The Utility Function: Ranking Return Against Its Risk

A utility function turns an expected return and a risk figure into one number. Two portfolios can then be put in order. The common form subtracts a penalty for variance from the expected return, and the size of that penalty is set by a chosen coefficient. Change that one coefficient and the order of two fixed portfolios can reverse.

The problem the device exists to solve is a genuinely awkward one. Consider two sets of figures. One thing returned more and moved about more. The other returned less and moved about less. Neither is better on both counts, so neither wins outright, and no amount of staring at four numbers will produce a third number that settles them. Something has to decide how much steadiness is worth. A utility functionA rule that turns several figures about one option into a single number, so that options can be sorted. The number is a sorting key, not a measurement of anything real. is the written-down answer to that question, and its only job is to produce an order.

The running example is the Anantara Multi-Asset Portfolio, an invented discretionary mandate of Rs 500 crore run by Faiz Ahmad Ansari for a charitable endowment whose investment committee is chaired by Rukmini Deshpande. The mandate is compared against an unnamed composite benchmark of 60 per cent a broad equity index and 40 per cent a broad bond index. For one stated twelve month period the Anantara Multi-Asset Portfolio returned 14.2 per cent with volatility of 11.8 per cent, and the composite benchmark returned 12.6 per cent with volatility of 10.4 per cent. All four of those figures belong to that single stated period, and all four are gross of the costs of running the mandate.

Volatility here is the ordinary two-sided measure: deviations are squared, so a rise and an equal fall count the same. The two-sided treatment is established under the measurement of volatility and carried forward rather than argued again. It matters twice below, so it is worth having in view from the start. The mean, the standard deviation and the variance are taken as already known, and how each is built is covered separately.

What does a utility function do that a return figure cannot?

A utility function settles ties that a return figure leaves open. A return figure answers exactly one question. How much came back. The return figure has nothing to say about how uncomfortable the year was, and when one candidate wins on return and loses on steadiness it simply refuses to arbitrate. The refusal is honest, and it is also useless if a decision has to be taken on Tuesday.

Think about how a household actually chooses between two jobs. One pays Rs 90,000/- a month, reliably, every month. The other pays Rs 1,20,000/- a month on average, but three months of the year it pays nothing because the work is seasonal. Nobody can rank those two by pay alone. The household ranks them anyway, and the way it does it is to ask what the steadier one is worth in rupees. The steadiness question, asked with numbers instead of feelings, is exactly what a utility function asks.

Now the narrow part, and it is narrow on purpose. The function is a ranking device and nothing more: it does not measure satisfaction, it does not say how much better one option is in any meaningful unit, and only the order it produces should be carried out of the room. Everything that goes wrong with this tool downstream goes wrong because somebody read the output as a quantity rather than as a position in a queue. The output has a decimal point and a per cent sign. Reading it as a quantity is almost irresistible, and it is still wrong.

What does the common form actually say?

The form used almost everywhere in portfolio work is one subtraction. The subtraction starts with the expected return and takes away a penalty, and the penalty is half of a chosen coefficient multiplied by the variance. Returns here are stated in percentage points. The Anantara portfolio's variance is 11.8 squared divided by 100, or 1.3924, and the composite benchmark's is 10.4 squared divided by 100, or 1.0816.

For the Anantara portfolio at a constructed coefficient of 4, the penalty is 0.5 times 4 times 1.3924, or 2.7848 percentage points. Subtracted from 14.2, that leaves 11.4152 per cent. The composite benchmark runs the same way: 0.5 times 4 times 1.0816 is 2.1632, and 12.6 less 2.1632 is 10.4368 per cent. Two portfolios, two numbers, one order. The machine is nothing more than that.

The coefficient is the only free part. The risk aversion coefficientThe single dial in this form. A larger value makes the subtraction for variability bigger. Somebody chooses the value when writing the rule down. No measurement of anybody produces it. is a parameter of a chosen form rather than a property of a person, a point established under risk aversion as a preference and taken as given here, so what follows works on what the parameter does. What is not in the subtraction matters as much: the risk-free rate, which was 6.5 per cent for the stated year, appears nowhere in this form. A ratio such as return over volatility cannot be read without it. The utility form can. The difference between the two tools is real, and it is worth knowing before either is reached for.

Why does the penalty attach to variance rather than to volatility?

Because the form is written that way, and the consequence is large enough that it deserves its own paragraph. Variance is volatility squared. So when volatility doubles, the penalty does not double: it goes up roughly fourfold. Hold the coefficient at a constructed 4 and work three cases. At a volatility of 5.9 per cent the penalty is 0.6962 points. At 11.8 per cent, twice as much, it is 2.7848 points, four times as large. At 23.6 per cent it is 11.1392 points, sixteen times the first.

The squaring is why this form treats a very volatile candidate harshly and a mildly volatile one almost kindly. The penalty grows faster than the risk does. The more volatile of two candidates loses ground quickly as the coefficient rises, and the gap between two candidates narrows from the very first step. It is a design choice with a real cost, because it means the ranking is far more sensitive at the volatile end than at the steady end. Two steady candidates will hold their order across a wide sweep of coefficients. A steady one and a wild one will swap far sooner than intuition suggests.

The street version is a stall that sells umbrellas. In a mild year the takings wobble a little and the stallholder barely notices. In a monsoon year they swing wildly, and the same stallholder starts turning down orders she would once have accepted, not because her average takings fell but because the swing itself became the thing she was paying attention to. The variance penaltyThe amount subtracted from an expected return in this form. Half the chosen coefficient multiplied by the variance gives the amount. Doubling the variability quadruples it. is that shift, written down as arithmetic.

What is a certainty equivalent, and why is it the readable output?

Consider what came out of the subtraction. A return in percentage points, less a penalty in percentage points, leaves a figure in percentage points. The matching units are not a coincidence. They are the reason this particular form survives despite everything wrong with it. The output lands in units a person can think in.

So it gets a name and a reading. A certainty equivalentThe steady, known return that would be treated as equally attractive to a risky one under a stated rule. The subtraction that produces it is done in per cent, so the figure arrives in per cent. of 11.4152 per cent is read as the steady, certain return that would rank equally with the Anantara portfolio's risky 14.2 per cent, under a constructed coefficient of 4, for the stated twelve month period. The risky 14.2 and the steady 11.4152 sit level. The 2.7848 points between them is what the variability costs under this rule.

The readability is why practitioners tolerate a form they can otherwise pick apart. A ranking key that arrives in per cent can be discussed in a meeting by people who have never seen the formula. Readability is the whole of its advantage. It is also the trap, because a number in per cent invites subtraction, and subtracting two certainty equivalents produces something that looks like a return difference and is not one. The subtraction failure has its own block below, and it is the most common thing that goes wrong with this tool anywhere.

What happens to the ranking as the coefficient rises?

Both penalties grow, so both certainty equivalents fall. The direction is obvious. The part that matters is that the two readings do not fall at the same speed. Per one unit of coefficient, the Anantara portfolio gives up half of 1.3924, or 0.6962 points. The composite benchmark gives up half of 1.0816, or 0.5408 points. The difference between those two rates is 0.1554 points, and that is the rate at which the gap between the two closes.

Watch it happen. At a constructed coefficient of 2 the readings are 12.8076 and 11.5184, so the portfolio leads by 1.2892 points. At 4 they are 11.4152 and 10.4368, a lead of 0.9784. At 8 they are 8.6304 and 8.2736, a lead of 0.3568. Not one input has changed anywhere in that sequence, and the lead has shrunk to a quarter of what it was. The lead was never a property of the two portfolios on their own.

The everyday shape is a race between two people where one is carrying a heavier load. On flat ground the stronger one wins easily. Put the two on a slope and the heavier load starts to matter. Steepen the slope and at some point the load decides the race. The coefficient is the slope. Nothing about either runner changed.

Where exactly do the two orders swap?

At the coefficient where the two certainty equivalents are equal, and that is a division rather than a search. Set the two expressions equal and the coefficient falls out: the return difference divided by half the variance difference. The return difference is 1.6 points gross of costs. The variance difference is 1.3924 less 1.0816, or 0.3108, and half of that is 0.1554. Divide 1.6 by 0.1554 and the answer is 10.2960.

Check it by substitution rather than trusting the algebra. At 10.2960 the Anantara portfolio reads 14.2 less 0.5 times 10.2960 times 1.3924, or 7.0319 per cent. The composite benchmark reads 12.6 less 0.5 times 10.2960 times 1.0816, also 7.0319 per cent. They meet. Below that value the portfolio is ranked first, above it the benchmark is, and at 20 the readings are 0.2760 and 1.7840, so the benchmark leads by 1.5080 points.

Now the property worth carrying away. The crossover coefficientThe single value of the coefficient at which two candidates produce the same ranking key. The value comes from the two candidates alone, and can be worked out before anybody states a preference. is a fact about the pair of candidates and not about any investor, so it can be computed before anybody's preference is known or stated. Nothing in the division used a preference. The division used two returns and two variances, all four locked by the record, and it produced a number that states exactly how much aversion it would take to flip the order. The same value is reached from the other direction under risk aversion as a preference, by asking what extra return the variability has to buy; here it is reached by asking where two rankings meet. Same pair, same crossing, two different questions.

Does landing near the crossing settle anything?

Less than might be hoped, and this is where stopping at the crossing would leave the analyst exposed. Near the crossing the two readings are so close that the order is decided by figures nobody would defend to four decimal places. At a constructed coefficient of 10 the Anantara portfolio reads 7.2380 and the composite benchmark reads 7.1920, a lead of 0.0460 points. At 11 they read 6.5418 and 6.6512, so the benchmark leads by 0.1094. One unit of a made-up parameter moved the answer from one candidate to the other.

An order produced inside that zone is not wrong, it is simply not robust, and the honest way to report it is to say that the crossing sits at 10.2960 and that the inputs are too close to separate the two candidates near it. A committee that hears the crossing value can see the width of the doubt for itself. A committee that hears only that one portfolio scored higher cannot.

What does this form assume, and where does that assumption fail?

The form assumes that variability is variability, whichever way it points. Variance squares the deviations, so a surprise of plus 11.8 and a surprise of minus 11.8 both contribute 139.24, and the penalty cannot tell them apart. Anyone who has actually held something knows those two experiences are not the same experience. The quadratic formThe name for a rule whose penalty is built on a squared quantity. Squaring is what makes an upward and a downward deviation of the same size count identically. makes the arithmetic tidy and pays for that tidiness here.

The symmetry itself is established in an earlier layer of the material and is carried forward here. The consequence is the part that belongs here. Because the penalty is symmetric, this form charges a candidate for its good surprises, and that is the known limitation of the whole construction. It is also precisely why alternative measures exist, ones built on downside deviation or on a threshold below which movement counts and above which it does not. Downside measures are covered separately.

Two smaller assumptions are worth naming. The form takes the expected return and the variance as given, so all the uncertainty in the estimates disappears the moment they are typed in. And it runs over one horizon, so a ranking of two candidates over one stated twelve month period says nothing about the same two over any other length of time.

What happens to the crossing once costs are counted?

The crossing moves down, and it moves down fast. A gross lead and a net lead are different claims about the same year, and everything so far has used gross figures, so the 1.6 point lead is a gross lead. Costs come out of the portfolio's return and leave the benchmark's alone, so any cost drag shrinks the return difference while leaving the variance difference untouched. The crossing is the return difference over 0.1554, so it shrinks in exact proportion.

Work the sensitivity rather than asserting it. A cost dragThe amount by which the costs of running a mandate reduce its return before the holder sees it. The drag lowers the return figure and leaves the variability figure alone. of 0.1 percentage points cuts the gross lead of 1.6 to 1.5, and 1.5 divided by 0.1554 is 9.6525. So each 0.1 point of drag pulls the crossing down by 0.6435. At a drag of 0.5 points the crossing falls to 7.0785, and at 1.0 points to 3.8610. At a drag equal to the full 1.6 point gross lead the crossing reaches zero. The benchmark is then ranked first at every positive coefficient there is.

Nobody has recorded what the drag actually was for this mandate in the stated year. The record locks the four gross figures and supplies no cost figure, so the honest output is the sensitivity above and a blank where the answer would go. The blank is not a gap in the teaching. The blank is the teaching: a certainty equivalent computed on gross figures ranks gross figures, and nobody holds a gross figure.

What will this form accept that a reader must not?

Anything shaped like a return and anything shaped like a variance. The arithmetic has no way of asking where the two numbers came from, and that indifference is the most dangerous thing about it in ordinary use.

Here is the trap with real figures. The Anantara mandate carries its own stated assumptions, chosen by the holder for planning rather than forecast by anybody: equity at 12.0 per cent expected with 18.0 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. At the policy weights of 60, 30 and 10 those assumptions give an expected return of 10.05 per cent and a volatility of 11.20 per cent. Feed that pair into the same form at a constructed coefficient of 4 and it returns 7.5426 per cent, obediently and without complaint.

Now put 7.5426 beside the 11.4152 computed from the stated year. One reading comes from a planning assumption about an unknown future and the other from a single realised twelve month period. The two look comparable and are not, and the form cannot see the difference. Ranking them against each other would be a category error dressed as arithmetic. The rule that keeps a comparison honest is simple to state and easy to skip: two readings may be compared only when both inputs answer the same question about the same period.

What does the ranking not settle?

Almost everything a decision actually needs. The ranking settles the order of two candidates, for one coefficient, over one stated period, on the inputs it was handed. Four separate limitations are hiding in that sentence, and each one has bitten somebody. Read it back slowly.

The coefficient was chosen rather than measured, so the ranking does not say the coefficient was the right one. The ranking does not say the holder's situation could absorb the outcome it cheerfully accepted. Whether a situation can absorb an outcome is a different question, and it is covered separately. A variance estimated from a short run is a shaky thing and the form treats it as a fact, so the ranking does not say the inputs were reliable. And it does not say anything about what anybody should hold. An order is an order, and turning an order into an instruction requires a whole apparatus of situation, constraint and obligation that this arithmetic never touched.

There is one more limit that is specific to a mandate. The Anantara mandate permits equity between 50 and 70 per cent of the Rs 500 crore total, a corridor of Rs 250 crore to Rs 350 crore. The policy weight of 60 per cent sits at Rs 300 crore inside it. A ranking device does not know that corridor exists. If a candidate scoring highest sat outside the permitted band, it would still score highest, and it would still be inadmissible. The constraint comes first and the ranking runs inside it, never the other way round.

How does a committee actually use this on a Tuesday?

Not by picking a coefficient and announcing a winner. The useful move is the opposite one, and it takes about four minutes with a spreadsheet. The crossing is computed first, from the two candidates alone, and the ladder of readings goes in front of the room before anybody has said what they prefer.

So the paper that reaches Rukmini Deshpande's investment committee says this. The two candidates order one way below a coefficient of 10.2960 and the other way above it. Here are the readings at a spread of constructed values, so the room can see how fast the lead is shrinking. The crossing is a property of these two candidates and would be the same whoever was sitting at the table. And the mandate's own coefficient is not recorded anywhere, so no reading in this paper belongs to this holder.

Putting the crossing first converts an argument about preferences into a question about one number, and a question about one number can be discussed without anybody having to claim they know their own aversion to a decimal place. A lender does the same thing when it asks how far a borrower's earnings would have to fall before a covenant breaks, rather than asking whether the borrower feels risky. An analyst does it when reporting the growth rate at which two valuations meet rather than defending a single growth assumption. The pattern is identical: locate the switching point, then talk about where the case stands relative to it.