Bond Price and Yield: Turning a Promise Into a Price

A bond price is the present value of everything the bond promises, discounted at one annual rate. Discount each interest payment and the face amount by that rate, add the results, and the sum is the price. Run it the other way and the yield is the single annual rate that makes those same payments add up to exactly what the bond costs.

The bond is the ten year 8.50 per cent bond, invented for teaching: face amount Rs 1,000/-, contracted coupon rate 8.50 per cent a year struck on that face amount, one coupon date a year and ten dates left to run. At a yield of 8.50 per cent a year applied once a year the ten discounted amounts come to Rs 78.34/-, Rs 72.20/-, Rs 66.55/-, Rs 61.33/-, Rs 56.53/-, Rs 52.10/-, Rs 48.02/-, Rs 44.26/-, Rs 40.79/- and Rs 479.88/-, and those ten add to Rs 1,000.00/-. Change the convention box alone to twice a year and the identical schedule prices at Rs 988.24/-.

A schedule of payments and a rate are enough to produce a price. A present valueWhat a sum falling due on some later date is worth today, once the waiting has been priced. It is always less than the sum itself, and how much less depends on the rate applied. is only a sum of shrunk-down amounts, and a bond has already stated what it will pay and when. The arithmetic cannot choose the rate, and it has no view. Every rate in this guide is annual and every discount is taken once a year unless the convention box is changed. A rate without its instruction is not yet usable, so the instruction is stated beside each rate rather than tucked into a footnote.

Which two runs come out of one calculation?

The commonest confusion here is not arithmetical at all. The confusion is that people think a bond price calculation and a yield calculation are two different tools, and they are not. There is one sum, it runs in two directions, and the same ten promised payments sit in the middle of both.

Run forwards, the arithmetic takes a yield and returns the price somebody wanting exactly that rate would hand over today for this schedule. Run backwards, it takes a price, perhaps one somebody has quoted, and returns the rate a buyer paying that price would earn, on the assumption that every payment arrives in full and on the date contracted for.

A fruit seller has a set of scales and a price board. Given a weight, the board gives the amount to pay. Given the amount paid, the same board gives the weight bought. Nobody imagines these are two separate machines, and nobody imagines the board has a view on whether mangoes are dear this week. The board converts. The arithmetic that turns a yield into a price stands in exactly the same relation to the bond.

Where is each of the five inputs actually found?

A calculation is only as trustworthy as the place each of its inputs came from. The table below is the field note version. The meaning of each input is settled elsewhere; the entry here is where to go to get it. Five inputs go in. Three are read straight off the document that carries the obligation, one is either chosen or read back out of a price, and one is the input nobody realises is an input at all.

The last row is the load-bearing one of the five. The other four announce themselves when they are missing: leave out the face amount and there is nothing to multiply the rate by, count nine dates instead of ten and the sum is short by an obvious lump. But a rate arrives as a bare number, it looks complete, and it slots into the arithmetic without complaint. The convention that says how to apply it does not sit inside the number. The convention sits beside the number, and the convention is the part that gets dropped in conversation, in a message, and in the third column of somebody else's spreadsheet.

The household version runs like this. Somebody says a loan costs one and a half per cent. One and a half per cent a month and one and a half per cent a year are two completely different obligations, and the number is identical in both. Nothing usable has been said yet. Nobody accepts the bare figure from a lender. The same discipline applies to a yield.

How do ten promised payments turn into one price?

Now the arithmetic, written out once. Take the first coupon of Rs 85.00/-, due one year from now, and discountTo shrink a later payment down to its worth today, by dividing it by one plus a rate raised to the number of years of waiting. it: divide it by one plus the yield. Take the second coupon, due two years from now, and divide it by one plus the yield squared. Carry on to the tenth date, the one that brings the face amount along with its coupon, and divide that by one plus the yield raised to the tenth power. Add the ten results. The sum is the price.

One structural property governs everything that follows. The yield sits in the denominator of every single term, so raising it makes every term smaller at once, and there is no arrangement of this sum in which a higher yield produces a higher price. The fall is not a market observation and not a behavioural claim about buyers. The fall is a property of division: nothing has to hold for it to be true, and no condition can suspend it.

The shrinking is easier to see on its own, before any rupee figures are attached to it. At 8.50 per cent a year, annual compounding, one rupee arriving a year from now is worth 0.9217 of a rupee today, one arriving in three years 0.7829, and one arriving on the tenth date 0.4423, less than half. The ten discount factors are the whole engine. Multiply the third by the Rs 85.00/- coupon and the answer is Rs 66.55/-, precisely what the third payment contributes to the price.

With the rupee amounts attached, the price builds itself. Nine of the ten dates carry Rs 85.00/-. The tenth carries Rs 1,085.00/-, being that year's interest with the face amount alongside it. Discounted at 8.50 per cent a year, annual compounding, all ten come to the calculation set out below.

Two readings are worth taking off that table before moving on. The nine ordinary coupon dates contribute Rs 520.12/- between them and the tenth date alone contributes Rs 479.88/-, so a single date carries 47.99 per cent of the price. The issuer has contracted to hand over Rs 1,850.00/- in total across the ten dates. The price is Rs 1,000.00/-. Discounting has removed Rs 850.00/- of that, and the gap is neither a loss nor a fee. The Rs 850.00/- is the cost of the waiting, priced at 8.50 per cent a year.

Why must the compounding convention travel beside the rate?

The convention is where a calculation earns or loses its keep, so it gets a part of its own rather than a line at the foot. Every rate worked with here is annual and every discount is taken once in the year: one payment a year, one division a year, and the exponent counting whole years. Annual compounding is the instruction, and it is stated beside every rate and inside the panel above.

Now watch what a different instruction does to the identical schedule. Take the same ten payments on the same ten dates, and take the same quoted number, 8.50 per cent, but apply it twice within each year instead of once. Nothing about the bond has changed. Nothing about the number has changed. A rate applied twice a year at 4.25 per cent a time really costs 8.68 per cent across the year, so the price comes out at Rs 988.24/- instead of Rs 1,000.00/-, a gap of Rs 11.76/-.

Rs 11.76/- on a thousand may sound like small change, and on one bond it is. The point is not the size of the gap but that no part of either calculation is wrong: no cell turns red, and both readers can show their working and be right. The convention used throughout is annual, and the right hand branch shows what a mismatch looks like from the inside. The right hand branch is not a second convention for this instrument.

What happens when the yield equals the contracted coupon rate?

Run the sum at 8.50 per cent a year and the price lands on Rs 1,000.00/- exactly, the face amount to the paisa. The coincidence looks like a number arranged for a textbook and it is not one. Whenever the yield wanted equals the contracted coupon rate, the sum returns the face amount, and the phrase at par is a name for that outcome rather than a rule imposed on bonds.

The intuition is short. The bond already pays 8.50 per cent a year on Rs 1,000/-. If 8.50 per cent a year is exactly the rate wanted, then handing over Rs 1,000/- delivers precisely that and there is nothing to adjust for. A lower price would earn more than the rate stated as wanted. A higher price would earn less. Only one amount leaves a buyer neither over nor under, and it is the face amount.

Par is also the sharpest place to see the labelling discipline at work. The contracted coupon rate reads 8.50 per cent a year and the yield reads 8.50 per cent a year: the same number, two entirely different objects. One is a term of the contract that will not move for ten years; the other is what a buyer today has decided to want. The two coincide once, and that is exactly why each is written out with its own name every time it appears.

Because the fall never reverses, that crossing happens once and once only. The crossing splits the whole range of yields into two clean regions with a single point between them. Above the contracted coupon rate the price sits below the face amount. Below it, above. There is no third case and no region where both hold.

The three runs, set down in full

The worked instance is three runs on the ten year 8.50 per cent bond, all three written out so that no control has to be touched. Face amount Rs 1,000/-, coupon Rs 85.00/- on each of ten annual dates, Rs 1,085.00/- on the last, annual compounding stated beside every rate.

One promise, unchanged in every single term, is worth three different amounts to three buyers who want three different rates, and the bond had nothing to do with it. The issuer has not renegotiated, the dates have not moved, and the Rs 85.00/- has not budged. The rate somebody on the other side decided to want is what changed, and the price is where that want gets expressed.

A rupee gap has no rounding in it, so quote the rupee gaps before the percentages: Rs 1,000.00/- less Rs 879.70/- is Rs 120.30/-, and Rs 1,143.78/- less Rs 1,000.00/- is Rs 143.78/-. Only then convert. Notice that the two moves are not the same size even though the rate moved by two percentage points in each direction: the fall is 12.030 per cent and the rise is 14.378 per cent. The price against yield drawing in the calculator carries that inequality as the bend in its curve. Why it exists is not settled here; it has a measure of its own and is covered separately.

How is a yield read back out of a price?

Now the sum runs from the other end. Somebody quotes Rs 1,143.78/- for this bond and asks what rate a buyer paying it would earn. The price is known, the ten payments are known, and the yield is wanted. The instinct is to rearrange the formula and put the yield on one side, as with almost any school equation.

The rearrangement cannot be done. The yield appears in ten denominators at ten different powers, so no rearrangement isolates it, and the answer is found by trying rates rather than by solving for one. The absence is not a gap in anyone's algebra. There is no closed form waiting to be discovered; the equation genuinely does not have one for ten dates, and it already resists at two.

The way through is a searchFinding a number by trying a value, seeing how far the result missed, and trying again in the direction that closes the gap, when no rearrangement isolates the unknown.. A rate is picked, the sum is run, the answer is set against the quoted price, and the rate moves in whichever direction closes the gap. Because the price falls steadily as the rate rises and never turns back, every trial says unambiguously which way to go next, and the search always closes.

The everyday version is a tap and a bucket. The water has to reach a mark, there is no formula for how far to turn the tap, so the tap is turned, the level is looked at, and the tap is adjusted again. The mark is the quoted price, the tap is the rate, and the process works because turning the tap one way always raises the level and never lowers it. A calculator does this in a fraction of a second with many more trials than three, and it is doing nothing cleverer.

What does the picture show that a single number does not?

A price is one number and it hides its own composition completely. Rs 1,000.00/- says nothing about where inside the ten years that value sits. One bar for each payment, each as tall as that payment is worth today, makes the composition memorable without effort.

The last date carries Rs 1,085.00/- while every other date carries only Rs 85.00/-, so nine small bars step gently downwards and the tenth towers over every one of them. Even after being discounted the hardest of the ten, the final bar stands more than six times the height of the first: Rs 479.88/- against Rs 78.34/-. Almost half the price of this bond, 47.99 per cent of it, sits on one date at the very end.

The claim that most of a bond's value sits at the end need not be taken on faith once the shape has been seen. The shape also explains why the drawing changes character as the rate moves: the tall bar is discounted over ten years and the short ones over one or two, so a change in the rate hits the tall bar hardest. The unequal effect of a rate change is worth handling rather than reading about.

Ten years of discounting is where a rate change does its heaviest work, so dragging the control left grows the tall bar faster than the nine short ones. Drag it right and the same happens in reverse. The face amount marker crosses at one point only, at 8.50 per cent a year.

How does anyone actually use this?

Four different people run this arithmetic for four different reasons, and knowing which reason applies keeps a reader from over-reading the answer. Everybody using this sum is really doing one of two things: converting a rate into a price so they can decide what to offer, or converting a price into a rate so they can compare it with something else.

A lender deciding what to offer for a bond somebody wants to sell starts from the rate it requires and runs the sum forwards. The output is a bid. A bid is not the worth of the bond but the most this lender will pay at the rate it wants. Two lenders wanting different rates produce two different bids from the identical schedule, and neither is wrong.

A price on its own is not comparable to anything, so an analyst handed a price runs it backwards. Rs 1,143.78/- and Rs 879.70/- may belong to different bonds with different coupons and different lengths, so side by side they say almost nothing. Converted to yields, 6.50 and 10.50 per cent a year, they become two numbers on the same scale. The backwards run turns prices into rates. Prices cannot be compared with one another and rates can, and that is why so much of the language around bonds is in yields rather than prices.

An investor deciding between two uses of the same money runs it backwards for the same reason, and then stops. The yield says what the schedule pays at that price if every payment arrives. The yield does not say whether the payments will arrive, and it does not say whether that rate is good. Why a shakier borrower is made to pay more, and what the rate ought to be in the first place, are both covered separately.

And a household does the same arithmetic without naming it. Somebody offers to buy out the remaining nine instalments on a chit for a lump sum today. Working out whether that lump sum is generous means discounting those nine instalments at the rate the holder would want, and comparing. The mechanics are the ones above; only the vocabulary is different.

What does this calculation not establish?

A number that comes out of a box invites a reliance it has not earned. The sum does not say the price is right. The sum does not say the payments will arrive, and it does not say what the yield will be tomorrow, or that today's yield is a sensible one to want.

The honest description of the output is that it is a translation between two ways of quoting the same schedule, and a translation is neither a valuation nor a view. Nothing inside the arithmetic is capable of adjudicating between a rate and a price.

Consider what the sum would have to know in order to say more. Whether this issuer will still be paying in year nine is a question about credit and is covered separately. Movements in rates elsewhere need a curve of rates and are covered separately. The other uses a buyer might have for the money are a question about that particular buyer. None of those three is anywhere in the formula. The formula holds ten cash amounts, ten dates and one rate and nothing else.

A price produced by this sum is an illustration and nothing stronger. The rate that went in was assumed, so the price that came out states only what that assumption implies and not what anybody would pay.

Who sets the conventions this is quoted under?

The conventions a price and a yield are quoted under are not part of the arithmetic, and they do not stay still.

Five separate requirements sit around the arithmetic above. The convention a bond price must be quoted in. The day count conventionThe counting rule that decides how much of a year has passed between two dates, which matters the moment a calculation stops landing on whole years. that interest accrues on between payment dates. The valuation norms deciding what a regulated holder must carry a bond at. How a yield must be computed and quoted for a given instrument. And the cycle on which a completed bond trade is settled. Every one of the five is set by an authority and every one of them is revised from time to time, so a copied value stops being true on the day it changes rather than merely growing old.

The cost of naming each authority rather than copying its value is the convenience of having the numbers on the same screen as the sum. The gain is arithmetic still correct after the next revision, and the habit of going to the authority for the parts that move. The arithmetic here needs none of the five in order to work.