Par Bond and Premium Bond: What the Price Difference Is
A par bond is priced at exactly its face amount and a premium bond above it. On the same borrower and the same payment dates, one input separates them: the coupon fixed at issue. A coupon matching what a fresh bond would carry leaves the price at face, and a coupon above that lifts it by the present value of the difference, nothing more.
Two bonds sit side by side. The same government borrowed both amounts. Both repay on the same day three years from now. Both repay the same Rs 1,000.00/-. One of them is quoted at Rs 1,000.00/- and the other at more than that, and a reader meeting this for the first time almost always reaches for the wrong explanation. The dearer one must be safer. The dearer one must be in demand. Somebody must think it is worth more. None of those is the reason, and the real reason is a single field written into one of the two documents years ago and never touched since.
The shape on a market street is identical to the shape in a bond market, and the shape is the whole lesson. Start on the street. Two shops stand side by side in the same building, the same size, the same footfall, the same three years left on each agreement. One shopkeeper signed his lease this morning at the rent the street charges today. The other signed hers four years ago, when the street was cheaper, and her rent is lower for the three years that remain. If both wanted to hand over their shops tomorrow, the second one would collect something for hers and the first one would collect nothing. The payment is not a statement that her shop is better or her business stronger. The payment is the value of a rent saving that runs for three more years and then stops.
Turned round, that is a bond at a premium. The bond is a promise to hand somebody cash on dated occasions. If the promise written into the paper is more generous than the promise a fresh piece of paper would carry today, the paper is worth more than the amount printed on it, by exactly the value of the extra generosity and by nothing else at all. The extra generosity here has a name, a size and an end date, and pinning down all three is arithmetic and nothing else.
What is held still in this comparison, and why that matters more than usual
A comparison is only worth reading when it has one moving part. If two things differ in four ways, the behaviour cannot be attributed to any one of the four, and watching them behave differently teaches nothing. So every other input is held fixed. One borrower. One set of three payment dates. One face amountThe sum the issuer repays at the end, and the base the coupon rate is applied to. It is printed in the terms and does not move. of Rs 1,000.00/- repaid at the end of the third year. One curve of rates, used to discount every single payment in both columns. After that freezing, exactly one input is left free to differ, and it is the coupon.
One invented curve of SPOT rates runs through this comparison, written down for the exercise rather than read off any market. The curve records the one year SPOT rate at 5.90 per cent a year, the two year SPOT rate at 6.25 per cent a year and the three year SPOT rate at 6.55 per cent a year. A SPOT rateThe rate for money placed today and returned at one stated future date, with nothing happening in between. is the rate for money placed today and returned at one stated future date. Each of a bond's payments has exactly that shape: one amount, one date. Rates of other kinds sit close enough to these three to be mistaken for them once a label slips, so every one of the three carries the word SPOT.
A sum cannot be reproduced without the compounding convention, so the convention sits inside the arithmetic rather than in a note underneath it. Both prices are struck on annual compoundingOne discounting period a year. An amount due in three years is divided by one plus the annual rate, three times over.: one discounting period a year, so an amount due in three years at the three year SPOT rate of 6.55 per cent a year is divided by 1.0655 three times over. The compounding basis is not housekeeping and not a footnote. The same three rates on a half yearly convention give different prices and a different set of everything downstream. Somebody who is not told which convention is running cannot reproduce a single one of these figures.
Two government bonds repay on the same date and one of them is priced above its face amount. Before any arithmetic at all, what is likely to be different about it?
What is a par bond, and what has to be true for one to exist?
A par bondA bond whose price equals its face amount, so a buyer hands over the same sum the issuer will repay at the end. is a bond whose price equals its face amount. Nothing more elaborate than that. Rs 1,000.00/- is handed over, coupons are collected along the way, and Rs 1,000.00/- comes back at the end. The reason it deserves this much attention is that being at par is not a property somebody grants a bond. Par is an accident of arithmetic that happens when one particular coupon meets one particular curve, and the fastest way to see that is to solve for the coupon that makes it happen.
Take a three year government bond on Rs 1,000.00/- of face, with a payment at the end of each of the next three years and the face amount returned with the last one. Its price is what its payments are worth today, and what a payment is worth today is the payment multiplied by a discount factorThe present value today of one rupee promised at a stated future date. Multiply a future payment by it to get what that payment is worth now.. A discount factor is what one rupee at that date is worth now.
| P | the price today, in rupees |
| C | the coupon paid at the end of each year, in rupees, fixed when the bond was issued |
| F | the face amount repaid with the last coupon, Rs 1,000.00/- in both columns |
| d1, d2, d3 | the discount factors for one, two and three years, each being one divided by one plus that year's SPOT rate, raised to the number of years, on annual compounding |
The three factors come straight off the invented SPOT curve and nowhere else. One divided by 1.0590 is 0.9442870633. One divided by 1.0625 squared is 0.8858131488. One divided by 1.0655 cubed is 0.8266842011. The three added together make 2.6567844132, a number that is going to do a surprising amount of work in what follows. Notice that each factor is smaller than the one before it. Money later is worth less than money now, and the three factors are that sentence written as numbers.
Now ask the question backwards. Instead of taking a coupon and finding a price, fix the price at Rs 1,000.00/- and find the coupon that produces it. Set P equal to F in the line above and the algebra falls out in one step.
| 1 less d3 | 0.1733157989, the part of the face amount that discounting has taken off the repayment |
| d1+d2+d3 | 2.6567844132, the present value today of one rupee a year for three years |
| C | the coupon in rupees that makes the price equal the face amount, and the only one that does |
The sum runs: Rs 1,000.00/- multiplied by 0.1733157989 divided by 2.6567844132 gives 0.0652351760 of the face amount. The par coupon is therefore 6.523518 per cent a year, or Rs 65.235176/- each year. The par coupon is not chosen and not quoted. Three recorded SPOT rates produce it and nothing else does, and a change in any one of the three moves it.
A formula shown and then taken on trust has taught nothing, so price that bond forwards and check the solve. Rs 65.235176/- multiplied by 0.9442870633 is Rs 61.600733/-. The same coupon multiplied by 0.8858131488 is Rs 57.786177/-. The last payment is the coupon plus the face amount, Rs 1,065.235176/-, and multiplied by 0.8266842011 that is Rs 880.613091/-. The three add to Rs 1,000.000000/-, to as many decimals as anybody cares to carry. The bond is at par because its coupon and the curve agree, and for no other reason.
Two things follow immediately and both are worth holding on to. The first is that par is a knife edge. There is exactly one coupon on this curve that lands on Rs 1,000.00/-, and every other coupon lands somewhere else. The second is that the par coupon of 6.523518 per cent a year is not any of the three SPOT rates. The par coupon sits above the two year SPOT rate of 6.25 per cent a year and just below the three year SPOT rate of 6.55 per cent a year, and it sits much nearer the second one because the overwhelming bulk of the money in the sum, the Rs 1,000.00/- of repayment, arrives at three years and is discounted at three years.
The coupon that prices this three year bond at par works out at 6.523518 per cent a year, while the recorded SPOT rates are 5.90, 6.25 and 6.55 per cent a year. Where does the par coupon sit among them, and why exactly there?
What is a premium bond, and what puts its price above the face amount?
A premium bondA bond whose price stands above its face amount, so a buyer hands over more than the issuer will repay at the end. is a bond whose price stands above its face amount. Now build one, and build it on the same borrower and the same dates so that nothing except the coupon has room to move.
The same government issued another bond some years ago. The older bond matures on the same day as the one above, repays the same Rs 1,000.00/- and pays on the same three remaining dates. When it was created, the rates of the day were higher, so the issuer wrote 8.50 per cent a year into the terms. On Rs 1,000.00/- of face that is Rs 85.00/- a year. The 8.50 per cent was a coupon fixed at issueA rate set when the bond is created and unchanged for its whole life, whatever happens to rates afterwards., and a coupon fixed at issue does not renegotiate. Whatever happened to rates afterwards, the document still says 8.50 per cent a year and will keep saying it until the last payment is made.
Using a different curve on the second bond would smuggle in a second moving part and destroy the comparison, so price it exactly as before, on the same three recorded SPOT rates and the same annual compounding. Rs 85.00/- multiplied by 0.9442870633 is Rs 80.264400/-. Rs 85.00/- multiplied by 0.8858131488 is Rs 75.294118/-. The final payment of Rs 1,085.00/- multiplied by 0.8266842011 is Rs 896.952358/-. Add them and the price is Rs 1,052.510876/-.
The price stands Rs 52.510876/- above the face amount, and a bond priced above its face amount is at a premium. Nobody decided the premium. Nobody granted it. The premium fell out of an addition in which every input was either written in the bond's own terms or read off the recorded curve.
Look at the two right hand columns for a moment before moving on. The year three rows are Rs 880.613091/- and Rs 896.952358/-, and they are enormous next to the year one and year two rows. The gap between the rows is not a discounting effect. The year three payment on either bond carries the Rs 1,000.00/- repayment along with the last coupon, and the repayment is the largest single amount a bond of this shape ever hands over. The premium bond's year three payment is Rs 1,085.00/- against the par bond's Rs 1,065.235176/-, and both are discounted at the three year SPOT rate of 6.55 per cent a year, three times over.
The premium bond's three discounted terms come out at Rs 80.264400/-, Rs 75.294118/- and Rs 896.952358/-. Why is the third one so much larger than the other two?
Where the premium actually lives in the document
The premium appears nowhere in the bond's own terms, and the omission is worth pausing on. The paper read end to end gives the borrower, the face amount, the coupon rate, the payment dates and the compounding basis. The document does not give the price. The price is the one quantity the document leaves free, and it is free precisely because everything else is nailed down. A buyer cannot renegotiate the coupon, cannot move the dates and cannot change the repayment, so the only lever available for making the exchange fair on a given day is what is handed over today.
The same document can therefore be a premium bond today and something else entirely on a different curve. Nothing about the paper would have changed. The rates it is being discounted at would have. A bond is not born a premium bond and does not stay one; it is at a premium on a particular day because of what the curve looks like on that day, and the document has no opinion about the matter.
What is the one input that differs, and how is it checked?
Lay the two bonds beside each other and count the inputs. The borrower is one government in both columns, so nothing about who is paying can explain the gap. The payment dates are the same three dates, so nothing about timing can explain it. The amount repaid at the end is Rs 1,000.00/- in both columns, so nothing about the size of the repayment can explain it. The rates used to discount are the one year SPOT rate of 5.90 per cent a year, the two year SPOT rate of 6.25 per cent a year and the three year SPOT rate of 6.55 per cent a year, applied to both, so nothing about the curve can explain it either.
One row is left. The par bond pays Rs 65.235176/- a year and the premium bond pays Rs 85.00/- a year. Both were fixed at their own issues and neither can be altered now. The coupon row is the entire cause of the entire gap, and because the coupon is the only difference, the gap can be computed from it rather than described around it.
The direction of the arrow, which is where most readers go wrong
The confusion that does most damage is a confusion about which way the causation runs. A reader sees a bond priced above face and concludes that the market must think well of it, that buyers must be competing for it, that the premium is a kind of applause. The causation runs the other way. The coupon was set first, years ago, and the price followed from discounting the cash that coupon promises; a bond does not become a premium bond by being wanted.
The everyday check is the lease again. The shopkeeper with the older, cheaper lease is not collecting a payment because her shop is nicer. She is collecting it because a specific, dated, countable saving runs for three more years and then stops. If somebody offered her double, that would be a different story about demand, and it would be a story about the price, not about the lease. Here nothing of the sort is happening: both bonds were priced by the same three factors, and the price gap is arithmetic all the way down.
There is a second reason to insist on the direction. A premium believed to reflect opinion is expected to move with opinion, and the surprise arrives when it shrinks steadily on a curve that has not moved at all. The surprise starts with an arrow pointing the wrong way, and it ends in the failure set out further down.
Somebody says a bond is at a premium because buyers want it more than the par bond. Is that the right account of what happened?
The premium bond pays Rs 19.764824/- a year more than the par bond, for three years. What is the premium, before the arithmetic below prices it?
What is the premium actually the price of, term by term?
A phrase can be nodded at and an addition has to be checked, so do the premium as an addition. The coupon difference is Rs 85.00/- less Rs 65.235176/-, or Rs 19.764824/- a year, arriving at the end of each of the next three years and then never again.
Discount those three amounts on the same three recorded SPOT rates. Rs 19.764824/- multiplied by 0.9442870633 is Rs 18.663668/-. The same amount multiplied by 0.8858131488 is Rs 17.507941/-. The same amount multiplied by 0.8266842011 is Rs 16.339268/-. Add them: Rs 52.510876/-. The total is not close to the premium and not approximately the premium. The total is the premium, to the last paisa. The premium was never anything else.
| Pprem | the premium bond's price, Rs 1,052.510876/- |
| Ppar | the par bond's price, Rs 1,000.000000/- |
| Cprem less Cpar | Rs 19.764824/-, the extra coupon each year, which is the only input that differs |
| d1+d2+d3 | 2.6567844132, the same sum of factors used to price both bonds |
Look at why the repayment vanished from that line. Both bonds repay Rs 1,000.00/- at the same date, so both carry the same term in exactly the same place in the sum: Rs 1,000.00/- multiplied by 0.8266842011, or Rs 826.684201/-. Subtract one price from the other and that term cancels, leaving only the coupon streams. The repayment is identical on both sides, and identical things do not survive a subtraction, so the premium cannot possibly be about the repayment.
The sliver in that drawing answers a question readers ask silently. Undiscounted, the extra coupons total Rs 59.294472/-. The premium is Rs 52.510876/-. The Rs 6.783596/- gap is not a discrepancy and not a fee. The gap is what waiting costs: Rs 52.510876/- placed today on this curve is precisely what it takes to produce Rs 19.764824/- at the end of each of the next three years, and the extra Rs 6.783596/- is the growth on the way.
The point is easy to feel and easy to lose. Take the household version one more time. Somebody offers to pay a household's electricity bill for the next three years, and the bill runs Rs 19,764/- a year. How much would that household accept today to give the offer up? Not Rs 59,292/-. None of that total would arrive today in any case. Something less, then, and how much less depends entirely on what money is worth over one, two and three years, which is the record a curve keeps.
Suppose the extra coupon of Rs 19.764824/- a year ran for six years instead of three, with everything else unchanged. What would happen to the premium?
The invented SPOT curve does not move at all for three years. What happens to the premium bond's price of Rs 1,052.510876/- over that time?
What happens to a premium as the payment dates run out?
Most holders of a premium bond have never been shown the behaviour that follows, and it is the behaviour that alarms them. A bond repays its face amount and nothing else at the end. The repayment is never the price. So whatever a holder paid above the face amount has to be gone by maturity, and since a price does not jump for no reason, it has to leave gradually. The gradual leaving has a name: the pull to parThe way a price standing away from the face amount moves towards it as the payment dates run out, even on a curve that has not moved..
The pull to par happens with the invented SPOT curve held perfectly still. Nothing in the movement can be blamed on rates. One year from today the bond has two payments left, at what are then one and two years away, and on an unchanged curve those two dates are discounted at the same one year SPOT rate of 5.90 per cent a year and two year SPOT rate of 6.25 per cent a year as before.
| P2 | the price with two payment dates remaining, in rupees |
| C | Rs 85.00/-, unchanged, because a coupon fixed at issue does not move |
| C+F | Rs 1,085.00/-, the last coupon travelling with the Rs 1,000.00/- repayment |
| d1, d2 | 0.9442870633 and 0.8858131488, the same two factors as before, reused because the curve is being held still |
Run it. Rs 85.00/- multiplied by 0.9442870633 is Rs 80.264400/-. Rs 1,085.00/- multiplied by 0.8858131488 is Rs 961.107266/-. The price is Rs 1,041.371667/-, so the premium has shrunk from Rs 52.510876/- to Rs 41.371667/-. A year later again, one payment remains: Rs 1,085.00/- divided by 1.0590 is Rs 1,024.551464/-, a premium of Rs 24.551464/-. And at maturity the bond pays Rs 1,000.00/- and the premium is nothing at all.
Notice the shape rather than just the numbers. The path is not a straight slide. The premium falls Rs 11.139209/- in the first year, Rs 16.820203/- in the second and Rs 24.551464/- in the third. The fall accelerates, and the acceleration has a plain cause: the premium is the present value of the extra coupons that are still to come, and each year one of them is paid away and stops being in the sum. In the last year the only extra coupon left is a single Rs 19.764824/-, one year away, and once that arrives there is nothing behind the price except the repayment.
On an unchanged curve a premium bond's price falls every single year, and the holder has given up nothing: the larger coupon arrived in the meantime and was exactly what the premium bought. The two halves are one arrangement. Reading either half alone gives a wrong answer, and reading the price half alone gives a wrong answer that looks alarming.
A holder watches the price move from Rs 1,052.510876/- to Rs 1,041.371667/- over a year while the curve stays exactly where it was. Have they lost Rs 11.139209/-?
Which of the two is the better buy, and why is that the wrong question?
The honest answer is that the arithmetic here has nothing to say. Both bonds were priced by discounting their own promised cash at the one year SPOT rate of 5.90 per cent a year, the two year SPOT rate of 6.25 per cent a year and the three year SPOT rate of 6.55 per cent a year. Each therefore costs exactly what its own cash is worth at those rates. The same three factors priced both of them, and a common measuring stick cannot make one thing a bargain. Neither is cheap against the other and neither is dear against the other.
A reader who insists on finding a difference will find one, and it is real, but it is a difference in timing rather than in value. Split each price into what it is buying and the point becomes visible in one line.
| C(d1+d2+d3) | what the coupon stream is worth today, Rs 173.315799/- for the par bond and Rs 225.826675/- for the premium bond |
| F d3 | the present value of the repayment today, Rs 826.684201/- for both bonds, because both repay Rs 1,000.00/- on the same date |
| P | the price, Rs 1,000.000000/- and Rs 1,052.510876/- respectively |
So the premium bond hands back more of its value along the way and the par bond hands back more of it at the end, relative to what each cost. The split is a difference about when cash arrives, and whether it matters to a particular holder depends on things that appear nowhere in the arithmetic.
Guessing about what a holder is treated as receiving does real harm. A coupon received and a gain or a shortfall on a sale or a redemption are not necessarily treated alike, and how each is treated is set by rule rather than by arithmetic. Treatment of both belongs to the Reserve Bank of India at rbi.org.in and to the Securities and Exchange Board of India (SEBI) at sebi.gov.in, and it moves on its own schedule.
There is one more reason the better buy question dissolves. A comparison with a single moving part can only answer questions about that moving part. The coupon difference has a value, and the arithmetic gives it to the paisa. The comparison cannot say which holder should hold which bond: nothing about a holder appears anywhere in the sum. An arithmetic that never contained a preference cannot produce one at the end.
So which of the two bonds is the better buy on this curve?
The error that gets made, and what it costs
A reader buys the premium bond at Rs 1,052.510876/- and settles in to wait for Rs 1,052.510876/- at the end. The bond repays Rs 1,000.00/-. The Rs 52.510876/- was never a deposit and was never going to come back as a lump. The premium was the price of Rs 19.764824/- a year of extra coupon for three years, and it returned inside those coupons instead.
Work the consequence rather than asserting it. A holder marking the position each year on an unchanged curve sees Rs 1,052.510876/-, then Rs 1,041.371667/-, then Rs 1,024.551464/-, then Rs 1,000.00/-, and reads three consecutive falls as three consecutive losses. Every one of them is the premium being consumed exactly as the arithmetic said it would be, and alongside each fall a coupon arrived that was Rs 19.764824/- larger than the par bond's for the same year.
Who makes this error: the reader who has been taught that a bond returns its face amount and has never been shown what happens to an amount paid above it. What it costs: three years of reading a perfectly ordinary instrument as a failing one, and at worst a position closed in alarm at a price that was behaving precisely as its own terms required.
The fix is one line. A bond bought away from its face amount splits what it returns between the price and the coupon, and neither column tells the story on its own. Read the two together.
The drawing carries one extra fact worth naming. In the third year the fall of Rs 24.551464/- is larger than the extra coupon of Rs 19.764824/-. The arrangement looks like it is finally turning against the holder. It is not. Over the three years together the falls add to Rs 52.510876/- and the extra coupons add to Rs 59.294472/-, and the surplus of Rs 6.783596/- is what the money handed over at the start earned along the way. Year by year the two halves do not match; across the arrangement they do. Being priced on a curve means exactly that.
How does somebody actually use this at a desk or at a kitchen table?
What a holder or a lender does with a price that is not the face amount
A bond bought away from its face amount behaves differently in a record from one bought at face, and the difference shows up every single reporting period. The work is done in this order.
- Read the coupon field first and write it down beside the price.The coupon is the fixed half of the arrangement and the price is the free half, so a note carrying only one of them has thrown away the ability to explain anything later. Two columns, always, from the first day.
Rs 85.00/- a year, fixed at issue. Rs 1,052.510876/- paid today.
- Price the bond off the curve rather than accepting the quote as a fact.Discount each dated payment at the SPOT rate for that date and add. A price rebuilt from the curve has a premium with a size attached to it, and a price merely read off a screen has a premium that cannot be decomposed.
Rs 80.264400/- plus Rs 75.294118/- plus Rs 896.952358/- is Rs 1,052.510876/-.
- Split the premium into its dated parts before anything else happens.Rs 18.663668/- of it belongs to year one, Rs 17.507941/- to year two and Rs 16.339268/- to year three. Once the premium has dates attached, the fall in each future year is already known rather than being a surprise when it arrives.
Rs 52.510876/- of premium, with a maturity date on every rupee of it.
- Write the expected price path down in advance and mark against it.Rs 1,041.371667/-, then Rs 1,024.551464/-, then Rs 1,000.00/-, on a curve that has not moved. Any difference from that path is news about rates. Anything matching it is not news at all, and a record that does not separate the two turns ordinary ageing into an event every year.
Falls of Rs 11.139209/-, Rs 16.820203/- and Rs 24.551464/-, all expected.
- Report the coupon and the price change in the same breath, never one alone.A period that shows only the price change on a premium bond reads as a run of losses, and a period that shows only the coupon reads as a run of unusually good income. Both are half of one arrangement that was fully priced on the day it was bought.
Rs 85.00/- received, Rs 11.139209/- of price given back, one arrangement.
A household does a rough version of this without any of the vocabulary. Somebody taking over a two wheeler loan from a cousin at an old, cheaper rate works out roughly what that saving is worth over the months that remain, hands over something for it, and does not then complain each month that the saving is smaller than it was. The months are running out, and everybody involved knew that at the start. The bond version differs only in that the arithmetic is exact and the dates are written down.
Named here, and deliberately left unwritten
Each item below is set by an authority and changes on its own schedule, so a figure stated from memory goes stale within a year or two. Only the compounding basis had to sit inside the sums above: a price cannot be reproduced without it.
| Item | Where it is settled |
|---|---|
| How a government security's price is quoted, and on what basis | Reserve Bank of India, rbi.org.in |
| The valuation norm that decides the price at which a holding is carried | Reserve Bank of India, rbi.org.in |
| The compounding convention a published yield is stated on | Reserve Bank of India, rbi.org.in |
| The day count convention a yield calculation must use | Reserve Bank of India, rbi.org.in |
| How a benchmark government curve is constructed and published | Clearing Corporation of India Limited, ccilindia.com |
| The treatment of a coupon received and of a gain on sale | Reserve Bank of India, rbi.org.in, and SEBI, sebi.gov.in |
| What an issuer must disclose in the terms of a bond it offers | SEBI, sebi.gov.in |
References
| Source | What it is named for | Where |
|---|---|---|
| Reserve Bank of India | Government securities, how a price is quoted and on what basis, the valuation norm deciding a carrying price, the compounding convention a published yield is stated on, the day count convention a yield calculation must use, and the treatment of a coupon received and of a gain on sale | rbi.org.in |
| Clearing Corporation of India Limited | How a benchmark government curve is constructed and published | ccilindia.com |
| SEBI | What an issuer must disclose in the terms of a bond it offers, and the treatment of a gain on sale | sebi.gov.in |
Both bonds, the government that issued them, the SPOT curve they are discounted on and Palash Cements Limited are invented.
Educational material. Not advice on any investment, tax, budget or market position.