Curve Strategy: Positioning on Shape, Not on Level

A curve strategy is a position built so that its outcome turns on the DISTANCE between two or more maturities rather than on the level of rates. On the invented SPOT curve used here, the ten year SPOT rate less the two year SPOT rate is 1.10 percentage points, and that distance, rather than the rates themselves, is what such a position is exposed to.

The level and the shape are two different questions, and almost every confusion in this material comes from answering one while believing the other has been answered. A whole term structureThe full set of rates for lending across different lengths of time, taken together rather than one at a time. can lift while the distances inside it stay exactly where they were, and it can hold its level while those distances open or close. A curve strategy is any position put together to answer the second question, and every one of them is built the same way: one part of the curve is held still in the arithmetic and another part is allowed to move.

What is a curve strategy actually a position on?

Start with two shops in the same shopping centre. Both sell the same bag of rice. One charges Rs 62/- and the other charges Rs 58/-, so there is a gap of Rs 4/- between them. Now ask two entirely separate questions about that shopping centre. The first: are groceries there expensive? The second: is the gap between those two shops going to widen or close?

Two questions can have opposite answers on the same day, and a position taken on one of them is not a position on the other. If both shops raise their price by Rs 3/- next month, the answer to the first question has changed and the answer to the second has not moved by a single paisa. If instead the cheaper shop holds at Rs 58/- while the dearer one goes to Rs 66/-, the gap has widened to Rs 8/- and the shopping centre is only slightly dearer on average. Somebody watching the gap and somebody watching the level are looking at the same two price tags and reading two unrelated stories out of them.

A yield curve works the same way, with one complication that makes it more interesting rather than harder. Instead of two shops, there are several lengths of time, and instead of a price for rice there is a rate for lending money over each of those lengths. A curve strategy is a position whose outcome is decided by the distance between named lengths of time rather than by where the rates sit.

A curve strategy is defined by what it is exposed to, and that settles nothing about whether the exposure is worth having. Which way rates go next is a separate question, and no arithmetic on a teaching curve can answer it. That answer needs real prices on a real day.

One more thing needs saying before any number appears. Every rate below belongs to an invented SPOT curve, built for teaching and matching no market anywhere, and every curve move is a scenario declared first and then applied to those six rates. A rate nobody ever quoted obeys exactly the same arithmetic as one somebody did, and that is what makes an invented curve safe to learn on.

What two numbers describe the shape of this curve?

The invented SPOT curve used across this material fixes six rates and nothing else. The one year SPOT rate is 5.90 per cent, the two year SPOT rate is 6.25 per cent, the three year SPOT rate is 6.55 per cent, the five year SPOT rate is 6.90 per cent, the ten year SPOT rate is 7.35 per cent and the thirty year SPOT rate is 7.60 per cent. Every one of them is written on ANNUAL compounding, and that convention is not decoration. The identical six figures on a half yearly clock would price the identical claims differently.

Two readings carry almost everything a person needs in order to describe that shape, and both are subtractions of rates that were already there. The first is the SLOPE, defined here as the ten year SPOT rate less the two year SPOT rate. Seven point three five less six point two five is 1.10 percentage points, or 110 basis points. A slope with no maturities attached to it is not a description of anything, so both maturities belong beside every reading of it. A slope taken between the one year and the thirty year points would be a different number describing a different thing, and neither reading is more correct than the other; they simply answer different questions.

The second reading is the BUTTERFLY READING, defined here as twice the five year SPOT rate less the two year SPOT rate less the ten year SPOT rate. Twice 6.90 per cent is 13.80, the two year SPOT rate of 6.25 per cent and the ten year SPOT rate of 7.35 per cent come to 13.60 together, and the difference is 0.20 percentage points, or 20 basis points. In words, the five year point on this curve sits a little above a straight line drawn from the two year point to the ten year point. If the middle of the curve sat exactly on that straight line, the reading would come out at zero.

Both readings are distances, and both are subtractions, so neither can carry an opinion about what happens next. Notice also that neither of them needs the one year, three year or thirty year SPOT rates at all. A person describing a curve is always choosing which part of it to describe.

What happens to those numbers when one end moves and the other does not?

Here is where the shopping centre pays off. Declare a scenario in which the two year SPOT rate stays exactly where the record fixes it, at 6.25 per cent, and the ten year SPOT rate sees a rise in the yield of 40 basis points, taking it to 7.75 per cent. The slope read 1.10 percentage points before and reads 1.50 percentage points after, so it has widened by 40 basis points. Only one end moved, and the whole of the move landed in the shape.

Now declare a second scenario in which every recorded rate sees a rise in the yield of 40 basis points together. The two year SPOT rate becomes 6.65 per cent, the three year 6.95 per cent, the five year 7.30 per cent, the ten year 7.75 per cent and the thirty year 8.00 per cent. Work the slope again: 7.75 per cent less 6.65 per cent is 1.10 percentage points, exactly what it read before anything moved. Every rate in the second scenario changed and the slope did not budge. The level and the shape are separate questions, and nothing shows it more cleanly.

Both scenarios are declared inputs rather than events. Forty basis points is large enough to see on a drawing and small enough to leave the curve recognisable, and a move of that size carries no likelihood with it either way.

The three structures people name get their meaning at exactly this point. A steepener is a position built so that its outcome improves as the slope widens. A flattener is a position built so that its outcome improves as the slope narrows. A butterfly is a position built on the third reading, the one that uses a short point, a middle point and a long point together. Each of those is a description of what a position is exposed to, not an instruction to put one on. Taking them apart and setting them against one another is covered separately.

Roll-Down: where does a gain come from when the curve stands still?

Everything so far has been about the curve moving. Now take the opposite case, and it is the more surprising one. Suppose the curve does not move at all for twelve months: the one year SPOT rate is still 5.90 per cent, the two year SPOT rate is still 6.25 per cent, every recorded rate is untouched. Does a claim held across that year earn nothing beyond the rate it was bought at? A claim held across that year earns more than the rate it was bought at, and the reason has a name.

Roll-down is what happens to a claim's price purely because the claim gets shorter while the curve stays where it is. A two year claim is a one year claim twelve months later. The shortening is not a financial insight, it is a calendar fact. But it has a consequence: if the curve has not moved, that claim is now priced off the one year SPOT rate rather than the two year SPOT rate, and on this curve the one year SPOT rate of 5.90 per cent sits below the two year SPOT rate of 6.25 per cent. A claim arriving at the shorter point is therefore discounted at a lower rate. A lower discount rate makes it worth more than the rate it was bought at alone would give. Roll-down is that difference and nothing else.

The everyday version is a queue. Somebody joins a queue at a counter with a numbered token, and the wait attached to that token shrinks every hour simply because time passes and the queue in front of it clears. Nothing about the counter changed and nobody granted a favour. The only thing that happened is that the token reached a place in the queue where the wait is shorter, and a shorter wait is worth more. A claim rolling down a curve is doing exactly that: it is arriving at a point on the curve where the rate is lower, without the curve itself having done anything.

The last part is the part that gets forgotten. Roll-down needs the shape. Roll-down exists because the shorter point on this curve carries a lower rate than the longer point. On a completely flat curve, where every length of time carried the same rate, a two year claim becoming a one year claim would be repriced at the identical rate and there would be no roll-down at all to have. On a curve where longer rates sat below shorter ones, an invertedDescribing a curve on which a longer stretch of time carries a lower rate than a shorter one, so the usual ordering is turned around. arrangement, the claim arriving at the shorter point would be repriced at a HIGHER rate and the roll would give ground rather than gain. Roll-down is a consequence of shape, so it inherits every property of the shape it came from.

Carry vs Roll-Down: which half is which, and why bother splitting them?

Carry is what a claim earns simply for being held, at the rate it was bought at, with the curve standing still and the claim getting shorter left out of the reckoning entirely. If a claim was bought to yield 6.25 per cent, carry over one year is 6.25 per cent. Carry is that and no more. The bare noun turns up everywhere without anybody saying which of several things they mean by it.

So a year on an unmoved curve has two parts with two different causes. Carry comes from the rate written on the claim, and roll-down comes from the shape of the curve the claim is sliding along. One of them would exist on a perfectly flat curve; the other would not. One of them depends on nothing except the purchase; the other depends on where the neighbouring point sits. Splitting them is worth the trouble precisely because they respond to completely different things, and a person who only ever sees the total has no way of knowing which of the two is doing the work.

Now the difficult sentence, and it is better said plainly than buried. Neither part is a payment that anybody makes. Nobody hands over carry and nobody hands over roll-down. Both are arithmetic run on a curve that has been assumed to stand still for twelve months, and a curve standing still for twelve months is a declared supposition rather than anything that has been observed. The split says where a year's outcome would come from under a stated assumption, and a stated assumption is not a promise.

What does a claim earn if the curve does not move at all?

Now a number, and it is the number the rest of the subject rests on. Take an invented zero couponDescribing a claim that pays nothing along the way: one single amount arrives on one single date and that is the whole of it. claim of Rs 1,000.00/- face, bought at the two year SPOT rate of 6.25 per cent on ANNUAL compounding. Discounting one thousand rupees across two years at 6.25 per cent gives a price of Rs 885.813149/-, and a reader can check that in one line: 1.0625 multiplied by itself is 1.12890625, and one thousand divided by that is 885.813149.

Hold it for twelve months against a curve declared to stand still. The claim is now a one year claim, so it is priced at the one year SPOT rate of 5.90 per cent: one thousand divided by 1.0590 is Rs 944.287063/-. The gain across the year is Rs 944.287063/- less Rs 885.813149/-, or Rs 58.473914/-. As a share of what was paid, that is 58.473914 divided by 885.813149, or 6.601157 per cent.

Now do something that looks unrelated. Derive the one year rate one year FORWARD from the two SPOT rates standing behind it. A FORWARD rate has no other source: it is what two SPOT rates already imply. Two years at 6.25 per cent grows one rupee to 1.12890625. One year at 5.90 per cent grows it to 1.0590. Divide the first by the second and 1.06601157 comes back, so the rate covering the twelve months that begin a year from today is 6.601157 per cent.

The two figures are the same figure, and that is not a coincidence: it is what a FORWARD rate is. Rolling down a curve that does not move earns the FORWARD rate covering the period held, not the SPOT rate the claim was bought at. The identity carries most of the subject on its own. The number is already sitting inside today's rates rather than being anybody's opinion about next year.

The whole year, split to the paisa

The split works in two units, and both are worth doing. The points close exactly and the rupees close only after rounding, and the small residue that rounding leaves behind confuses nobody who has seen it once.

In points first. The whole year came to 6.601157 per cent. Carry is the rate the claim was bought at, 6.25 per cent. Roll-down is everything left over: 0.351157 percentage points, or 35.1157 basis points. Add them and 6.601157 per cent comes back with nothing unaccounted for.

Now in rupees, rounded to the paisa so it can be added up by hand. Carry is 6.25 per cent of the Rs 885.813149/- that was paid, or Rs 55.363322/-, and that rounds to Rs 55.36/-. Roll-down is 0.351157 percentage points of the same base, or Rs 3.110593/-, and that rounds to Rs 3.11/-. The two come to Rs 58.47/-, and the whole gain of Rs 58.473914/- rounds to Rs 58.47/- as well, so the two routes agree at the paisa.

To show that this is a method rather than one lucky answer, run it again a year further out. An invented zero coupon claim of Rs 1,000.00/- face bought at the three year SPOT rate of 6.55 per cent costs Rs 826.684201/-. Twelve months later, on a curve declared to stand still, it is a two year claim worth Rs 885.813149/-, the same price the first claim was bought at. The gain is Rs 59.128948/-, or 7.152544 per cent of what was paid, and that figure is the one year rate two years FORWARD derived from the three year SPOT rate of 6.55 per cent and the two year SPOT rate of 6.25 per cent. Carry is 6.55 per cent and roll-down is 0.602544 percentage points, or 60.2544 basis points. In rupees, carry is Rs 54.15/- and roll-down is Rs 4.98/-, coming to Rs 59.13/-, exactly what Rs 59.128948/- rounds to.

Two things are worth noticing in that table. The three year claim has the larger roll-down of the two, 0.602544 percentage points against 0.351157, and yet the step it takes down the curve is the shallower one: the three year SPOT rate of 6.55 per cent to the two year SPOT rate of 6.25 per cent is a drop of 30 basis points, while the two year SPOT rate of 6.25 per cent to the one year SPOT rate of 5.90 per cent is a drop of 35 basis points. The larger roll-down on the smaller drop looks backwards until the second thing is noticed: roll-down depends on the drop AND on how many years of discounting that drop is being applied across, and the three year claim has two years left to run against the two year claim's one. Neither the step nor the remaining length settles it on its own.

Which maturities can be rolled on this curve, and which cannot?

A one year roll needs a rate at a length of time one year shorter than the claim held. The requirement sounds small until the contents of this record are counted. Six lengths of time carry a rate and nothing between them carries anything, so a claim can only roll if the point one year shorter happens to be one of the six.

Exactly two one year rolls exist on this curve: a two year claim rolling onto the one year point, and a three year claim rolling onto the two year point. A five year claim would need a four year SPOT rate. A ten year claim would need a nine year SPOT rate. A thirty year claim would need a twenty nine year SPOT rate. None of those three rates is fixed anywhere in this record, and a rate that is not fixed cannot be rolled onto.

The temptation to manufacture one is real, so it deserves a name. Drawing a straight line between the three year point and the five year point and reading a four year rate off it is called filling in a value that is not held, and to interpolateTo fill in a value between two values that are held, by assuming something about what lies between them. The assumption is doing the work, and it is usually invisible. across a gap is a decision with an assumption hidden inside it rather than arithmetic. The line drawn is a claim about what the curve does between the two points, and this record makes no such claim, so the four year slot stays empty.

The prices themselves are perfectly available at the six recorded points, and setting them out makes the gap precise. On ANNUAL compounding, an invented zero coupon claim of Rs 1,000.00/- face costs Rs 716.327252/- at the five year SPOT rate of 6.90 per cent, Rs 492.016324/- at the ten year SPOT rate of 7.35 per cent and Rs 111.078974/- at the thirty year SPOT rate of 7.60 per cent. Every one of those is a plain discounting of one thousand rupees at a recorded rate. The missing figure is not the price today but the rate the claim would be repriced at twelve months from now.

How does anybody actually use two readings and a split?

Start with a household. The shape is easier to see when the amounts are small. A person who has kept some savings in a one year fixed deposit and the rest in a three year one is holding a shape, whether or not they have ever thought of it that way. Their outcome over the next twelve months depends on two separate things: what happens to deposit rates generally, and what happens to the gap between the one year rate and the three year rate. If every deposit rate lifts by the same amount, the person with money in both is affected in one way. If the three year rate lifts and the one year rate does not, they are affected quite differently, and the difference lands entirely on the longer deposit. Nobody in that household set out to take a position on shape. Holding two different lengths of time is what taking a position on shape consists of. The household took one anyway.

A person running the treasury of a mid sized manufacturer does the same thing with more zeroes and more deliberation. The treasurer has money that is not needed for eleven months and money that is not needed for four years, and every placement made sits at some point along the curve. The two readings above are how such a person describes what they already hold before deciding anything at all. Working the slope tells them how much of their outcome depends on the distance between their short placements and their long ones. Working the carry and roll-down split on each placement tells them how much of next year's expected outcome comes from the rate they locked in and how much comes from a shape they have assumed will still be there.

The carry and roll-down split is where the honesty lives. A person who has separated carry from roll-down knows which part of next year's arithmetic survives the curve moving and which part does not. Carry is fixed at purchase and cannot be taken away by a shape change. Roll-down is entirely a consequence of shape, so it can be enlarged or wiped out by one, and a plan that quietly counts on it has counted on the curve behaving. Somebody analysing a set of holdings for a client, or writing up a set of holdings for a committee, earns their keep by saying which of the two is carrying the plan.

And the same discipline sets a limit. Naming an exposure is not the same as knowing what to do about it, and no reading taken on a curve settles which structure or which length of time a particular holder should want.

Which number can this curve not produce?

A reader who has followed everything above wants one more step, and it is a perfectly reasonable thing to want. Fine, the slope widens by 40 basis points. What does that earn or cost in rupees on something somebody actually holds?

The record standing behind this material holds one curve, no history of any curve, and no outcome for a holding under any move other than a parallel one. There is no second curve to compare against, no series of past moves to lean on, and no measured frequency of anything. So the cell where a rupee figure would sit is drawn empty, with the reason written inside it.

One number is out of reach, and only one: what a holding gains or loses in rupees when the shape changes. Every scenario above is arithmetic on figures the record does contain, so the slope readings, the butterfly reading, the roll-down splits and the two prices all stand in full. The subject is not missing. One number is, and nothing in the record supports it.

An empty cell with a reason inside it teaches more than a plausible number does. A plausible wrong number does the most damage of any failure. A reader has no way of telling it apart from a right one, carries it away, and builds on it. An empty cell is impossible to misread.

The SPOT curve used here, the zero coupon claims priced against it and every scenario applied to it are invented.
Educational material. Not advice on any investment, tax, budget or market position.