Risk Aversion: Why Two Investors Choose Differently

Risk aversion is how much extra return a holder demands before accepting an extra unit of variability. Risk aversion is a preference rather than a mistake, so two holders reading identical figures can reach opposite choices without either being wrong. Written as a number, it fixes the point at which a higher returning, more variable portfolio stops being the preferred one.

Every quantity so far has belonged to the portfolio. The ratios belonged to it, the return and risk arithmetic belonged to it, and the assumptions feeding both belonged to the mandate that wrote them down. Risk aversion is the first quantity that belongs to somebody else entirely: the holder. Two holders can be handed the same sheet of figures, agree on every digit, and walk out of the room wanting different things. Nothing has to be wrong with either of them for that to happen.

The running record 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. For one stated twelve month period the record locks four numbers that this whole guide turns on: the portfolio returned 14.2 per cent with a volatility of 11.8 per cent, and the unnamed composite benchmark returned 12.6 per cent with a volatility of 10.4 per cent. The offer is a gross excess return of plus 1.6 percentage points bought with plus 1.4 points of extra volatility, and nothing in those four numbers says what would make the trade worth taking.

One property of volatility matters before the arithmetic starts. Deviations are squared, so volatility is symmetric: a rise and an equal fall enter the calculation identically. Everything below prices variability, not the risk of loss, and the two are not the same thing.

What is risk aversion, and is it a mistake?

Risk aversionThe extra reward somebody wants before they will accept a more uncertain outcome instead of a steadier one. Risk aversion describes what a person wants, not what they know. is the extra return a holder wants before accepting more variability. The definition is that short, and its shortness hides how often risk aversion is misread. Risk aversion is not fear, not innumeracy, and not a failure to understand what the figures say. A preference is a statement about what somebody wants, sitting alongside, and quite separate from, what they believe.

Economics treats it as a primitive. The word primitive carries weight: a primitive is something a model starts from rather than something it explains or corrects. Beliefs can be wrong. The world eventually says so. A preferenceA ranking of outcomes that somebody holds. Preferences can be described and compared. Unlike a forecast, they cannot be checked against the world and found false. cannot be wrong in that sense, because there is no later observation that contradicts it. An account describing a cautious holder as making a mistake has confused a preference with a defect, and that confusion is where most poor writing on this subject begins.

Here is the everyday version. Two people are offered the same job: a fixed salary of Rs 90,000/- a month, or a commission arrangement averaging Rs 1,10,000/- a month that swings between Rs 40,000/- and Rs 1,80,000/-. One takes the fixed salary. Nobody thinks that person cannot do arithmetic. The person taking the salary has looked at the same two numbers, agreed the average is higher on the commission, and decided that the swing is not worth Rs 20,000/- a month. Risk aversion is exactly that, stated in the only terms that matter: a price.

How does a preference become a number without anyone measuring it?

One step is usually skipped, and skipping it is how a perfectly ordinary parameter gets mistaken for a reading off an instrument. Nobody measures risk aversion the way a thermometer measures temperature. The real procedure is quieter and much less impressive. A form is chosen, that form contains a penalty for variability, and the size of the penalty is written as a coefficientA number that sets how strongly one term in a chosen expression pulls on the result. A coefficient belongs to the expression it was written into, not to the thing being described..

The form that runs everywhere in portfolio construction subtracts a multiple of varianceThe squared spread of a set of readings around their own mean. Volatility is its square root, so variance always moves faster than volatility does. from expected return. Harry Markowitz, in Portfolio Selection, 1952, is where the return against variance trade-off enters the subject in this shape, and the whole of mean variance construction rests on it. The coefficient is simply how much of the variance gets subtracted.

The uncomfortable consequence is that the coefficient is a parameter of the chosen form and not a property of the person, so the same holder written into a different form carries a different number. A form that penalises variance and a form that penalises the square root of variance are both perfectly respectable, and they will not agree on the digit. Neither form is flawed for that. A coefficient quoted without its form is not a fact about anybody. Such a coefficient is a digit that has lost the sentence it belonged to.

How much extra return does extra variability actually cost?

Now the arithmetic, and it is one line. Under the variance penalising form, the extra return required to accept a more variable holding is half the coefficient multiplied by the increase in variance. Written with the coefficient as A and the increase in variance as the difference between the two squared volatilities, that sentence is the whole mechanism. Everything else follows from that one line.

In the record, the portfolio's volatility of 11.8 per cent squares to a variance of 139.24, and the benchmark's 10.4 per cent squares to 108.16. The difference is 31.08 in those squared units. Held as decimals, the same two figures give 0.003108. Halving it gives 0.01554 per cent squared, and multiplying by the coefficient turns it into the risk premium demandedThe amount of extra expected return somebody wants before they will accept a more variable outcome. The premium demanded is a demand, not an observation of what any market pays., expressed in percentage points. At a coefficient of 1 the demand is 0.1554 points. At 4 it is 0.6216. At 8 it is 1.2432. At 12 it is 1.8648.

Notice what the requirement scales with: variance, not volatility, and that single fact makes variability far more expensive than most people expect. Volatility is the number everybody quotes because it is in the same units as return, but the demand is built on its square. Take the benchmark's 10.4 per cent and double it to 20.8 per cent. The variance goes from 108.16 to 432.64, four times as much rather than twice. At a coefficient of 4 the required extra return goes from nothing to 6.4896 percentage points. Doubling the movement did not double the price. The price quadrupled.

Why can two households agree about a market and still want different things?

Set beside the arithmetic, the human version arrives at the same place by a much shorter route. Two households in the same street save for the same purpose over the same period, both reading exactly the same sheet of figures and agreeing on every one of them.

The first household runs on one salary. If the year goes badly, the shortfall has to come out of savings that also stand behind the rent, the school fees and the medical cover, and there is no second income to lean on while the market takes its time recovering. The second household runs on two salaries in unrelated lines of work. A bad year is unpleasant and it is absorbed; the standing costs are still covered while the recovery happens.

Now ask them the same question. Both households believe the more variable holding will average more over the period. Both are right about that belief, or wrong about it, together. And yet one of them wants the steadier holding and the other does not. The two households differ in preference and in circumstance, not in information, and arguing harder about the market will never resolve a difference of that kind. Both households could be sent the same research pack for ten years and would still want different things at the end of it, because the pack was never the thing they disagreed about.

Where exactly do two holders part company?

The mechanical result is unusually clean, and worth stating carefully. The extra return on offer is fixed by the record: 1.6 percentage points gross, and no amount of preference moves it. The increase in variance is fixed too, and the coefficient simply multiplies half of it, so the extra return demanded rises in a straight line with the coefficient. One quantity stands still while the other climbs. The two quantities must therefore cross at exactly one coefficient, and that crossing is the entire explanation of why identical figures produce opposite choices.

Solve for it. The demand equals half the coefficient times 0.003108, and the offer is 0.016 in the same decimal units. Divide 0.016 by half of 0.003108 and the answer is 10.2960. Below that coefficient the demand is smaller than the offer, so the more variable portfolio ranks ahead. Above it the demand is larger, so the steadier benchmark ranks ahead. At the crossover pointThe value of a parameter at which two options that were ranked one way start being ranked the other way. Everything about the options themselves stays exactly the same across it. itself the two are level and nothing separates them.

Put two constructed readings against it. A holder written at a coefficient of 4 demands 0.6216 points and is offered 1.6, so the portfolio ranks ahead by 0.9784 points. On the Rs 500 crore mandate base that is Rs 4,89,20,000/- of gross margin over what was demanded. A holder written at 12 demands 1.8648 against the same 1.6, falls short by 0.2648 points, and ranks the benchmark ahead by Rs 1,32,40,000/- on the same base. Identical returns, identical volatilities, identical period, opposite conclusions.

There is a second finding here, and it is less comfortable than the first. A coefficient above 10 is a high aversion by any conventional reading of this form. On these particular figures, therefore, the crossing sits a long way out, and almost any ordinary setting of the dial ranks the portfolio ahead. Saying that plainly is more useful than pretending the case is finely balanced when the arithmetic says it is not. The crossing is what makes disagreement possible; it does not make disagreement likely at every set of figures, and the distance to it is itself a fact worth reporting.

What has the endowment's own record actually written down?

Every coefficient used so far was constructed, and the reason lies in the record. The record for this endowment locks no risk aversion coefficient at all, and a missing coefficient is reported as missing rather than supplied. The 4 and the 12 above are settings of a dial, chosen to show what the dial does. Neither is a reading taken from Rukmini Deshpande's committee.

The record does carry the place where a preference of that kind actually gets written down: the mandate. The Anantara Multi-Asset Portfolio is a Rs 500 crore mandate. Its equity band runs from 50 to 70 per cent, or Rs 250 crore to Rs 350 crore on that base, with the policy weightsThe share of a portfolio the holder decided in advance that each asset class should carry. Actual weights drift away from them as prices move between one rebalancing and the next. placing equity at 60.0 per cent, or Rs 300 crore. No single holding may exceed 5 per cent of the portfolio, a cap of Rs 25,00,00,000/-. There are no unlisted holdings, and the fixed income sleeve carries a minimum credit standing stated as a policy rather than as a rating symbol.

Read those four lines again and notice what they are. The four lines are not forecasts and not estimates. Nobody can check them against a later year and find them false. The limits are a committee stating, in advance and in writing, how much variability it is willing to have arranged on its behalf. A written limit is a preference, recorded in the only form an institution has: a constraint that binds the person running the mandate.

Does the corridor price variance the same way the stated year did?

Here is a computation the record permits and that the earlier return and risk arithmetic never ran. The mandate's own assumptions put equity at an expected return of 12.0 per cent with 18.0 per cent volatility, fixed income at 7.5 and 5.0, cash at 6.0 and 0.5, with a correlation of 0.20 between equity and fixed income and cash taken as uncorrelated. The assumptions are the holder's own choice, not forecasts, and a different set gives a different answer.

The record locks the weights at the policy point but says nothing about how fixed income and cash are set when equity sits at either edge of its band. So the two edge portfolios below are constructed: cash is held at its policy 10.0 per cent and fixed income takes the rest. A constructed weight set that goes unlabelled becomes a recorded one within about two readings.

At the policy point the variance is 125.3725, a volatility of 11.1970 per cent, the 11.20 the record carries, on an expected return of 10.05 per cent. At the lower edge, 50 per cent equity and 40 per cent fixed income, the variance is 92.2025 and the volatility 9.6022 per cent, on 9.60 per cent expected. At the upper edge, 70 and 20, the variance is 164.8025 and the volatility 12.8375 per cent, on 10.50 per cent. The band the committee wrote is, in volatility terms, a span of 3.2353 points, and that span is the clearest statement of accepted variability anywhere in the record.

Running that arithmetic across the span, something useful falls out. Moving from the lower edge to the upper edge offers 10.50 less 9.60, or 0.900 percentage points of extra expected return. The move costs 164.8025 less 92.2025, or 72.6000 in extra variance. Half of that, divided by a hundred to reach percentage points, is 0.3630 points of demand per unit of coefficient. Dividing the 0.900 on offer by 0.3630 puts the crossing at 2.4793.

Each comparison sets its own price. The same holder can therefore sit on opposite sides of two different crossings, and the two crossings have nothing to do with each other. A constructed setting of 4 sits below the stated year crossing of 10.2960, so it ranks the portfolio ahead of the benchmark. A setting of 4 also sits above the corridor crossing of 2.4793, and on the mandate's own assumptions it ranks the lower edge ahead of the upper one. Neither reading contradicts the other. The stated year offered 1.6 points for 31.08 of extra variance, a generous price. The corridor offers 0.900 points for 72.60, a much thinner one.

Where does risk aversion come from, and does it move?

Risk aversion comes from circumstances, obligations and experience, in roughly that order of durability. An endowment with a committed annual disbursement is placed differently from one without. A household with a fixed rent and a single income is placed differently from one with two incomes and a paid-off home. Underneath both sits whatever the holder has actually lived through. Nobody controls that part, and that part moves.

Here is the specific and awkward finding, stated plainly and without moralising. Measured risk aversion tends to read lower after a strong period and higher after a weak one. A preference recorded immediately after either is recorded at its least reliable moment. The same questionnaire, the same person, the same wording, three months after a sharp fall against three months after a strong run: the score moves, and the person did not.

The drift is not a reason to distrust the exercise. The drift is a reason to record the exercise properly. A score that travels with its date and the conditions it was taken in is a usable observation. A bare score with no date is treated within a year as a fixed property of somebody it never described, and the mandate built on it inherits a moment rather than a preference.

What does risk aversion never tell anybody?

Two things fall outside anything a coefficient can say. The first is how much variability the holder's situation can actually absorb. Capacity is a question about obligations, timing and what happens if a bad year arrives at the wrong moment, and it has a separate answer arrived at by a separate route. Willingness and capacity can point the same way or opposite ways, and when they point opposite ways the arithmetic here has nothing to say about which one should govern. Willingness against capacity is covered separately, and the two make a comparison precisely because they are not the same quantity.

The second is whether any portfolio is well built. A coefficient prices variability. A coefficient says nothing about whether the variability was necessary, whether the exposure was concentrated in one shared driver, whether the fees ate the difference, or whether the result came from anything the holder was paying for. All of those are properties of the construction and the monitoring, and the arithmetic here reaches none of them.

Which leaves one boundary: an account that slides from describing a preference to telling a reader what theirs is, or what to do about it, has stopped teaching and started advising.

The easiest confusion available on this subject sits in one gap worth naming explicitly. The mandate's assumptions put the weighted average of the three volatilities at 12.35 per cent while the policy portfolio itself computes to 11.20, a difference of 1.15 points. Correlation of 0.20 rather than 1.00 between equity and fixed income produces those 1.15 points, and they would appear at any coefficient whatsoever. Preference decides where inside the band the holder sits. Correlation decides what the holder gets for sitting there. The correlation arithmetic is set out in full under the assumptions feeding a portfolio.

How does a committee use any of this on a Tuesday?

The first surprise is that nobody computes a coefficient. In an actual room the arithmetic here does three jobs, and none of them involves anybody stating a number for themselves.

The first job is pricing a proposal before anyone argues about it. When Faiz Ahmad Ansari brings the committee a proposal that moves equity towards the upper edge of its band, the honest way to present it is as an offer: this much extra expected return, this much extra variance, and here is the coefficient at which the two become level. On the mandate's own assumptions that figure is 2.4793 for the full width of the band. Rukmini Deshpande's committee then has something specific to discuss instead of the words more or less aggressive, and those two words never resolve anything.

The second job is catching a mismatch between what a mandate says and what a proposal implies. If a proposal only makes sense at a setting far outside anything the mandate's written limits suggest, that is a flag before it is a disagreement. The limits are already on paper. The proposal has to live inside them.

The third job belongs to a lender or an analyst reading somebody else's mandate from outside. The corridor, the cap and the listing requirement are public facts about the arrangement, and they bracket accepted variability without anybody having to volunteer a preference. Computing the band's volatility span, as done above to get 3.2353 points, turns four lines of legal text into one comparable number. A written limit is a preference somebody has already stated, and reading it as one costs nothing and takes a single calculation.

The household version of all three is the same exercise done with a pen. The household writes down what a bad year would actually reach: the rent, the fees, the cover. Beside that goes what the steadier option gives up. The question is then whether the gap between them is worth the reach. Nobody needs a coefficient to run that; the coefficient is only what emerges when the answer is insisted on as a digit.