Treasury Bills: Short-Dated Government Borrowing Priced

A treasury bill is government borrowing over a short horizon, and it pays once: a single stated amount on a single stated date, with no coupon at any point before it. A buyer parts with less than the sum it repays, so the whole of the return is that lone gap, whose width follows from the SPOT rate recorded for its horizon.

What does the holder of a treasury bill actually hold?

The starting point is a familiar one. A supplier hands a buyer an invoice for Rs 1,000.00/- and says it may be paid a year from now, or settled today for Rs 944.00/-. There is no interest line on that invoice, no monthly instalment, no schedule of any kind. There is one number today, one number later, and the whole of the arrangement lives in the space between them. Nobody in that room finds this confusing.

A treasury billShort-dated government borrowing that makes exactly one payment, at maturity. It carries no coupon, so nothing at all arrives before that single date. is that invoice, issued by a government. The bill is a promise to pay one stated amount on one stated date, and nothing before it. Everything else about the instrument falls out of that single property.

Watch what disappears when a security makes only one payment. There is no coupon, so there is no payment schedule to read and no dates to diarise. No money arrives part way through, so during the life of the security there is no question about what to do with it. There is no along the way, so there is no split between what the holder collects on the way and what the holder collects at the end. A security that pays once has no interior at all.

Watch also what has to be true about the price. If the bill were sold for exactly the amount it repays, a lender would be handing over money and getting the same money back later, having waited for nothing. Nobody does that willingly. So the price must sit below the repayment, and the size of the shortfall is the whole of the compensation for waiting. In an instrument with a coupon, compensation arrives partly as cash along the way and partly as the difference between price and repayment. Here there is only one channel left open, and every rupee of return has to travel down it.

A single channel of return is why this instrument makes one particular mistake visible more cleanly than any other. When a return arrives through several channels, a reader who measures one of them against the wrong base still gets a roughly sensible answer, and the error hides. When the whole return sits in one number, choosing the wrong thing to divide that number by produces a figure that is wrong by a size that can be pointed at. Most of what follows is about that division.

Where does the price come from, if nobody quotes one?

Every price below comes from one schedule of rates, invented for teaching. The schedule records a SPOT rateThe rate for money placed today and returned at one stated future date. Every rate in this guide is labelled either SPOT or FORWARD, because the two are different objects. at six horizons and nothing else: a one year SPOT rate of 5.90 per cent a year, a two year SPOT rate of 6.25 per cent a year, a three year SPOT rate of 6.55 per cent a year, a five year SPOT rate of 6.90 per cent a year, a ten year SPOT rate of 7.35 per cent a year and a thirty year SPOT rate of 7.60 per cent a year.

Take an invented bill repaying Rs 1,000.00/- once, one year from today. Its horizon lands exactly on the one year SPOT rate, so no judgement is needed about which rate applies. One division does all the work.

Notice what that division did and did not need. The division did not need a market screen, a dealer, an auction result or anybody's opinion. One repayment amount, one horizon and the recorded rate at that horizon were the whole of it. The price was derived rather than asserted, so a reader who doubts it can rerun the division and find out. Deriving a price rather than asserting one matters more than it sounds. A quoted price teaches nothing: a reader cannot tell whether it is right, and a reader who cannot check does not check.

Now subtract, and look at what is left over.

What follows leans on two habits of naming, so both are worth stating once, plainly. The face amountThe single sum repaid at maturity. A discount is usually measured against it, which is exactly why it must be named whenever it is used as the number underneath a ratio. is the Rs 1,000.00/- that comes back. The price paidWhat the holder actually parted with. A return on money put in has to be measured against this and against nothing else. is the Rs 944.287063/- that went out. The face amount and the price paid are different amounts, they answer different questions, and a reader who lets them blur has already lost the argument.

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

Every rate and every price above is struck on annual compoundingOne discounting period a year. An amount due in three years is divided three times by one plus the three year rate, rather than six times by half of it.: 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 times, and an amount due in one year at the one year SPOT rate of 5.90 per cent a year is divided by 1.0590 once. Annual compounding is the whole of the convention.

The compounding convention is not housekeeping tucked into a footnote. On a semi-annual convention the same six recorded rates give a different price everywhere, and a different FORWARD rate everywhere too. A reader who is not told which convention is in use cannot reproduce a single sum, and a figure a reader cannot reproduce is decoration. The convention therefore sits inside the arithmetic itself, beside the division it governs.

Why does one gap in rupees read as two different rates?

The central point takes about ninety seconds to see and rather longer to stop getting wrong.

There is one quantity in rupees, Rs 55.712937/-, and it has to be expressed as a rate. A rate is a ratio, and a ratio needs something underneath it. There are exactly two amounts on this security that could reasonably go underneath: the Rs 944.287063/- that left the holder's hands, and the Rs 1,000.00/- that came back. Both divisions are legal arithmetic. Both produce a number that looks like a yearly percentage. The two answers are not the same number.

Run the first division. Put the gap over the price.

Now run the second division. Put the same gap over the amount repaid.

Subtract one reading from the other and the distance is 0.328706 percentage points. One basis pointOne hundredth of a percentage point. So 0.328706 percentage points is 32.8706 basis points, and the two units are never swapped for each other. is one hundredth of a percentage point, so the same distance is 32.8706 basis points. Hold on to how ordinary the setting is: one year, one government promise, a rate under six per cent, and no cleverness anywhere. The two readings still land a third of a percentage point apart, and the entire cause is that one division used the price as its base and the other used the face amount.

The everyday version sits in any shop that offers two per cent off for paying cash. Two per cent off the marked price and two per cent added to the discounted price are not the same amount of money, and everybody knows it in the shop and forgets it in the spreadsheet. A ratio is not a fact about a quantity. A ratio is a fact about a quantity and the baseThe number underneath a ratio. It fixes what the ratio means, so the same numerator over two different bases gives two different and equally correct answers. it was measured against, and detaching it from the second half of that pair is how a correct number becomes a wrong statement.

Which of the two readings is the return on the holder's own money?

The question in its plainest form: what did the holder part with? Rs 944.287063/-. Not Rs 1,000.00/-. The face amount was never in the holder's possession, never left the holder's account and never sat idle waiting to be repaid. The face amount is simply the size of the promise bought.

So the return on money put in is the gap over the amount put in: Rs 55.712937/- over Rs 944.287063/-, or 5.900000 per cent a year. Only one of the two readings answers the question a lender actually asks, and 5.900000 per cent a year is the one.

Look now at where that figure came from. The check that follows is the satisfying part. The price was built by dividing Rs 1,000.00/- by 1.0590, and 1.0590 came from the one year SPOT rate of 5.90 per cent a year. Reading the gap back against the price returns 5.900000 per cent a year. The rate went in at the start and the same rate came back out at the end. Closing the ring is the check that the whole derivation holds and that nothing was quietly added or lost in the middle.

The other reading is not a mistake and it is not useless. Its 5.571294 per cent a year answers a genuine question: of the Rs 1,000.00/- eventually repaid, what fraction was surrendered up front as the discount? The share given up is a perfectly sensible thing to want to know, particularly for anybody thinking in terms of the size of the promise rather than the size of the outlay.

Neither of the two readings wins outright. The habit is what matters: neither figure may be written down without its base written into the same sentence. Say that the return on the price paid was 5.900000 per cent a year, or say that the discount on the face amount was 5.571294 per cent a year. Both sentences are complete and both are true. The sentence that fails is the one that says the rate was 5.571294 per cent and stops. A rate with no base attached is not yet a statement about anything.

Where does a bill sit on the same borrower's schedule?

A treasury bill is not a separate species of animal that happens to live near government bonds. A bill is the near end of the same borrower's schedule, and reading it that way removes most of the mystery at once.

Here is the organising idea, and it is worth having before any arithmetic touches it. The schedule is not six rates. The schedule is one borrower at six different horizons. Nothing about the borrower changes between the one year point and the thirty year point: same promise, same source of repayment, same everything. The length of the wait is the only thing that changes. The schedule is the price of waiting, quoted at six lengths of wait, and the shape of it is the thing to read before any single level on it.

Run the shape yourself rather than taking it. The thirty year SPOT rate of 7.60 per cent a year less the one year SPOT rate of 5.90 per cent a year is 1.70 percentage points, or 170 basis points, across the whole recorded stretch. The ten year SPOT rate of 7.35 per cent a year less the two year SPOT rate of 6.25 per cent a year is 1.10 percentage points, or 110 basis points. The bill sits at the low end of that stretch, and it sits there because its wait is the shortest recorded rather than because it is a different kind of promise.

Now the smallest and most useful observation of all. The bill priced above at Rs 944.287063/- is the identical object priced at the opening of this sequence as a one year zero coupon claimA promise of one amount on one date and nothing else. A treasury bill is one, which is why its price falls out of a single division.. Same Rs 1,000.00/-, same one year horizon, same one year SPOT rate of 5.90 per cent a year, same annual compounding, same division by 1.0590 and therefore the same Rs 944.287063/-. Nothing whatever changed except the word on the front of it.

The identical arithmetic under two names is worth pausing over because a great deal of confusion in this subject is vocabulary wearing a costume. The market has a name for a short government zero coupon claim and the name is a bill. The market also has names for longer ones, and names for the coupon-bearing kind, and the names carry real information about the conventions attached to them. The names do not carry different arithmetic. When two objects discount the same amount at the same rate over the same horizon, they are the same object, and no amount of separate naming will make one of them behave differently from the other.

How short a bill can be priced from this schedule, and what stops it going shorter?

Now the refusal, and it is the sharpest one in this sequence for a reason that is easy to feel. A bill is exactly the instrument a reader wants priced at a short horizon. Somebody is going to ask for three months.

The schedule records six nodesThe horizons at which a rate is actually recorded. Here there are six of them, and no value exists at any horizon that is not one of the six. and nothing else: one, two, three, five, ten and thirty years. There is no four year SPOT rate on it, no nine year SPOT rate, no twenty nine year SPOT rate, and nothing at all shorter than a year. The one year node is the shortest thing on the record.

So a one year bill can be priced and everything shorter is declined, without drawing a line between the recorded points and reading a value off it. The reason is not fussiness. Interpolation is a method, and methods disagree: a straight line drawn between two nodes and a curve fitted through all six give different answers at the same horizon. Two treatments working honestly from the same record would then print two different numbers for the same object, and a reader would have no way to tell which one to believe.

There is a tempting way round it and it is worth naming so that it can be declined out loud. Somebody will suggest scaling the one year SPOT rate of 5.90 per cent a year down onto a shorter stretch, halving it for six months or quartering it for three. Scaling is interpolation wearing a different hat. Scaling manufactures a rate at a horizon the record never carried, and it silently picks a convention for how a rate is spread across part of a year. Choosing that convention is a matter for an authority rather than for arithmetic. Where a reader expects a rate in between, the honest move is to draw the cell empty with the reason inside it and stop. Drawing the gaps as gaps is more truthful than filling them, and it is the single thing most pictures of a rate schedule get wrong.

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

One discipline runs through this whole subject area and it earns its keep right here. Every rate carries the word SPOT or the word FORWARD. A SPOT rate 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 and FORWARD are different objects that happen to be measured in the same unit, and on this schedule they sit close enough together to be merged by anybody reading quickly.

See how close. Derive the one year one year FORWARD rate from the two SPOT rates that already contain it: the rate for money placed a year from now and returned a year after that.

Now set that beside the three year SPOT rate of 6.55 per cent a year. The one year one year FORWARD rate of 6.601157 per cent a year stands 0.051157 percentage points from it, or 5.1157 basis points. Two numbers that close together, both quoted in per cent a year, both about the same borrower, and they are completely different objects: one is the price of a two-stage wait beginning a year from now, the other is the price of a single three year wait beginning today. Nothing has been moved apart to make life easier. The label does the entire job of keeping them separate. No rate above appears without the word SPOT or the word FORWARD beside it.

Which parts of a bill are set by an authority rather than by arithmetic?

Most of what a reader wants to know next about a bill cannot be worked out from arithmetic at all. All of it is rule, and rule belongs to whoever writes it.

The list is longer than people expect. At what lengths this borrowing is offered. The mechanic by which a new one first reaches a holder. The schedule on which offerings are made. How the price is quoted, and on which basis. Which day count convention a yield calculation has to use. The convention deciding when a purchase is paid for and delivered. Who may hold and deal in these securities, and under what conditions. The valuation norm deciding the price at which a holding is carried on somebody's books. Every one of those is set by the Reserve Bank of India, at rbi.org.in, and every single one of them moves.

Each of them is named above, and each is set by an authority elsewhere. A treatment that wrote any of those out would not merely become old when the rule changed, it would become wrong on the day of the change, and a reader who had learned the figure from it would carry it into a decision without knowing it had moved. Naming the item and naming the address is the treatment that survives. The arithmetic above is written free of every convention except the compounding basis, and that one sits inside the sums because no sum can be reproduced without it.

What does a bill not do?

A short list, because the things people quietly assume about bills cost more than the things they get wrong on the arithmetic.

A bill does not pay along the way. A holder who wants money before maturity has to sell the security. The sale price then depends on the rates ruling at that moment, and no such rates are recorded above. There is no schedule to fall back on and no instalment to collect early.

A bill does not shield anybody from a move in rates while it is held. After a rise in the yield the price of an unsold bill falls, exactly as any other dated claim's price falls, and after a fall in the yield the price rises. The holder who intends to wait to maturity is unaffected by that movement in the sense that the repayment is unchanged. The holder who has to sell is affected in full. The two are different people and must be kept apart.

And a bill does not settle what to do with money once it arrives. A bill moves that decision. A security paying along the way hands back cash on several dates and asks, on each of them, what should now be done with the instalment. A bill hands back Rs 1,000.00/- once, and asks the same question in full, on that one date, with nothing spread out to soften it. Concentrating a question is not the same as answering it, and a reader who treats one payment as one less thing to think about has confused the shape of the arrangement with the size of the decision.

Whether a bill is safer than a bond is covered separately, and in any case the word safe hides which risk is meant: a security can be steady in price and awkward to sell, or easy to sell and sharp in price, and one word cannot carry both. A bill is not immune to a fall in its price either.

One note on how rate movements are written looks like pedantry until it bites. Up and down are useless words for a rate. A move is written as a rise in the yield or a fall in the yield, every single time. The reason is that up and down mean the price in one sentence and the yield in the next, and a treatment that lets them alternate can state the exact reverse of its own arithmetic without a single reader noticing.

What can honestly be claimed about a treasury bill, and what cannot?

Less than the length of the treatment might suggest, and enough to be worth the reading. A treasury bill is short-dated government borrowing that pays once, at maturity, and every other property of the instrument follows from that. The price is one division away from a recorded rate, and the division can be rerun by anybody. The gap between price and repayment is the entire return. And the one gap reads as 5.900000 per cent a year against the price paid and 5.571294 per cent a year against the amount repaid, standing 0.328706 percentage points or 32.8706 basis points apart, with the whole of the difference living in the choice of denominator.

Arithmetic cannot say at what lengths this borrowing is offered, or how one first reaches a holder, or how a price is quoted, or which day count applies. All four belong to an authority. No horizon shorter than a year can be priced from a record that stops at one year. Whether a bill should be held is a decision rather than a division. The habit that outlives the instrument entirely is the small one: a rate without its base written beside it is not yet a statement about anything at all.