Reinvestment Risk: The Risk That Coupons Earn Less

Every payment a bond still owes can be shrunk back to today. A yield to maturity is the single shrinking rate at which those amounts add up to the price paid. Turn the same arithmetic forwards and it says each coupon received goes straight back to work at that rate until maturity. Reinvestment risk is the risk that it cannot. The price never said what those coupons will earn.

One feature makes this worth working through in its own right. A yield is quoted as a rate for the whole life of a bond, but the money a bond pays does not turn up at the end of that life. The money turns up in instalments. Bond A, the invented ten year instrument used throughout below, hands over Rs 85/- at the close of each of ten years and Rs 1,000.00/- of face amount on the last of them. Every one of those early instalments has to sit somewhere between the day it lands and the day the bond finishes. Something happens to it. And the arithmetic that produced the yield has already decided what that something is, without ever saying so out loud.

So the open question is not whether coupons get placed back into the market. The open question is what the yield assumed about the rate they earn when they do, where that assumption is hiding, and what the quoted number becomes when the assumption turns out to be wrong. One input moves and everything else stays still: same bond, same price, same coupons, same dates, nothing missed and nothing defaulted. Only the rate the coupons go back at changes, and the rate the holder ends up with changes with it.

Where inside the arithmetic does the assumption actually sit?

An assumption stated but never shown is an assumption a reader drops. The reinvestment assumption is not a footnote bolted onto the yield. The assumption is the yield, read in the other direction.

A yield to maturity is defined by discounting. Take each payment the bond still owes, shrink it by a factor of one plus the yield, applied once per year of waiting, and add up what comes out. The rate that makes the addition come out at the price is the yield to maturity. Nothing else is smuggled into the definition.

Now turn the operation over. Shrinking a year three coupon by that factor three times over is a statement that the coupon, standing at year three, is the same object as a smaller amount standing today. Run the equivalence the other way and the same statement says that an amount standing today, grown at the yield for three years, becomes that coupon. Which means the yield treats a rupee at year three and a rupee today as convertible in both directions at one rate, and it does that for every date at once.

Pushed one step further, that gives the assumption. If today and year three convert at the yield, and today and year ten convert at the yield, then year three and year ten are joined through today. Year three and year ten therefore convert at the yield too. The Rs 85/- arriving at the end of year three is therefore being treated as Rs 85/- grown at the yield for the seven years remaining. Not because anybody promised it, but because that is the only way the conversions stay consistent across eleven different dates.

Dropping the reinvestment assumption means dropping the arithmetic that produced the yield in the first place. The assumption cannot be detached from a yield to maturity at all. The yield and the assumption are kept together, or both are dropped. There is no third option in which the number survives and the condition attached to it does not.

And the shape of the exposure is uneven in a way worth seeing before any figure is quoted. Bond A's first coupon lands at the end of year one with nine years still ahead of it. Its ninth lands with one year ahead. Its tenth lands on the final date itself, with nowhere left to go. One single rate therefore reaches ten separate amounts and does ten unequal amounts of work on them. The early coupons carry almost all of the exposure and the last one carries none.

What clock is the placing back running on?

Before a single total is worked out, one detail has to be pinned down. Without it, none of the arithmetic below can be rebuilt by anybody checking it. A rate is not a number. A rate is a number attached to a clock, and 8.50 per cent means two different things on two different clocks.

Every figure below runs on one tick a year. Rs 85/- placed back at 8.50 per cent a year is multiplied by 1.085 once for each full year it sits there. The Rs 85/- is not multiplied by 1.0425 twice. The distinction sounds like a bookkeeping detail until the difference is priced.

Read the same 8.50 per cent as two ticks of 4.25 per cent and a rupee left alone for a year becomes 1.0425 times 1.0425, or 1.08680625. Growth of 1.08680625 is 8.680625 per cent for the year, not 8.50, and the extra 0.180625 percentage points came from nowhere except the shape of the clock. Feed that into Bond A's ten coupons and the money held at year ten is Rs 2,271.8789/- rather than Rs 2,260.9834/-. The same words, 8.50 per cent, are worth Rs 10.8954/- more on one clock than on the other, and the holder's realised rate reads 8.552172 per cent instead of 8.500000. The gap of 5.2172 basis points comes from the shape of the clock alone.

The twice a year figure demonstrates what happens when two clocks get mixed, and is not a second convention for the instrument. Bond A has one clock and it ticks once a year. The mismatch is worth working anyway. A reader who copies a rate from one source and applies it on a different clock lands somewhere else and assumes an error was made. The source never said which clock it meant.

So what is reinvestment risk, and who is carrying it?

With the assumption located, the definition costs one sentence. Reinvestment risk is the risk that money a bond pays before it matures cannot be put back to work at the rate the yield assumed. Nothing about the bond failing. Nothing about the borrower being short. Purely about what happens to cash that has already left the borrower and arrived with the holder.

Now the part that surprises people. The holder carries every bit of this, and the borrower carries none of it at all. Consider what the borrower actually agreed to: Rs 85/- on ten named dates and Rs 1,000.00/- on the last one. The list of eleven amounts was fixed at issue and does not lengthen, shorten or change when rates move. Whether the holder puts the Rs 85/- back into the market at 11.00 per cent, at 3.00 per cent, or into a tin under the bed, the borrower pays the identical eleven amounts and has met the contract in full. The uncertainty sits entirely on the holder's side of the table, and it starts the moment the money arrives.

Here is the everyday version, and it is worth holding on to. A small trader lends Rs 1,00,000/- to a supplier and is repaid in ten yearly instalments. Each instalment turns up, on time, in full, exactly as agreed. But the trader now has to do something with instalment one for the next nine years, and instalment two for the next eight, and so on. Whatever the trader manages to do with them is not part of the agreement, was never negotiated, and cannot be enforced against anybody. The supplier has kept every promise. The trader's outcome still depends on nine separate decisions the supplier had nothing to do with.

Two things follow. The first is that the risk runs only from the date money arrives to the horizonThe date somebody is actually measuring their outcome to, which need not be the date the instrument itself ends., the date the holder is finally measuring to. Bond A's holder below is measured at year ten, so a coupon arriving at year one has nine years of exposure and a coupon arriving at year ten has none. The second is that this is a different animal from a borrower failing to pay. A defaultA borrower failing to pay what the contract said, either in full or on the date named in it. Covered separately. is a broken promise; reinvestment risk turns up when every promise is kept.

What happens when the assumption holds exactly?

The case where nothing goes wrong comes before the case where something does. A failure is only readable against it.

Bond A, invented, ten yearly dates, Rs 1,000.00/- of face amount, 8.50 per cent a year written into its terms, bought at the face amount for Rs 1,000.00/-. One tick a year. Every coupon that arrives goes straight back into the market at 8.50 per cent a year and stays there until the final date. Ten coupons of Rs 85/- form a run of equal amounts on equal spacing, the shape an annuityA run of equal amounts falling due at equal spacing, which is the shape a set of level coupons makes. makes. What each one turns into by year ten depends only on how many years it has left.

The ten grown coupons add to Rs 1,260.9834/-. The Rs 1,000.00/- face amount lands on the final date and grows for nothing. Adding it leaves the holder standing on Rs 2,260.9834/- at year ten, the future valueWhat an amount grows to by a later date once a rate has been applied to it for the whole stretch in between. The mirror of bringing an amount back to today. of the whole arrangement.

Now check it by a route that never touches a coupon. Take the Rs 1,000.00/- handed over on day one and compound it at 8.50 per cent a year for ten years. Compounding gives Rs 2,260.983442/-. The two routes agree to four decimal places, and they have to. The agreement is the reinvestment assumption written as arithmetic rather than as a caveat. This is forced, not a coincidence: multiply the pricing statement through by 1.085 to the tenth and one side collapses into the other.

One number is still missing, the rate the holder actually got. Rs 1,000.00/- goes out on day one and Rs 2,260.9834/- stands there ten years later, so the question is which single yearly rate turns the first into the second. The answer is a root, not an average.

Run it on the base case. Rs 2,260.9834/- divided by Rs 1,000.00/- is a multiple of 2.2609834, the tenth root of that is 1.085, and subtracting one leaves 8.500000 per cent a year. A holder of Bond A actually earns 8.50 per cent a year across the whole ten years in exactly one setting, the setting in which the coupons went back at 8.50 per cent.

What happens when only the placing back rate changes?

Here is where the risk stops being an adjective and becomes a figure. Keep everything about Bond A exactly as it was. Same Rs 1,000.00/- paid. Same Rs 85/- on the same ten dates. Same Rs 1,000.00/- of face amount on the last one. Nothing missed, nothing late, nothing renegotiated. Change one input only, the rate the coupons go back at, and work the total again.

The bottom row comes first. Showing only the loss teaches a warning rather than a mechanism. At 11.00 per cent the holder ends with Rs 2,421.3708/-, Rs 160.3873/- more than the quotation implied, and the Rs 1,000.00/- actually earned 9.246147 per cent a year. Set against the quoted 8.50 it stands 0.746147 percentage points over, or 74.6147 basis points. The bond did nothing special to earn it.

Now the top row. At 6.00 per cent the holder ends with Rs 2,120.3676/-, Rs 140.6159/- less, and the Rs 1,000.00/- paid earned 7.805549 per cent a year. The realised rate sits 0.694451 percentage points below the quoted yield, or 69.4451 basis points a year, on a bond that paid every single rupee it promised on every single date it promised. Nobody broke anything. The shortfall came entirely from what happened to money the bond had already handed over.

Where exactly did the Rs 140.6159/- go?

A total that drops by Rs 140.6159/- is easy to state and hard to feel, so break it apart. Nothing was taken from any coupon: all ten arrived at Rs 85/- as promised. The change is in what each coupon turned into by the final date, and the damage is spread very unevenly.

Look at the last column before anything else. The first coupon alone gives up Rs 33.5220/-, nearly a quarter of the whole shortfall from one payment out of ten. The ninth gives up Rs 2.1250/- and the tenth gives up nothing whatever. The money that arrives earliest has the most time to be affected, so the exposure is concentrated there. A cut in the placing back rate is not a haircut applied evenly to ten coupons. The cut is a heavy charge on the first few and almost nothing on the last few.

One honest wrinkle, and it is worth printing rather than smoothing. Add that last column exactly as it appears above and it comes to Rs 140.6157/-. The shortfall worked from the two unrounded totals is Rs 140.6159/-. The Rs 0.0002/- difference is nothing but ten roundings to four places, each nudging its own figure a fraction. The parts are printed as they round and the total is worked from the unrounded amounts. No other version of this arithmetic reproduces itself when somebody checks it.

And if nothing at all is placed back?

The 6.00 per cent case is not the floor. One case sits below it: the holder who receives each Rs 85/- and simply keeps it, earning nothing on any of it. Keeping the cash is not exotic. Keeping the cash is what happens by default to somebody who is not paying attention, and it states most cleanly how much the assumption is worth.

Ten coupons of Rs 85/- kept as cash come to Rs 850.00/-, and with the Rs 1,000.00/- face amount the holder is standing on Rs 1,850.00/- at year ten. The tenth root of 1.85 less one is 6.345024 per cent a year. A bond whose terms say 8.50 per cent, bought at its face amount, delivers 6.345024 per cent a year to a holder who lets the coupons sit, and the gap of 215.4976 basis points a year is the entire value of the placing back. Put the other way: of the Rs 1,260.9834/- the coupons are worth at year ten in the base case, Rs 850.00/- is the coupons themselves and Rs 410.9834/- is what placing them back earned.

Which bonds carry more of this, and why?

The rule falls straight out of the arithmetic above. The more of a bond's value arrives before the last date, the more of it has to be put somewhere at a rate nobody knows yet. Rather than assert that in the abstract, Bond A's own price is split below to show which parts are actually exposed.

The word exposed is easy to get wrong by one date. Be precise about it. A coupon is exposed if it arrives and then has to sit somewhere. Bond A's coupons at years one through nine all do. Its tenth coupon does not. The tenth coupon arrives on the final date itself, alongside the face amount, at the very moment the holder stops measuring. So the exposed part of Bond A is the nine dates before the end, and the unexposed part is the single last date carrying Rs 1,085.00/- of cash.

Bring each of those back to today at 8.50 per cent and the Rs 1,000.000000/- price splits in two. The nine dates before the end account for Rs 520.120325/-. The final date, coupon and face amount together, accounts for Rs 479.879675/-. The two amounts close on the price with nothing left over, and as shares of the price they are 52.0120 and 47.9880 per cent, closing on 100.0000. Slightly more than half of what is being paid for Bond A is money that will have to be put back at a rate that has not been set yet.

Two directions follow from that split, and both can be stated without inventing a second instrument to prove them. A higher rate written into the terms puts more of the value into the coupons and therefore earlier, so it raises the exposure. A longer life gives the early coupons more years to be affected, so it raises the exposure too. Anything that moves value forward in time moves exposure up with it. A bond that pays a lot along the way is therefore more dependent on the assumption than one that pays little.

Why does a bond that pays nothing until the end carry none of it?

Bond B is the invented zero, maturing in 7.1191 years and read at 8.50 per cent a year, one tick a year. Bond B hands over nothing on the way. No coupon, no partial repayment, nothing at all until the final day, when the Rs 1,000.00/- of face amount arrives and the arrangement is over.

Follow the consequence rather than asserting it. There is no cash arriving early, so there is nothing to put back. There is nothing to put back, so there is no rate to be wrong about. There is no rate to be wrong about, so the price the holder pays and the amount the holder receives are the only two numbers in the exercise, and the rate joining them is fixed the moment the price is agreed. A holder who buys Bond B and keeps it to the end earns exactly its yield, and needs no assumption about any future rate to do so.

Pricing it settles the matter. Rs 1,000.00/- brought back 7.1191 years at 8.50 per cent a year means dividing by 1.085 raised to 7.1191, a divisor of 1.787425, and the result is Rs 559.4640/-. Carrying the maturity out further, to 7.119062643353 years, the same operation gives Rs 559.4657/-. Both figures appear above and neither is a slip: the whole of the Rs 0.0017/- between them is how many decimals went into the exponent, and none of it is anything about the bond. Somebody who keys 7.1191 into a calculator, lands on Rs 559.4640/-, and then meets Rs 559.4657/- alongside it will decide a mistake was made. Arithmetic whose only claim is that it can be rerun has no worse outcome available, and the two figures therefore stand side by side. The part that matters more than either figure: each price put back through the same expression at its own maturity reads exactly 8.500000 per cent a year. Decimals in the exponent shift the price by under two thousandths of a rupee and shift the realised rate not at all.

Now the sentence that has to come in the same breath. Leaving it out would turn arithmetic into a suggestion. Removing one uncertainty does not make an instrument better, it changes which uncertainties are left. Bond B carries no reinvestment exposure. Bond B's other exposures, its cost relative to Bond A, and which of the two a holder should prefer are separate questions, settled on evidence held elsewhere. Bond B is the control case in an experiment about one variable. The control case is not a conclusion.

Is there one setting where the quoted rate is the earned rate?

There is exactly one, and finding it is the fastest way to see where the assumption has been hiding all along. Push the placing back rate across a wide stretch and plot what the holder actually realises against it. More on the coupons means more at the end. The result is a rising line, and it crosses the quoted 8.50 per cent at precisely one point.

The single crossing carries the whole idea: a quoted yield to maturity is the rate actually earned only at the one setting where the coupons go back at the quoted yield itself, and a different number at every other setting. Everywhere left of the crossing the holder does worse than the quotation; everywhere right of it, better. Nothing about the bond changes anywhere along that line.

Two positions of that control are worth taking. At the far left the nine right hand blocks shrink and the dark block on the left refuses to budge. At 3.00 per cent the holder ends on Rs 1,974.4297/- and realises 7.039524 per cent a year, 146.0476 basis points under the quotation. At the far right the stack pushes well past the dashed reference: Rs 2,643.6701/- and 10.209930 per cent, or 170.9930 basis points over. The Rs 1,085.00/- arrives on the last day and is never placed back at anything, so the dark block reads Rs 1,085.00/- at every one of the 221 settings.

How does anybody actually use this?

Start with the version closest to home, the one that costs real people real money. A household decides to put Rs 1,000.00/- aside for something ten years out: a wedding, a fee, a deposit on a place to live. The household finds something quoting 8.50 per cent a year, does the arithmetic once, and writes Rs 2,260.98/- in a notebook as the amount that will be there when the date arrives. Everything about that calculation is correct except one thing. The calculation is not a plan for one decision but a plan for ten. There is the decision to buy, and then nine more decisions about what to do with each coupon as it lands, spread across nine years in which nobody knows what will be on offer.

So the first practitioner move is simply to say the condition out loud whenever the number is written down. Not "8.50 per cent a year", but "8.50 per cent a year if each coupon goes back at 8.50 per cent a year". The two sentences are different, and only the second one is true. A household that writes the second version has already stopped treating a quotation as a promise, and the habit is most of the protection available.

The second move is to stop planning on a point and start planning on a stretch. Somebody reading Bond A properly does the arithmetic three times, not once: at a rate well below the quotation, at the quotation, and at a rate well above it. Three passes produce Rs 2,120.3676/-, Rs 2,260.9834/- and Rs 2,421.3708/- instead of a single figure, and a household looking at a spread of roughly Rs 300/- on Rs 1,000.00/- committed will make a different decision about how much to set aside than one looking at Rs 2,260.98/- alone. The useful output of this arithmetic is not a better single number, it is the width of the stretch that single number was hiding.

The third move belongs to anybody comparing two things to hold. A quoted yield to maturity is a fair comparison between two instruments only where the assumption behind it is equally reasonable for both, and the split above shows it usually is not. A bond with more than half its value arriving early depends heavily on nine future decisions. A bond that hands over everything on one date depends on none. Ranking those two on quoted yield alone compares a conditional number against an unconditional one and calls the result a difference in value.

The fourth move is where this stops being an individual's problem. Somebody managing money against a date that is already fixed, such as an amount owed at a known point in the future, can attack the exposure directly by arranging what they hold so that money lands on the dates money is wanted rather than earlier. The approach is called cash flow matchingArranging holdings so that money lands on the dates money is actually needed, rather than earlier or later. Covered separately., and it is worked out separately. The exposure has a structural answer as well as a warning.

What can never be settled in advance?

The limits come last rather than first, and they mean more once the arithmetic has moved.

No arithmetic settles what a coupon will actually earn on the day it lands. The rate is set years later, by a market that has not opened yet. Every case above holds one rate fixed for the whole ten years, and no real market does that. How rates behave over stretches of time is a separate subject, settled on measured series rather than on a worked example. The 6.00 and 11.00 per cent settings used throughout are inputs chosen to make the arithmetic legible, and they are not predictions, estimates, central cases or the edges of a likely range. Neither is the 3.00 to 14.00 per cent span on the control, picked to put the crossing comfortably inside it with room on both sides.

The restraint is not modesty. A shortfall of 69.4451 basis points shown alongside a claim that 6.00 per cent is what to expect would replace checkable arithmetic with a guess that cannot be checked, and the guess would look authoritative precisely because the arithmetic beside it was sound. The narrower result is the more useful one. The assumption has a name, a location inside the yield, a figure for what it does to the realised rate in either direction, and a split of the price into the part that is exposed and the part that is not.

Bond A and Bond B are invented.
Educational material. Not advice on any investment, tax, budget or market position.