The Efficient Frontier: The Best Trade-Off, in Theory

The efficient frontier is the set of portfolios offering the highest expected return available at each level of expected volatility. Anything below it is dominated. Some other combination offers more return for the same risk. The curve is drawn entirely from the expected inputs, so it is a picture of the assumptions rather than a picture of any market.

The running example is the Anantara Multi-Asset Portfolio, an invented Rs 500 crore discretionary mandate run by Faiz Ahmad Ansari for an invented charitable endowment whose committee is chaired by Rukmini Deshpande. Its policy shape is equity 60.0 per cent at Rs 300 crore, fixed income 30.0 at Rs 150 crore and cash 10.0 at Rs 50 crore, and the mandate keeps equity between 50 and 70 per cent. The holder also wrote down its own input setThe assumed numbers a curve is computed from: an expected return and an expected volatility for each holding, plus a correlation for each pair.: equity at 12.0 per cent expected return and 18.0 per cent volatility, fixed income at 7.5 and 5.0, cash at 6.0 and 0.5, with an equity to fixed income correlationA number between minus one and one saying how closely two things have moved together. One means in step, nought unrelated, minus one opposite. of 0.20 and cash taken as unrelated to both. The nine numbers are assumptions the holder chose, not forecasts, and a different nine would draw a different curve.

What is the efficient frontier actually a boundary of?

The set comes before the curve. Plotting every combination a mandate is able to build produces a region rather than a line: the achievable setEvery combination the available holdings can make, plotted by expected return and expected volatility.. At a fixed volatility, say 11.20 per cent, a line running straight up the region meets its top edge. The highest point on that line is the best return available at that volatility. Repeated at every volatility and joined up, those points trace the frontier.

The frontier is a boundary of what is available given one input set, and calling any point on it optimal without naming that input set is where almost all of the misuse of this picture begins. Harry Markowitz, in a 1952 paper called Portfolio Selection, is where this boundary enters the subject, and what mattered was not the drawing but the argument underneath it: what a holding does to a portfolio depends on how it moves with everything already there.

The same idea without any arithmetic. At a Sunday vegetable market the best quality on offer at each price traces a boundary, and everybody knows that boundary is a fact about which stalls turned up rather than a law about vegetables. The same reader forgets exactly that in front of a chart.

Notice what is not in that figure. No history, no price series, no measurement of anything: the picture has the finish of a measurement and the content of an assumption.

What does it mean for a portfolio to be dominated?

A combination is dominatedBeaten by another available combination on both counts at once: at least as much expected return and no more expected volatility, with at least one of the two strictly better. when another available combination offers more expected return at the same expected volatility, or the same expected return at lower expected volatility. Being better on one count and worse on the other is not enough. The two combinations are then simply trading, and no ranking has been established.

Score the policy shape against a curve built from the holder's own assumptions. At 60, 30 and 10 the expected return computes as 0.60 times 12.0 plus 0.30 times 7.5 plus 0.10 times 6.0. The sum is 7.20 plus 2.25 plus 0.60, or 10.05 per cent. The variance is 116.6400 from equity, 2.2500 from fixed income, 0.0025 from cash and 6.4800 from the equity to fixed income cross term. The four sum to 125.3725, and the square root of 125.3725 is 11.20 per cent. Hold that volatility fixed and ask what an equity and fixed income pair alone could return at it: the equity weight that produces a volatility of 11.20 is 58.9 per cent, and it returns 10.15 per cent. Under the holder's own stated assumptions the policy shape is beaten by 0.10 points.

Two cautions. A beaten shape is not automatically a wrong shape. The assumption set carries no liquidity requirement, no settlement need and no view of what the cash sleeve is for, so a mix beaten on two axes may be doing work neither axis records. And dominance is a comparison inside one set of assumptions, so a combination beaten under one input set can sit on the boundary under another.

The third point in that figure is the one people skip: neither it nor the policy shape beats the other. Two combinations that cannot be ranked are the ordinary case rather than the exception, and that is why dominance clears away far less than a frontier chart suggests.

How is the curve actually traced out?

One point at a time, and the method is duller than the picture. A level of expected return is fixed, the optimiser returns the combination that reaches it with the least variance, the result is recorded, then the target moves and the run repeats. How one such run works, and where it is fragile, is settled under mean variance optimisation. The curve is what two hundred of those runs look like joined.

Read the third row across: a target of 10.0 per cent forces an equity weight of 55.6 per cent, at a volatility of 10.67 per cent. Nothing about that row knows about any other. Every point on the curve is one full run of the optimiser, so the input sensitivity established for a single run belongs to the whole curve rather than to some awkward corner of it. Two hundred points look like a lot of evidence. One assumed input set has simply been used two hundred times.

Nothing was computed on the dashed stretches. Joining the recorded points assumes a run taken between two of them would land between them. The assumption holds for this arithmetic and for nothing outside it.

Why is the frontier a curve rather than a straight line?

Because expected return is a weighted average and expected volatility is not. Mix the equity and fixed income assumptions in any proportion and the expected return moves in a straight line from 7.50 to 12.00: half and half returns 9.75, exactly halfway. Nothing bows.

Now do the same to the volatility. If the two moved exactly together, a half and half mix would carry the weighted average, 11.50 per cent. At the holder's assumed correlation of 0.20 the volatility comes out at 9.81 instead. The variance is 324 times 0.25 plus 25 times 0.25 plus 36 times 0.25, or 81.00 plus 6.25 plus 9.00. The total is 96.25, and the square root of 96.25 is 9.81. The 1.69 point difference is the whole of what the correlation assumption bought.

A two holding variance carries three parts: each holding's own variance scaled by the square of its weight, and a covarianceThe part of a variance that depends on two holdings together rather than on either alone. It is the correlation multiplied by both volatilities. term that depends on the pair. A correlation cannot move the first two. The size of the bow is set entirely by the correlation term, so a frontier drawn from highly correlated inputs is nearly a straight line and one from weakly correlated inputs bows hard to the left.

Both end points are pinned in that figure, and the only thing changing between the five curves is one assumed number. A room shown one of the five and not told which correlation drew it has been shown almost nothing.

Where is the leftmost point, and what if it cannot be reached?

The leftmost point of the curve is the minimum variance portfolioThe lowest expected volatility the available holdings can produce, and not necessarily reachable., the combination with the lowest expected volatility the pair can produce. Only the part of the curve above that point is efficient. Everything below it offers less return at a volatility also available higher up, so the lower branch is beaten from end to end.

For two holdings the leftmost weight has a closed form worth running rather than quoting. Take the variance of the safer leg, subtract the covariance, and divide by the sum of the two variances less twice the covariance. Run it on the only two entities the record carries. Over one stated twelve month period the Anantara portfolio returned 14.2 per cent at a volatility of 11.8, its unnamed composite benchmark returned 12.6 at 10.4, and the beta was 1.08. The four figures are an assumed input set rather than a forecast. The correlation those figures imply is the beta times the benchmark volatility divided by the portfolio volatility. The arithmetic is 1.08 times 10.4 over 11.8, or 0.9519. The covariance is the beta times the benchmark variance. On these figures that is 1.08 times 108.16, or 116.8128.

Now the leftmost weight. The numerator is 108.16 less 116.8128, or minus 8.6528. The denominator is 139.24 plus 108.16 less twice 116.8128, or 13.7744. The answer is minus 0.6282, or minus 62.8 per cent in the portfolio leg. A negative weight is a real result rather than an error: at this assumed correlation no long onlyA mandate that may buy holdings but may not sell what it does not hold, so every weight sits between nought and one hundred per cent. combination carries less volatility than the benchmark leg held on its own.

An interior leftmost point appears only where the assumed correlation falls under the ratio of the lower volatility to the higher, a test established under mean variance optimisation. On this pair that ratio is 10.4 over 11.8, or 0.8814, and the implied 0.9519 sits over it, so there is no turning point inside the range and the leftmost reachable point is the benchmark leg by itself.

Substitute a correlation of 0.50 into the same two end points and the picture changes. The same closed form now gives 46.80 over 124.68, or 37.5 per cent in the portfolio leg, at a volatility of 9.52 per cent, below both legs. A beaten branch appears between 9.52 and 10.40. The benchmark leg alone returns 12.60 at a volatility of 10.40, and 75.1 per cent in the portfolio leg returns 13.80 at exactly the same volatility. The same two holdings and the same four figures produce a curve with no interior leftmost point at one assumed correlation and a clear one at another.

What does the whole reachable stretch look like on the record's own numbers?

Awkward, and the awkwardness is the finding. Because the leftmost weight came out negative, volatility rises without interruption from 10.40 and 12.60 at one end of the long only range to 11.80 and 14.20 at the other, and every combination between sits on the efficient part. The half and half mix carries a variance of 34.81 plus 27.04 plus 58.4064, or 120.2564. Its volatility is 10.97 per cent and its return 13.40.

Every reachable point here sits on the efficient stretch, so dominance rules nothing out and the choice between them is entirely a matter of the holder's own ranking rule. A drawing that appeared to be doing the deciding has handed the decision back, and the holder's own ranking rule is all that remains.

Measure what the spreading actually bought. A weighted average is what the volatility would be if the parts moved exactly together. On the holder's three asset class assumptions that average is 0.6 times 18.0 plus 0.3 times 5.0 plus 0.1 times 0.5. The sum is 10.80 plus 1.50 plus 0.05, or 12.35 per cent, against a real 11.20. The 1.15 point difference is the diversification, and it exists only because the correlation was assumed at 0.20. On the record's two legs, at 0.9519, the same measure gives 0.13 points.

What happens when a holding with no variance is available?

The shape of what is reachable changes, and so does the structure of the decision. Suppose a holding exists whose expected volatility is nought. Mixed with any point on the curve, that leg contributes no variance of its own and no covariance with anything, so the volatility of the mix is the fraction held in the risky point multiplied by that point's volatility. The return is a weighted average as always. Two plain weighted averages plotted against each other draw a straight line, so the reachable set gains a straight edge from that holding up to whichever point was chosen.

The steepest such line touches the curve at a single place, called the tangencyThe single point where a straight line drawn from a holding with no variance just touches the curve rather than cutting through it. point. The line itself is the capital allocation lineThe straight line of combinations available between a holding with no variance and one chosen risky combination., and its steepness is the risky point's return above the risk-free rate divided by its volatility. James Tobin set out in 1958 what this does to the decision. The separation converts one question into two that are taken separately: which risky combination to hold, and how much of it to hold against the holding with no variance.

Run it on the record's own two legs, every figure of which belongs to the same stated twelve month period. The risk-free rate is 6.5 per cent. The portfolio leg returns 14.2, so its excess is 7.7, and 7.7 over 11.8 is 0.6525. The benchmark leg returns 12.6, so its excess is 6.1, and 6.1 over 10.4 is 0.5865. The touching weight computes at 171.4 per cent in the portfolio leg. A weight above one hundred per cent means selling the benchmark leg short by 71.4 per cent. Where a mandate forbids that, the reachable corner is the leg with the higher ratio, the portfolio leg at 0.6525.

Now the honest part. The straight line above needed a rate, and got one because 14.2, 12.6, 11.8, 10.4 and 6.5 all belong to the same stated year for one invented mandate. The holder's three asset class assumptions are a different block of the record and contain no holding with no variance: cash there is assumed at 6.0 per cent with a volatility of 0.5, small but not nought. So no straight line and no touching point belongs on the assumption set curve. Importing the 6.5 per cent figure from the realised year would manufacture a number nobody wrote down.

How far does the curve move when one input moves?

Further than almost anyone expects, and that is what decides whether a chart is safe to read a gap off. Lift the portfolio leg's assumed return from 14.2 to 16.2 per cent, leaving both volatilities and the correlation as they were, and ask again what is available at a volatility of 10.97 per cent.

No expected return appears anywhere in a variance, so the weight that produces a volatility of 10.97 has not moved and cannot have. The weight is still half and half. But the return that weight delivers has moved. The old figure was 0.5 times 14.2 plus 0.5 times 12.6, or 13.40, and the new one is 0.5 times 16.2 plus 0.5 times 12.6, or 14.40. A two point change in one assumed return moved the available return at a fixed volatility by a full percentage point, with no market having changed at all.

The horizontal positions are identical under both assumed returns, to the last decimal, and every vertical position moved. On Rs 500 crore a point of expected return is Rs 5,00,00,000/-, so the difference between the two drawings is not a rounding matter in anybody's reporting.

What does a stated mandate band do to the curve?

A stated band removes most of the curve, before anybody looks at the picture. The Anantara mandate keeps equity between 50 and 70 per cent, and on its equity and fixed income pair the leftmost point lands at 7 divided by 313, or 2.2 per cent equity, at a volatility of 4.98 per cent and a return of 7.60. The mandate cannot reach its own leftmost point, and no amount of accuracy in drawing the curve changes that.

The band leaves one short arc. At 50 per cent equity the variance is 313 times 0.25 less 14 times 0.5 plus 25, or 78.25 less 7.00 plus 25, giving 96.25. The volatility is 9.81 per cent and the return is 9.75. At 70 per cent the variance is 313 times 0.49 less 14 times 0.7 plus 25, or 153.37 less 9.80 plus 25, giving 168.57. The volatility is 12.98 per cent and the return is 10.65. Everything outside 9.81 to 12.98 is closed by the mandate rather than by the arithmetic.

Two consequences follow. The floor forces 4.83 points of volatility the arithmetic did not require and lifts the expected return from 7.60 to 9.75 in exchange, so the band is a trade made in advance rather than a cost carried by accident. And because the admissible arc sits entirely right of the leftmost point, the band removed the whole beaten branch before the optimiser was asked a question.

Score the five optimiser runs traced earlier against the band. The runs at 9.0 and 9.5 per cent need 33.3 and 44.4 per cent equity, under the floor, and the one at 11.0 needs 77.8, over the ceiling. Three of five points on a perfectly correct curve are unreachable, and nothing in the drawing says so.

Put both into rupees. A committee argues in percentages and signs off on money. On Rs 500 crore a point of expected return is Rs 5,00,00,000/-, so a gap of 0.1009 points is about Rs 50,45,440/- and a movement of 1.1782 points about Rs 5,89,09,085/-. The number the room is asked to act on is under a twelfth of the number it is not being shown.

What does the frontier assume, and what is it a picture of?

Every point on it is a consequence of the expected inputs and of nothing else. Not of history, except where somebody chose to build an expectation out of history, and not of any market. A variance formula and a minimisation run perfectly happily on numbers nobody has checked. The smoothness and precision of the curve are properties of the arithmetic rather than evidence about anything outside it.

There is a household version. Somebody builds a spreadsheet of what a home will cost over twenty years, with a growth rate in one cell, and the answer comes out to the rupee. Change that cell by half a point and the answer moves by more than anybody would call rounding. Nobody thinks the spreadsheet knows the future, and yet the same arithmetic with a curve on it acquires authority it never earned.

A frontier drawn from an input set that is not stated cannot be checked by anybody looking at it, and a claim nobody can test is not yet a claim. Show the assumptions beside the picture and it becomes an argument somebody can engage with. Leave them off and it is decoration with a marker on it.

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

Backwards from the way it is usually presented. The nine numbers are the only things anybody can disagree with, so a committee like Rukmini Deshpande's starts with them rather than with the chart. The first question is the expected return on equity, over what horizon, and who set it. The second is the correlation assumed between the sleeves, and when it was last looked at. By the time those are answered, most of the useful conversation has happened.

An analyst does the same more narrowly: before quoting any distance from a curve, they recompute it with one input moved by a plausible amount and quote both numbers together. A lender reads a frontier chart in a pack as a statement about the borrower's assumptions rather than its portfolio, and asks for those assumptions. A household with modest savings can do it with a pen: writing down the assumed return for each thing shows how much of the plan turns on a number invented on a Wednesday.

The habit under all four: never quote a position relative to a curve without also quoting how far the curve moves. A report carrying only the first of those two numbers has told its reader almost the opposite of the truth.

Three of the questions a frontier raises are settled by the arithmetic above, three are settled under neighbouring subjects, and one is answered nowhere.