The Coupon and the Yield: Why They Match Only at Par

Nobody can change a coupon rate after the day it was written down, so it sits still. A yield is worked backwards out of whatever price is being paid right now, so a yield does not sit still. Put the two side by side and they read the same number in exactly one case, the case where the price has come out equal to the face amount.

One of those two numbers was settled on the day the instrument came into existence and cannot be touched by anybody since. The other is the output of an arithmetic search run on today's price, and it is a different output every time the price is different. Nothing joins them together except one special case. The two numbers share a unit, they do not share a nature, and almost every confusion on this subject starts with somebody treating the shared unit as evidence of a shared nature.

Every sum below is set up with every term showing, evaluated, then evaluated again with exactly one input changed. The addition is shown rather than the total. Somebody who has watched eleven amounts pile up into Rs 1,000.00/- can rebuild that pile on their own paper. Somebody handed the Rs 1,000.00/- on its own cannot check it and, in practice, will not try.

Four terms are worth pinning down before the arithmetic starts. A spot rate runs from today out to one stated date and stops there. A forward rate does not start today at all. A forward rate covers a stretch of time beginning on some future date, so a forward rate is a different object even when it reads a similar number. Take one percentage pointThe unit a rate is measured in. Going from 7 per cent to 8 per cent is a move of one percentage point, whatever the starting level happens to be., cut it into a hundred equal slices, and one slice is a basis point; several of the gaps below are quoted in slices rather than in whole points. And where annual compounding is stated here, the discounting clock ticks a single time in a year. Every price further down is struck that way, and each of them repeats it on the spot.

What is a coupon rate, and what is the base it sits on?

Bond A, an invented ten year bond, carries an 8.50 per cent annual coupon on Rs 1,000.00/- of face. Read that sentence again slowly. Two things in it decide everything that follows. The rate is 8.50 per cent. The base the rate is applied to is the face amount of Rs 1,000.00/-. Not the price. Not what the last buyer paid. The face amount, and nothing else.

Multiplying the two does not give another rate, it gives a rupee amount: 8.50 per cent of Rs 1,000.00/- is Rs 85/-, and Rs 85/- is what the issuer hands over once a year. That is the crucial move, and it is worth being slow about it. The moment the rate meets its base it stops being a rate. The rate becomes cash, on a date, of a stated size. Rs 85/- is Rs 85/- if the bond changes hands tomorrow at Rs 700.00/-, and it is the same Rs 85/- if it changes hands at Rs 1,400.00/-. The buyer, the price and the calendar have no vote in it.

Now think about what that means for the person on the other side of the arrangement. Suppose a shopkeeper agrees, in writing, to pay a fixed Rs 85/- a year to whoever is holding a particular receipt, and to hand over Rs 1,000.00/- with the tenth of those payments. The receipt then changes hands four times over the decade, at four wildly different prices, in four different moods. How much does the shopkeeper pay in year seven? Rs 85/-. The prices were conversations between other people. A coupon rate is a term of an agreement, and a price is an opinion about that agreement, and the agreement does not read the opinion.

The word rate can be unhelpful here for exactly that reason. Everybody says a bond pays 8.50 per cent, and that phrasing quietly hides the base. Read it out in full every time and the trap disappears: 8.50 per cent a year of Rs 1,000.00/- of face amount. The period is a year, the base is the face, and the product is Rs 85/-. Every one of those three has to be said out loud to make the number mean anything at all.

What is a yield, then, and where does it come from?

Here is a definition that will hold up under pressure. A yield is the one rate at which every payment still to come, each pulled back to today by however many years it has to travel, adds up to precisely the price being paid. The definition ends there. A yield is not a feature of the bond, it is printed on nothing, and the issuer has never once heard of it.

The direction of that sentence matters. Reversing it is the commonest confusion on the subject: the price comes first, and the yield is dug backwards out of it. Nobody sets a yield. Two people agree on a price, and once they have, a yield exists as a consequence, in the same way that a speed exists once a distance and a time are known. The driver never chose the speed. The driver chose to drive somewhere, and the speed fell out.

How the yield is found is worth a moment. The yield cannot be made the subject of that expression and read off. There is no formula waiting to be transposed. The search is a hunt: a rate is guessed, the eleven amounts are priced with it, the total is compared against the price paid, and the guess is adjusted for another attempt. A machine does this in microseconds and a person can do it in about four guesses. Either way, it is a hunt, not a calculation, and the fact that it is a hunt is the clearest evidence that a yield is an answer rather than a term.

Everyday version, and it transfers exactly. A shop's lease has ten years to run and the rent written into it is fixed. Somebody now offers to take that lease over from the holder for a lump sum. Whatever they pay, the rate a year that lump sum has earned them against the rent they will collect can be worked out afterwards. A different lump sum gives a different rate. The rent in the lease never moved once through any of it, and neither did the landlord's cheque.

Which one is fixed, and which one gets worked out today?

The contrast is worth laying out property by property rather than describing. The contracted rate of Bond A was settled at issueOn the day the bond was created and its terms were written down. Nothing settled at issue can be revised afterwards, by the issuer or by anybody else.. The contracted rate is applied to the face amount. The contracted rate comes out as cash. The contracted rate is identical in year one and year nine. Nothing that happens in any market reaches it. The yield of Bond A is computed today, from today's price. The yield never meets a base and never becomes rupees, so the yield is applied to nothing whatsoever. The yield exists only as a rate. The yield is a different number every time the price is different. And the yield is manufactured entirely out of what somebody was willing to hand over.

The only thing those two columns have in common is that both are quoted as a rate a year. The shared unit is exactly why they are confused, and exactly why the base has to be named beside each one. Say the base out loud and the confusion cannot survive. The contracted rate is 8.50 per cent a year of the face amount. The yield is 7.3500 per cent a year of nothing: it is the rate that balances a sum, and there is no pile of rupees anywhere that it is a percentage of.

The everyday version is a shop again, and it is worth running because it is the same structure with the finance stripped off. A shop signs a ten year lease at a rent fixed for the whole ten years. The rent is a rate a year, written into the document, and it will read the same in year eight as it did in year one. Meanwhile, what somebody would pay today to step into that lease moves every few weeks, with the street, the footfall and the mood of the market. Converting that lump sum into a rate a year gives a second number, also expressed as a rate a year. Two rates, same unit. One is in the document. The other is an opinion that changes on Thursday.

The extreme case, which settles the argument on its own

The cleanest possible proof that these are two different objects is to take one of them away and see whether the other survives. Bond B, an invented zero coupon bond maturing in 7.1191 years, has no coupon at all. Its contracted rate is 0.0000 per cent of face. There are no payment dates before the last one, and no cash arrives at any point in between. On the terms card of Bond B, the coupon line reads zero.

And yet Bond B has a yield of 8.5000 per cent a year. The gap against a contracted rate of nothing at all is 8.5000 percentage points, or 850 basis points. The gap is not a defect of Bond B and it is not a warning about it. The gap is simply what happens when the coupon is deleted and the price left alone. The yield had no dependence on the coupon in the first place, so removing the coupon did not remove the yield. Every rupee of Bond B's earning is the climb from what is paid today up to the face amount that arrives at the maturity dateThe date the final payment falls due and the arrangement ends. On a zero coupon bond it is the only date anything arrives..

Notice also what Bond B cannot do. Bond B cannot trade at par on any date before its last one. A bond priced at its face amount hands back exactly the money that was handed over, plus whatever cash arrives on the way, and Bond B has no cash arriving on the way. So a price equal to face would mean a yield of zero. A yield of zero describes a different bond from the one on the terms card. The one case where the two numbers agree is unavailable to an instrument that has only one of them. What matters about Bond B is its shape rather than its rupees.

Why do the two read the same number at par?

Two objects just established as unrelated are now sitting there reading 8.5000 and 8.5000. The match deserves better than a shrug, and it is not a coincidence of invented figures. The match is forced, and the forcing can be watched step by step.

Do the addition. Take Bond A's eleven amounts: ten payments of Rs 85/- on years one to ten, plus the Rs 1,000.00/- face amount arriving with the tenth. Pull each one back to today at 8.50 per cent a year, with the discounting clock ticking a single time in each year, so an amount at year seven is divided by 1.085 seven times over. The multiplier in the third column is the discount factorThe multiplier that turns one future rupee into what it is worth today. The factor gets smaller the further out the date is. for that year, and the fourth column is the second column multiplied by it. The fourth column is then added up.

Rs 1,000.000000/- is the face amount, and the sum landed on it exactly. A price of Rs 1,000.00/- and a yield of 8.50 per cent a year are therefore the same statement written two different ways. One detail is worth pausing on, and it does not always hold. The eleven amounts above are printed to four places, and adding the printed column on a calculator gives Rs 1,000.0000/- with nothing left over. The exact agreement is unusual. Rounded parts normally leave a residual against an unrounded total, and on other yields they do. Here the eleven roundings happen to cancel, so the printed column and the true total agree, and the whole thing can be checked without any hedging.

Now the same argument in words. Somebody hands over Rs 1,000.00/-. The buyer collects Rs 85/- a year for ten years, and on the last date gets the Rs 1,000.00/- back. At the end they hold their own money, returned intact, plus ten coupons. The only thing they ever earned beyond their own money is the coupon, so the rate describing what they earned can only be the contracted rate, and that equality is the whole meaning of the words at par.

The claim is worth testing on yourself. Change the contracted rate to 5 per cent, or to 14 per cent. Change the ten dates to three, or to thirty. As long as the yield being used equals the contracted rate, the eleven amounts still add to the face amount, every time, with no exceptions and no near misses. The identity has nothing to do with 8.50 and nothing to do with a decade. The identity is what happens whenever a stream is discounted at the rate that generated it.

One more thing falls out of the same block of arithmetic, and it looks like a coincidence until it is checked. The eleven amounts, before any discounting, come to Rs 1,850.00/-: ten coupons of Rs 85/- plus Rs 1,000.00/- of face. Discounting knocks that down to Rs 1,000.00/-, so discounting removed Rs 850.00/-. And Rs 850.00/- is exactly the ten coupons. The match is not a coincidence either: the pile is ten coupons plus the face, the price is the face, so the difference has to be the ten coupons, whatever the rate happens to be. The match is forced by the subtraction, not discovered in the numbers. Watch the two come apart the moment the price is anything other than the face amount. At any other price the subtraction is no longer forced.

Two things are worth doing with that control. Dragged slowly through Rs 1,000/-, it pinches the shaded band shut and opens it again on the other side; that pinch is the entire subject of this guide. Dragged as far as it will go in either direction, it leaves the bottom panel exactly as it was. Ten bars of Rs 85/-, one block of Rs 1,000.00/-, unchanged. The yield has moved across more than ten percentage points without a single rupee the issuer owes moving at all.

What happens to the pair when the price moves?

Work it twice, changing exactly one input each time and leaving everything else where it was. Take Bond A to a price of Rs 1,079.4804/-. Run the hunt on that price and the yield comes out at 7.3500 per cent a year. Now look at the contracted rate. The contracted rate is 8.50 per cent struck on a face amount of Rs 1,000.00/-. The cash comes to Rs 85/-. The payments fall on the same ten dates as before. Nothing there moved. The two numbers were sitting on top of each other a moment ago and are now 1.1500 percentage points apart, or 115 basis points, with the contracted rate on top.

Now go the other way. Take Bond A to a price of Rs 931.2325/-. The yield comes out at 9.6000 per cent a year. And the contracted rate is 8.50 per cent of Rs 1,000.00/- of face, or Rs 85/-, on ten dates. Still. The gap is now 1.1000 percentage points, or 110 basis points, and this time the yield is the one on top.

Read the second column downwards and it never changes; read the third column downwards and it changes every time; and the only row where the fourth column reads zero is the row where the price came out equal to the face amount. That zero is not a rounding artefact and it is not a near miss. The gap reads 0.0000, and the row it sits on is what the words at par mean, written as a table entry instead of as a sentence.

An intuition sits underneath the table, and having it in words saves rerunning the arithmetic every time. Paying more for the same set of payments puts more money in to collect the identical cash out, so the rate earned on the money must come down. Paying less pushes it up. The sloping line in the control above was never doing anything more than that. The contracted rate draws a flat line not because it is stable in some admirable way, but because the price was never one of its inputs.

The household version is a fixed deposit receipt that a relative wants to buy off the holder. The bank will pay whoever holds it a fixed sum on a fixed date, and no negotiation between holder and relative can alter that sum by a rupee. If the relative pays generously, they will earn a low rate a year on their money. If they drive a hard bargain, they will earn a high one. The bank's cheque does not know which of those two conversations happened, and the printed rate on the receipt does not either.

Does the issuer pay any more when the yield goes up?

The question deserves a head on answer, and most treatments of the subject skip straight past it. The yield on Bond A moves from 8.5000 per cent a year to 9.6000 per cent a year. Ask what the move costs the issuer.

Nothing. Not one rupee, not on any date, not ever. The issuer hands over Rs 85/- a year and Rs 1,000.00/- with the last of them, which is precisely what it handed over before, and precisely what it will hand over if the yield goes to 14 per cent tomorrow afternoon. The obligation is a fixed set of rupee amounts on fixed dates, settled at issue, and a yield is not one of the levers attached to it.

Something else entirely changed, and naming it correctly settles the question. The price somebody would have to pay today to step into that stream of payments is what changed. Somebody stepping in now pays Rs 931.2325/- rather than Rs 1,000.00/-, and they will collect exactly the same Rs 85/- a year that the previous holder was collecting. The movement happened between the buyer and the seller, and the issuer was not a party to that conversation.

The everyday version is one most households have lived through. A household takes a fixed rate loan and the instalment is set. Rates in the wider world then move, in whichever direction. The instalment was fixed, so it does not move, and the household writes the same cheque it has always written. The lender's position has genuinely changed: it is now stuck lending at yesterday's rate when it could be lending at today's, or the reverse. The two facts sit side by side without contradicting each other, and separating them is the same separation as the one between a contracted rate and a yield.

The direction of that comparison is easy to overrun, so here is one caution. A yield movement does not make the issuer better off or worse off, and a bond bought above the face amount is not thereby a better or worse thing to hold than one bought below it. The arithmetic on both is identical and only the price differs. The narrower and firmer claim is this: price sensitivityHow far a price shifts when a rate shifts. Duration and convexity are the tools that measure it. is a question about the price, and the payments are not part of it.

Why does the compounding convention have to sit beside both numbers?

Everything above has been struck on annual compounding: one discounting tick a year, so an amount is divided by 1.085 once for every year it has to travel. The convention has been stated beside every price for a reason. The two numbers being compared are quoted in the same unit, and a unit with no convention attached to it is not yet a number, so the convention is load bearing here in particular.

The coupon side comes first. Bond A's terms say Rs 85/- once a year. They could have been written instead to pay Rs 42.50/- twice a year. Added up either way, the cash comes to Rs 85/- a year, every year, ten years running, Rs 850.00/- in total. Identical. So the two are the same arrangement?

The two are not the same arrangement, and here is the arithmetic. Value that half yearly stream on the identical once a year clock at 8.50 per cent a year, with the face amount still arriving at year ten, and it prices at Rs 1,011.6098/-. The price sits Rs 11.6098/- above the face amount, produced by nothing whatsoever except half the cash arriving six months earlier than it used to. Same rupees, same rate, same ten years, same discounting clock. Only the dates moved, and the dates were worth Rs 11.6098/- on a Rs 1,000.00/- face amount.

Now the other side, the yield. A published figure of 8.50 per cent a year can mean 8.50 applied once, or it can mean 4.25 applied twice. The two are not the same rate. Apply 4.25 per cent twice and a rupee becomes 1.0425 squared, or 1.08680625, so the year has actually cost 8.6806 per cent. The half yearly reading sits 0.1806 percentage points above the annual reading of the very same printed figure, from a number that looks identical in print.

Here is the part that surprises people, and it is worth working rather than asserting. Ask what happens to the par identity if the convention changes on both sides at once. Pay Rs 42.50/- on twenty half yearly dates and discount at 4.25 per cent for each half year, with the Rs 1,000.00/- arriving at the twentieth. The answer comes out at Rs 1,000.000000/-. Exactly. The identity survives the change of convention untouched. The identity was never about the size of the tick: a stream discounted at the rate that generated it comes back to the base it was struck on.

Which leaves a clean and slightly uncomfortable summary. The one place the convention does not bite is the one place both numbers already agree. Everywhere else it bites hard, and the two figures here most exposed to it are the two that are not at par. So a source that publishes 8.50 per cent with no convention beside it has handed over a figure that can be quoted and cannot be used. Such a figure can be repeated. Pricing with it, checking with it and comparing it against a figure struck somewhere else are all out of reach. semi-annual compoundingTwo ticks of the discounting clock in a year instead of one, so a rate is halved and applied twice. The halving gives a different price from the same printed figures. and annual compounding are not two ways of saying the same thing.

A third number lurks in that failure and deserves its own warning. It is the one people reach for when they realise 8.50 is wrong. Rs 85/- a year against Rs 1,079.4804/- of money handed over comes to 7.8742 per cent a year. The figure is honest as far as it goes, and it goes about half the distance: it counts the cash and ignores the Rs 79.4804/- that never comes back. Sitting 0.5242 percentage points above the yield of 7.3500 per cent, it is the current yield, a real measure covered under its own name, and it is not the number being compared with the contracted rate. Two wrong readings of the same holding, one too high by 115 basis points and one too high by 52.42 basis points, and both of them are more available than the right one.

How does anybody actually use this in practice?

Start with the household. The shape is identical and the sums are smaller. Somebody is offered a chance to take over a relative's deposit receipt or small savings certificate at an agreed price. The certificate has a rate printed on it. The printed rate does not decide whether the deal is any good. The comparison that decides it is between the agreed price and the amount the certificate will pay back. Above, and the printed rate flatters the arrangement; below, and it understates it; equal, and the printed rate is telling the exact truth. The same three cases hold at a kitchen table, and telling which one applies needs no arithmetic at all.

Now the same habit inside an institution. Somebody handling a portfolio of bonds is asked what it earns. The tempting answer is to average the contracted rates. Contracted rates are printed on every position and never move, so they are delightfully easy to add up. The average of the contracted rates is wrong on every position not bought at its face amount, and it is wrong in a known direction: it flatters everything bought above face and understates everything bought below. An analyst who wants a defensible figure has to go back to the prices actually paid and solve each position from its own price. Solving each position is slower, unpopular, and the only version that survives a question.

A lender looking at a borrower reads the same pair the other way round. The borrower pays out the contracted rate on the face amount, and the cheque is what goes into the cash flow forecast. The lender's own position is worth a yield instead, and the yield depends on what the lender paid rather than on what the borrower promised. The two sit in different columns of different documents, and the discipline that prevents most of the confusion is refusing to let one number do both jobs.

The habit that transfers out of all three, and it is one line: before writing down any rate at all, look at the price paid beside the face amount and notice whether they are the same figure. If they are, the contracted rate is the yield and there is nothing to do. If they are not, the contracted rate is a fact about the bond and a fiction about the holder's position, and only a solved number will do.

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