Steepener, Flattener and Butterfly: What Each Reads

A steepener and a flattener are attached to one distance and differ only in which way that distance has to travel; a butterfly is attached to a second distance. On the six invented SPOT rates worked here, the first distance is 1.10 percentage points, the two year SPOT rate taken off the ten year SPOT rate, and the second is 0.20 percentage points, twice the five year SPOT rate less both outer ones.

Three names, and almost everybody meets them one at a time, months apart, from three different explanations. Meeting them separately is what makes the three names feel like three unrelated machines. In fact one idea sits under all three. Each one is a position built so that a single number decided in advance settles what happens to it, and once that number has been identified, everything else about the structure falls out without further explanation.

The comparison only works in one order: all three get defined before any of them is contrasted, and then the same moves are put through all three to see which readings notice. Somebody who follows that order can hear the word steepener and immediately ask the question that makes it mean something.

Every rate that follows belongs to an invented SPOT curve, built to teach with and matching no market on any day. A SPOT rateThe annual rate at which one single amount, due on one named future date and nothing else, is discounted back to today. belongs to exactly one horizon and discounts one dated amount. A curve move worked out below is a scenario somebody declares and then applies, and declaring one is how a shape gets tested without waiting years for a real move to arrive.

What do these three structures actually have in common?

Start away from bonds entirely. A woman runs a tailoring shop and buys her cloth from two wholesalers on the same street. The one at the near end charges Rs 240/- a metre. The one at the far end, who stocks the heavier grade, charges Rs 268/-. She has been in the trade twenty years, and what she watches is not either price. Her whole business model is buying the cheaper grade and finishing it well enough to sell against the dearer one, so what she watches is the Rs 28/- between them. If both wholesalers raise their price by Rs 15/- next season, she has not lost her business model. If the gap closes to Rs 9/-, she has, and both prices could have fallen while that happened.

Every structure in this guide is built the way that tailor thinks: exposed to a distance between two named things and not exposed to where either of them sits. The distance is measured between rates on a curve rather than between two cloth merchants, and the arithmetic is a subtraction rather than a business, but the shape of the exposure is identical, and so is the discipline it demands. The two things have to be named.

Here are the levels every reading in this guide is taken from. Six horizons carry one, and nothing between them carries anything at all, so no gap below gets filled in. Each of the six is a nodeA length of time at which this record actually fixes a rate. Anything between two nodes is simply not written down here., and each level is stated on annual compoundingInterest reckoned once a year rather than in half yearly or quarterly slices. The identical figures on a different clock would not produce the same prices., a convention repeated inside every subtraction below rather than parked in a footnote.

Now the two readings. The first is the SLOPE, defined here by taking the two year SPOT rate off the ten year SPOT rate. The subtraction runs: 7.35 less 6.25 leaves 1.10 percentage points. The same distance written another way is 110 basis pointsA percentage point split a hundred ways, and one of those hundredths. Rates are quoted in them because a hundredth of a per cent is a real amount of money at scale.. The second is the BUTTERFLY READING, defined here by counting the five year SPOT rate twice over and then removing the two year SPOT rate and, after it, the ten year SPOT rate. Twice 6.90 is 13.80; 6.25 plus 7.35 is 13.60; the difference is 0.20 percentage pointsA whole one per cent of difference. A basis point is a hundredth of it, and the two are never interchangeable., or 20 basis points. Both subtractions run on annual compounding, and both are numbers anybody can take off this curve in the time it takes to write them down.

Two things are worth pinning down before the contrast starts. Neither reading is a recommendation, and neither is a forecast. A slope of 1.10 percentage points does not say the curve should be steeper, and a butterfly reading of 0.20 percentage points does not say the middle of the curve is due to move. Both readings describe a shape, in the same sense that the Rs 28/- between the two cloth wholesalers describes a street rather than a plan for the season.

What is a steepener exposed to, and what is it blind to?

A steepener is a structure at two named maturities, put together so that what it gains from is a widening of the distance between those two. On the nodes above that distance is 1.10 percentage points, or 110 basis points, and it is the entire exposure. Nothing else in the definition of the structure appears anywhere.

The level of rates has been designed out of it, and the fastest way to feel that is to put two different curves side by side. Take a curve carrying 6.65 per cent at its two year node and 7.75 per cent at its ten year node. Subtract: 1.10 percentage points. Now take the recorded nodes, 6.25 per cent and 7.35 per cent. Subtract: 1.10 percentage points. Every borrower on the first curve is paying 40 basis points more than on the second, at both horizons, and a steepener between those two named maturities cannot tell the two situations apart.

Somebody who wanted exposure to the level of rates would never have gone looking for a distance in the first place, so the blindness is a design feature and not a limitation. Somebody exposed to the level cares enormously whether the curve sits at 6.25 and 7.35 or at 6.65 and 7.75; that is a completely different position with completely different arithmetic, and it is covered separately. Here the two situations are the same situation.

Two guesses about a steepener come up constantly, and both are wrong. A steepener is not a view that rates in general will see a rise in the yield: the second curve above has seen exactly that at both nodes and the reading has not shifted. And it is not a view about any single rate on its own: the ten year node could see a rise in the yield worth 40 basis points and the reading would widen, but so would it if a fall in the yield of the same size landed at the two year node instead, and the structure treats those two as the same event.

How is a flattener different, and how is it exactly the same?

A flattener measures the same distance. One sentence carries the whole of the difference and the whole of the sameness, and everything else follows from it. A flattener is a structure at those same two named maturities, put together so that what it gains from is a narrowing of that distance instead of a widening. There is no second measurement involved, no separate arithmetic and no additional rate to look up.

The two words are usually taught as though they named two different instruments, one bought on one view and one bought on the opposite view. A steepener and a flattener are one reading with two signs, not two objects. The slope of 1.10 percentage points is the single number both are attached to, and the names record only which direction of change each is built around.

There is a practical consequence worth carrying away from this. If somebody describes holding a steepener and a flattener at the same two named maturities at the same time, they have described holding nothing: the two are attached to one number and are built around opposite directions of it, so put together they read the shape and then cancel their own reading of it. An account that treats the two as separate objects has no way to explain that cancellation. Explaining it is a decent test of whether the definition has landed.

What does a butterfly read that the other two cannot?

A butterfly is a structure at three named maturities, built around the middle one measured against the outer two. The middle maturity is called the BODY and each outer one is called a WING, and those two words carry no arithmetic of their own; they are just names for positions in the structure. The butterfly reading, again, counts the five year SPOT rate twice over and then removes both outer levels, the 6.25 per cent standing at the two year node and the 7.35 per cent standing at the ten year node. What is left is 0.20 percentage points, or 20 basis points.

Why write it with a two in front? Because what the reading actually measures is how far the body sits from a straight line drawn between the wings. The average of the two wings is 6.25 plus 7.35 halved, or 6.80 per cent. The body is at 6.90 per cent. The gap between them is 0.10 percentage points, and the butterfly reading is exactly twice that. Doubling it is a convention rather than a discovery, and it keeps the reading in the same units as the rates without a stray halving in every sum.

Now watch what that third node buys. Declare a scenario in which only the body moves. Pin the two year SPOT rate at 6.25 per cent and pin the ten year SPOT rate at 7.35 per cent, then let the five year node see a rise in the yield worth 30 basis points. That SPOT rate now stands at 7.20 per cent. Recompute the slope. The slope subtracts the two year SPOT rate from the ten year SPOT rate, and neither of those two levels is any different, so it reads 1.10 percentage points before and 1.10 percentage points after. Recompute the butterfly reading: twice 7.20 is 14.40, less 13.60 leaves 0.80 percentage points. The butterfly reading has widened by 60 basis points.

The curve plainly moved, and the slope reported absolutely nothing about it. Not a small change, not a rounding, nothing. An observer whose only reading was the slope would have written down that the shape was unchanged, and would have been wrong by 60 basis points on a quantity that was never being measured. Seeing a class of curve move that the slope is structurally deaf to is the single most useful thing a butterfly does.

Notice also how the 30 becomes a 60. The body enters the reading twice and each wing enters once, so a change at the body is felt at double strength. The doubling is not a special property of this curve or these numbers. It falls straight out of the way the reading is written, and it is why a body move of 30 basis points shows up as a widening of 60.

Push the control to the left and something appears that a single positive number would have hidden. At a five year SPOT rate of 6.80 per cent the butterfly reading is exactly zero, and the picture shows why: 6.80 is the plain average of the two wings, so all three nodes sit on one straight line and there is no gap left to measure. Below that setting the reading turns negative, and at the bottom of the declared range it reads minus 2.80 percentage points while the slope still reads 1.10.

What are both readings blind to?

One move now, touching every node at once. Declare a rise in the yield worth 40 basis points at each of them together. The two year SPOT rate then stands at 6.65 per cent, the five year SPOT rate at 7.30 per cent and the ten year SPOT rate at 7.75 per cent. Every borrower on this curve pays more at every horizon. A curve can hardly undergo a larger change and still have the same shape afterwards.

Recompute both readings. The slope is 7.75 less 6.65, or 1.10 percentage points, exactly as before. The butterfly reading is twice 7.30, or 14.60, less 6.65 less 7.75, leaving 0.20 percentage points, exactly as before. Neither structure sees a parallel move at all, and that is not a weakness anybody works around; it is the reason these structures were built this way.

Go back to the tailor for a second. Her model was never about the price of cloth. If both her wholesalers raise cloth by Rs 15/- a metre, her costs have changed a great deal and her business model has not changed at all. Somebody who cannot separate those two things will look at her books after a bad season and blame the wrong thing entirely. The same separation is what these structures are for, and the parallel move shows it most clearly of all: everything moved and neither reading blinked.

Which move touches one reading, and which touches both?

Block four showed a move at the body that changed the butterfly reading and left the slope perfectly still. Assuming the reverse also holds is natural, and the reverse does not hold. Most short comparisons of these three leave that asymmetry out.

Declare a different scenario. Pin the two year SPOT rate at 6.25 per cent, pin the five year SPOT rate at 6.90 per cent, and let the ten year node see a rise in the yield worth 40 basis points. That SPOT rate now stands at 7.75 per cent. One wing has moved and nothing else has. The slope is now 7.75 less 6.25, or 1.50 percentage points, so it has widened by 40 basis points. But the butterfly reading does not sit still either: twice 6.90 is 13.80, less 6.25 less 7.75 leaves minus 0.20 percentage points, a change of 40 basis points in the other direction.

Each wing sits inside both subtractions and the body sits inside only one, so a move at the body touches one reading and a move at a wing touches both. Read the definitions again and it is obvious: the ten year SPOT rate appears in the slope and again in the butterfly reading, so anything that happens to it is felt twice over, in two different places. The five year SPOT rate appears in one of the two and nowhere else. A butterfly can therefore be built so that it sits out a change in the slope; a steepener has no way to be built so that it sits out a change at its own end.

One more thing that scenario reveals. The butterfly reading did not merely shrink; it went past zero and came out the other side, from 0.20 percentage points to minus 0.20 percentage points. A reading with a side of zero is a different object from a reading that only gets smaller, and any later description that treats the butterfly reading as a positive quantity has quietly narrowed it.

Why does a structure named without its maturities describe nothing?

Six nodes are fixed above, and any two of them define a distance. So the word steepness has several different values on this one curve at this one moment, and they are genuinely different numbers rather than roundings of each other.

Take the ten year SPOT rate and remove the two year SPOT rate from it: 1.10 percentage points. Take the thirty year SPOT rate and remove the five year SPOT rate: 0.70. Remove the ten year SPOT rate from that same thirty year SPOT rate instead: 0.25. Remove the two year SPOT rate from the five year SPOT rate: 0.65. Four readings, one curve, one instant, and the largest of the four is more than four times the smallest.

Now put the scenario from the block above through all four of them, the one where a rise in the yield worth 40 basis points landed at the ten year node alone. The first reading widens to 1.50 percentage points. Neither the thirty year SPOT rate nor the five year SPOT rate was touched, so the second does not change by anything whatever. Nor does the fourth, for the same reason. And the third crosses zero. The declared move has lifted the ten year node above the thirty year node, so removing the ten year SPOT rate from the thirty year SPOT rate now gives minus 0.15 percentage points rather than 0.25.

Two people saying the word steepener to each other while meaning different pairs are not disagreeing about the curve; they are describing two different quantities with one word. One of them can be right that steepness widened by 40 basis points while the other is right that it did not change by anything, on the same afternoon, off the same six rates, with neither of them having made an arithmetic mistake. The word carries no information at all until the two maturities are attached to it. Every reading above names both.

The third pair shows what a scenario declared on one node does at the far end. Holding the thirty year SPOT rate at 7.60 per cent while lifting the ten year SPOT rate to 7.75 per cent puts the ten year node above the thirty year node. The inversion is an honest consequence of the declared move rather than an error in the arithmetic. A scenario that moves one node and pins the others will do that sooner or later, and saying so plainly saves a reader wondering whether something broke.

All four moves and both readings in one place

Here is every scenario declared above, with both readings recomputed from the nodes in each row, on annual compounding throughout. The recorded row comes first so that everything below it can be compared against something. The slope column and the butterfly column read down together: that pairing is the whole comparison, and the last column is the part of it the arithmetic cannot supply.

Every reading in that table is a subtraction anybody can check in under a minute, and every scenario in it was declared rather than observed anywhere. Two rows carry a slope of 1.10 percentage points for two completely unrelated reasons, and the same two rows carry different butterfly readings. The pairing is what makes reading the columns together worth the trouble.

Can the comparison say which of the three did better?

A reading is arithmetic on rates that are written down, and every scenario above is declared outright, so the readings can be compared. The outcomes cannot be compared, and the difference between those two sentences is the honest boundary of the whole comparison.

One curve sits here, no run of past moves sits beside it, and nothing on record says what a holding did through a change of shape, so which structure did better is a question with nothing available to answer it rather than an answer being kept back. The last column of the table above therefore carries no figure, and the reason for the blank sits inside the cells themselves.

A reader who wanted that column filled has understood the argument rather than missed the point. Wanting it filled means arriving at the edge of what this arithmetic can support, and noticing an edge is a more useful skill than being handed a number manufactured to sit where the edge is.

Who reads these numbers on an ordinary working day?

Somebody at a debt fund keeps a shape log. Once a week they write down the day, the two readings, and, in the same line, the maturities each reading was taken between. A log without those maturities is a column of numbers that cannot be compared with itself six months later, once a colleague has quietly started using a different pair. The colleague who switched pairs is the reason the discipline exists. The log is a record of shape, not a record of what anything earned, and the two get kept in separate columns for exactly the reason the last column of the table above stays empty.

An analyst writing a note about a curve move uses the pairing rather than either reading alone. If the slope moved and the butterfly reading did not, a wing moved and the body followed it in step. If the butterfly reading moved and the slope did not, the middle went somewhere on its own. If both moved, one wing went somewhere alone. Each of those three sentences says what already happened, without reaching for words like twist and hump that mean different things to different people.

Somebody running a household meets the same arithmetic without the vocabulary. A person with a fixed deposit maturing in a year and a home loan running for another eighteen has two ends of their own small curve, and what matters to them is not whether rates in general are high. The distance between what their savings earn at the near end and what their borrowing costs at the far end is what matters. The distance can widen in a year when every rate saw a fall in the yield, and hold perfectly still in a year when every headline said rates had risen. Most confusing commentary is about one while sounding like it is about the other, so a great deal of it sorts itself out once the distance and the level are held apart.

None of those three uses is a position. Money decisions belong to the person whose money it is, worked through with an adviser they have chosen themselves, against live prices on the day in question. A curve built for a lesson is not that.

The six SPOT rates worked here, the four moves applied to them and the two cloth wholesalers are invented.
Educational material. Not advice on any investment, tax, budget or market position.