How to Compare Government Security Maturities on a Curve

Two government security maturities are two dates on one borrower's schedule, so the comparison is arithmetic rather than opinion. Subtract the two recorded SPOT rates and write both units. Divide by the years the step covers. Derive the FORWARD rate those two SPOT rates already imply. Multiply the path back out. Price it. Then refuse any horizon the record leaves blank.

Somebody sets two government securities side by side. One matures in two years, the other in ten. The question sounds like a matter of judgement and it is not one. Everything that separates those two securities is the length of the wait, so the comparison reduces to a short run of subtractions, divisions and one check, and every figure it produces can be argued with by anybody holding the same schedule.

Here is the everyday shape of it before any rate appears. A landlord quotes two rents for the same flat: one figure for a tenant signing for a single year, another for signing for ten. The flat has not changed. The landlord has not changed. The length of the commitment changed, for the landlord and for the tenant alike. Finding what the ten year commitment really costs against the one year commitment takes no opinion about the landlord. The finding takes a subtraction, and then a question about what the subtraction comes to per year.

A government's schedule of borrowing rates works the same way, with one useful simplification: the borrower on every line is the same borrower. One borrower on every line is what makes the comparison a procedure rather than an argument. Worked in order, the steps leave two people who disagree about everything else landing on the same numbers.

What is being compared when two government maturities are compared?

Not two securities. One borrower, read at two dates. Nothing about who is doing the borrowing changes between the one year point and the thirty year point, so nothing about creditworthiness, recovery or willingness to pay is inside the difference about to be computed. The whole of the difference is the wait.

The single borrower is what licenses the arithmetic. Two different borrowers side by side means two things moving at once, so no subtraction can say which one moved. One borrower at two horizons means exactly one thing moving. Subtraction then does honest work, and each step below adds a reading rather than an opinion.

The schedule below, an invented one, records a SPOT rateThe rate for money placed today and returned at one stated future date, with nothing paid in between. at six horizons and at no others.

Six rows and nothing between them. There is no four year SPOT rate here, no nine year SPOT rate, no twenty nine year SPOT rate and nothing at all shorter than one year. Each of those six horizons is a nodeOne of the horizons at which a rate is actually recorded, as against a horizon read off a line somebody drew between two recorded points., and the last step of this procedure exists to stop a rate being read anywhere else.

The compounding convention sits 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 three year rate, three times over.: 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. The convention is not housekeeping and does not belong in a footnote. The same six numbers read on a semi-annual convention produce different prices and a different set of derived rates, so a reader who is not told which convention is in use cannot reproduce a single sum here.

The growth factorOne plus the rate, raised to the number of years. Compounding multiplies by exactly that number. column matters more than it looks. Almost every step further down is a ratio of two of those factors, so once the column is built the rest of the procedure is division.

What are the six steps of the comparison, in order?

Six steps, run in sequence, with nothing skipped and nothing reordered. Each one is stated here as an instruction only; the reasoning behind each sits in the section that runs it. Read the list once as a list. The order is the method, and the last step is the one everybody drops.

1. Take the gap between two named horizonsSubtract the near horizon's recorded SPOT rate from the far horizon's recorded SPOT rate. Write the answer in percentage points and in basis points, with both maturities beside it.
   Checking: are both maturities written on the line, in both units?
2. Divide the gap by the years it coversTake the gap from step one and divide it by the far horizon less the near horizon. Record the result as percentage points a year.
   Checking: is the divisor the number of years between the two, rather than the far horizon itself?
3. Derive the FORWARD rate the two SPOT rates implyDivide the far growth factor by the near growth factor, take the root of the number of years in the window, subtract one. Write both parent SPOT rates beside the result.
   Checking: are the two SPOT rates it came from on the same line as the answer?
4. Multiply the whole derived path back outGrow one rupee through every recorded window in turn and compare the result against growing one rupee at the far horizon's SPOT rate alone.
   Checking: do the two routes land on the same figure, digit for digit?
5. Price the comparison in rupeesDivide the same promised amount by each horizon's growth factor and put the two prices side by side.
   Checking: is the promised amount identical at both horizons, so only the wait differs?
6. Refuse the comparison at any horizon not recordedCheck both horizons against the recorded list before anything else is written. Where either is absent, state the absence and stop.
   Checking: is every horizon used in steps one to five actually on the record?

What does the gap between two named horizons establish?

Step one is a subtraction, and the discipline sits in what gets written beside it rather than in the sum. A slopeThe gap between the SPOT rates at two named horizons. Without both names it is a number nobody else can check. is not a property that a schedule of rates has. A slope is a distance between two horizons that somebody chose, and a distance quoted without its two endpoints cannot be reproduced by the person reading it.

Run on the two year and ten year pair used all the way through, 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. Run on a different pair, the answer changes completely: 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. Same schedule, same borrower, same moment, and two answers that differ by more than half a percentage point.

A basis pointOne hundredth of a percentage point, so 110 basis points and 1.10 percentage points are the same quantity written two ways. is one hundredth of a percentage point. The single most common defect in a comparison like this is a gap written in one unit and read in the other, so the definition is worth stating flatly. A schedule where the two year and ten year points sit 110 percentage points apart would be an extraordinary object; one where they sit 110 basis points apart is the ordinary one on this record. Write both units on every gap, every time, and the mistake becomes impossible rather than merely unlikely.

Why must the gap be divided by the years it covers?

Because a gap bought with one extra year of waiting and a gap bought with twenty extra years are different objects even when the two read the same size. Step two turns each gap into a rate of pay per year of waiting. Almost everybody skips that step: the raw number already looks like an answer.

The everyday version of this is easier to feel. Two shops quote a surcharge for delivery. One charges Rs 60.00/- to hold an order for a day. The other charges Rs 90.00/- to hold it for a month. The second surcharge is larger and the second shop is far cheaper for waiting, and that stays invisible until the surcharge is divided by the wait. Comparing the two surcharges as they stand says nothing about which shop charges more for time.

Run it on every step of the invented schedule. From one year to two years the schedule adds 0.3500 percentage points across one year, so 0.3500 percentage points a year. Two to three adds 0.3000 across one year, so 0.3000 a year. Three to five adds 0.3500 across two years, so 0.1750 a year. Five to ten adds 0.4500 across five years, so 0.0900 a year. Ten to thirty adds 0.2500 across twenty years, so 0.0125 a year.

Now put the last two side by side. The five to ten step against the ten to thirty step is the whole of this step. On the raw reading, 0.4500 against 0.2500 looks like one step paying not quite twice the other. Per year of waiting it is 0.0900 against 0.0125, seven times over, and the raw reading was not slightly off. The raw reading was the wrong shape of answer.

How do two SPOT rates give up the FORWARD rate between them?

Steps one and two both describe the two endpoints. Neither of them says anything about the stretch of time in between, and that stretch is usually the thing the question was really about. Step three gets it out, and the figure is already inside the two recorded rates rather than being added to them from outside.

The reasoning is a no-free-lunch argument and it takes one sentence. If the two routes did not agree, one of them would be plainly better and nobody would take the other. So money placed for two years at the two year SPOT rate of 6.25 per cent a year has to end up where money placed for one year at the one year SPOT rate of 5.90 per cent a year and then rolled for a second year ends up. So the second year's rate is whatever makes the two routes agree, and it comes out by division.

Run it on the shortest window the record carries, where the root is over a single year and disappears. Two years at the two year SPOT rate of 6.25 per cent a year gives 1.0625 squared, or 1.1289062500. One year at the one year SPOT rate of 5.90 per cent a year gives 1.0590. Divide: 1.1289062500 over 1.0590 is 1.0660115675. Subtract one, and the one year one year FORWARD rateThe rate for money placed at one future date and returned at a later one. A SPOT rate starts today instead. is 6.601157 per cent a year.

The check runs in the other direction before the figure is written down. 1.0590 multiplied by 1.0660115675 returns 1.1289062500, exactly 1.0625 squared. A FORWARD rate that has been checked backwards is arithmetic; a FORWARD rate that has only been divided out is a number somebody is hoping about.

Why every rate carries the word SPOT or the word FORWARD

Every rate written anywhere below 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 rates and FORWARD rates are different objects doing different jobs, and on this schedule they sit close enough to be merged by anybody reading quickly.

Look at how close. The one year one year FORWARD rate derived above is 6.601157 per cent a year. The three year SPOT rate recorded on the table is 6.55 per cent a year. The two figures sit 0.051157 percentage points apart, or 5.1157 basis points. They are not remotely the same thing: one is the price of a single year that has not started yet, the other is the average annual price of the next three years starting today. On any smooth schedule of rates FORWARD rates land near SPOT rates, and the label is what does the work of keeping them apart.

The second reason the label is compulsory is that a FORWARD rate looks exactly like a forecast and is not one. Shown as a quotient of two SPOT rates it is plainly arithmetic pulled out of today's schedule. Quoted bare, as a single number with no parents beside it, it reads as somebody's opinion about what rates will be, and that reading is wrong in a way that is very hard to argue somebody out of afterwards.

What happens when the same step is run on every window?

One quotient is a trick. Five quotients run on every windowThe stretch of time between two recorded horizons. A FORWARD rate is the price of one such stretch. the record allows is a procedure, and the fifth one produces a result most readers do not expect.

Each FORWARD rate below is the ratio of the two growth factors on either side of its window, taken to the root of the number of years the window spans, and each is shown with the two SPOT rates it came from because it cannot honestly be shown any other way.

Read the last two rows again. Every SPOT rate on the record is higher than the one before it, without exception, all the way from 5.90 to 7.60 per cent a year. And yet the derived path climbs to 7.801894 per cent a year over the five to ten window and then comes back down to 7.725218 per cent a year over the last twenty. The path turned while every recorded rate was still rising.

The turn is not a contradiction and not an error in the schedule; it is what an average does. A SPOT rate is one average annual rate covering its whole length, and an average can keep rising while the newest figures being averaged into it have stopped rising. A batsman whose career average is still climbing after a modest innings has not scored more than ever in that innings; the average climbed because the innings was still above what came before. The thirty year SPOT rate is higher than the ten year SPOT rate, and the twenty years it added to the average were nonetheless charged at less than the five before them.

How is it checked that the five derived rates are one identity?

Five derivations look like five separate claims, and a reader is entitled to ask why they should trust any of them. Step four answers that in one multiplication. The five FORWARD rates are not five additional facts bolted onto the schedule; they are the same schedule written as a run of windows instead of as a run of endpoints, and if any one of them were an opinion the product would not close.

Take one rupee. Grow it for a year at the one year SPOT rate of 5.90 per cent a year. Grow the result for one more year at the FORWARD rate of 6.601157 per cent a year, then one more at 7.152544 per cent a year, then two years at 7.427157 per cent a year, then five years at 7.801894 per cent a year. Every one of those came from two of the recorded SPOT rates: 5.90, 6.25, 6.55, 6.90 and 7.35 per cent a year. The path covers ten years of growth, and it lands on 2.0324528891.

Now take one rupee and grow it for ten years at the ten year SPOT rate of 7.35 per cent a year and nothing else. The rupee lands on 2.0324528891. Same figure, to as many places as anybody cares to carry.

The cancellation is the whole guarantee of the method, and also the reason a mistake in any single window is caught immediately. Change one derived rate by a basis point and the product stops landing on the ten year growth factor. The check is a working test rather than decoration.

What does the same promise cost at each horizon in rupees?

Step five exists because a difference of 1.10 percentage points is easy to nod at and hard to feel. Convert the comparison into what somebody actually hands over today and the size of it stops being abstract.

Rs 1,000.00/- promised by this one borrower costs Rs 885.813149/- if the promise falls due in two years, discounted at the two year SPOT rate of 6.25 per cent a year. The identical Rs 1,000.00/- costs Rs 492.016324/- if the promise falls due in ten years, discounted at the ten year SPOT rate of 7.35 per cent a year. Same borrower, same promised amount, same moment, and a price that has fallen by almost Rs 394.00/-.

Push it to the ends of the record and it is starker still. At the one year SPOT rate of 5.90 per cent a year the promise costs Rs 944.287063/-. At the thirty year SPOT rate of 7.60 per cent a year it costs Rs 111.078974/-. The rates across the whole schedule span 1.70 percentage points, and the prices span nearly the entire promised amount. A modest annual difference compounded over a long wait does exactly that.

What happens when the horizon wanted is not recorded?

Step six is a refusal, and it is the step a reader is most tempted to skip because skipping it always produces an answer. The record carries six horizons and nothing whatever between them. There is no four year SPOT rate, no nine year SPOT rate, no twenty nine year SPOT rate and nothing shorter than a year at all.

So the gaps are left as gaps rather than bridged by a line drawn between the recorded points. InterpolationReading a value off a line drawn between two recorded points. The figure it produces depends on the line somebody chose to draw. would produce a figure whose value depends on the method chosen rather than on anything recorded: a straight line between two points and a smooth curve through several will disagree, and neither disagreement can be settled by the record. Two people working from the same six numbers would then print two different rates for the same object and both would be able to defend themselves. Drawing the gaps as gaps is more honest than filling them, and it is the thing most pictures of a schedule of rates get wrong.

Concretely, two ordinary requests have to be turned down. Neither the four year rate nor the six year rate is recorded, so the gap between those two horizons cannot be had. The nine year SPOT rate is not recorded either, and a window needs a recorded rate at each end, so the ninth year on its own cannot be priced. Available instead is the FORWARD rate covering the fourth and fifth years together, 7.427157 per cent a year, derived from the three year SPOT rate of 6.55 per cent a year and the five year SPOT rate of 6.90 per cent a year, and that is a rate for a two year window rather than for either year inside it. Naming which windows exist and which do not is the last step of the procedure rather than a footnote under it.

Can the whole procedure be run on any recorded pair?

Fifteen pairs are reachable on six recorded horizons and there is no sixteenth. The two controls below move three readings at once on whichever pair is selected: the gap, the gap per year of waiting, and the FORWARD rate for the window between the two dates. The three readings do not move together, and watching them come apart is the fastest way to learn that they answer three different questions.

Drag the far horizon out from five years to thirty and the raw gap grows while the per year figure shrinks, and the derived FORWARD rate rises and then turns back. The controls open on the two year and ten year pair. The gap there reads 1.10 percentage points, or 110 basis points, over eight years, so 0.1375 percentage points a year. The FORWARD rate for the eight year window between those two dates is 7.626775 per cent a year, derived from the two year SPOT rate of 6.25 per cent a year and the ten year SPOT rate of 7.35 per cent a year. Inside that window the record carries rates at three and five years as well, giving FORWARD rates of 7.152544, 7.427157 and 7.801894 per cent a year across the three recorded windows within it.

Which of the three readings answers which question?

The three are not competing estimates of one thing. Each answers a different question, so picking between them is a matter of what was asked rather than of which is best, and reporting one when another was wanted is the quiet way this material goes wrong in a meeting.

No one of the three substitutes for another. A comparison reporting only the first is the one nobody stops. The gap is the reading almost everybody stops at: a single subtraction that sounds like an answer. The gap answers a narrower question than the one usually being asked.

Who actually runs this procedure, and when?

Somebody runs it at the moment two maturities are set on the same sheet and a sentence has to be written about them. A funding desk inside a bank runs it before deciding where along the schedule to raise money. The per year reading tells the desk what each additional year of committed funding is being charged for. The FORWARD reading tells it what the stretch between two dates is charged for, a different question and often the one that matters. Neither reading tells it what to do, and the desk knows that.

An analyst writing up a move in the schedule runs it because the alternative is writing a sentence nobody can check. Reporting that the schedule steepened is not a finding until the two horizons are named, and the moment they are named the reader can rerun the subtraction and either agree or not. Step one has exactly that social function: it turns an impression into something falsifiable.

A household runs a smaller version of it without the vocabulary. A deposit for one year pays one rate, a deposit for five years pays another, and the interesting question is almost never which rate is bigger. The live question is what the extra four years of being tied up is being paid for, step two, and whether that stretch of four years is being paid better or worse than the first year was, step three. The procedure does not change size when the amounts get smaller; only the vocabulary does.

The horizon somebody asks about is very frequently not one that anybody records, so a person on any of those three desks runs step six most often of all. The correct output there is a sentence naming what is missing. The sentence looks like an unfinished job and is the finished one.