Portfolio Return and Risk: Why Only One Is an Average

A portfolio return is the weighted average of what its parts returned, exactly, with no correction term. Its volatility is not. Two holdings that do not move in lockstep combine into a volatility below the weighted average of their separate volatilities, and that gap widens as their correlation falls. Return adds in a straight line. Risk does not, and that asymmetry is what makes spreading worth anything.

Everything the instrument starts with comes from one place: the Anantara Multi-Asset Portfolio and the composite benchmark it is measured against, both invented, both reported for one stated twelve month period. The two sets of figures are set out below, and they do not change anywhere in this guide. The Anantara Multi-Asset Portfolio returned 14.2 per cent for the stated year with a volatilityThe standard deviation of a series of returns, expressed for a stated period. Volatility measures how widely the readings sat around their own mean. of 11.8 per cent. The composite benchmark, 60 per cent a broad equity index and 40 per cent a broad bond index, neither named here, returned 12.6 per cent with a volatility of 10.4 per cent over the same year. The beta of the portfolio against that benchmark was 1.08. Deviations from the mean are squared before they are added, so volatility here is symmetric: a rise and an equal fall count the same. Squaring before adding is set out under standard deviation.

On those figures, held half and half, the blend returns 13.40 per cent with a volatility of 10.97 per cent. The return is half of 14.2 plus half of 12.6, or 7.10 plus 6.30. The volatility is the square root of 34.81 plus 27.04 plus 58.41, a variance of 120.26. The weighted average of the two volatilities is 11.10 per cent, so the blend sits 0.13 points below it. The worked example the rest of this guide takes apart is exactly these numbers, set out here so they can be read without touching a control.

How is a portfolio's return built from the returns of its parts?

Multiply each part's return by its weight. Add the results. The whole operation is those two steps, and there is nothing else in it. No correction term, no residual, no adjustment for how the parts moved against each other. The return of a portfolio is a weighted averageEach input multiplied by its share of the total, with the products added. The shares must sum to one for the result to mean anything. of the returns of its parts, and it is that exactly rather than approximately.

One consequence falls straight out, and it catches errors for free. A weighted average can never land outside the range of the things being averaged. If the parts returned 18, 9 and 6 per cent over the same period, then whatever the weights are, the portfolio returned something between 6 and 18 per cent. At 40, 35 and 25 per cent it returned 11.85 per cent. However far the weights are pushed around, the answer stays inside the fence. So when a report shows a portfolio return above every one of its parts, nothing clever has happened: a weight is wrong, a period has been mixed, or a part has been left out of the list.

The fence is easier to see in a household. Hold that version beside the portfolio one for a moment. Three earners bring home different amounts and the figure per earner sits somewhere between the smallest packet and the largest one, always. No arrangement of the three can push it past either end, and the same fence sits under every portfolio return.

What do the weights look like on a portfolio of Rs 500 crore?

The Anantara Multi-Asset Portfolio runs Rs 500 crore for a single institutional holder, an invented charitable endowment, under a discretionary mandate. Its stated policy weightThe share of the portfolio that the mandate states for an asset class. A policy weight is a written instruction, not a measurement of what is currently held. set is 60 per cent equity, 30 per cent fixed income and 10 per cent cash. In rupees that is Rs 300 crore, Rs 150 crore and Rs 50 crore, on a base of Rs 5,00,00,00,000/-.

The two checks worth running before any of the arithmetic in this guide are that the weights sum to 100 per cent and that the rupee amounts sum to the total, and a weight set that fails either is the commonest arithmetic error on this subject. Sixty and thirty and ten make one hundred. Rs 300 crore and Rs 150 crore and Rs 50 crore make Rs 500 crore. Both hold, so the weight set can be used. A set reading 60, 30 and 15 sums to 105, and every output computed from it is inflated by five per cent of something, silently, with no error message anywhere. The control at the top marked mis-key the weights does exactly that to the two holding case, putting 60 and 45 into the two weight fields: the blended return then reads 14.19 per cent against a true 13.51, and the blended volatility reads 11.62 against a true 11.07, each of them five per cent too high and neither of them flagged by anything except the sum line.

The weights were not chosen here. The mandate set them, alongside the constraints that everything else sits inside: equity between 50 and 70 per cent, no single holding above 5 per cent of the portfolio, no unlisted holdings, and a minimum credit standing on the fixed income sleeve stated as a policy rather than as a rating symbol. The weights are an input here, exactly like the returns.

What is the difference between a policy weight and an actual weight?

A policy weight is what the mandate says. An actual weightThe share an asset class occupies right now, worked out from current values. An actual weight changes on its own as prices move, without anybody trading. is what the portfolio is actually holding on a given day, worked out from current values. On the day the mandate is funded these are the same. The three sleeves do not move together, so the two figures separate immediately afterwards. If equity rises faster than fixed income for a few months, the equity share climbs above 60 per cent without anybody buying a single share.

The separation has a name: rebalancing driftThe gap that opens between stated weights and held weights as prices move, until trades are made to close it.. Drift is not a failure of discipline, it is the arithmetic of holding two things that move differently, and it is the entire reason rebalancing exists as an activity. A household that keeps three months of expenses in a savings account and the rest in a long term holding faces exactly this. Nobody moves the money, and the split moves anyway.

The consequence here is narrow and precise. A return computed on policy weights and a return computed on actual weights are two different numbers from identical holdings, so a report has to say which one it used. The record here locks the portfolio total for the stated year and does not lock the three sleeve returns, so the difference can be named but not computed. Naming what a figure cannot be pushed to say matters as much as computing what it can.

A refusal stated in words is easy to mistake for a shrug, so it is worth being exact about what this one costs and what it does not. The record does lock one equation, and the equation can be written out in full: the three sleeve returns, taken at the policy weights, add to the 14.2 per cent the portfolio reported for the stated year. What is missing is not the structure, it is three of the four numbers inside it.

The drift from 60, 30 and 10 to 63, 28 and 9 is plus 3, minus 2 and minus 1 points, and those three add to zero. So the difference between a return on policy weights and a return on actual weights is not driven by the level of the sleeve returns at all: it turns entirely on the spreads between them. If all three sleeves had returned the same figure, the two weight sets would give the identical answer to the last decimal, and it is only because they did not that the question exists. The record carries no sleeve spread, so the size stays unknown.

The same refusal reads more plainly in rupees, so here it is that way round.

Why is volatility not a weighted average?

The hinge of the whole subject sits here. For two holdings with weights that sum to one, the varianceThe square of the standard deviation. Squaring is what lets the pieces be added together. A standard deviation on its own does not permit that. of the blend is the first weight squared times the first variance, plus the second weight squared times the second variance, plus twice both weights times the covarianceA measure of how two series moved together, in the units of both. Divide it by the two standard deviations and it becomes the correlation. between them. The square root of that variance is the volatility.

Two things in that sentence do the work. First, the weights are squared. A weight of 0.5 becomes 0.25, so the first two terms alone are already smaller than a plain weighted average of the two variances would be. Second, the third term carries the covariance. The covariance is the only quantity in the expression that knows anything about how the two holdings moved against each other, and it is where the size of the effect actually lives.

The missing half is not a fixed quantity either. The coefficient on the cross term depends on the weights, and it is at its largest exactly where the two holdings are held in equal amounts. The half and half blend used throughout this guide sits at that maximum.

Run it on the case. Half the Anantara Multi-Asset Portfolio and half the composite benchmark, both for the stated year. The correlationCovariance rescaled to sit between minus one and plus one, so that two pairs measured in different units can be compared. is not given directly. The beta of 1.08 times the benchmark volatility of 10.4, divided by the portfolio volatility of 11.8, gives 0.9519. Three figures in, one out, and the three used are named. The covariance is then 0.9519 times 11.8 times 10.4, or 116.81.

The blended return, meanwhile, took one line: half of 14.2 plus half of 12.6 is 13.40 per cent, and it is exactly 13.40 because a return is an average and nothing else. The weighted average of the two volatilities is half of 11.8 plus half of 10.4, or 11.10 per cent. The computed volatility is 10.97 per cent. The gap of 0.13 percentage points is the whole diversification effect available from this pair at this correlation, and it is small for a reason that the next block makes exact.

What exactly does correlation do to the answer?

Covariance is correlation multiplied by the two volatilities. So the third term of the variance can be written as twice the weights, times the correlation, times both volatilities. The two volatilities are fixed by the record. The weights are chosen. Only the correlation is free to change the answer, and when it falls, the third term shrinks with it. At a correlation of zero the third term disappears entirely and the variance is just the two squared weight terms.

The third term is the only place in the whole calculation where the relationship between the holdings enters. Diversification is therefore a statement about correlation and never a statement about the number of holdings. Ten shops in one shopping centre are ten holdings with one driver. When the centre closes for repairs, all ten stop trading on the same day. Counting them gives ten. The arithmetic that matters was the correlation, and it was near one the whole time.

Harry Markowitz set out this argument in Portfolio Selection in 1952, and it is why the covariance between holdings, rather than the risk of each holding on its own, became the thing a portfolio is built around. The rest of this guide is that one term, examined at different values.

One more thing follows from the same term, and it is the reason the two holding case is worth working through for a portfolio of twenty eight. Every holding brings one term of its own. Every pair of holdings brings a cross term. Pairs grow much faster than names do, so past a handful of holdings almost the whole expression is cross terms and almost nothing in it is about any single holding on its own.

What happens at a correlation of exactly one?

The boundary case establishes what everything else is being measured against. At a correlation of 1.00 the variance becomes 34.81 plus 27.04 plus 61.36, a total of 123.21. The square root of 123.21 is 11.10, and 11.10 is the weighted average of 11.8 and 10.4 to the decimal. The expression has collapsed into a plain average.

So a weighted average of volatilities is not merely a rough answer. A weighted average is the exact answer at a correlation of one, the single worst case available, and every point of volatility below that line is bought with correlation and with nothing else. Holdings that move together offer no reduction whatever their number. The boundary is useful for exactly that reason: it converts a vague word into a measurable distance from a line that can be drawn.

Now walk the same blend down. At the case correlation of 0.9519 the volatility is 10.97 per cent, 0.13 below the line. At 0.50 it is 9.62 per cent, 1.48 below. At zero it is 7.86 per cent, 3.24 below. Same two holdings, same weights, same returns of 13.40 per cent throughout, and a volatility anywhere between 7.86 and 11.10 per cent, with the correlation the only thing that moved.

The calculator above accepts a correlation below zero because the arithmetic does, and the left hand end of that picture is where the pair very nearly cancels. Nothing in this record sits there. A range the arithmetic allows and a range the record contains are two different statements, and only the second one describes anything that happened.

What does this tool compute, and what does it refuse to compute?

The calculator above takes two returns, two volatilities, two weights, a correlation and a beta, and returns the blend of both. Every field note says which document and which line the number is read from, not what it means. Choosing weights is a decision for a holder and a mandate rather than an output of a formula.

The field notes guard against one particular confusion. The record for this portfolio carries two different kinds of number and they are not interchangeable. The portfolio and benchmark figures above are realised: they are what the stated twelve months actually produced. The holder also has its own stated assumptions for the three asset classes, chosen when the mandate was written, and those are not measurements of any period at all. The chosen assumptions belong to allocation work, not to this arithmetic.

Feed a realised figure into one field and a chosen assumption into the next, and the output is neither a description of what happened nor a statement of what was assumed. Nobody can place a number like that. This guide runs entirely on the realised pair. Where an analyst substitutes other figures, they should all be of one kind, and which kind should be written down on the same sheet.

The discipline runs backwards too. Reading other people's papers is where it earns the most. A report that gives a blended volatility and stays quiet about the correlation need not be accepted as given. The correlation is the only unknown left once the two volatilities and the weights are on the table, so a reported volatility can be turned inside out and made to confess what it assumed.

What is the output, and what is it not?

The volatility computed here is a spread figure for a stated period. The figure says how widely the readings sat around their own mean over those twelve months, and nothing else. A spread figure is not a loss estimate, not a worst case and not a probability of anything, and reading it as any of those three is the most expensive misreading available on this subject.

The loss question is a different question with different machinery behind it, and it is taken up under market and liquidity risk. The figure supports comparison and little else: this blend against that blend, this year against the stated year, on inputs of one kind.

One observed year also does less to a designed mandate than it appears to. The mandate was built to a policy volatility that the assumptions implied, and the stated year came in at 11.8 per cent for the portfolio. One year is one draw. Neither the closeness nor the gap is evidence of skill or of error, and saying that plainly is the finding rather than a hedge.

Who actually uses these two figures, and how?

Rukmini Deshpande chairs the investment committee of the endowment that holds the Anantara Multi-Asset Portfolio. Faiz Ahmad Ansari runs the mandate. When the two of them sit with a quarterly pack, this arithmetic is doing three jobs at once, and none of them is deciding what to hold.

The first job is a check on the pack itself. Any return line that sits outside the range of its own components is an error, and it takes ten seconds to spot. The second is placing the risk number. If a paper reports a volatility, the first question is what correlation it used. At a correlation of one the answer is arithmetic anyone can do in their head, and any figure below that line has to have come from somewhere specific. The third is the one that survives longest. A committee that has understood the covariance term stops asking how many holdings there are and starts asking what the holdings have in common. Only that second version of the question has an arithmetic answer.

A lending officer does the same thing with a borrower. Five income streams from five tenants in one industrial estate are not five streams. A household with two salaries from the same employer is closer to one salary than to two. The formula is identical, the correlation is near one, and the count is not the answer.

The Anantara Multi-Asset Portfolio, the charitable endowment that holds it, Rukmini Deshpande and Faiz Ahmad Ansari are invented.
Educational material. Not advice on any investment, tax, budget or market position.