Portfolio Weighted Averages: What Averages and What Does Not

A weighted average multiplies each weight by each part's figure and adds. Return is linear in the weights, so on the Anantara Multi-Asset Portfolio's stated weights and the holder's own assumed figures the return comes out at 10.05 per cent exactly. Volatility is not: the weighted average of 12.35 per cent sits above the computed 11.20 per cent, and the 1.15 point difference is arithmetic rather than error.

The arithmetic in this tool is three multiplications and an addition. The arithmetic can be done on the back of an envelope, and most people who need it do. The difficulty is not the working but the question of which figures may be put through it. Feed the wrong kind of figure into that working and the answer looks perfectly reasonable, sits in a table beside figures that are right, and is wrong by an amount nobody in the room can see.

Every number here belongs to the Anantara Multi-Asset Portfolio, an invented discretionary mandate of Rs 5,00,00,00,000/- run for a single institutional holder, an invented charitable endowment. The stated policy weights are 60 per cent equity, 30 per cent fixed income and 10 per cent cash. In rupees that is Rs 3,00,00,00,000/-, Rs 1,50,00,00,000/- and Rs 50,00,00,000/-. The expected returns, volatilities and correlations used below are the holder's own stated assumptions, chosen when the mandate was written. The assumptions are not forecasts, not market expectations and not anybody's published estimates. Quietly treating a chosen input as a measured one turns an assumption into a claim about the world, so the label travels with these figures every time they are used. The three sleeves were bought for Rs 2,40,00,00,000/-, Rs 1,46,00,00,000/- and Rs 50,00,00,000/-, so the book cost of the whole portfolio is Rs 4,36,00,00,000/- against a market value of Rs 5,00,00,00,000/-, and that difference is what the calculator uses to show what weighting by the wrong base does.

What does this calculator take in, and what does it hand back?

The calculator takes a list of weights and a list of per part figures, and it returns the sum of weight times figure. That is all. The tool is trivially simple. The entire difficulty lies in knowing which figures may be fed into it, and that is a question about the quantity rather than about the arithmetic. The field notes under each input say where to find that number in a pack, not what it means, because the meaning belongs with the measures themselves.

Two things have to be true before the arithmetic means anything. The weights must sum to one hundred, and every weight must be struck against the same denominator. Fail the first and every output is scaled by a factor nobody wrote down. Fail the second and the sum usually stops being one hundred as well. The broken sum is the only stroke of luck in this subject.

Which portfolio quantities are exact weighted averages of their parts?

One rule decides every case. A quantity is an exact 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 its parts when it is linear in the weightsA quantity is linear in the weights when doubling a part's weight doubles that part's contribution and nothing else changes. No squares, no roots, no products of two weights., and it is not when the parts interact. A portfolio's return is literally the sum of what its parts contributed, so return and beta are linear. Two holdings can move against each other and cancel, and cancelling is an interaction rather than a sum, so volatility is not.

Take an everyday version first. A household with three earners has a household income that is exactly the three salaries added together, whatever the three people do for a living and whatever happens to any one of them. Nothing about the relationship between the three jobs changes the total. Now ask how unpredictable that household income is from month to month. Suddenly the relationship is the whole question. If all three work at the same factory on the same shift, a bad month is a bad month for all three at once. If one drives an auto, one teaches and one runs a shop, a slow month for one is often an ordinary month for the others. The three salaries add the same way in both households. The wobble does not.

Why is a portfolio return exactly the weighted average of its parts?

Because of what a return is. A return is a change in value divided by the value at the start. There is nowhere else for value to come from or go, so the portfolio's change in value is the sum of the changes in value of its parts. Dividing that sum by the portfolio's opening value divides each part's change by the same figure. Dividing by one common figure is the same thing as multiplying each part's own return by that part's share of the total. No step in that has an assumption in it.

Run it on the case. The sleeveOne asset class portion of a portfolio managed as a unit, such as everything held in equity. A convenient word for a slice of the whole. at Rs 3,00,00,00,000/-, on the holder's assumed 12.0 per cent, contributes Rs 36,00,00,000/-. The fixed income sleeve at Rs 1,50,00,00,000/-, on an assumed 7.5 per cent, contributes Rs 11,25,00,000/-. Cash at Rs 50,00,00,000/-, on an assumed 6.0 per cent, contributes Rs 3,00,00,000/-. Added, they come to Rs 50,25,00,000/-. Divided by Rs 5,00,00,00,000/-, that is 10.05 per cent. The short way, 0.6 times 12.0 plus 0.3 times 7.5 plus 0.1 times 6.0, is 7.2 plus 2.25 plus 0.6, and it gives 10.05 per cent again. The two routes agree to every decimal place because they are the same arithmetic written twice, and the word exactly here is not a compliment to the method but a statement about it.

Does the same reasoning carry over to a portfolio beta?

A portfolio beta averages the same way, and the reason is worth spelling out. A portfolio betaHow sensitive a portfolio's return is to the return of a stated benchmark. A beta of 1.08 means a one point benchmark move went with a 1.08 point portfolio move on average over the stated period. is the sensitivity of the portfolio's return to the benchmark's return. The portfolio's return is already established as a weighted sum of the parts' returns. Sensitivity passes straight through a weighted sum: scaling up one part's weight scales that part's response with it, and nothing else in the expression changes. So the portfolio's sensitivity is the weighted sum of the parts' sensitivities. Beta averages exactly for the same structural reason return does. A portfolio beta may therefore be built up from its holdings while a portfolio volatility may not, and those two facts sit uncomfortably close together in the same table.

The composite benchmark is 60 per cent a broad equity index and 40 per cent a broad bond index, both left unnamed. Set the three sleeve betas at 1.65, 0.30 and 0.00 against that benchmark. The three sleeve betas are chosen rather than measured, and they are chosen so the arithmetic lands on the beta of 1.08 the record carries for the stated year. The working runs 0.60 times 1.65 for 0.99, plus 0.30 times 0.30 for 0.09, plus 0.10 times 0.00 for nothing. Total 1.08. One line, no correlations, no square root.

Why is portfolio volatility not the weighted average of its parts?

Because the quantity that adds up is not the volatility but the varianceThe square of the volatility. Squaring is what makes the arithmetic add up, and it is also what stops the volatility itself from adding up., and it adds up in a shape that has products of two weights in it. Portfolio variance is the sum of each weight squared times each part's variance, plus twice every pair of weights times their correlation times their two volatilities. The statistics layer settled that expression, and it is applied here rather than derived again. The shape of that expression is what matters: it has two groups of terms, and only the second group knows that the parts have anything to do with each other.

Work the case. The three squared weight terms come to 116.64, 2.25 and 0.0025, each built in the table below. There is a single covariance termThe part of a variance sum that pairs two different holdings. The term is twice their two weights, times their correlation, times their two volatilities.. The holder takes cash as uncorrelated with both other sleeves, so only the equity and fixed income pair produces one. The cross term is twice 0.6 times 0.3 times 0.20 times 18.0 times 5.0, or 6.48.

Now the comparison the whole calculator exists for. The weighted average of the three volatilities is 0.6 times 18.0 plus 0.3 times 5.0 plus 0.1 times 0.5. The three products are 10.8, 1.5 and 0.05, and they add to 12.35 per cent. The correct working gives 11.20 per cent. The weighted average of volatilities is not an approximation of portfolio volatility but its upper bound. The weighted average is exactly the figure that would arise if every pair of parts moved in lockstep, and nothing moves together more than lockstep. Push the correlationA number between minus one and one saying how closely two series moved together. One means lockstep, zero means no relationship in the readings, minus one means opposite. as high as it will go and the correct figure rises to meet the average from below; it never comes down to it from above.

How large is the gap, and what decides its size?

Push the correlation to its ends and read off what happens. At zero the cross term vanishes altogether, the variance is 118.8925 and the volatility is 10.90 per cent. At one the cross term reaches its maximum of 32.4, the variance is 151.2925 and the volatility is 12.30 per cent. The holder's assumed 0.20 sits between the two at 11.20 per cent, and every setting in between is on the same short climb. The gap is entirely a function of the correlations and the weights, it is widest where large weights sit on parts that move weakly together, and computing it is the only way to say anything concrete about how far the parts fail to move as one.

Now the detail worth catching. The detail shows that the rule is exact rather than roughly right. Even at a correlation of one the portfolio volatility is 12.30 per cent and not the 12.35 per cent of the weighted average. The remaining 0.05 points is there because cash was taken as uncorrelated with the other two sleeves throughout, so one pair in the portfolio is still not moving in lockstep even when the other pair is. The weighted average is reached only when every pair moves together, and the holder's assumptions never let that happen. Upper boundA ceiling a quantity can approach but not pass. An upper bound is not a best guess of the quantity; the true figure is at or below it, and often well below. is the accurate word for what 12.35 per cent is.

How does the choice of base change the answer?

A weight is a ratio, so it has a baseThe denominator a share is struck against. Change the denominator and the same rupee amount reads as a different percentage, without either figure being wrong. underneath it, and moving that base moves the weight. The largest single holding in the Anantara Multi-Asset Portfolio is Rs 23,00,00,000/-. Against the whole portfolio of Rs 5,00,00,00,000/- that is 4.6 per cent. Against the equity sleeve of Rs 3,00,00,00,000/- the very same position is 7.7 per cent, or 7.67 before rounding. Both figures are correct, they answer different questions, and a weight list that mixes the two does not sum to one hundred, at which point every output of this calculator is scaled by a factor nobody wrote down.

The base is not a pedantic point about tidy tables. The mandate's cap on any single holding is written against the portfolio, and 4.6 per cent sits inside it. Read against the sleeve, the same holding is 7.7 per cent. The sleeve reading would breach a 5 per cent sleeve cap, and no sleeve cap exists. The two readings would put the same position on opposite sides of a line. Whoever writes the row heading decides which question the number answers, so the row heading is not decoration.

The base selector above works through the two mistaken bases that actually get used, rather than leaving their size to be taken on trust. The equity sleeve was bought for Rs 2,40,00,00,000/- and now stands at Rs 3,00,00,00,000/-, so weighting the three sleeves by what they cost instead of by what they are worth makes the weights 55.05, 33.49 and 11.47 per cent. The answer falls from 10.05 to 9.81 per cent, or Rs 49,02,52,294/- of return rather than Rs 50,25,00,000/-. Equal weighting is cruder and costs more: three weights of 33.33 per cent give 8.50 per cent, or Rs 42,50,00,000/-, understating the year by Rs 7,75,00,000/-. Both of those lists sum to one hundred. The sum to one hundred catches a base that was mixed and cannot catch a base that was consistently wrong. The base therefore belongs in the row heading rather than in somebody's head. On the report date the market value weights happen to land on 60, 30 and 10 per cent as well, and that agreement is worth checking rather than assuming. The agreement stops holding the moment one sleeve outruns the others.

What does the calculator get wrong if it is fed the wrong quantity?

Nothing, in its own terms, and that is exactly the hazard. The calculator cannot tell a volatility from a return, so handed three volatilities and three weights it returns 12.35 per cent without hesitating. The arithmetic it ran was correct. The quantity it ran that arithmetic on was not one the arithmetic applies to. A simple tool used on a quantity that breaks its assumption fails in exactly this way, and the only defence is an output that names the quantity and says whether averaging it is exact, an upper bound, or meaningless.

The quantity selector on the calculator therefore changes nothing in the arithmetic and everything in the label. The ratio setting is the one worth dwelling on. Averaging the parts' ratios gives a figure that is not a ratio of anything, so a ratio of two portfolio quantities has to be rebuilt at the portfolio level from its own numerator and denominator.

What does a monitoring pack do with these two figures?

Rukmini Deshpande chairs the investment committee of the endowment that holds the Anantara Multi-Asset Portfolio, and Faiz Ahmad Ansari runs the mandate. The pack that reaches her committee carries a risk section, and her first job with it is not to read the risk figure but to work out how it was made. There are only two possibilities and they are 1.15 points apart on these assumptions.

The check she can run in her head is the one built up here. Any figure in the risk section that is a plain weighted average of sleeve figures is either exact or a ceiling, and which of the two it is depends only on the row heading. A return line, a beta line and any rupee line she can verify by weighting. A volatility line, or anything with a volatility underneath it, she cannot, and if the pack does not show the working she has to ask for it. A lending officer looks at a borrower the same way: five tenants paying rent into one building give a rent roll that adds up exactly, but how reliable that rent roll is depends on whether the five tenants are five unrelated businesses or five outlets of the same chain, and the addition says nothing about which.

The fix Faiz Ahmad Ansari can make to the pack is small and mechanical. Label every quantity in it as linear in the weights or not. Compute the ones that are not from their full working. Then print the weighted average beside the computed figure rather than instead of it, so the gap between 12.35 and 11.20 per cent appears in the pack as a number the committee can discuss instead of a difference nobody knows is there. Printing both figures costs one column and removes the entire class of error.

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.