The Swap Curve: A Benchmark Built From Swap Rates

A swap curve is the set of fixed rates at which arrangements of each maturity are being agreed on a given day, plotted against those maturities. A curve is built out of quoted rates rather than out of anything a single agreement contains. One rate at one maturity is a point, and no curve can be drawn through a single point.

Everything under that answer sits on one fact about how these arrangements come into existence. Each one that gets agreed on any given day carries a fixed rate inside it, and that rate was settled for an arrangement running a particular length of time. Gather up the rate being agreed for each length, put length along the bottom of a chart and rate up the side, and a curve appears. There is nothing else in the object. A curve also cannot exist for anybody holding a single arrangement, however carefully they read it: one agreement carries one length and one rate, and a collection needs many of both.

One warning shapes everything that follows, so it belongs at the top rather than tucked in near the end. The floating benchmark has one reading, taken for the opening period, and no schedule of readings after that. Nobody built one, nobody borrowed one from a neighbouring subject, and nobody guessed one. Two consequences follow, and they are strict. No arrangement can be given a worth today. No figure can be quoted for any period after the first. Where a value would ordinarily appear, the question stands instead with the answer left blank. An absence like that would elsewhere read as a limitation. With the schedule missing it happens to be the lesson.

What is a swap curve, before anybody does anything with it?

Start with the object itself and leave what people do with it for later. A swap curve is a set of points. Each point is two readings taken together: one maturityThe stretch of time an arrangement runs before it finishes. A maturity is a length, not a date, and two arrangements of the same length starting on different days share one. along the bottom, and one fixed rate up the side. A point has no other anatomy. Many of those points on the same pair of axes make up the object that gets called a curve.

A swap curve is a collection of separately agreed numbers rather than a formula, and every point on it got there because two parties sat down and settled on a figure. Nothing generates the points. No equation produces them. The points arrive one at a time, from real negotiations, and there are exactly as many of them as there are maturities somebody is willing to deal at. If nobody is dealing at a particular length on a particular day, that length has no point on that day, and no amount of drawing will conjure one.

Here is the everyday version, and it is worth holding on to. The everyday version does something the abstract description cannot. Think of the board outside a currency counter at an airport, the one listing a different figure for every note it will take. Each figure on that board was arrived at separately. Nobody derived the figure for one note from the figure for another. The figures sit on one board because printing them on one board is convenient, and a reader who assumed the board was generated by a rule would be reading a rule into a list. A swap curve is that board. The board is a list that happens to be drawn as a picture.

The drawing carries no line. A line is the very thing worth setting aside for a moment. Once the line is on the paper it looks like the primary object and the points look like decorations scattered on it. The truth runs the other way round. The points are the data and the line is the decoration, and getting that order the right way up is what makes every later question here answerable.

Where does a single point on one come from?

A point at a given maturity is the fixed rate at which arrangements of that length are being agreed today against the floating benchmark. Somebody walking in wanting to enter an arrangement of that length would expect to be shown roughly that rate. The point means that and nothing more. A number whose origin cannot be pictured tends to get treated as though it fell out of the sky, so this one is worth tracing back one step further, to the room where it was settled.

Two parties talk. The two of them are discussing an arrangement of one particular length. Between them they settle on a fixed rate for that length, quoted against the floating benchmark, and that settled rate is the point. Repeat the whole business at every length anybody is dealing at, and the set of settled rates is the collection. There is no stage in that sequence where a formula appears. Every stage is two parties putting a number to something.

Every point is quoted against the same floating benchmark, and that single property holds the collection together. The points then differ only in the length of the arrangement, so the collection is a picture of what length costs rather than a picture of several unrelated things. Take that property away and the picture stops meaning anything. Two rates quoted against two different benchmarks are answers to two different questions, and putting them on one pair of axes does not make them comparable, it just makes them adjacent.

Two things a point is not, and both of them are easy to slip into. A point is not a rate that anybody paid in the past. A point is a rate at which arrangements are being agreed now, and a different tense is a different fact. And a point is not a rate anybody is obliged to offer. A quoteA rate somebody is offering to deal at, as against a rate two parties have already shaken hands on. The first is an invitation and can be withdrawn; the second is settled and cannot. is an invitation to deal, and an invitation can be withdrawn, revised or simply not extended.

What does one point actually say?

Less than most readers expect, and this is where it helps to work the only point this material has in front of it.

Two invented parties sit behind every rupee figure here. On a notional of Rs 1,000 crore, Chitrakoot Cements Limited has undertaken to hand over 7.20 per cent a year, the fixed rate typed into the agreement, and to take the floating benchmark back the other way. Facing it across the same agreement, Saranga Capital Limited hands over whatever the floating benchmark shows and takes the fixed rate in return. For the opening period that benchmark showed 6.00 per cent a year. After that opening period the record simply stops, and the consequences of that stop show up in nearly every part of this guide.

So the rate typed into the Chitrakoot Cements agreement reads 7.20 per cent a year. The 7.20 per cent a year is one point. The point says something narrow and precise: on the day this agreement was struck, these two parties agreed that this rate, against this floating benchmark, for an arrangement of this length, was an even exchange for both of them. Not a good deal for one of them. Not a view either of them held about anything. An even exchange, in the judgement of the two people settling it, on one morning.

Now the three things it does not say, and it is worth being blunt about each. The point was never a statement about any other length, so it says nothing about arrangements of any other length. The rate was settled on the day it was settled, and days move, so the point says nothing about what that same length would be quoted at today. And where the floating benchmark goes next was exactly the question the two parties declined to answer and chose to exchange instead, so the point says nothing whatsoever about that either.

Then the harder point, stated plainly rather than worked round. Which maturity that 7.20 per cent a year belongs to is nowhere in the record, so the single point cannot be placed along the bottom axis at all. Saying it that way is sharper than any sentence about missing data could manage. A point needs two readings before it can be put anywhere. The record supplies one of them.

Why is a set of these rates used as a reference at all?

Step back from this particular arrangement for a moment and ask why anybody bothers assembling these rates into a collection. The answer is a combination of four ordinary properties, and no single one of them would be enough on its own.

Fixed rates on these arrangements get agreed continuously rather than once a quarter. Fixed rates get agreed at many different lengths rather than at one conventional length. Fixed rates get agreed between many different parties rather than inside one institution. And each of them is a number two parties were willing to commit real money to, rather than an opinion somebody expressed in a note or a level somebody thought looked about right. Put those four together and the collection turns into something genuinely useful to people who are party to none of the agreements in it: a running record of the level at which length has actually been exchanged.

The last phrase, has actually been exchanged, is doing the work. When somebody commits to pay a fixed rate for five years in exchange for a floating benchmark, they have put a price on the difference between being fixed and being floating over five years. When somebody else does the same over two years, they have priced the two year version of the same trade. Enough of those collected together show what the market charges for length, and they show it in the only way that ever really convinces: by what people actually paid rather than by what they say they think.

Describing why a collection like that is used as a reference is one job, and using one is a different job that takes the collection itself. The distinction is worth keeping in view throughout. Explaining a mechanism costs nothing. Operating it costs data.

The abstraction gets slippery, so here is the household version of the same idea. A street vendor deciding what to charge for a plate of food does not run a model. The vendor walks past four other stalls, sees what those stalls are charging, and sets a figure. The other stalls have been treated as a reference level: not because the stalls are authoritative, but because those figures are prices somebody actually accepted. A curve of swap rates is that walk past four stalls, written down, at a scale where nobody could walk it.

Which three readings of a curve are wrong, and why do they arrive together?

The three arrive together because they all come from one shared error, and killing the three together beats meeting them one at a time six months apart.

The first wrong reading is that an upward sloping curve is a forecast that rates will rise. An upward sloping curve is nothing of the kind. The curve is a set of rates being agreed today for arrangements of different lengths. Why a longer arrangement gets agreed at a different rate from a shorter one is a real and interesting question, and it belongs to the term structureThe way a rate changes as the stretch of time it covers gets longer. Why that happens is worked through in the fixed income material rather than here. material rather than here. The shape is a record of agreements struck, and a record of agreements struck is not a prediction, however much it looks like one when it slopes.

The second wrong reading is that a curve says what a particular agreement is worth. On its own it does not. Working out what an existing arrangement is worth needs more than a set of rates, and what else it would need is set out below. The record behind these figures holds none of it, so no arrangement in it can be marked to marketRestating something at what it would fetch if it were closed out now rather than at what was originally paid for it. Restating anything needs a current level for everything inside it. or given a worth today.

The third wrong reading is that the line between two points is data. The line is not data. Being the most visually prominent thing on the chart, the line is also the wrong reading people find hardest to give up. The line is a drawing convention. A maturity where nobody is quoting has no point, and it is a gap in the record however smooth the picture over it looks. Filling that gap by interpolationFilling in a value that sits between two known ones by applying a rule, rather than by observing it. The filled value is a construction and never an observation, however reasonable the rule. produces a number, and the number is a construction rather than an observation, which is a distinction that survives no amount of smoothing.

All three collapse into one sentence: a curve is a record of rates agreed, not a statement about what comes next, and every wrong reading of one comes from forgetting that.

Why does a quoted swap rate carry no amount beside it?

A quoted swap rate is a percentage a year and nothing else. There is no rupee amount attached to it, and that absence is not an oversight in how quotes are displayed. The absence says something about what kind of thing a rate is.

The notional gets supplied later, by whoever actually enters the arrangement. Rs 1,000 crore, or ten times that, would leave the quoted rate exactly where it was. Size is a term two parties settle between themselves once they have decided to deal, and it has no place at all in a level that people outside the deal are meant to be able to read and compare.

The reverse of that is the sentence worth carrying away: two arrangements at wildly different sizes can be struck at the same rate, and the rate is identical because the rate was never about the size in the first place.

There is a practical consequence for anybody reading a screen of quotes for the first time. Nothing on that screen says how big anything is, and a reader who goes looking for the size is looking for a thing that was deliberately left out. The amount moving between the two parties is worked out afterwards, from the rate and from the notional the two of them settled privately, and the notional never reaches the screen because the screen is not about any particular pair of parties.

Why is no curve drawn here?

Here is the sharp edge of the whole thing, and it is easier to reach after the arithmetic than before it, so take the arithmetic first.

The arrangement between Chitrakoot Cements and Saranga Capital runs on a notional of Rs 1,000 crore, or Rs 10,00,00,00,000/- written out in full. Over one complete opening period at a day count fraction of 1.0000, Rs 1,000 crore run at 7.20 per cent a year puts Rs 72,00,00,000/- on the fixed side, or Rs 72.00 crore. The same notional at 6.00 per cent a year settles the floating side at Rs 60,00,00,000/-, or Rs 60.00 crore. Set one against the other and the gap measures Rs 12,00,00,000/-, Rs 12.00 crore, and Chitrakoot Cements is the side handing that difference across.

A second route arrives in one step instead of three, and it is worth having because it exposes what the two legs are really doing. Skip both of them and go straight at the difference between the two rates, a gap of 1.20 percentage points or 120 basis pointsOne hundredth of a percentage point, so a hundred basis points make one percentage point. Rates get quoted in them because a hundredth of a point on a large notional is a real sum of money.. Rs 1,000 crore multiplied by 120 basis points also lands on Rs 12,00,00,000/-. Take that Rs 12.00 crore back through the notional of Rs 1,000 crore and 1.2 per cent falls out, the difference between the two rates and nothing else.

Now the sharp edge. Drawing a swap curve takes a fixed rate for each maturity, each one quoted with a date beside it and a source behind it. The record carries one such rate at one maturity: the 7.20 per cent a year inside the agreement. One rate at one maturity is a point. A curve drawn through a single point is any curve at all. A shape that can be anything says nothing, so any curve at all is no curve.

A plausible shape could have been drawn in about ten minutes. A drawn curve with made-up numbers on it teaches the shape of somebody's guess, and no reader has any way of telling that shape from a real one afterwards. A shape nobody has quoted is a shape somebody has guessed.

The empty pair of axes above appears instead: labelled on both sides, carrying no numbers, with the one rate this material holds shown floating beside them as a level rather than plotted as a dot. The maturity that rate belongs to is not recorded, and a point with only one of its two readings cannot be placed anywhere.

What would have to be obtained before one could be drawn?

A list of what is missing turns an absence into an errand somebody can actually run. Complaining that it is missing does not. There are four things, and every one of them is ordinary.

First, a fixed rate for each maturity, all of them quoted on the same day as each other. Second, the date those quotes were taken. A curve is a snapshotA set of figures all captured at the same moment, so that differences between them come from what is being measured rather than from when each one was taken. and two dates mixed together produce a picture of nothing in particular. Third, the floating benchmark every one of them is quoted against. Rates quoted against different benchmarks do not belong on the same set of axes. Fourth, the source that published them, for anybody who wants to go back and check.

All four are ordinary and obtainable, and all four have to come from a live source rather than from a teaching record. A teaching record can supply one rate that a reader recomputes on the spot; a made-up set of rates at many maturities would be a shape rather than a set of figures.

What does reading a curve not license?

Being able to say what a swap curve is, and where the points on one came from, is a reading skill and it stops there. The skill is not a reason to be in an arrangement, and whether anybody should be is settled nowhere above.

Four things would need answering before that question could be put to anybody at all, and every one of them is about the party asking rather than about the curve: what a party already owes and on what basis it owes it; what it is actually trying to change about the shape of its own cash flow; over what stretch of time it wants that change to hold; and what it intends to do in the periods where the arrangement runs against it. A screen full of quotes supplies none of those four, and no other set of quoted rates supplies them either. Quoted rates hold no outcome for any party, have followed no arrangement through to its end, and say nothing at all about how likely any particular ending was.

Understanding how a mechanism works has never been a reason to be inside it, and the gap between the two is exactly where reading stops and deciding begins.

Chitrakoot Cements Limited, Saranga Capital Limited and the floating benchmark they are quoted against are invented.
Educational material. Not advice on any investment, tax, budget or market position.