Treasury Bill vs Sovereign Bond: One Payment or Many

A treasury bill makes exactly one payment, on its maturity date, so a single division prices the whole arrangement. A sovereign bond makes several: a coupon on every date and the face amount on the last one. The borrower behind both is identical, so the count of payment dates is the only structural difference, and every other contrast below falls out of that count.

A reader usually arrives holding one of those two terms and not the other. Arriving with one term and not the other is the ordinary way into a comparison, and it settles the order of the work: both instruments are defined in full, one at a time, before a single word of contrast is written. A comparison that starts contrasting before it finishes defining leaves the reader with two half-formed pictures and a verdict. Two half-formed pictures and a verdict are worse than nothing at all.

Begin where the arithmetic is not yet involved. Two neighbours borrow from the same lender on the same day, for the same reason, and both are equally good for the money. The first says: nothing at all will be handed over until next Diwali, and then the whole amount comes in one go. The second says: a small sum will be paid at the end of each year, and the balance on the last one. Nothing about who is repaying has changed between those two sentences; the only thing that changed is how many times something is handed over. The count of handovers is the entire subject below, dressed in rupees.

The finance version keeps the borrower fixed too, and that is what makes the comparison worth running as arithmetic rather than as opinion. Both instruments here are issued by the same government. Nothing about the identity of the borrower differs between them, so nothing about the borrower can explain any difference found. Whatever separates the two must therefore live in the shape of the payments. The shape of the payments is a much smaller and much more checkable object than it first looks.

Every rate used below comes from one invented schedule of SPOT rates, a SPOT rate being the rate for money placed today and returned on one stated future date. The schedule carries the one year SPOT rate at 5.90 per cent a year, the two year SPOT rate at 6.25 per cent a year, the three year SPOT rate at 6.55 per cent a year, the five year SPOT rate at 6.90 per cent a year, the ten year SPOT rate at 7.35 per cent a year and the thirty year SPOT rate at 7.60 per cent a year. The schedule is an input to arithmetic rather than an observation about any market anywhere.

What is a treasury bill, before anything is compared with it?

A treasury billShort-dated government borrowing that makes exactly one payment, at maturity, with no coupon along the way. is short-dated government borrowing that makes exactly one payment, at maturity, and no payment before it. There is no coupon. There is no interim date on which anything happens. The holder pays something today, waits, and receives a stated amount on one stated day.

Because there is nothing in the middle, the entire arrangement is priced by one division. Rs 1,000.00/- repayable in one year is priced on the invented SPOT curve used throughout this sequence at the one year SPOT rate of 5.90 per cent a year. Dividing once by 1.0590 gives Rs 944.287063/-. The holder hands over Rs 944.287063/- today and is handed Rs 1,000.00/- at the end of the year. The gap of Rs 55.712937/- is not part of the return; it is the whole of it, because no other rupee ever changes hands.

Notice what the word discountThe gap between the amount an instrument repays and the price paid for it today, on an instrument that makes only one payment. is doing there. On an instrument with one payment date, the gap between what is repaid and what is paid today is the only quantity there is to talk about. There is no coupon competing with it for the reader's attention, and there is no schedule of interim receipts to keep track of. Everything the holder will ever learn about this instrument is settled the moment those two amounts are written down.

The compounding convention, stated inside the arithmetic rather than under it

Every rate and every price in this guide is struck on annual compoundingOne discounting period a year, so an amount due in three years is divided by one plus the annual rate three separate times., meaning one discounting period a year. 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 separate times, and not by anything else. The convention is not housekeeping tucked into a footnote. The same six recorded rates read on a half yearly convention give different prices and a different set of FORWARD rates, so a reader who is not told which convention is in use cannot reproduce a single sum in this guide. A price without its compounding convention is not a price, it is a number that happens to have a rupee sign attached.

The convention on which a published government yield is stated, the day count a yield calculation must use and the basis on which a price is quoted are all set by the Reserve Bank of India at rbi.org.in. Every sum below states its own convention inside itself. Stating the convention inside the sum lets the arithmetic stand on its own, and lets a rule set be attached to it later without rewriting a line.

What is a sovereign bond, before anything is compared with it?

A sovereign bondGovernment borrowing that makes several payments: a coupon on each of a series of dates, and the face amount on the last one. is government borrowing that makes several payments: a coupon on each of a run of dates, and the face amount on the last of them. The holder is handed something more than once, and the last handover is much larger than the ones before it because it carries the face amount along with the final coupon.

Take a three year government bond on Rs 1,000.00/- of face, issued at parA price equal to the face amount, which is where both bonds in this guide are struck. on the same invented schedule. Issuing at par is a constraint, not a decoration: it fixes the coupon rateA rate applied to the face amount, which fixes the size of the payments. It is not a return, and the two coincide only in particular cases. at whatever level makes the price come out exactly at the face amount. On this schedule that level is 6.523518 per cent a year, so Rs 65.235176/- falls due at the end of year one, Rs 65.235176/- at the end of year two, and Rs 1,065.235176/- at the end of year three.

Now price it back. Pricing it back is the only way to know the coupon was solved correctly rather than asserted. Each of the three amounts is divided by one plus the SPOT rate for its own date, raised to the number of years until that date, and the three answers are added.

Read the two right hand columns against each other and the shape of the instrument becomes visible without any theory. The bond will hand over Rs 1,195.705528/- in total across three dates, and all of that is worth Rs 1,000.0000/- today. Money arrives twice before maturity. Something has to be decided about it twice, and that is the first real consequence of having more than one payment date. The four decimal figures in the last column are each rounded on their own and still add to Rs 1,000.0000/- exactly; carried further the sum is Rs 1,000.00000000/-.

What happens when both are run at one shared horizon?

Here is the move that most comparisons of these two instruments skip, and skipping it is why so many of them end in nonsense. Before asking how a one year instrument differs from a three year one, put both instruments at the same horizon and see what is left. Set a one year treasury bill beside a one year government bond issued at par, same borrower, same date, same invented schedule.

A one year bond pays its coupon and repays its face amount on the same day, so it has exactly one payment dateA date on which something is actually handed over. Counting these is the whole of the structural difference in this guide.. With only one date, the coupon that puts the bond at par has to be the one year SPOT rate of 5.90 per cent a year exactly, and nothing else will do. Rs 1,059.00/- divided once by 1.0590 is Rs 1,000.000000/-.

Set the two sums beside each other. The bill: Rs 1,000.00/- divided once by 1.0590 gives a price of Rs 944.287063/-. The bond: Rs 1,059.00/- divided once by 1.0590 gives a price of Rs 1,000.000000/-. The two sums are the same operation at the same rate, and the only thing that separates them is which of the two amounts had the round number written on it. The round number written on the repayment gives a bill. The same number written on the price gives a bond at par.

A reader who thought a bill and a bond were built on different principles has just watched that idea collapse, and it is worth sitting with the collapse for a moment. Nothing was hidden to make it happen. No approximation was used. At a horizon where the bond has only one date, a bond is a single dated amount, and the vocabulary that surrounds it is the only thing that made it look otherwise.

What is the one thing that differs, and where does it first show up?

The difference is the number of payment dates, and a difference in the number of dates can only reveal itself where there is more than one. The difference stayed invisible at the one year horizon for exactly that reason, and appears the moment a third year is added.

At three years the bond has three dates, so it is priced against three different recorded rates: the one year SPOT rate of 5.90 per cent a year for the first coupon, the two year SPOT rate of 6.25 per cent a year for the second, and the three year SPOT rate of 6.55 per cent a year for the closing amount. A three year claim paying once is priced against the three year SPOT rate of 6.55 per cent a year alone, and on Rs 1,000.00/- repayable it costs Rs 826.684201/-.

A coupon bond is not exposed to one point on the schedule of rates; it is exposed to every point its payments fall on. The consequence of that sentence is immediate, and it can be checked. The three year bond issued at par carries a coupon of 6.523518 per cent a year. The three year SPOT rate reads 6.55 per cent a year. The coupon sits below the rate for its own final date by 0.026482 percentage points, or 2.6482 basis points.

Why below and not above? Because two of the bond's three payments are pulled forward onto earlier dates where the recorded rate is lower. The first coupon meets the one year SPOT rate of 5.90 per cent a year and the second meets the two year SPOT rate of 6.25 per cent a year, and both of those are beneath the three year SPOT rate of 6.55 per cent a year on this schedule. Averaging in two cheaper discountings drags the level coupon down slightly from the rate that would apply if the whole thing arrived on the last day.

What does each instrument do about money that comes back?

Here the difference in the count of dates stops being a curiosity and starts costing somebody a decision. The three year bond hands back Rs 65.235176/- at the end of year one and again at the end of year two. Each time, the holder is standing there with money in hand and has to do something with it, at whatever rates are ruling on that day rather than at the rates recorded today. The one year bill hands back nothing at all until it matures, and then hands back the whole Rs 1,000.00/- in one movement.

The everyday shape of this is familiar to any household. Money that arrives in small instalments through the year tends to get absorbed into the month it lands in. Money that arrives as one lump on one day is a decision impossible not to notice. Neither pattern is better; they are different problems. The instalments never present themselves as a decision and so are rarely treated as one. The lump presents itself as a decision on a single day, ready or not.

Neither instrument removes the reinvestment questionWhat to do with money once it has been handed back, which is a question both instruments raise and neither of them settles.; they schedule it differently. The honest version stops there. The comfortable version is only a sentence away and is worth naming: the bill avoids the problem because nothing comes back early. The bill does not avoid it. The bill concentrates the question, postpones it, and hands the holder the whole of it on one day.

The rate the returned money then earns cannot be read off this schedule or off anything behind it. Answering that needs the rates ruling on the dates the money arrives, and this schedule records rates for money placed today only. A rate for money placed in one year, or in two, is set on that future day by lenders who have not yet met. The row where a reader expects a reinvestment figure is therefore drawn empty, with the reason written inside it, rather than filled with something invented for the sake of having a number there.

Which of the two is the better holding?

A comparison is expected to answer this question, and leaving it unanswered is not modesty. The refusal is arithmetic. Saying which of the two is better needs three things, and none of the three is available.

1. When the holder actually wants the money.An instrument maturing in one year and an instrument maturing in three are answers to different questions about dates. Which one fits is a fact about the holder's own calendar, and no schedule of rates carries a calendar.
   A fact about the holder, not a property of either instrument.
2. What the holder will do with each amount as it arrives.The bond hands back Rs 65.235176/- twice before maturity. Whether that gets spent, held or placed again changes what the whole arrangement comes to, and the instrument does not decide it.
   A behaviour, not a term written into either contract.
3. What rates will be ruling on the dates money comes back.The ruling rate is the one thing no amount of care can supply. Today's schedule prices money placed today. The schedule contains no rate for money placed at any future date, and a view about future rates is not arithmetic on anything recorded.
   Not present in this schedule at all, and not obtainable from it.

Two sentences sound like conclusions and both are wrong, for reasons worth stating plainly. The first is that a bill is safer because it is short. The borrower is identical in both instruments, so a shorter wait is not a different borrower; the word safer also hides which risk is meant, and hiding that is exactly how a comparison turns into a recommendation without anybody noticing. The second is that a bond is better because it pays more. The sentence compares a coupon rate against a discount without saying what either quantity is measured on, and the failure block below takes it apart line by line.

The plain factual version of each may be written. A bill's holder waits less time to be repaid: that is a statement about dates, it is checkable, and it makes no claim about safety. A three year bond hands over more rupees in total than a one year bill of the same face amount: Rs 1,195.705528/- against Rs 1,000.00/-. The comparison is checkable, and it says nothing at all about which is the better holding because the two totals sit across different lengths of time.

Why a move is always a rise or a fall in the yield

A rate does not simply go up or down on its own. The yield is what moves, and the price moves with it in the opposite direction, so a move is named as a rise in the yield or a fall in the yield every single time. The reason is not fussiness. A writer who means the price in one sentence and the yield in the next has given one name to two opposite movements, and the prose then asserts the reverse of its own arithmetic while looking perfectly reasonable. The same discipline puts the word SPOT or the word FORWARD on every rate, a distinction the closing sections return to.

Which comparisons does this schedule support, and which does it refuse?

A comparison earns its keep partly by what it declines to run. Sorting the comparisons is mechanical rather than a matter of taste: either every quantity a comparison needs is recorded here, or it is not.

The last row stops the comparison a reader will most want to run, and it deserves its own paragraph. The invented schedule used throughout this sequence records six horizons and nothing at all between them: one, two, three, five, ten and thirty years. There is no four year SPOT rate here, no nine year SPOT rate, no twenty nine year SPOT rate, and nothing shorter than one year in any form.

No line is drawn between the recorded points to read a value off it. Reading between them produces a figure that depends entirely on the method chosen, so a straight line reading and a curved reading would disagree, and two accounts working from the same schedule would then print two different numbers for the same object. Where a reader expects a rate in between, the rate is simply not recorded. Drawing the gaps as gaps is more honest than filling them, and it is the single thing most pictures of a schedule of rates get wrong.

The practical consequence lands directly on this comparison. The one year SPOT rate is recorded, so a one year bill against a one year bond at par can be run in full. No rate for any horizon inside the first year is recorded, so a bill maturing at any point inside the first year cannot be priced at all. A blank inside the first year is not a limitation to apologise for; it is the difference between a figure somebody can reproduce and a figure somebody invented.

The labelling rule, without which none of this can be written

Every rate in this guide carries the word SPOT or the word FORWARD, and the insistence is worth explaining. 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. A FORWARD rate is the rate for money placed at one future date and returned at a later one. SPOT rates and FORWARD rates are different objects, and on this schedule they sit close enough to be mistaken for each other by any reader who meets them without labels.

Derive one and watch how close it lands. Two years of growth at the two year SPOT rate divided by one year of growth at the one year SPOT rate leaves the growth over the second year alone, and that is the one year FORWARD rate, one year from now.

The one year FORWARD rate reads 6.601157 per cent a year. The three year SPOT rate on the same schedule reads 6.55 per cent a year. The two figures are 0.051157 percentage points apart, or 5.1157 basis points, and they are completely different objects: one is a rate for a single year that has not started yet, the other is a rate for three years starting today. The label is the only thing that separates them, and moving the numbers apart to spare the reader the confusion would only teach the reader that the confusion was not real.

How does somebody actually use this distinction at a desk?

What survives the comparison once the verdict is refused?

A verdict would have expired the moment the holder's own circumstances changed. Two things survive instead, and between them they are worth considerably more.

The first is the borrower. The same government stands behind both instruments, so any difference a reader feels between them is a difference in the shape of the promise rather than a difference in who is making it. Most casual comparisons of these two instruments are secretly comparisons of borrowers that nobody checked, and that single observation dissolves a great deal of loose thinking.

The second is the arithmetic, and it is the same in both cases. Divide each payment by one plus the SPOT rate for its own date, raised to the number of years until that date, and add the answers together. For the bill that is one term. For the three year bond it is three. There is no second method anywhere in this guide; there is one method applied a different number of times.

A way of reading the whole schedule follows from that, and it is worth saying before any single level on it is discussed. The six recorded rates describe one borrower at six different horizons, not six borrowers, and the differences between the rates are what that single borrower is charged for money returned on six different dates. Nothing about the borrower changes between the one year point and the thirty year point. Only the length of the lender's wait changes.

The treasury bill, the one year bond at par, the three year bond at par, the three year single payment claim and the schedule of SPOT rates used throughout are invented.
Educational material. Not advice on any investment, tax, budget or market position.