The Yield Curve: Its Shapes, Its Types and How It Moves

A yield curve is a set of interest rates read at one moment, one rate for each length of time, all belonging to a single quality of borrower. Each point says what an amount placed today earns if it comes back on that one date. The invented curve here fixes six rates and nothing between them, and its shape, its type and its movements are all read off those six.

Everything that follows rests on one small idea. An interest rate is a price for time, and once prices for different amounts of time are allowed to differ from one another, they can be set out in order of time and looked at together. A yield curve is that arrangement and nothing more exotic than that. The arrangement is worth study because it carries information no single rate carries: an order, a set of distances, and a set of holes.

What does one point on a yield curve actually report?

A point is a rate, not a bond and not a price. The claim carries more weight than its length suggests. The commonest way to misread every shape set out below is to arrive believing that each point is some particular instrument's yield. It is not. Each point on the curve used here is a SPOT rate: the rate governing an amount that goes out today and comes back as one single payment on one named future date, for one credit qualityHow likely the particular borrower standing behind a rate is to pay what was promised. A set of rates read for one borrower quality says nothing at all about any other., read at one moment, on ANNUAL compounding.

Six of them are fixed on the invented curve used throughout this guide, and the word invented is not a formality: these are teaching numbers, and no rate below was taken from any market. At one year, 5.90 per cent. At two years, 6.25 per cent. At three years, 6.55 per cent. At five years, 6.90 per cent. At ten years, 7.35 per cent. At thirty years, 7.60 per cent. Each of those six is a SPOT rate and carries that word wherever it is written below. Six numbers is the whole object, and every shape, distance and movement described below is read off those six and nothing else.

The cleanest way to feel what one of those points reports is to give it a rupee amount and watch what happens. Take the ten year SPOT rate of 7.35 per cent. Rs 1,000/- placed today at that rate, with interest added once a year for ten years, comes back as Rs 2,032.452889/- in one payment on one date. Read the same rate the other way and it discounts instead: Rs 1,000/- falling due in ten years is worth Rs 492.016324/- today. Notice everything the point leaves out of both of those statements. The rate does not mention a borrowing instrument, a seller, a price, a coupon, or a single one of the nine dates in between.

Run that across all six recorded lengths of time and the object stops being abstract. The table below is the whole curve read as money in both directions, and the second column is what makes the shape matter: Rs 1,000/- left for thirty years at the thirty year SPOT rate comes back as Rs 9,002.603850/-. Compounding is not addition, so thirty years is not six times what ten years produces.

Software and spreadsheets use that last column constantly, and it has a name worth carrying. Divide it by the Rs 1,000/- and what remains is a discount factorWhat one rupee falling due on a stated date is worth today. Multiply any future amount by it and out comes the present amount, which is why software stores these rather than storing rates.: 0.4920163244 at ten years, 0.7163272523 at five. Nothing new is being claimed there. The discount factors are the same six rates written in a second unit. Because the choice of unit turns out to change an answer, that second unit matters two sections from the end.

Which Yield Curve Types does one market draw at the same moment?

Four of them, at least. Which of the four is in hand has to be settled before any of the other questions, and the question is almost never asked. A picture of a rising line labelled with maturities looks self explanatory. The same market, read at the same instant, for the same quality of borrower, produces several different curves at once, and they carry different numbers. A curve handed over without its type named is not one figure but four figures wearing one coat.

The first is the SPOT curve, sometimes called the zero curve because it is the curve a zero couponA borrowing instrument that pays nothing at all until it ends and then pays one single lump. Its entire return sits in the difference between what was paid for it and what comes back. instrument would carry. One rate for one dated repayment: exactly the object taken apart in the section above. The SPOT curve is the one drawn throughout, and it is drawn because its points alone need no assumption about anything else to be interpreted.

The second is the par curve. For each length of time it gives the coupon rateThe rate written into the terms of a borrowing instrument, fixing what it pays each period. It is stated against the amount printed on the instrument rather than against whatever somebody later paid for it. that would price a borrowing instrument of that maturity at its face amountThe amount printed on a borrowing instrument, which is what falls due at the end of its life. It is a term of the instrument and does not change when the price changes. exactly. The par curve is a curve of coupon rates, not of rates for single dated repayments, and the two agree only in the special case where a single payment is all there is.

The third is the FORWARD curve, giving rates for periods that do not begin today but at some stated later date. Two of them recur throughout: the one year rate one year FORWARD, and the five year rate five years FORWARD. Neither carries a number here. A FORWARD rate is arithmetic sitting inside the SPOT rates already recorded, so it has to be produced from the two SPOT rates behind it rather than announced, and that production is covered under forward rates.

The fourth is a curve of yields to maturityOne rate standing in for a whole set of dated payments on one instrument, so two instruments falling due on the same date can carry different ones. What that single rate quietly assumes is settled where pricing is taught. on actual borrowing instruments, and it is the one most likely to be shown: nothing is easier to assemble from things that trade. The curve of yields to maturity is also the one that misbehaves. Each of its points is one blended rate covering a whole schedule of payments rather than one rate for one date, so two instruments maturing on the same day with different coupon rates land at two different heights on it. Same date, same borrower, two points. Two points at one date cannot happen on a SPOT curve, where a date has exactly one rate.

The abstraction hides how ordinary the mistake is, so here is the everyday version. A grocer, a wholesaler and a warehouse can all quote a price for rice on the same morning. All three prices are real, and all three differ because each one includes a different set of things. Nobody would put the three numbers on one chart and argue about the trend. With rates the equivalent chart gets built weekly. All four curves are drawn in the same shape, with maturities along the bottom and per cent a year at the side, and the drawing gives no clue which one it is.

So the discipline is a label, not a calculation. Every rate in this guide and in the material beside it carries the word SPOT or the word FORWARD attached to it, permanently, even where the sentence reads clumsily as a result. On the curve used here the one year rate one year FORWARD and the three year SPOT rate happen to sit remarkably close together, as they do on any smooth curve. A reader who meets both without labels will merge them into one number and never find out.

What shape is this yield curve, and how is a shape settled rather than eyeballed?

By subtraction, one adjacent pair at a time. A shape is a statement about the order the recorded rates stand in, and an order is settled by taking differences rather than by looking at a picture. The picture is helpful and it is also where the mistakes come from. The vertical scale of a drawing can be chosen to make 25 basis points look dramatic or invisible, and a reader who forms an impression before doing the subtraction has already been steered.

Six recorded points make five adjacent pairs, so doing the subtraction on this curve means taking five differences. One year to two years, 35 basis points. Two years to three years, 30. Three years to five years, 35. Five years to ten years, 45. Ten years to thirty years, 25 basis points. All five are rises, so the curve is rising at every recorded node. The five steps add to 1.70 percentage points, exactly the distance separating 5.90 per cent at the one year node from 7.60 per cent at the thirty year node. Nothing has gone missing.

The second column of that list is the one that gets skipped, so look at it now rather than at the first. Each of those steps covers a different number of years. The 35 basis point step at the front covers one single year. The 25 basis point step at the far end covers twenty. The front of this curve therefore travels further in one year than the back of it travels in twenty, and any comparison of the two step sizes that ignores the years each one spans is not a comparison of pace at all.

The fourth row is where an eye goes wrong most reliably, and it is worth pausing on. Forty five basis points is the biggest number in the second column, so a reader scanning the column picks the five to ten year stretch as the fastest moving part of the curve. The five to ten year stretch is nearly the slowest. The step is the biggest because it has the second longest run to make it in, and the two facts have to be read together or the conclusion inverts.

Two further things fall out of that drawing, and both matter later. The first is that the six points are not evenly spaced in time, so any drawing that spaces them evenly is telling a different story from the numbers. The second is that the dashed line joining them is a drawing aid and nothing more. The dashed line has been added so an eye can follow the order, and it asserts no rate anywhere along its length. The failure block below is about nothing else.

What shapes can a yield curve take, and what does each one say about the ordering?

Four names cover almost everything a reader is likely to be shown, and every one of them is a statement about ordering rather than about appearance. The difference between ordering and appearance is the reason the definitions below are written as comparisons and not as descriptions of a picture.

A curve is rising when each rate stands above the one at the shorter length of time. A curve is flat when the rates being compared are the same. A curve is inverted when each rate stands below the one at the shorter length of time, so the ordering is reversed completely. And a curve is humped when the rates climb to some middle length of time and are lower after it: the case where none of the first three statements is true of the whole thing at once.

An inverted curve is the one readers find least intuitive, so here is the everyday version. Being kept waiting is a cost, so a shopkeeper will normally charge more to take goods on a longer credit than a shorter one. Suppose instead the shopkeeper says the six month terms cost more than the two year terms. Dearer short terms are not a statement that a longer wait is cheaper in some general sense. The quote is a statement about the six months specifically: something about that near window has become expensive. An inverted curve carries exactly that flavour, and it is a statement about the near end far more often than the far one.

Because all four are ordering claims, something follows that surprises people. A single curve can be inverted between one pair of lengths of time and rising between another pair at the very same moment, and no contradiction is involved. A curve might sit lower at ten years than at two, and higher at thirty years than at ten. Ask whether that curve is inverted and the only correct reply is another question: between which two lengths of time?

Every shape claim in this guide therefore names its two lengths of time. The invented curve here is rising at every recorded node, and the claim is strong precisely because it has been checked at all five adjacent pairs rather than asserted from a glance. Weaker claims are available and are perfectly respectable, so long as they say what they cover: rising across the five year and ten year nodes, for instance, a true statement and one fifth of what the stronger claim says.

One more point of housekeeping saves confusion later. A curve does not have to be inverted from end to end to be interesting, and in practice it usually is not. The commonly discussed cases are inversions between two particular lengths of time somewhere in the near half, with the far half still rising. Such a curve is the humped case wearing everyday clothes, and it is only describable at all if the reader is willing to name pairs instead of reaching for one word to cover the whole drawing.

Which Yield Curve Movements account for nearly everything a curve does?

Three readings, taken in a fixed order: level, slope and curvature. Level, slope and curvature are not three descriptions of the same thing, they are three separate measurements, and each one is deliberately blind to what the next one sees. The blindness is the reason all three get taken rather than just the first, and it is also why an argument about whether a curve has moved usually turns out to be an argument about which of the three somebody was looking at.

Level is the easiest and is read at a named node. On the invented curve, 7.35 per cent is recorded at the ten year node. A node and its rate, stated together, is a complete level reading. A bare number without its node attached is not a level reading of anything, and it is astonishing how often one is quoted that way.

Slope is one named difference between two named lengths of time, stated in basis points. On this curve, taking the two year SPOT rate away from the ten year SPOT rate leaves 7.35 per cent less 6.25 per cent, or 1.10 percentage points, and the same distance written in the other unit is 110 basis points. Note that both units name the same thing. A hundred basis points make one percentage point, so the two figures are one distance said twice, and swapping the units for one another states a distance a hundredfold wrong without any arithmetic changing.

Curvature is the third reading and it exists because slope cannot see the middle at all. Slope is built out of two rates, one at each end of the pair, so anything happening between them is invisible to it by construction. The five year SPOT rate can move anywhere at all and the ten year less two year figure of 1.10 percentage points does not budge. A measurement blind to the middle needs a partner that is not, and curvature is that partner.

The calculation takes three rates rather than two: twice the middle one, less each of the two outer ones. On this curve that is twice 6.90 per cent, less 6.25 per cent, less 7.35 per cent: 13.80 less 13.60, giving 0.20 percentage points, or 20 basis points in the other unit.

The second reading of the same formula turns an odd looking expression into a picture, and it is the one worth carrying. Draw the two outer rates, join them with a straight line, and ask where the middle rate sits relative to that line. Above it means the middle is dearer than a plain average of its neighbours, and this curve sits above. The gap is 0.10 percentage points and the curvature reading is twice it, by construction rather than by coincidence.

Put the three readings side by side and their independence becomes concrete. Add the same amount to all six recorded rates and the level readings all change while the slope of 110 basis points and the curvature of 20 basis points both stay exactly as they were. Raise only the far end and level and slope both change while curvature may not. Raise only the five year SPOT rate and curvature changes on its own, with level at the other nodes and the ten year less two year slope both untouched. Three readings, three different things seen, and no one of them able to report what the other two report.

Two things are worth doing deliberately with that control before reading on. First, slide it slowly through 6.25 per cent and watch the label change at the exact moment the one year SPOT rate passes the two year SPOT rate, not at the moment the picture starts to look odd. A shape is settled by ordering, and the control makes that literal. Second, keep sliding all the way past 7.60 per cent, at which point the one year SPOT rate stands above every other recorded rate on the curve. Nothing at ten years or thirty years moved at any point during either of those, and the ten year less two year slope reading of 110 basis points never changed once. Blindness to the front node is exactly what that means.

What anchors the front of a yield curve, and what is left to the market beyond it?

One point of it is decided, and the rest is priced. The anchored front is the only part of a yield curve that is not produced by arithmetic, and it is worth naming carefully because a reader who thinks the whole curve is set by somebody will read every movement on it as a message. At the extreme front sits an administered rate, announced by an authority as an instrument of policy and applying to borrowing measured in nights. Around it sits a corridorA pair of administered rates placed above and below a central one, so that overnight borrowing has a ceiling and a floor. What those two are, and how far apart they stand, belongs to the authority that fixes them., and alongside both sits a set of market operations that carry the announced level into overnight borrowing, together with a requirement about how much of what a bank takes in has to be kept aside. The requirement shapes how much money is looking for a home each night.

In India every single one of those is fixed by the Reserve Bank of India, at rbi.org.in. Every single one of them is revised from time to time, and a level typed into a text is stale from the moment it changes while giving no signal at all that it has become wrong.

Where the setting stops and the pricing starts can be said exactly. The rate that is announced applies overnight. The rate paid at ten years on this invented curve is 7.35 per cent, and 145 basis points separate it from the 5.90 per cent recorded at the one year node. Nobody signs off that distance. The 145 basis points are not an instrument, they are not announced, and there is no meeting at which they are fixed. The distance is a residue: whatever survives after an announced overnight level has travelled outward through everything that stands between a night and a decade.

Here is the everyday version. An announcement is made over one loudspeaker at the entrance of a large railway station. The announcement itself is a decision, made by a person, at one place. Passengers on platform nine hear something that is not a decision at all. Platform nine hears what is left of the announcement after distance, noise, echo and the crowd have all had a turn at it. Measuring what platform nine hears is a completely different exercise from reading the announcement, and confusing the two is the error this block exists to prevent.

One consequence is worth stating plainly, and it disciplines a lot of loose reading. If the announced level at the front changes and the ten year SPOT rate does not, nothing has malfunctioned. The two are different objects with different mechanisms behind them, and a change in one carries no obligation on the other. Whether a change at the front usually travels outward, and how far, is a measured question, and the route to a measured series is the data site of the Reserve Bank of India at dbie.rbi.org.in.

What does this yield curve say about the maturities it does not fix?

Nothing whatsoever, and that answer is not a dodge but the most useful sentence in this guide. The invented curve fixes a SPOT rate at one, two, three, five, ten and thirty years, and at no other length of time at all. There is no four year SPOT rate here. There is no seven year SPOT rate here. There is no twenty year SPOT rate here. Four, seven and twenty years are gaps, and the honest treatment of a gap is to call it a gap and leave it alone.

A reader can still say something honest about a gap without inventing a number inside it, and the wording matters. Between the five year and ten year nodes the recorded rise is 45 basis points in total, spread across five years. The recorded rise of 45 basis points is a true statement about the pair. Any sentence about what a single year inside that stretch did is the reader's method talking, not the curve, and the two have to be distinguishable in the sentence or the reader downstream cannot tell which one they are being handed.

The red line does something the other two never do, so it is worth a second look. Fill the gap by drawing a straight line through the discount factors and the six year figure that comes out is 6.863509 per cent, below the five year SPOT rate of 6.90 per cent that the route started from. A method handed a rising pair of rates has produced a small inversion inside them, purely as a by-product of which quantity it chose to straighten. Nobody asked for that inversion and no recorded rate implies it, and a reader given the filled series without being told how it was filled would go looking for a reason it exists.

Note also what all three methods agree on. The agreement disciplines the criticism. All three were built to pass through the recorded points, so at five years and at ten years all three land on exactly 6.90 and 7.35 per cent. The methods disagree only where the curve is silent, and that is precisely the region where the disagreement cannot be settled by looking at anything. No one of the three is defective for that. The disagreement is the honest signature of a gap.

How is a yield curve read, in what order, and which step gets skipped?

Seven steps, and the last one is the one that gets left out. Each step assumes the one before it has been answered, so the order matters as much as the content. Asking for the slope before establishing which curve is in hand produces a confident number about nothing in particular.

Run the invented curve through all seven once and the exercise takes under a minute. Step one: it is a SPOT curve. Step two: one credit quality throughout, and it is not Palash Cements Limited, an invented issuer named elsewhere and carrying no rating anywhere. Step three: ANNUAL compounding, without which not one of the sums above can be reproduced. Step four: 7.35 per cent at the ten year node, the two named together. Step five: 110 basis points separate the ten year node from the two year node. Step six: twice the five year less the two year less the ten year is 20 basis points. Step seven: one, two, three, five, ten and thirty years are fixed, and every other length of time is a gap.

Who actually reaches for a yield curve, and what do they take off it?

Three quite different readers, wanting three quite different things, and none of them wanting the whole curve.

The first is anybody who has to put a value on a dated obligation. A pension liability falling due in ten years, an arbitration award payable in five, a deposit maturing in three: each of them needs one number, the rate that turns an amount on a stated future date into an amount today. The valuer takes a single node and the discount factor behind it and ignores the shape entirely. For that use the curve is not a shape at all, it is a lookup table, and step seven of the procedure above is the whole job: is my date one of the six, or am I about to fill a gap?

The second is anybody comparing two quotes. A treasurer offered a three year facility and a five year facility wants to know what the extra two years are being charged at, and the honest answer on this curve is the recorded step of 35 basis points across the three year and five year nodes. The treasurer takes a difference rather than a level, and cares intensely about which curve type the two quotes came from. A difference between a rate off one curve and a rate off another is not a difference between anything.

The third is anybody watching the same curve across time. The watcher takes the three movement readings and nothing else: the level at a named node, the slope between two named lengths of time, and the curvature from three. Reporting those three every time is what makes two readings on different days comparable at all, and reporting only the first is why so many descriptions of a curve moving turn out, on inspection, to be descriptions of a level changing with the shape completely unexamined.

All three are less exotic than they sound, so here is the household version at once. A person deciding between a one year deposit and a three year one is doing the second job. A person working out what a lump sum promised at a child's admission date is worth today is doing the first. And a person who checks the same set of deposit rates every few months, and notices that the near ones changed while the far ones did not, is doing the third, whether or not they would call it a slope reading. None of those three needs a view about what rates will do next.

The six node SPOT curve and Palash Cements Limited are invented.
Educational material. Not advice on any investment, tax, budget or market position.