Weighted Average Life: One Number for When Money Returns

Every rupee of principal carries a number of years until it lands, and those years are averaged using the rupees as the weights. Each amount is multiplied by its year, the products are added, the total is divided by the total principal, and what comes out is measured in years. No rate is needed and nothing is discounted. The result is one number, and the schedule behind it cannot be read back out of it.

A repayment schedule is a list. Rs 300 crore here, Rs 300 crore a year later, and so on down the column. Lists are honest and useless for comparison: two of them set side by side are two tables to read rather than an answer to a question. Collapsing a list to one number makes the comparison possible, and it does that by throwing information away. A tool of this kind is only safe in the hands of somebody who knows exactly which information went out with the collapse. The arithmetic runs four different ways below, and what the arithmetic cannot say takes far longer to set out than the arithmetic itself.

One thing decides how every figure below is read, so fix it before anything else. The order in which losses reach the pieces of a structure is settled by the payment waterfall and is untouched by everything here. A calendar and a queue answer different questions. Who absorbs before whom is not changed by moving a payment date. No likelihood is attached to any outcome either. Attaching one takes a distribution of losses, and a repayment calendar is not one.

What is actually being averaged?

The times. Not the amounts. Averaging times rather than amounts sounds like a small distinction and it is the whole measure.

The starting point is a list of principal amounts, each with a date. Principal is the only thing being counted: the interestWhat a borrower pays for the use of money, on top of handing the money back. a borrower pays for the use of the money sits outside this arithmetic entirely. Every amount carries a number of years, counted from today to the day it lands, and the measure averages those year numbers by letting each amount vote in proportion to its size. A large amount arriving early drags the average towards the early years, and a small amount arriving late barely shifts it. The answer comes out in years, it can never land before the earliest date on the list nor after the latest one, and the midpoint of the calendar has no claim on it unless the amounts happen to sit symmetrically either side of it.

A household has lent Rs 1,20,000/- to a cousin setting up a workshop, and it comes back in three yearly instalments of Rs 40,000/-. When the money is back starts an argument at the table. One person says year three, the date the last note arrives. Another says year one, the date the first note arrives. Both are answering a different question from the one asked. Weight the three years by the rupees standing at each of them and the average is 2.00 years. The average is neither answer, and it is the only figure using all the information on the table.

The same thing scaled up. Sarvani Receivables Trust holds Rs 1,200 crore of principal. Asked of Rs 1,200 crore instead of Rs 1,20,000/-, the same question is answered by the same arithmetic in the same unit. Size does not enter the measure at all, a point the ladder of four schedules and the shortcut below both make again.

What are the four lines?

Four lines, short enough to run on the back of a receipt. One, write every principal amount against the number of years until it arrives. Two, multiply each amount by its number of years. Three, add those products together. Four, divide the total by the total principal. The result is in years.

Notice what never appears in those four lines. No rate, no price, nothing discountedShrinking a future amount down to what it is worth today, using a rate. The four lines do none of it., and no assumption about where the money goes after it arrives. The inputs are amounts and dates, and the shortness of that list is why the panel above can exist while its better known cousin cannot. The comparison with MODIFIED duration is set out below.

The unit on the third line is where people lose confidence. Multiplying rupees by years gives rupee-years. Nobody quotes rupee-years, and the pile of 3,000 crore-years sitting in the middle of the calculation looks like a mistake. It is not. The fourth line divides those rupee-years by rupees, the rupees cancel, and years survive on their own. An intermediate total carrying a unit nobody uses is usually a sign the arithmetic is on track rather than off it.

Is there a second route that checks the first?

There is, and it is worth having for two reasons: it uses none of the same steps, so it catches a slip rather than repeating one, and it turns the measure into a picture that can be pointed at.

Instead of tracking money arriving, track money still outstanding. Sarvani Receivables Trust owes the full Rs 1,200 crore through year one. The first instalment lands, so Rs 900 crore is outstanding through year two, then Rs 600 crore through year three and Rs 300 crore through year four, after which nothing is owed. Add those four balances: 1,200 plus 900 plus 600 plus 300 is 3,000. Divide by the Rs 1,200 crore the pool started with. The answer is 2.50 years again, from a route that never multiplied anything by a year number.

The two routes agree because they are the same sum read in the other direction, not because two independent methods happened to corroborate each other. An amount that arrives at the end of year three was outstanding through years one, two and three, so it is counted three times in the staircase and multiplied by three in the procedure. Forced arithmetic, not luck. A reader who thinks two separate strands of evidence agreed will trust the answer more than the evidence deserves. Say so out loud whenever two routes meet.

The staircase adds a shape. The area under the outstanding balance line, divided by the height it started at, is the width of the rectangle holding that same area. The rectangle is the answer drawn rather than worked: the whole pool held flat for 2.50 years, then gone.

What happens when one pool is returned four different ways?

Hold the pool fixed and change only the calendar. Sarvani Receivables Trust has Rs 1,200 crore to return in equal yearly instalments at the end of each year. Three instalments means Rs 400 crore a time, four means Rs 300 crore, five means Rs 240 crore and six means Rs 200 crore, and every one of those four schedules hands back exactly the same Rs 1,200 crore.

Four schedules, one total, four different answers, and the whole difference is the calendar. That sentence is the reason the measure exists. A reader handed only the total cannot tell these four apart at all. A reader handed only the last date can separate all four, but would call them three, four, five and six years. Those dates say when each schedule finishes rather than where its weight sits.

One caution about a figure in that table. Rs 240 crore is the instalment when the pool is repaid in five equal parts. The same amount turns up elsewhere in this material as a rupee total belonging to the ordering of the pieces, a different subject entirely. Two unrelated things wear one number, so every figure here is labelled with what it is rather than left to stand alone.

Where does the shortcut hold, and where does it die?

Read the four answers again as a sequence: 2.00, 2.50, 3.00, 3.50, against three, four, five and six instalments. Every rung up that ladder costs half a year, exactly half, with no drift. The half year step is not a pattern somebody noticed and hoped would continue. The step falls out of the arithmetic in two lines.

Call the number of instalments n. Each one is the pool divided by n, so call it A. The years are 1, 2, 3 and so on up to n, and the whole numbers from one to n add to n times n plus one, all over two. So the products add to A times that quantity. The divisor is the total principal, A times n. The A cancels. So does one n. Left standing is n plus one, over two, and the size of the pool has vanished from the answer entirely.

The shortcut is genuinely worth committing to memory. Six instalments, 3.50 years, and the size of the pool never entered it, whether Rs 1,200 crore or Rs 12 lakh. The shortcut is also the narrowest result set out here, and the fence around it has three rails. The instalments must be equal. The instalments must be annual. The first one must land at the end of year one. Break any single one of those and the shortcut is worth nothing at all, not approximately right, and the only route left is the four lines.

Can two different schedules give the same number?

Two different schedules can share one number, easily, and this is the limit that matters more than every other limit on the measure.

Schedule A is the three instalment case from the ladder: Rs 400 crore closing year one, Rs 400 crore closing year two, Rs 400 crore closing year three. Schedule A balances at 2.00 years. Now build Schedule B deliberately. Rs 600 crore lands at the end of year one. Then Rs 200 crore at the end of year two, Rs 200 crore at the end of year three, and Rs 200 crore at the end of year four. Total returned, Rs 1,200 crore. Run the four lines on it: 600 times one is 600, 200 times two is 400, 200 times three is 600, 200 times four is 800, and those four products add to 2,400 crore-years. Divide by Rs 1,200 crore. The answer is 2.00 years.

Same number. Not close, not roughly, the same. And the two are not alike in any other respect: A is finished at the end of year three while B is still paying at the end of year four, A hands back a third of the pool in its first year against B's half, and A has three payment dates against B's four. The heavy front payment in B pulls the balance point in exactly as far as its long tail pushes it out, and the two cancel.

So the measure cannot be run backwards. A weighted average life narrows the field of schedules it could have come from, and it never identifies one of them. The number alone does not say when the last rupee arrives, how many payment dates there are, or whether the money comes in evenly or in one lump followed by a dribble.

Is 2.50 years the date the money is back?

No. The number looks so much like a date, and that resemblance is the confusion that costs people the most.

Go back to the four instalment schedule. Four instalments of Rs 300 crore balance at 2.50 years. Its final Rs 300 crore is still outstanding until year four closes. Between the balance point and the final payment there is a year and a half in which money is still coming in, and anybody who read 2.50 years as the moment the pool is cleared has closed the file eighteen months early. The balance point of a schedule and the end of a schedule are two different places on the same axis, and they coincide only in the single case where every rupee arrives on one date.

The single date case is worth holding on to. The case is the only one where the two numbers agree, and the panel above shows it: with everything loaded into year one, both readings are 1.00 year. Spread over anything at all, the balance point moves in front of the last date and stays there, never later than the final date, never earlier than the first, and strictly between them the moment more than one date carries money.

Why can one of these two measures be computed here and not the other?

Sooner or later somebody asks for the MODIFIED durationA price, a rate standing on it, and the question of how much of the first is given up per point of movement in the second. Both have to exist before it can be worked out. of one of these schedules, having heard the two described in the same breath, and it is worth knowing exactly why that request cannot be met here rather than fumbling it.

The two measures consume different inputs. A MODIFIED duration reports the sensitivity of a price. Per point of movement in a rate, so much of the price given up. Producing one therefore takes two things that have to be sitting there already: a rate standing on the thing being measured, and a price for that rate to move. A weighted average life needs neither. The measure is arithmetic on amounts and dates, and a schedule carries both already. The missing input is a yieldThe one rate at which a price and a set of promised payments agree with each other. A yield belongs to something priced, not to a bare list of amounts. on the instrument being measured, and none of the pieces here has been given one. The pieces of this structure are carrying no rate anywhere in this material. Without a rate there is no price either, so a MODIFIED duration cannot be produced for any of them. The weighted average life gets computed here four times over on a pool that carries no rate at all.

One middle relative sits between the two, and it is worth naming before it gets mistaken for either. A MACAULAY durationA waiting time that shrinks every payment first and weighs it afterwards, so a rate has to be picked before the sum can even begin. also averages waiting times, but it shrinks each payment before weighing it, so a rate has to be chosen before the first line of the sum. Settle it in the rate sensitivity material rather than here.

The practical consequence: anybody who quotes one of these as the other has not made a rounding slip. The swap changes what a sensitivity is measured against: a figure answering when money comes home is passed off as one answering how much a price moves. Those are not adjacent claims.

What does it look like as a picture?

Lay a plank along a line marked out in years. At every year where money arrives, stand a weight on the plank, and make each weight as heavy as the payment landing there. Now find the point where the plank balances on a single support. The balance point is the weighted average life.

Once that image is in place, the whole subject is visible at once. A heavy weight moved towards the near end takes the balance point with it, never as far as the weight itself travels. The other weights are still holding their side down. One small weight left far out pulls the point away from the crowd, so a long thin tail matters more than its size suggests. And two completely different arrangements of weights can balance at exactly the same point. The limit set out above becomes one image rather than one argument.

The frozen row is the part worth a minute. The row sits unchanged at every one of the six positions, and at exactly one of them, three instalments, the moving marker lands on top of the row that never moves. Two rows with different bar heights, different numbers of bars and different finishing dates, reporting one answer, and the marker is seen arriving at it rather than the agreement being asserted.

Who actually reaches for this number?

Three kinds of reader, and they want it for three different reasons.

A lender that has put money into a structure has its own repayments to meet, and those fall on dates. The lender cannot line up what it owes against a two sheet repayment table for every holding, so it asks each holding for a single number saying roughly when the money comes home, and stacks those numbers against its own calendar. The honest use of the measure is a first sort, on one dimension, before anybody opens the schedules. The moment two candidates come out close, the schedules go back on the desk, because on this measure close says nothing about shape.

An analyst reading a fact sheet meets the number the other way round, already computed, with the schedule that produced it nowhere on the sheet. The useful instinct is to treat the figure as a question: what pattern of arrivals produced this, and would a different pattern with the same figure change what I am about to write. Somebody holding the schedule can check the number in four lines. Somebody holding only the number cannot get back to the schedule at all, and that traffic runs one way.

And the household version, the same job at a different scale. A household offered two repayment patterns by a relative it has lent to, or choosing between two chit style arrangements, is running exactly this arithmetic when it asks which one gets the money back sooner. Weighting each promised date by the rupees standing at it makes the two offers comparable in a way that two lists of dates never are. The caution travels with it: sooner on average is not the same as finished sooner, and a household that has understood that difference has understood the whole of this material.

What does the number refuse to carry?

The tool answers when, for a list of amounts handed to it. Which list to hand it is the part it cannot supply, and nothing standing behind this material puts one list ahead of another. Every schedule here was written into this guide or entered into a panel. An answer computed off any of these lists carries whatever the list was built on. In every case above the list was a writer's decision rather than a repayment record, a measured speed of repayment or an observed pattern.

Two things follow, and they are worth separating. First, the arithmetic is exact while the input was written for teaching, a combination that reads as more solid than it is: 2.50 years is precisely correct about a schedule nobody has ever seen a pool follow. Second, and people slide past this one, the answer to when says nothing whatever about the order in which losses reach the pieces of a structure. The senior piece keeps 20.0 per cent of the pool standing below it at every setting of either panel above, because moving repayment dates around does not reorder a queue. A calendar and a queue answer different questions, and neither is evidence about the other.

Two behaviours are covered separately. PrepaymentMoney coming home sooner than the contract asked for, because borrowers settled early. Prepayment is covered separately. is money coming back sooner than the contract asked for, and extensionThe slower case. Repayment drags, and the far end of a schedule stretches past where it was drawn. Also covered separately. is the slower case. The tail stretches out past where it was drawn. Both of them change the schedule that goes into the four lines, and both are covered separately. The four lines are narrower than either, and worth stating plainly: they run on the schedule as given, whatever produced it.

Attaching a likelihood to an outcome takes three inputs the four lines never see: a distribution of losses, a statement of how the receivables move together, and a schedule of periods. Lacking all three, how often a given piece of a structure is reached cannot be worked out from a weighted average life, and neither can how safe that piece of it is. This guide supplies a way of turning a list into a number, plus a clear account of what the list had to say that the number cannot.