a fact about the future. Variance is treated as what a holder dislikes, so a holder who minds a fall more than an equal rise is represented by a measure that cannot tell the two apart. And correlations are treated as holding across conditions. Capital market expectations shows they do not.

The inputs do not degrade independently, so the third assumption compounds with the first two. Where an allocation decision is actually being tested, expected returns are being revised, volatilities are rising and correlations are converging at once. An optimiser fed a stable correlation set is being handed the input most likely to be wrong precisely when its answer is being leaned on hardest. An input that is simply noisy would be the kinder failure.

What is done about the fragility in practice?

Three responses are ordinary, each a mechanism rather than a recommendation: a constraintA floor and a ceiling written into the run in advance, so the search cannot return an answer outside them., which acts on the output; shrinkagePulling every expected return part of the way toward a common value before the run., which acts on the input; and resamplingRerunning the procedure over many slightly different input sets and reading the spread of the answers., which acts on how the answer is read.

All three work in the same way, by declining to treat the inputs as known to the precision the arithmetic assumes, and none of them makes the inputs any better. They are presented as rival techniques, and they are three ways of admitting the same thing.

Shrinkage can be watched on the numbers already computed. The two expected returns are 14.2 and 12.6 per cent, and their common value is 13.4. Pull both a quarter of the way toward it and the weight falls from 82.4 per cent to 46.1; pull them all the way and it is minus 62.8 per cent, the lowest variance combination computed earlier. Shrinkage walks the answer from whatever the return views said toward the answer that ignores them.

Resampling makes a different admission. The procedure runs many times over input sets drawn around the starting ones, and the spread is reported. With the honest resolution of a forward return view taken as half a percentage point, the first leg runs from 13.7 to 14.7 per cent. The 90.8 point range that comes back is the honest object; the 82.4 per cent printed is one reading from inside it.

The output is quoted in steps of 0.1 of a percentage point of weight. The input can honestly be argued only in steps of about half a percentage point of return, and half a point of return is worth 45.4 points of weight. The smallest honest step in the input is 454 times the smallest step the output is quoted in, and the decimal place is the part everybody reads.

Once the limits bind, the optimiser is not choosing the weight; it is choosing which limit to sit on, and everything the three inputs were supposed to contribute has been reduced to the sign of a comparison. None of that is an argument for writing limits, and none of it is an argument against them. Knowing which of the two produced a number is the difference between a decision and a ritual.

Where does the mandate band bite on the holder's own assumptions?

The run above used the portfolio against its own benchmark, the extreme case. Two genuinely different asset classes allow one check the earlier run cannot support: the answer is an equity weight, so it can be laid against the mandate's stated band of 50 to 70 per cent.

The Anantara mandate states an equity band but no risk aversion coefficient, and no coefficient can be recovered from a band without assuming the very thing being tested. The coefficient here is NOT SUPPLIED, so the sweep shows what each possible coefficient would have implied rather than claiming which one applies.

Read the arithmetic rather than the curve. The unconstrained equity weight sits inside the mandate's band only for coefficients between about 2.12 and 3.01, and at the coefficient of 8 used earlier it is 20.2 per cent, far under the floor. A band and a coefficient are two statements about the same appetite. A committee that writes a band in one document and a coefficient in another has written two answers to one question, and the optimiser will obey whichever one it was handed.

The equity run is also far better behaved: the same one point error moves the answer 4.0 points here against 90.7 in the first run. The fragility is worst exactly where the assets are most alike. An optimiser is least worth listening to at exactly that point.

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

An investment committee like Rukmini Deshpande's will not rerun the arithmetic in the meeting. A committee can refuse to receive the output in the shape the software prints it. Three habits do almost all of the work.

First, the return assumptions get argued before any weights are shown. Once a table of weights is on the screen the conversation is about the weights rather than the argument they came from. Faiz Ahmad Ansari circulates the expected returns on their own, to be signed off before anything is optimised. Second, the output arrives as a range, so the committee sees 37.0 to 127.8 rather than 82.4. Third, somebody states out loud whether the number came from the arithmetic or from a limit. The two are different objects wearing the same digits.

A household does the same thing without the vocabulary. Somebody splits a recurring deposit and an equity plan sixty forty on a belief about which will do better over ten years. The useful question at the kitchen table is not whether sixty forty is right, but how much it would change if that belief moved a little. If it would flip completely, the split was a decision about a guess. The test is the same at Rs 500 crore and at Rs 5,000/- a month: move the assumption a little and see whether the answer survives.

What is mean-variance analysis not?

Mean-variance analysis is not a way of finding out what will happen, not a source of any information, and not improved by being run again on the same inputs. The framework is, and this is the part worth keeping, the clearest statement available of what a set of beliefs about returns, volatilities and correlations implies if taken completely seriously. A weight of 173.1 per cent is therefore information about the beliefs rather than about the portfolio.

The covariance contribution, not the fragility, is the part that survived, so it takes the last word. Every piece of portfolio arithmetic in this sequence, from the 1.15 point diversification benefit to the way a risk budget is split, rests on the idea that a holding counts through what it does to the whole. The framework that idea arrived in is fragile. The idea is not.