Portfolio Risk and Attribution: Gross Excess to Net

This calculator takes a portfolio's gross return, its benchmark return, its beta, the risk-free rate and the fee terms of the arrangement, and returns four things: the gross excess, the share of it that is only market exposure, the residual, and the excess that survives the fees. For the Anantara Multi-Asset Portfolio's stated year it returns plus 1.6 points gross and minus 0.28 net.

Beta and the ratios are covered separately, the split of the stated year under the monitoring sequence, and the commercial terms of this arrangement under the mandate that carries them.

Here is the record it arrives holding. The Anantara Multi-Asset Portfolio is an invented discretionary mandate of Rs 5,00,00,00,000/-, or Rs 500 crore, run for one institutional holder. Over one stated twelve month period it returned 14.2 per cent gross of feesA return struck before the cost of the arrangement has come out of it. against a composite benchmark, 60 per cent broad equity and 40 per cent broad bonds with neither index named here, that returned 12.6. The risk-free rateThe rate the calculation treats as available without exposure to the benchmark. Here it is 6.5 per cent for the stated year. was 6.5 per cent, the beta 1.08, and the arrangement charges 1.25 per cent of assets plus 15 per cent of the return above a 10 per cent hurdle.

On those eight figures it returns a gross excess of plus 1.600 points, an exposure part of plus 0.488, a residual of plus 1.112 and a net excess of minus 0.280, after total fees of Rs 9.40 crore which are 1.88 per cent of assets. The rest of this walkthrough is that run taken apart one output at a time.

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

The four come in a fixed order, and the first, the gross excessThe portfolio return less the benchmark return, both struck before fees. Taken after fees it is the net excess, a different figure., is simply the portfolio return less the benchmark return.

Then it stops, and the stopping is deliberate rather than an omission. The calculator does not say whether one stated year is evidence of anything, and it does not say whether the arrangement was worth having. Worth turns on what an alternative would have returned and what it would have cost, and this platform's record holds no alternative at all. The inputs a tool holds here cannot reach the question, so a tool that produced a verdict would be producing it out of nothing.

Where does each of the seven fields come from?

Seven fields drive the arithmetic; the eighth, the asset base of Rs 500 crore, changes no output and only turns percentage points into rupees.

Three of the seven fields are measurements, one is an estimate, and three are terms somebody negotiated, and the four outputs mix all three kinds without saying so on their face. A reader told the residual was 1.112 points hears a measurement, when what produced it was three measured figures and one estimate, and the estimate carries the least certainty of the four.

The four outputs do not all consume the same fields either, and knowing which eats which is what lets a reader see, without running anything, that a dispute about the beta cannot touch the net figure.

How much of the gross excess was simply market exposure?

The mandate ran at a betaThe sensitivity of the portfolio's return to the benchmark's, fitted over a stated window. Here it is 1.08. of 1.08, so it carried more exposure than the benchmark did, and exposure of that size calls for a return of its own before anything else is credited. Take the benchmark's 12.6 per cent, subtract the 6.5 per cent risk-free rate to leave 6.1 points, multiply those 6.1 by 1.08 to get 6.588, and add the 6.5 back. The result is 13.088 per cent, the return the exposure alone called for.

The portfolio returned 14.2 per cent gross, so against the 13.088 per cent the exposure called for, the residualWhat is left of a gross return once the return the market exposure called for has been subtracted. The residual carries Michael C. Jensen's name. is plus 1.112 points; and against the 12.6 per cent benchmark, the exposure itself contributed 13.088 less 12.6, or plus 0.488 points. Of the plus 1.6 points of gross excess, 0.488 points is what carrying a beta of 1.08 called for and 1.112 points is everything else, and the two add back to 1.600 exactly. The build-up above draws those two figures as its first two bars, and the check printed under it is that sum.

A second decomposition of the same 1.6 points sits elsewhere in this platform's record, and one warning about it belongs on the same screen as that split. The second decomposition asks where the excess came from across decisions, reports an allocation part of plus 0.35 points and a selection part of plus 1.25, and also sums to 1.60. Neither is more true: the questions differ and so do the bases. The calculator here runs the exposure split only, it prints that fact beside its output, and no term from the other decomposition may be carried into a sentence with a term from this one.

Why do the two fees behave completely differently?

The management feeA charge struck as a rate on the value of the assets, the same figure in a good year and a poor one. is a rate applied to the value of the assets. At 1.25 per cent on Rs 500 crore it is Rs 6.25 crore, whether the year returned 20 per cent or nothing at all. Nothing in that calculation knows what the return was.

The performance feeA share of whatever the return exceeded a stated level by, nil where the return did not reach that level. is the opposite: a share of whatever the gross return exceeded the hurdleThe level the performance share is struck above. Here it is 10 per cent, a term of the arrangement rather than anything an authority sets. by, and below the hurdle it is nil, not a reduced figure but genuinely nothing. Above it the fee climbs in a straight line: each further point of gross return puts Rs 5 crore above the hurdle, and 15 per cent of that is Rs 0.75 crore of extra fee. The stated year sat 4.2 points above the hurdle. Those 4.2 points are Rs 21 crore on Rs 500 crore, and 15 per cent of Rs 21 crore is Rs 3.15 crore.

The management fee does not know what the return was and the performance fee is nothing but a function of it, so adding the two into one percentage hides the only property of them worth knowing. The combined 1.88 per cent is correct and misleading at once: it looks like a rate, and a rate is what only two thirds of it is.

What does the gross return look like once the fees come out?

Rs 6.25 crore of management fee plus Rs 3.15 crore of performance fee is Rs 9.40 crore. On Rs 500 crore that total is 1.88 per cent of assets. Take that out of the 14.2 per cent gross return and the net returnThe return after the cost of the arrangement has come out. The net return is what the holder experienced. is 12.32 per cent, against a benchmark of 12.6 over the same twelve months. The net excess is 12.32 less 12.6, or minus 0.28 percentage points.

The portfolio beat its benchmark by plus 1.6 points gross and the holder finished 0.28 points behind it net. Both sentences describe the same portfolio, the same twelve months and the same benchmark. Neither is the excess return on its own. There is no such thing without the label, and the two figures differ only in whether the cost of collecting the return has come out.

Run the same chain on the return itself. A gross 14.2 per cent on Rs 500 crore is Rs 71.00 crore. The management fee takes that to Rs 64.75 crore and the performance fee to Rs 61.60 crore. Rs 61.60 crore is the net 12.32 per cent in money, against the benchmark's Rs 63.00 crore. The Rs 9.40 crore of fees was 13.24 per cent of everything the securities produced.

Two risk adjusted lines settled earlier in the reading order behave the same way and are worth recomputing on the net figure. Return over volatility, the measure carrying William F. Sharpe's name, is 7.7 over 11.8 gross, or 0.653, and 5.82 over 11.8 net, or 0.493, against the benchmark's own 6.1 over 10.4, or 0.587: above the benchmark gross, below it net. The information ratio does the same: plus 1.6 over a tracking error of 3.7 is 0.43 gross, and minus 0.28 over the same 3.7 is minus 0.076 net.

Written as a ladder the chain has three rungs: 14.2 per cent gross, 12.95 after the management rate of 1.25, and 12.32 after the performance share of 0.63 per cent of assets. The arrangement crossed the benchmark on its second rung, and a report that stopped at the first would have shown the holder ahead.

The same chain in rupees is blunter, and rupees are what a committee argues about. A gross excess of 1.6 points on Rs 500 crore is Rs 8.00 crore against fees of Rs 9.40 crore, a difference of minus Rs 1.40 crore. On Rs 500 crore that difference is the minus 0.28 per cent already computed.

A single year is not the shape. Generalise it. Below the hurdle the net return is the gross return less 1.25, so the two move together point for point. At or above the hurdle it is the gross return less 1.25 less 15 per cent of whatever the gross exceeded 10 by. The same expression rearranges to 0.85 times the gross plus 0.25. The two expressions meet at a gross 10 per cent, where both give 8.75, so the line is continuous and only its slope changes. Above the hurdle each further point of gross return adds only 0.85 of a point to the net. The arrangement has a bend in it at the hurdle. At a gross 14.2, 0.85 times 14.2 plus 0.25 is 12.32, reproducing the worked year exactly.

Where the bend sits is a field the tool will move. With the hurdle set to 12.6 instead of 10 and everything else left alone, the performance share falls on 1.6 points instead of 4.2, so it is 0.24 per cent of assets rather than 0.63, the total fee becomes 1.49 per cent, and the net excess turns from minus 0.28 points to plus 0.11. The change of sign is arithmetic.

Setting 0.85 times the gross plus 0.25 equal to the 12.6 per cent benchmark and solving gives 14.53 per cent, the gross figure at which the holder finishes exactly level on this fee schedule. The level figure sits 0.33 points above the 14.2 the year delivered, and the gap between them is the difference between beating a benchmark and reaching a holder.

How does the fee compare with the residual it was charged against?

Both are percentages of the same Rs 500 crore over the same twelve months, so they sit beside each other directly: the residual of 1.112 points is Rs 5.56 crore against fees of Rs 9.40 crore. The cost of the arrangement exceeded the residual it was charged against by 0.768 points, or Rs 3.84 crore for the stated year. A holder asking whether the fee was reasonable is asking for that comparison, and it is sharper than the benchmark comparison. The residual is the part of the year the arrangement could plausibly claim.

Elsewhere this platform's record carries the residual rounded to 1.11 points and puts the gap at Rs 3.85 crore. The calculator holds the unrounded 1.112 and prints Rs 3.84 crore. The difference is one lakh of rupees on Rs 500 crore, and it matters only in that a committee paper should say which rounding it used.

Every quantity here is a percentage of the same asset base, so moving between points and rupees is one multiplication: one percentage point of Rs 500 crore is Rs 5 crore. A committee arguing in points and a treasurer signing in rupees rarely notice they hold the same figure.

Which base a fee is quoted against changes the figure by two orders of magnitude, and the same Rs 9.40 crore honestly carries four. The fee is 1.88 per cent against the Rs 500 crore of assets, 13.24 against the Rs 71.00 crore gross return, 117.5 against the Rs 8.00 crore of gross excess and 169.1 against the Rs 5.56 crore residual. All four are correct and answer four different questions, so the base has to travel in the same sentence as the figure.

At household scale, for every Rs 100/- the arrangement collected in fees, Rs 59/- of residual stood behind it. The Rs 59/- is Rs 5.56 crore divided by Rs 9.40 crore.

The residual is the gross return less 13.088, and above the hurdle the fee is 0.15 times the gross return less 0.25; setting those equal gives 15.10 per cent, where both stand at about 2.02 points. Below that the fee is the larger of the two, and the stated year sat below it.

What does the residual inherit from the beta input?

Three of the four outputs use no beta at all: the gross excess is two returns subtracted, the total fee is a schedule applied to a return and an asset base, and the net excess is the gross return less the fee, less the benchmark. Only the split between the exposure part and the residual consumes a beta, and it consumes it heavily.

Work the sensitivityHow far an output moves when one input changes and everything else is held still. Sensitivity is a property of the calculation, not a measurement. directly. The residual is the gross return less the risk-free rate less the beta times the benchmark's 6.1 point margin over that rate, or 7.7 less 6.1 times the beta. Every 0.01 of beta moves the residual 0.061 points the other way. At a beta of 1.00 the exposure part is nil and the residual is the whole 1.60 points, at 1.08 it is 1.112, and at 1.15 it is 0.685. Raise the beta and more of the gross excess is called exposure. Lower it and more is called residual. The gross excess and the net excess do not move at all.

Run the same test on the net excess and the picture inverts. A point on the benchmark moves it a full point the other way, a point on the gross return moves it 0.85 because of the bend, a quarter point on the management rate moves it a quarter point, five points on the performance share moves it 0.21, and two points on the hurdle moves it 0.30. The beta and the risk-free rate move it not at all, and that pair of zeroes is the finding.

Run the tool at more than one beta and read the residual as a range with the estimate printed beside it, rather than as a single figure with the estimate left out of the sentence. A residual of 1.112 points reads like a measurement to three decimals; a residual of somewhere between 0.69 and 1.60 across betas from 1.15 to 1.00 reads like what it is, and it is the same year.

What happens to the outputs when the year comes in below the benchmark?

One run on a different year shows the outputs behaving the same way when the signs turn over. Hold everything and set the gross return to 9 per cent. The gross excess is 9 less 12.6, or minus 3.600 points. The beta, the benchmark and the rate did not move, and the exposure part does not either. It stays at plus 0.488. The residual is 9 less 13.088, or minus 4.088, and the two add back to minus 3.600 exactly. The year sat under the hurdle, so the performance fee is nil, the fee is the Rs 6.25 crore management charge alone, and the net excess is 7.75 less 12.6, or minus 4.850.

The nine per cent run can be made in the calculator above by entering 9 in the gross return field, and the two checks printed under the bars report the sums balancing as they do here.

What can this calculator never settle?

Whether one stated year is evidence: it is one draw, and a single observation cannot separate a process from an outcome. Whether there is a durable edge: that needs many periods. Whether a different arrangement would have done better: that needs an alternative with its own return and its own cost, and this platform's record contains neither.

A tool that answered a question its inputs cannot reach would hand a committee a figure with no visible route back to where it came from, and a figure like that is worse than no tool at all. The four refusals are printed on screen beside the four outputs for exactly that reason. Stating the comparison and declining the verdict is the whole of the honest position here, and it is a smaller answer than a reader wants.

Who runs this arithmetic, and what do they do with the output?

Rukmini Deshpande chairs the investment committee of the endowment that holds the mandate, and Faiz Ahmad Ansari runs it. The first thing Deshpande does with the annual pack is check which of the two excess figures it leads with. A pack leading with plus 1.6 points and putting the fee schedule in an appendix has told the truth and buried the consequence; a pack leading with minus 0.28 has told the same truth in the order the holder experiences it.

The second thing she does is ask what beta the residual was struck at and over what window, not because the estimate is suspect but because the figure moves so far with it that a residual quoted without its beta is a sentence with a term missing. Ansari uses the same output the other way, setting the residual against his own fee schedule and knowing before the meeting that Rs 9.40 crore of fee stood above Rs 5.56 crore of residual. Bringing that comparison himself is a different meeting from having it produced across the table.

The same arithmetic turns up far from a committee room. A lender looking at a small manufacturer computes what the business earned before its interest cost and then after it, and treats only the second as the figure that services anything. A household comparing two savings arrangements does the identical thing when it asks what the charges were, and so does a street vendor weighing a supplier's credit terms on a few thousand rupees. The structure is always a gross figure, a cost of getting it, and a net figure that is the only one anybody lives on.

Shrink the arrangement to a household and the arithmetic does not change. On a holding of Rs 10,00,000/- the management fee is Rs 12,500/-, the 4.2 points above the hurdle are Rs 42,000/- and the performance share on them is Rs 6,300/-, so the fee is Rs 18,800/- and the gross return of Rs 1,42,000/- delivers Rs 1,23,200/- against a benchmark of Rs 1,26,000/-. The household finishes Rs 2,800/- behind, the same minus 0.28 points.

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