Liability-Driven Investing: Building Around What You Owe

Liability-driven investing builds a portfolio against what the holder owes rather than against a return target or a market benchmark. The managed quantity becomes the surplus: assets less liabilities. Risk becomes the volatility of that difference. A portfolio can be steady on its own and volatile against an obligation, or the reverse.

The reversal in that last sentence is the whole of it, and it is the part that catches people. A portfolio does not become riskier by being looked at differently, and yet the number that measures its risk can go up by more than a point the moment somebody writes an obligation on the other side of the ledger. Nothing was bought. Nothing was sold. The arithmetic did not change either. The quantity the arithmetic was pointed at is what changed.

Liability-driven investingBuilding and monitoring a portfolio against a stream of payments the holder must make, so that the thing being managed is the difference between the assets and that stream rather than the assets on their own. is the ordinary machinery of expected return, volatility and correlation applied to a difference instead of to a level. Every tool used here is settled elsewhere: means, variances, correlations and the fact that a portfolio return is the weighted average of its parts while its volatility is not. The habit worth building is asking what quantity a number is being computed on.

The running example is the Anantara Multi-Asset Portfolio, an invented discretionary mandate of Rs 5,00,00,00,000/- run by Faiz Ahmad Ansari for an invented charitable endowment whose investment committee is chaired by Rukmini Deshpande. Its policy mix is equity 60.0 per cent at Rs 3,00,00,00,000/-, fixed income 30.0 per cent at Rs 1,50,00,00,000/- and cash 10.0 per cent at Rs 50,00,00,000/-, summing to Rs 5,00,00,00,000/- exactly. The policy weights are the shape the holder chose in advance, and actual weights drift away from them between one rebalancing and the next.

One fact comes before everything else: the record for that mandate carries no payment schedule, no spending rule and no discount basis, so no surplus can be computed for it at all. The absence is not a gap being papered over. The record is honestly that thin, and working around it openly teaches the subject better than inventing four numbers would. Every liability quantity below is either marked NOT SUPPLIED or is a labelled stand-in introduced to make an identity visible.

What is liability-driven investing?

Liability-driven investing is one move, and the move is small enough to describe in a sentence. Everything up to this point on the path measures a portfolio against itself or against a benchmark. Liability-driven investing measures it against an obligation. The change ends there.

Notice what did not move. The variance of a difference is computed the same way it always was. Correlation still means what it meant. Volatility is still the square root of a variance, and it is still symmetric, and it still is not a measure of the chance of loss. The change sits in the series the volatility is computed on rather than in how it is computed, so nothing about the arithmetic changes and everything about the question does.

Say it in kitchen terms. A household counts the money in the tin and calls the count its position. The count is asset-only riskRisk measured on the assets by themselves, with no obligation on the other side. It is correctly computed and it answers a narrower question than most holders think it does.: how much does the tin move? Then somebody remembers that a wedding has been fixed for a date eighteen months out, and the question quietly becomes a different one. Not how much does the tin move, but how much does the gap between the tin and the wedding bill move. The tin and the gap are two different quantities. The wedding bill has a life of its own, so the gap can be jumpy while the tin is calm.

The distinction matters more than it sounds because the asset-only number is not wrong. The number is correct, carefully computed, and answering a question nobody asked. An error of that kind is the most durable one in this subject, and it survives review precisely because there is nothing to find wrong in the calculation.

What is the surplus, and why does it become the managed quantity?

The surplusAssets less the value placed on what the holder must pay. It is a difference, so it can move when only one of the two sides moves, and it can fall in a period when the assets rose. is assets less liabilities. The definition is one anyone can hold in their head, and the consequences of taking it seriously run through everything that follows.

Once the surplus is the managed quantity, an asset movement stops being a result on its own. The movement becomes half of a comparison. A portfolio can rise and the surplus can fall, and once the surplus is what is being managed that is a bad period regardless of what the return column says.

Take a constructed holder, unrelated to the Anantara mandate and introduced only to make a difference visible. The holder has Rs 2,00,00,00,000/- of assets against a set of obligations valued at Rs 1,60,00,00,000/-, so the surplus is Rs 40,00,00,000/- and the assets cover the obligations 1.25 times over. A period passes. The assets rise 4.0 per cent to Rs 2,08,00,00,000/-. Any report would print that as a good period. The value placed on the obligations rises 7.0 per cent to Rs 1,71,20,00,000/-. The surplus is now Rs 36,80,00,000/-, down Rs 3,20,00,000/- or 8.00 per cent, and the cover has slipped from 1.25 times to 1.215 times.

Read that column of numbers again. Nothing went wrong on the asset side. The portfolio did what it was supposed to do. The holder is nevertheless further from meeting the obligation than it was, and a report that stopped at the 4.0 per cent would have said the opposite. The reversal is not a trick of the example. A difference between two moving quantities behaves this way ordinarily, and that behaviour is the reason the surplus frame exists at all.

What actually moves the value of an obligation?

A liability looks as though it sits still. The obligation has a number on it, the number was written down, and numbers that were written down do not look like market quantities. The assumption that a written number sits still is the source of most of the trouble in this subject.

A promised set of future payments is valued by discounting them back to today. The moment that is true, the value of the obligation depends on the discount rateThe rate used to bring a future payment back to a value today. A higher rate makes a distant payment worth less now, and a lower rate makes it worth more. used to do the discounting, and the dependence runs the other way round from the rate. Rates fall, the value of the obligation rises. Rates rise, it falls. Nobody chose that. The direction follows from the arithmetic of discounting, settled under fixed income and applied here rather than rebuilt.

Here is a constructed stream, belonging to no portfolio on this platform and introduced only to put digits on the movement. Five payments of Rs 10,00,00,000/- each, one at the end of each of the next five years. Discounted at an illustrative 6 per cent, the five present values are Rs 9,43,39,623/-, Rs 8,89,99,644/-, Rs 8,39,61,928/-, Rs 7,92,09,366/- and Rs 7,47,25,817/-, summing to Rs 42,12,36,378/-. Discount the same five payments at 5 per cent instead and the total becomes Rs 43,29,47,667/-, higher by Rs 1,17,11,289/- or about 2.78 per cent. Discount them at 7 per cent and the total becomes Rs 41,00,19,744/-, lower by Rs 1,12,16,634/- or about 2.66 per cent. The three rates are illustrative choices made to show the movement and are not observations of any market anywhere.

So an obligation is a position with a duration, whether or not anybody wanted one, and that is exactly what makes matching possible in the first place. The liability durationThe weighted average time until the payments in an obligation fall due, which governs how much the value of that obligation moves when the discount rate moves. of the constructed stream above works out at about 2.88 years, the point where its payments sit on average once each one is weighted by what it is worth today. Longer streams move more for the same change in the rate. How that sensitivity is derived belongs to the fixed income layer and is not re-argued here.

The consequence for construction is the whole reason the frame matters. If the obligation moves with the discount rate, and some of the assets also move with the discount rate, then the two sides share a driver. A shared driver is a correlation. And a correlation is the only thing standing between a holder and a surplus that swings more than the portfolio does.

How is surplus volatility actually computed?

With the same identity that computes the volatility of any difference. Surplus volatilityThe volatility of the difference between the assets and the value of the obligation, computed with the same variance identity used on any other combination of two series. is the square root of the asset variance plus the liability variance less twice the correlation times both volatilities. Written out in words: take the asset volatility and square it, take the liability volatility and square it, add the two, then subtract twice the correlation multiplied by the asset volatility multiplied by the liability volatility, and take the square root of what is left.

The liability enters with a minus sign in front of the correlation term, and the shape of that expression is the whole subject. An obligation that moves with the assets reduces surplus risk; one that moves against them raises it. Everything below is a consequence of that one sign.

Before that identity can be run on the Anantara mandate, the asset side has to be computed rather than quoted. The holder's own stated assumptions are 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 a correlation of 0.20 between equity and fixed income and cash taken as uncorrelated. The holder chose those assumptions; they are not forecasts and not anybody's published estimates, and a different set gives a different portfolio.

At the policy weights of 60, 30 and 10 the variance comes out as 116.64 from equity, plus 2.25 from fixed income, plus 0.0025 from cash, plus 6.48 from the single cross term between equity and fixed income. The four terms sum to 125.3725, and the square root of 125.3725 is 11.196986, printed as the 11.20 per cent everything below measures against. The cross term is the only place the 0.20 correlation appears, and it contributes 5.17 per cent of the total variance while the cash term contributes 0.002 per cent.

Why can the same portfolio look riskier once an obligation is in view?

Now the identity gets run, and it needs a liability volatility the mandate record does not hold. So one is borrowed openly. Take a stand-in figureA number introduced only to make an identity visible, labelled as such on every appearance, standing in for a quantity the record does not carry. It describes nobody and must never be read as an observation. of 5.0 per cent for the liability volatility, taken from the fixed income assumption because the record supplies nothing better, and carry that label on every appearance of it below. The 5.0 per cent is not this endowment's obligation. The record holds no obligation at all.

At a correlation of nought the third term disappears entirely, so the two variances simply pile up. The asset variance is 125.3725 and the stand-in liability variance is 25, giving 150.3725, whose square root is 12.262647, printed as 12.26 per cent.

Stop on that before going any further: the assets carry 11.20 per cent volatility, the stand-in obligation is assumed to move independently, and the surplus carries 12.26 per cent, a full 1.07 points higher than the assets alone. No holding was bought. None was sold. The mandate corridor was not touched, the 5 per cent cap was not touched, and the measured risk went up by more than a point because a second uncertain quantity entered the calculation and nothing was there to cancel it.

The rise is the counterintuitive result, and it is the reason the distinction matters. A holder who moves from a return frame to a liability frame on a Monday can find on the Tuesday that their risk figure is higher, with an unchanged portfolio and an unchanged spreadsheet. Whether that is a discovery or a shock depends entirely on whether anybody explained the arithmetic first.

How much co-movement does it take before the surplus becomes the steadier number?

Walk the correlation up from nought and run the identity at each step. At 0.25 the surplus volatility is 11.06 per cent. At 0.50 it is 9.72. At 0.75 it is 8.15. At 1.00 it is 6.20. Every one of those readings is below the 11.20 asset figure except the first, of 12.26 at a correlation of nought. The crossing therefore happens somewhere in the first quarter of the range.

Solve for it rather than guessing. The surplus volatility equals the asset volatility exactly when the liability variance equals twice the correlation times both volatilities. Rearranged, the crossing correlation is the liability volatility divided by twice the asset volatility. On these figures that is 5.0 divided by 22.393972, or 0.223274, about 0.22.

So a correlation of about 0.22 is enough to turn the sign of the whole exercise on these figures, and a very small amount of shared behaviour between the assets and the obligation is all it takes to make the surplus the steadier of the two numbers. The crossing point is the claim worth carrying, not any single height. Matching does not have to be good to be worth doing.

The tidy form of that rule is worth carrying away. The crossing correlation is always the liability volatility divided by twice the asset volatility, so it rises in a straight line as the obligation gets jumpier. At a stand-in of 2.5 per cent the crossing is 0.1116. At 7.5 per cent it is 0.3349. At 10.0 per cent it is 0.4465. When the obligation has the same volatility as the assets the crossing sits at exactly 0.50. And when the obligation is twice as volatile as the assets, at 22.393972 per cent here, the crossing sits at exactly 1.00. An obligation jumpier than that can never make the surplus steadier than the assets at any correlation whatsoever.

What does a perfectly matched portfolio give up?

Push the correlation all the way to 1.00 and the surplus volatility falls to 6.20 per cent, exactly the asset volatility of 11.196986 less the stand-in liability volatility of 5.0. The match is not a coincidence of these figures. At a correlation of one the identity collapses to the plain difference between the two volatilities. A perfect correlation on its own therefore does not finish the job.

To reach nought the two volatilities also have to converge. If an obligation had exactly the same volatility as the assets, 11.196986 per cent, and moved with them perfectly, the surplus volatility would be exactly nought and the holder would be certain of covering the obligation whatever happened. Such a portfolio is the matched portfolioA portfolio built so its value moves with the value of the obligation it exists to meet, so the difference between the two barely moves at all. in its pure form, and it needs no stand-in to describe because it is defined by an equality rather than by a number.

The price of getting there is every other reason for holding anything: a portfolio built to move with an obligation is built around somebody else's discount rate, and each holding in it was chosen to track that rather than to earn. The expected return of such a portfolio is whatever falls out of the tracking, and the holder has stopped asking what a holding earns. Asset allocation otherwise turns on exactly that.

The corridor of the Anantara mandate is recorded, so the mandate makes the trade-off concrete without needing any invented obligation at all. Equity must sit between 50 and 70 per cent, being Rs 2,50,00,00,000/- and Rs 3,50,00,00,000/-. So even at the floor of the corridor, half the portfolio is in a class that is held for what it earns and not for how it tracks anything. A full match is simply inadmissible under this mandate, and that would be true whatever obligation the record eventually carried.

The corridor moves the asset figure too. At the 50 per cent equity floor, with fixed income at 40 and cash held at 10, the portfolio volatility works out at 9.6022 per cent and the crossing correlation rises to 0.2604. At the 70 per cent ceiling, with fixed income at 20, the volatility is 12.8375 per cent and the crossing falls to 0.1947. Both are computed from the same assumption set as the 11.20 per cent, and both sit inside the mandate. Quoting them shows how far the asset figure travels without any breach.

What does this approach need before it can start?

Four things. The list itself is the argument, so it is worth setting out rather than describing. A schedule of the obligations, meaning what is payable and when. A rule for valuing them, meaning the basis on which they are discounted. An estimate of their volatility. And a correlation between them and each asset class held.

A return-driven mandate never has to produce those four inputs, and missing any one of them stops the arithmetic completely rather than degrading it. Without the schedule there is nothing to discount. Without the basis the value of the obligation is undefined, so the surplus has no level. Without the volatility there is no liability variance term, and without the correlations there is no third term. The identity then cannot be evaluated at all.

Precision about the correlations matters: a single number will not do. Treating the assets as one block, as the identity above does, is fine for teaching. In practice a holder needs one correlation for each class it holds. Equity, fixed income and cash respond to a discount rate in quite different ways, and a single blended figure hides exactly the differences that matching depends on.

What does the mandate record hold, and what does it not?

The record for the Anantara mandate is unusually complete on the asset side and completely silent on the other. The record holds the Rs 5,00,00,00,000/- total, the policy weights, the corridor, the 5 per cent cap on any one holding, an assumed volatility of 11.20 per cent computed above, and a realised portfolio volatility of 11.8 per cent for one stated twelve month period. The record also holds a drawdown of 9.7 per cent measured peak to trough within that window, against 8.1 per cent for the composite benchmark. A different window would give a different figure, so the window is quoted every time.

Missing from the record are a payment schedule, a spending rule, a discount basis, a liability volatility, and a correlation between an obligation and any class. Five absences. So the honest surplus figure for this mandate is not a number at all: it is NOT SUPPLIED, and saying so is the finding rather than an evasion of it.

The distinction matters because the temptation runs the other way. Ten minutes would be enough to invent a plausible spending programme, discount it at a plausible rate and print a surplus volatility to two decimal places. The result would look exactly like a computed figure and would carry none of its standing.

How does a committee use any of this in a room, on a Tuesday?

By printing two risk figures instead of one, and never letting the second replace the first. The asset-only figure answers how much the portfolio moves. The surplus figure answers how much the gap between the portfolio and the obligation moves. A committee that has both can tell which of the two it is worried about; a committee with one can only argue about it.

An investment committee like Rukmini Deshpande's would ask three questions in order, and the order matters. Is there an obligation at all, written down, with dates. If there is, what basis values it, and who chose that basis. And what is the correlation between that obligation and each class the portfolio holds? The answer to that one decides whether the surplus figure comes out above or below the asset figure. On the Anantara record all three questions currently return the same answer: the paperwork has never been written.

An analyst reading a mandate does the same work from outside. The asset-only volatility is usually printed. The next step is to look for anything on the other side of the ledger. If there is an obligation and no surplus figure anywhere in the reporting, the reporting is answering the narrower question, and the analyst then knows which question is going unasked rather than guessing at an answer.

A household runs the identical test with a pen and no arithmetic at all. Write down the savings on one line. Write down the fixed commitments and their dates on the next: a wedding in eighteen months, a deposit due at a known date, a course fee that falls in a particular term. The savings moving is one thing. The gap between the savings and those dated commitments moving is a different thing, and it is the second one that decides whether the commitment gets met. Anyone who has watched a wedding budget rise faster than the tin it was being saved into has already met surplus volatility without the name.