Prepayment Risk and Extension Risk: When Cash Flows Move

Prepayment risk is the chance a borrower hands the money back early, arriving just as the rates it could be put back to work at have fallen. Extension risk is the chance repayment slows instead, leaving the holder with a low paying instrument just as better rates appear. One shortens the schedule, one lengthens it, and neither moves in the holder's favour.

Underneath both sits a single assumption that usually goes unstated. Every price, every duration and every convexity worked so far treated the list of dated amounts as immovable, and let the discount rate do all the moving. Drop that, and the measurement itself starts responding to the rate it was supposed to be measuring.

What was holding perfectly still, and nobody mentioned it?

Consider any priced bond covered so far. There was a list of dated amounts, and there was a rate used to discount them. When the rate changed, the price changed. The demonstration was correct as far as it went. The list itself sat there without flinching through every one of those repricings, and nobody said so.

The stillness of the schedule was an assumption the arithmetic needed, not a property that bonds have. A contract fixes ten dates and ten amounts for the ten year bullet bond and gives nobody the right to alter them, and the schedule really does stand still. A still schedule is a feature of that particular instrument. Plenty of lending does not work that way at all.

The difference lives in ordinary lending. Consider a housing loan on any residential street. The borrower has a schedule, the schedule has dates, and every one of those dates is written down. The borrower also has the right to walk into the branch with a lump sum and close the loan on a Tuesday afternoon in year four. Nothing about that is exotic and nothing about it is a default. Early repayment is a right the borrower bought with the loan, and the right makes the list of dated amounts on the lender's side a forecast rather than a contract.

Some borrowers can choose when they repay, and a borrower who can choose will choose the moment that suits the borrower. Everything that follows falls out of that sentence, including the part that turns a comfortable measurement error into an uncomfortable one.

What exactly is prepayment risk, for a reader meeting the term for the first time?

Prepayment risk is the risk that the money comes back to the holder earlier than the schedule said it would. The definition stops there. Nothing has gone wrong, nobody has failed to pay, and no promise has been broken. The borrower simply used a right they always had, and the dated amounts counted on for years five through ten arrive in a single lump in year four instead.

A borrower has a reason to use the right only after rates have fallen. Prepayment arrives at that moment and at no other. Seen from the borrower's side of the table: the borrower is paying 8.50 per cent a year on a loan taken out some time ago. The lender across the road will now write the same loan at 6.50 per cent. The borrower takes the cheaper loan, uses it to close the expensive one, and is two percentage points a year better off for the rest of the term. Nobody needs a theory of interest rates to do that. The same instinct makes a household move a balance from one card to another.

Back on the lender's side, where the holder of the instrument sits. The cash lands in the holder's hands at exactly the moment the rates it can be put back to work at are lower than the rate that was being paid. The holder loses an instrument paying 8.50 per cent and can replace it only with something paying less. The loss is not the lump sum. The lump sum arrives in full. The loss is everything that was going to be earned between year four and year ten and now cannot be.

The everyday version is a tenancy. A landlord has a tenant on a long lease at a rent the landlord is happy with. The tenant gives notice and leaves early, and leaves precisely in the year when the rent the place could be re-let at has dropped. The property comes back, nothing owed has been lost, and the landlord is unambiguously worse off. Nobody wronged the landlord. The timing did.

And what is extension risk, taken on its own?

Extension risk is the risk that the money comes back to the holder later than the schedule said it would. Same structure, opposite direction. The borrower who was expected to close in year four does not close in year four, does not close in year five either, and stays exactly where they are until the contract makes them leave.

A borrower sitting on cheap borrowing has every reason to sit still, and extension risk arrives once rates have risen. Back in the borrower's chair: the borrower is paying 8.50 per cent on a loan taken out some years ago, and the same loan today would cost 10.50 per cent. Refinancing would be an act of self harm. The borrower keeps what is already in hand, makes the minimum payment, and stays for the full term. Again, no theory required.

Back on the holder's side, the holder is now locked into an instrument paying less than the alternatives that have just appeared, and locked in for longer than planned. The money expected in year four is somewhere else until year ten. The money is not free, so whatever is being offered elsewhere at the higher rate is out of reach.

The same tenancy makes the point again, with the terms reversed. Rents in the area have climbed a long way since the tenant signed, so the lease has become a bargain. The tenant is not going anywhere. The lease will be held to its last day, and on every one of those days the landlord is collecting a rent below what the market would pay. Once more, nobody has done anything wrong.

What four questions separate the two risks?

Before running the comparison, the questions have to be fixed. A comparison that changes its questions halfway through is not a comparison at all, and this is one of the places where a reader who half remembers the material will merge the two risks back together.

Four questions do the work, and the fourth is the one that matters most. Which way did the rate move. Which way does the schedule move in response. Which loss does the holder actually take. And what happens to the duration that was measured before any of this began. The first three separate the two risks cleanly and are more or less common sense once the definitions are in place. The fourth is where the trouble is. For both risks the answer points the same unhelpful way.

Run them in that order, on both risks, without reordering them. The grid below does exactly that.

How do the two risks answer those four questions?

Take them one at a time. On the rate, they are opposites: prepayment follows a fall in the yield, extension follows a rise. On the schedule, they are opposites again: prepayment takes dates off the far end of the list, extension puts dates back on. On what the holder gives up, they are opposites a third time, though the two losses are shaped differently. Prepayment removes a good instrument and offers only a worse replacement. Extension keeps a poor instrument and blocks the better one.

Then comes the fourth question, and the pattern breaks. The measured duration moves the wrong way in both directions, and that is the row worth carrying away. Where the yield falls, the schedule shortens, so the measured duration shortens, so the holder carries less sensitivity at precisely the moment a longer position would have gained the most. Where the yield rises, the schedule lengthens, so the measured duration lengthens, so the holder carries more sensitivity at precisely the moment a shorter position would have lost the least.

The fourth row bears rereading. Three of the four questions give two clean opposites. A comparison usually delivers exactly that. The fourth gives one shared disadvantage wearing two costumes. Prepayment risk and extension risk have in common neither the direction of the rate nor the direction of the schedule. The shared thing is the direction of the harm.

Which instruments actually carry this, and which only look as though they do?

A test that gets used a lot, and gets used wrongly, is whether the instrument repays its principal gradually. Many readers file amortisingRepaying the principal in slices across the life of the loan rather than in one block at the end, so each payment is part interest and part capital. instruments under prepayment risk on sight, and stop there. The gradual repayment test is not the right one, and it will sort instruments into the wrong boxes.

Gradual repayment does not create either risk. Somebody else holding the right to decide the timing does. A loan that repays a slice of principal every year on a schedule written into the contract has a moving balance and a completely still list of dates. Nobody can bring a rupee of it forward and nobody can push a rupee of it back. The only thing that can change is the discount rate, so its duration is measurable today and will be the same measurement tomorrow.

Turn the test around and it becomes reliable. Ask whether anybody on the other side of the arrangement can change a date. If a borrower may repay early, or a borrower may be permitted to defer, or an issuer holds an embedded optionA right written into the instrument itself, held by one of the two sides, letting that side change something about the deal later. Callable and puttable structures are covered separately. to retire the instrument early, the dates are a forecast. If nobody can, the dates are a contract. The question about who may change a date does the sorting, and it puts an ordinary fixed schedule loan that pays down principal every year into the same box as the ten year bullet bond, exactly where such a loan belongs.

What does a moving schedule do to the shape being measured?

Now the payoff. A duration laid against a price curve is a straight edge held against something bent. On a fixed schedule bond the miss that produces has a very particular shape. The straight edge says the price will fall by a certain amount where the yield rises, and rise by the same amount where the yield falls. The bond does neither. The price falls by less than the line promised, and rises by more.

On the ten year bullet bond, the straight line is wrong in a direction that flatters the holder on both sides of the starting yield. The holder is pleasantly surprised where the yield falls and less badly hurt where the yield rises. The comfortable one sided error has a name, settled separately: the price curve lies above the straight line on both sides. The size of that comfort is worked out below in rupees and in points.

The price scale has to be honest about a bond that starts at Rs 1,000.00/-, so the gap at each end of that drawing is real but small. So the next drawing does the only fair thing available: it keeps the same yield axis and plots nothing but the distance between the two lines, on a scale of its own. The distance is magnified, and neither the prices nor the rates are touched.

Now change one thing and watch the whole picture invert. Suppose the borrower refinances when the yield falls and stays put when the yield rises, so the schedule shortens on one side and lengthens on the other. The gain is cut short exactly where the fixed schedule bond was collecting its bonus, and the loss runs on exactly where the fixed schedule bond was being spared.

Take the two halves separately. Where the yield falls, a longer position gains more and is the one worth having. Instead the instrument is handed back and the position gets shorter. The upward part of the curve gets clipped. Where the yield rises, a shorter position loses less and is the one worth having. Instead the borrower settles in and the position gets longer. The downward part of the curve keeps going.

Put those together and the price curve now lies below the straight line on both sides rather than above it. The shape has a name of its own, and the name is negative convexity. The effect on the measuring tools is the sentence to carry away. The straight line estimate on a fixed schedule bond is wrong in a known direction that works in the holder's favour. On an instrument whose dates can move, the same estimate is wrong in a direction that does not.

Why can the fixed schedule bond finish ahead on both sides?

One side can be worked. The ten year bullet bond used throughout this material has Rs 1,000.00/- of face amount and pays 8.50 per cent a year across ten annual dates, compounding once a year. Bought at par, its yield is 8.50 per cent as well. Every reading below comes out of those ten dated amounts and nothing else.

Now move the yield 200 basis points each way and reprice the same ten amounts. A basis point is a hundredth of a percentage point, so 200 basis points is a move of 2.00 percentage points, and the two repricings are done at 10.50 per cent and at 6.50 per cent a year.

Smaller loss, bigger gain, ahead of the straight line on both sides, and the structural reason is that not one of those ten amounts moved. The contract fixes them. No borrower can bring a rupee forward and no borrower can push a rupee back. Only the discounting responds to the yield in that whole calculation, and a bond whose ten amounts hold still can beat the estimate in both directions at once.

How wide is that comfort? There are two ways to work it and they do not agree. The disagreement is worth stopping on. Work the miss from the unrounded repricings, never by subtracting the two printed percentages.

Look at the two rows before the total. The second row agrees exactly, so a reader who reached for the subtraction on that side got the right figure and has no way of knowing they took a route that does not generally work. The first row misses by two ten thousandths of a point. The miss is small enough to look like nothing and is not nothing: it is the whole method failing, visible on one row only. A right answer standing over working that never produced it is exactly the thing a checking reader has to catch. The printed 12.030 and the printed 13.123 are display figures. Neither is an input.

What figure can be given for an instrument whose dates move?

None. Not a duration, not a convexity, not a price move, not a single number of any kind. The blank is a decision rather than an oversight, and the reason is worth more than the figure would have been.

Putting a number on either risk requires a prepayment speedAn assumed rate at which borrowers in a group repay ahead of schedule, usually stated as a fraction of the balance retiring each period. Nothing in this record supplies one.: an assumption about how fast borrowers actually repay ahead of time. A speed is a statement about human behaviour, not a term written into any contract, and nothing in this record supplies one. There is no such speed here, no series from which one could be estimated, no schedule that repays principal in slices to apply it to, and no distribution of losses for the pooled structure that does appear.

The shape is the teachable part, and the shape survives without a speed. Consider the alternative honestly. Choose a speed, and out drops a duration, a convexity and a neat pair of price moves, printed in the same typeface, sitting in the same table, indistinguishable in print from the figures above that came out of ten contracted amounts. A reader has no way to tell them apart. An invented input produces an output that wears the clothes of a measurement, and that is a worse outcome than a blank.

The second half of the refusal is a different kind of blank. Authorities set what a securitisationThe practice of pooling many small loans together and funding the pool by issuing instruments against it, so holders receive what the pool collects. must set out about its expected payments, and what prepayment assumption a supervised holder has to apply if any, and they revise that wording on their own timetable. Neither of those is a gap in the arithmetic. Both are rules, and rules move. A recital from memory does not go stale. A recital goes false, on whatever morning the keeper picks.

Who actually has to act on this, and what do they do with it?

Three people meet this in a working week, and none of them meets it as a shape on a chart.

Start with a lender who writes housing loans and funds them with deposits. The loans run for twenty years and the deposits run for three, so the lender is already managing a mismatch in the ordinary way. Prepayment risk makes the shorter side of that mismatch unpredictable, and unpredictable in a correlated direction: when rates fall, a great many borrowers refinance in the same few months, and the lender is handed a great deal of cash at once, all of it to be redeployed at the new lower rates. The problem is not that some borrowers repay early; it is that they all do it in the same quarter. A lender who has planned for an average speed and not for a bunched one has planned for the wrong thing.

Then an analyst reading a fact sheet. There is a line on it saying the holding has a MODIFIED duration of some figure, and that line looks exactly the same whether the underlying instruments have contracted dates or forecast ones. The analyst's job here is not to recompute anything. The job is to ask one question of the person who produced the figure: what was assumed about early repayment, and where did that assumption come from? A duration quoted without its schedule assumption is a number with a hole in it, and the hole is invisible in print.

Then a household. Prepayment and extension are not only institutional subjects. A household with a home loan, in a year when rates fall, is the borrower in every paragraph above, and the right to repay early is the valuable thing it holds. A household with money in a long dated instrument that can be called away early is the lender, and should expect the money back at the least convenient moment. Both sides of this subject are already in an ordinary Indian household's balance sheet. The vocabulary is the only thing that is new.

What can a pooled structure teach here, and what can it not?

One pooled structure sits behind this material, and it is worth using carefully because it is easy to over read. A pool of Rs 1,200 crore of receivables is funded by three pieces: a senior piece of Rs 960 crore, a mezzanine pieceThe middle claim in a stack of claims on one pool. The middle claim is met after the claim below it has been wiped out and before the claim above it is touched. of Rs 180 crore, and an equity piece of Rs 60 crore. Against the Rs 1,200 crore base those are 80.0 per cent, 15.0 per cent and 5.0 per cent.

The three percentages are an order in which losses are met, and they are not shares of the pool. The distinction between an order and a share does all the work. Losses reach the equity piece first, then the mezzanine piece, then the senior piece. Rs 180 crore and Rs 60 crore set against the Rs 1,200 crore pool come to Rs 240 crore, or 20.0 per cent of the pool, and 20.0 per cent has to be destroyed before the senior piece gives up a single rupee. A pool loss of 5.0 per cent is Rs 60.0 crore on that base, and wipes the equity piece exactly without touching anything above it.

Here is the part to hold back on. The originatorThe firm that made the underlying loans in the first place and then sold them into the pool. and the servicerThe firm that collects the instalments from the underlying borrowers month by month and passes what it collects into the pool. both keep records of what the pool collects and when. None of that reaches this material. There is no distribution of losses here, no assumption about how the underlying borrowers move together, and no schedule of collections period by period. So the order alone supports no probability that any piece is touched, no grade for any piece, and no assertion that the senior piece is safe. The order is the teaching. The odds are not in it, and no behaviour about early repayment is attributed to that pool anywhere either.

What would a control over prepayment speed have required?

A control over these curves would have meant choosing a speed at which borrowers repay ahead of schedule, then choosing how that speed responds to the rate, then drawing the curve those two choices produce. All three would have been invented, and a reader dragging a slider across them would come away believing they had watched a relationship rather than a pair of guesses. Naming the missing speed teaches more than a slider built on an invented one would have. The ninth question above asks the reader to perform the distinction instead, on a pool no odds are attached to.

The ten year bullet bond and the pool funded by its senior, mezzanine and equity pieces are invented.
Educational material. Not advice on any investment, tax, budget or market position.