The Four Assumptions That Decide a Fixed Income Answer

Before any arithmetic runs on a bond, four things have been settled by the writer rather than looked up: the call assumption, on whether the borrower repays early; the reinvestment assumption, on what each payment earns after it lands; the prepayment assumption, on how fast a pool of loans repays; and the liquidity assumption, on whether the holder can leave before the end. None is observable. Each decides an answer rather than adjusting one.

Underneath all four sits the same small fact about how fixed income arithmetic is built. An answer here is a sum of dated amounts, so it has two kinds of ingredient and not one: how much, and when. Move a date and the sum moves. Change what becomes of an amount after it has arrived and the sum moves again. Every one of these four suppositions is a statement about a date or about an exit, and a statement of that kind reaches deep into a figure that looks like nothing but calculation.

What Does a Fixed Income Assumption Actually Do to an Answer?

An example away from bonds altogether makes the point. Somebody asks a driver how long it takes to reach a cousin's wedding two districts away, and the driver says four hours. Sitting inside that four is a view about the traffic on the day. The driver did not measure it, could not look it up, and supplied it out of experience. The person who wrote it in their diary now has a number. The person with the number has no way of telling that part of it came out of the driver's head rather than off a map.

Most readers arrive with the wrong picture of what an assumption is, and the driver's four hours is exactly the shape of the problem. The common picture is a hedge: a cushion added around the edge of a calculation, a little slack, a polite acknowledgement that things might go otherwise. An assumption in fixed income is nothing of the sort. An assumption is an input, sitting in the calculation in precisely the slot a looked-up figure would occupy, and the arithmetic cannot tell the two apart.

Look at what goes into pricing or reading an instrument and sort the ingredients by where each one comes from. Some are written down in a document somebody signed. Some are printed on a statement. And some have not happened yet.

The first three rows can be argued about only in the sense that somebody might have copied them wrongly. There is nothing to copy the last two from, so they cannot be argued about in that way at all. The last two arrive from a person, and then something quietly bad happens: they stop looking like they came from a person. The output is a figure with two decimal places, sitting in a sentence, printed exactly like a figure somebody looked up.

Leaving no trace is what makes everything that follows necessary. An answer produced with an assumption inside it looks identical to an answer produced without one. There is no shading, no widened range, no asterisk that the arithmetic itself puts there. If a reader is going to know, somebody has to tell them in words.

What Is the Call Assumption, and Which Answer Does It Decide?

A repayment right is an embedded optionA right written into an instrument itself, belonging to one side of the deal, letting that side change the schedule. Part of the contract, not something bought separately.: a clause letting the borrower give the money back before the end, on terms fixed in advance. Households meet this from the other side of the table all the time. A home loan the borrower is allowed to clear early is the same clause pointing the other way: the household holds the right, the lender holds the schedule, and on the morning the household decides to clear it the lender's schedule simply stops. Whether it will ever be done is not written anywhere. Clearing early depends on what happens to rates, to the household's income, and to a dozen things nobody has a record of.

The lender's problem, put into a research note, becomes the call assumption. Because nobody can observe whether a repayment right will be used, an analysis has to state a view about it, and that view fixes the date the money comes back. Fixing that date is not a small act. The date is one of the two ingredients every fixed income sum has.

All four of the suppositions taken apart below are declared rather than measured. A declared quantity sits in a calculation with no evidence standing behind it, and the reader has nothing to check it against.

With that said, here is the supposition, declared. Take the ten year instrument used throughout this material. The instrument pays 8.50 per cent a year on a face amount of Rs 1,000.00/-, it was bought for Rs 1,000.00/-, and the compounding clock ticks once a year. Now suppose that written into it, from the day it was drawn up, sat a clause allowing the borrower to clear the whole face amount once the fifth year had finished. The instrument as described carries no such clause. Everything below follows from supposing one was written in.

Why Does That Supposition Leave the Yield Alone and Move the Risk Reading?

Both readings worked out show that the difference does not land where most people expect. The headline comes first. The buyer paid the face amount. Under the supposition the face amount comes back. In between, every year, the buyer collects the contracted rate calculated on that same face amount. There is no discount and no premium anywhere for the passage of time to work on. So whether the giving back happens after five years or after ten, the rate earned across the time the instrument was held is the contracted rate. The reading to the supposed repayment date and the reading to the final date are the same number, 8.50 per cent a year, so the supposition touches the headline figure not at all.

The result is a genuinely uncomfortable one and it is worth sitting with. The one figure most likely to be quoted, most likely to be compared against another instrument, and most likely to end up in somebody's spreadsheet is completely blind to a clause that changes when half the money comes back.

Now the risk reading. Under the supposition the payments stop after year five, so there are five dated amounts instead of ten, and the weight of the schedule shifts hard towards the present. MACAULAY durationThe average waiting time for the money, in years, with every date weighted by how much of today's price is sitting on it. drops from 7.1191 years to 4.2756 years, and MODIFIED durationNot a number of years at all. It reads how far a price shifts, in per cent, when a yield moves by a hundred basis points. drops along with it.

The larger MODIFIED duration divided by the smaller gives 1.6650. One instrument, one price, one yield, and two answers on how far the price moves, sitting 1.6650 times apart. Carried through to money it stops being abstract. Repricing both schedules properly, rather than leaning on the duration line, shows what a declared 200 basis point move does.

On the rise, the two suppositions are 4.544 percentage points of price apart. On the fall, the distance worked from the unrounded prices comes to 6.066 points. Both of those roundings happened to fall the same way, so subtracting one printed figure from the other gives 6.067, one thousandth higher. Take the unrounded route, and say which route was taken. A reader checking the subtraction against the printed figures will otherwise find a residual and assume the error is theirs. The person holding the instrument has not done anything, the borrower has not done anything, and no rate quoted anywhere has changed between the two rows. The entire distance between those answers is a supposition that nobody can check, sitting inside a calculation that presents itself as arithmetic.

What Does the Reinvestment Assumption Assume, and Where Is It Hiding?

Take a household with a five year deposit at a bank that pays the interest out every year into the savings account. The bank quotes a rate. The household spends the interest as it arrives, on school fees and a wedding and a hospital bill. At the end of five years, did the household earn the rate on the poster? No, and not because the bank did anything wrong. The poster rate was worked out on the understanding that each year's interest goes back in and earns too. Spend it instead and the finishing pile is smaller. The poster rate stays exactly where it was printed.

The poster rate's hidden condition is the reinvestment assumption, and in fixed income the condition is not written on a poster at all. The reinvestment assumption hides inside the yield to maturity, the single figure most readers of a note trust most and inspect least. A yield to maturityOne rate, applied to every payment still to come, that brings their discounted total back to the price on the screen. It summarises a price. It forecasts nothing. is whatever single rate brings the discounted payments back level with the price. Getting from that rate to an actual amount of money in the holder's hands at the end takes something extra, and the extra thing is never printed anywhere near it: every coupon has to go back to work at that same rate, for however many years it has left, right through to year ten.

Work it and the hiding place opens up. The instrument pays Rs 85.00/- a year for ten years, Rs 850.00/- of coupons in all, and returns Rs 1,000.00/- of face amount at the end. Three declared positions on what happens to those coupons, and one unchanged instrument underneath all three.

Read the last column downward. The yield never claimed to know what happens to a coupon after it lands, so the yield did not move once across three completely different outcomes. What moved is the only thing that was ever going to move: what actually became of the money once it arrived.

The first row explains the whole arrangement. With every coupon put back to work at 8.50 per cent, the realised annual return comes out at 8.500000 per cent. The yield hands itself back. Coincidence has nothing to do with that, and neither has checking the sums. The definition is showing its face. The yield is the rate that works if, and only if, everything gets put back at the yield.

Now look at the size of what is being assumed. Of the Rs 2,260.983442/- in the first row, Rs 850.00/- is coupons and Rs 1,000.00/- is the face amount. The remaining Rs 410.983442/- is neither: earnings on earnings. Earnings on earnings are 18.1772 per cent of everything the holder finishes with. Nearly a fifth of the pile at the end was never promised by anybody; it was assumed by the arithmetic. And the distance between the first row and the second, between putting everything back and putting nothing back, is 215.4976 basis points a year on an instrument whose terms never changed.

The realised annual return in that middle column is a geometric meanThe single rate that, applied the same number of times over, would arrive at the same finishing amount as a run of varying rates did.: the one rate which, applied ten times over to Rs 1,000.00/-, lands on the finishing amount. The geometric mean is why putting nothing back to work still gives 6.345024 per cent rather than nothing at all. Money did arrive; it just stopped growing the moment it did.

Could the curve settle the reinvestment assumption?

The sharp question at about this point deserves a straight answer. There is a SPOT curveA set of rates read off today, one for each maturity, each covering the whole stretch from now out to that maturity. in this material, invented like everything else here, with a rate recorded at each of several maturities. Out of any two of those recorded rates it is possible to work out a FORWARD rateA rate for a period beginning on a future date, worked out of two rates that can be read today rather than guessed at.: for instance the rate that covers twelve months beginning once year one has closed. So why not simply use the FORWARD rate covering each future year as the rate that year's coupon goes back to work at, and stop assuming?

Because a FORWARD rate is what a curve implies today, not a promise about what any coupon will actually earn. Using it is a perfectly respectable choice and it is a better documented one than picking a number out of the air. But it swaps one supposition for another with a better pedigree; it does not remove the supposition. The rule that follows is small and it holds everywhere: naming where a rate came from is worth doing, and it is not the same act as making the rate a fact. The pair of recorded rates that would produce such a number does not appear anywhere above, so no number is put on any FORWARD rate and there is nothing to check one against.

Even the clock the reinvestment runs on is a supposition

Here is how deep this goes. Everything above assumes the coupons grow on a once-a-year clock, matching the compounding convention used throughout this material. Suppose instead that each coupon, once it arrives, earns 8.50 per cent a year credited in two half-yearly steps of 4.25 per cent. Nothing about the instrument changes. The coupons are the same size, they arrive on the same dates, and the rate written on them is the same 8.50 per cent.

The finishing amount becomes Rs 2,271.878887/- instead of Rs 2,260.983442/-, a difference of Rs 10.895445/-, and the realised annual return becomes 8.552172 per cent instead of 8.500000 per cent. The gap is 5.2172 basis points, produced by nothing but the frequency at which the earnings are credited. A reader handed only the finishing amount could not tell which of those two clocks produced it. The problem is the one met above, arriving one level further down.

None of that makes the yield a bad measure. The yield is an excellent measure of exactly what it measures: the relationship between a price and a promised schedule. The trouble starts at the moment somebody reads the yield as a statement about what will be in their hands in ten years. Ten years of unknowns sit inside that second question.

What Does the Prepayment Assumption Fix, and How Far Does It Move an Answer?

A shopkeeper who sells on instalments to two hundred regular customers knows roughly what he is owed. The shopkeeper does not know when it turns up. Some customers clear the whole balance the month a bonus lands. Some pay the minimum for a year. He cannot ring two hundred people and ask, and even if he did, what they tell him in March is not what they will do in September. His problem is not the size of what is owed. His problem is the shape of the coming year.

Package a great many loans together and fund them with an instrument, and that shopkeeper's problem becomes the prepayment assumption. Where the money behind an instrument is a set of loans that pay themselves down over time, the speed of that paying down decides when money comes back, and the speed cannot be observed before it happens.

The set of receivables used in this material, Rs 1,200 crore in size, is read under two declared repayment schedules. Under the first, principal comes back in three equal yearly instalments. Under the second, it comes back in six. Nothing else is different. The pool is the same pool, the borrowers are the same borrowers, and the total that comes back is identical.

The two right-hand columns are each year multiplied by the share of the pool coming back in that year, worked from the unrounded share rather than from the printed one, and both columns happen to add to their totals with nothing left over. A column like this will not always add up with nothing left over, so it is worth checking rather than assuming. Added, they give the weighted average lifeThe average number of years before principal is back, with each repayment counting in proportion to its size. of each schedule.

So the weighted average life under the first schedule works out at 2.0000 years. Under the second it works out at 3.5000. The two sit 1.5000 years apart. Nothing about the set of loans is different between those two answers, and nothing on this platform chooses between them, so on this one the supposition is not part of the answer, it is the whole of it. The dependence is more naked in the prepayment assumption than anywhere else, and it is the reason a repayment speed quoted without the schedule behind it tells a reader almost nothing.

What Does the Liquidity Assumption Govern, and Why Does It Sit Under the Other Three?

Somebody lends four lakh to a cousin for a daughter's wedding, on a clear understanding: paid back over three years, a little each Diwali. Six months in, the lender's own roof needs replacing. The four lakh is not gone and it is not in doubt, but it is not available either. Compare that with a deposit at a bank which can be broken tomorrow at the cost of a penalty. The two arrangements differ in one thing only: whether the lender can get out early. Same money, same borrower quality, and a completely different situation for the person who lent it.

Getting out early is the liquidity assumption, and its subject is different in kind from the other three. The call, reinvestment and prepayment assumptions are all statements about what happens on the instrument's own terms, playing out over the whole life of the thing. The liquidity assumption is a statement about whether the holder is still there when any of that happens. So the liquidity assumption decides whether the other three matter at all.

The claim is a stronger one than it first sounds. A holder who cannot leave is exposed to every one of the other three suppositions, all the way to the last date, with no way of stepping out of any of them. A holder who can leave is exposed to none of them beyond the moment of leaving; whatever was going to happen in year seven is somebody else's problem from the day the position changes hands. The other three describe a road. The liquidity assumption decides how much of the road the holder is actually going to be on.

Two things then combine here in a way that is worth sitting with. The liquidity supposition governs the reach of the other three, so it reaches the furthest of the four. And it has the least evidence behind it of any of the four. No volume, no bid and no offer is recorded anywhere in this material, so there is not one observation to lean on. The supposition with the widest consequence is the supposition with the thinnest support. Widest consequence on thinnest support is exactly the combination most likely to get written down as though it were a fact. Separating that supposition from an observation, item by item, is covered separately.

The difference is worth putting in plain structural terms. The other three suppositions are about the instrument's whole life, from the day it was written to the day it ends. The liquidity supposition is about the holding periodThe stretch between buying something and letting go of it, which may be far shorter than the instrument's own life, and usually is.. The holding period belongs to the holder rather than to the instrument, and the instrument's documents say nothing about it at all.

What Do the Four Have in Common That Makes Them So Easy to Miss?

Four properties are shared by all four suppositions, and if the detail fades in a month, these are the four lines worth keeping.

Leaving no trace is the property that makes all four dangerous rather than merely uncertain. Uncertain would be honest: uncertain would show up as a range, or a wider band, or a second figure beside the first. The four produce instead a single confident number with nothing attached, and the confidence is real even though the support underneath it is a supposition.

Which of the four actually bites is settled by the instrument in question, and more than one can bite at once. A plain bullet with no clauses and no pool behind it still carries two of them. Add a repayment clause and there are three. Fund it with a set of loans and all four are live at the same time.

How Is Each of the Four Written So a Reader Can Argue With It?

Writing the supposition down is what turns everything above into something an analyst can do on a Tuesday afternoon. The job takes one sentence, the sentence has three parts, and an analyst who has written it twice does not go back.

Part one names the supposition: the word assumed belongs inside the sentence written, not in whatever the reader infers from it. Part two spells out the conditions under which the supposition holds, in ordinary language, with none of the arithmetic in them. Part three gives the answer under a second supposition, so the reader can see how much of the figure is the supposition and how much of it is the instrument. Almost nobody writes part three. The third part is what converts a claim into something a reader can measure their disagreement against, and without it the first two parts are decoration.

Here it is on the call assumption, written out. The instrument is assumed to run to its final date, which requires the borrower never to use the repayment clause, giving a MODIFIED duration of 6.5613; assumed instead to be cleared after year five, the same instrument reads 3.9406. A reader who thinks the borrower will almost certainly clear early now knows, without opening a spreadsheet and without asking the writer a question, that the disagreement between the two of them is worth 1.6650 times on the risk reading. Handing over the size of the disagreement is the entire purpose of writing it that way, and it costs one clause.

All four go the same way. Here is each of them written so a reader can push back.

Read the fourth row against the other three. Three of them end with a second number. The fourth ends with a stated absence, and that is not a weaker sentence than the other three. The fourth row is the same sentence doing the same job in a case where no second number can be produced. Saying so plainly is the honest version of the practice rather than a failure of it.

Who Actually Uses This, and What Changes for Them?

Four people meet these four suppositions in four different ways, and it is worth seeing each of them, because the abstraction above lands differently depending on which side of the table a person sits.

A lender writing a loan with an early clearing clause is selling that clause, whether or not anybody prices it. The clause is worth something to the borrower and it costs the lender something, and the cost is not a fee. The cost is a reshaping of the schedule the lender was counting on. A lender who prices a loan as though the schedule were fixed and then hands the borrower a right to change it has given something away for nothing. Nothing in this material puts a number on that right, so no number is offered here; what is offered is the observation that the number is not zero.

An analyst writing a note uses all of this as a discipline rather than as a calculation. The practical move is small: before the note goes out, go through it and find every figure that came from a date rather than from an amount. Beside each one, ask what fixed that date, and if the answer is nobody, that figure is carrying a supposition and the sentence around it has to say so. The check takes about ten minutes on a note, and it is the difference between a document somebody can argue with and a document somebody has to take on trust.

Somebody deciding where to put money runs into the reinvestment assumption most often, and usually without knowing it. Two instruments are compared on their quoted yields; one is higher; the higher one is picked. But a quoted yield is a promise about a price and a schedule, not about a finishing amount, and the two instruments will only realise their quoted yields if their coupons find their way back to work at those same rates. If the money is going to be spent as it arrives, neither instrument will do what its yield says. The comparison is still useful, but it is a comparison of two prices against two schedules, not a comparison of two outcomes.

And a household meets the same thing in the simplest form there is. A monthly income scheme, a bank deposit paying interest out, a small savings account with an annual credit: every one of these quotes a rate that quietly assumes the interest goes back in. If the interest is paying school fees, the household will end up with less than the rate suggested, and nobody misled them. The way to hold this is not suspicion of the rate. The better move is one question, asked before signing: does this number assume the interest comes back in, and is the household going to put it back in?

Palash Cements Limited is invented.
Educational material. Not advice on any investment, tax, budget or market position.