The Butterfly Trade: Betting on Curvature

A butterfly takes one middle SPOT rate and sets it against two outer ones: twice the middle, less each of the outer two. On the six invented rates used here that is twice 6.90 per cent, less 6.25 per cent, less 7.35 per cent. The answer comes to 0.20 percentage points. The reading measures how far the middle sits from the average of the outer pair, and it belongs to those three maturities and to nothing else.

Here is what it rests on. Any two rates, one subtracted from the other, give a distance. The distance is the whole of what a pair can report, and a pair is silent about everything happening between the two maturities chosen. A third rate opens a new question. Where does the third one sit relative to the other two? No distance can answer that. The butterfly reading is that question compressed into a single number, and this guide is about what the number can and cannot see.

What is curvature as a number, and why does it take three rates to get one?

Consider a plank laid across two stools. With a third stool set between them, the only interesting question in the room is whether the plank rests on the middle stool, hangs above it, or is pushed up by it. The question cannot be asked with two stools. Measuring how far apart the two outer stools are cannot answer it either. The measurement of that distance is finished before the middle stool is even carried in. The middle stool is the entire subject, and a third measurement is the only way to get at it.

A yield curve sets the same problem. The curve used throughout was built for teaching, and it fixes exactly six horizons and nothing at all between them: 5.90 per cent at one year, 6.25 at two, 6.55 at three, 6.90 at five, 7.35 at ten and 7.60 at thirty. Every one of those rates is a SPOT rate, meaning that for a single future date, the SPOT rate for that horizon is the rate at which a lump sum due on that date and on no other is discounted today. All six rates were built for teaching, and none was read off a market. Every sum below compounds once a year, and the compounding has to be stated inside the sum: the identical six numbers on a twice-a-year clock give different prices.

Three of those six are taken. Which three is a choice, and the same one is made all the way down, so here it is as a row rather than as a sentence:

Now the arithmetic, short enough to do standing up:

Twenty basis points, and a basis point here is a single percentage point cut into a hundred equal slices, so 0.20 percentage points is twenty of those slices. Percentage points and basis points are used side by side all the way through this guide, and swapping one for the other silently multiplies an answer by a hundred. The two words are worth keeping straight.

Now the picture behind that sum. A picture makes the formula something that can be rebuilt rather than something that has to be remembered. The two outer rates averaged: half of 6.25 plus 7.35 is 6.80 per cent. The five year rate of 6.90 stands 0.10 percentage points clear of that average, and the reading of 0.20 percentage points is exactly twice that gap, every time, on any three rates whatever.

A reader who skips one thing about 6.80 per cent will spend the rest of this guide half-wrong, so here it is flatly. The 6.80 per cent figure is the average of two SPOT rates, and it is not a rate on this curve. No rate anywhere here is read out of the space between two recorded maturities. The record fixes six horizons; between them it fixes nothing, and 6.80 per cent is a reference level to measure against rather than a rate anybody could ever be paid.

Which rate is the body, which are the wings, and why is the middle one doubled?

The three rates get names, and the names are worth learning because everything downstream uses them. The middle maturity is the body. The two outer maturities are the wings. On the reading used throughout, the body is the five year SPOT rate and the wings are the SPOT rates at two years and at ten. Nothing about which is which depends on the numbers: it is purely about which maturity sits between the other two.

So why double the body? Not because somebody decided to. In this corner of finance arithmetic the reason is visible in about four seconds. Set the body at exactly 6.80 per cent, the average of its wings, and work it through. Twice 6.80 is 13.60. The two wings add to 6.25 plus 7.35, and that also comes to 13.60. The two sides cancel and the reading comes to nothing at all.

Weights of one, two and one are the only weights that make the reading vanish whenever the body sits at the average of its wings, and land on exactly twice the gap whenever it does not. The cancellation is the whole design. Doubling anything else, or doubling nothing, gives a number that moves around when the entire curve moves and reports the level of rates rather than the shape between three points.

The second half is where the usefulness lives, so check it too. Add half a percentage point to all three rates and see what happens: twice 7.40, less 6.75, less 7.85, comes to 0.20 percentage points again. Every term picked up the same addition, and the doubling on one side cancels the two single subtractions on the other. A parallel shiftEvery rate on a curve shifting an identical distance, all of them the same way, which leaves each gap between any two of them untouched. moves the whole curve and leaves the reading exactly where it was. Anything meant to describe shape has to behave in precisely that way.

What does a butterfly reading catch that a slope reading walks straight past?

The result surprises most people the first time, so a prediction is worth committing to before the working.

The slope here is one subtraction, and both maturities go into its name every time. The slope begins at the ten year SPOT rate and deducts the two year SPOT rate. On the recorded nodes that is 7.35 less 6.25, giving 1.10 percentage points, or 110 basis points. Inside that subtraction: two rates. Outside it: everything else on the curve, including the entire middle of it.

So the move is declared. Two year SPOT rate: 6.25 per cent, untouched. Ten year SPOT rate: 7.35 per cent, untouched. Five year SPOT rate: a declared fall of 15 basis points in its yield, from 6.90 down to 6.75 per cent. Every one of those three values is a declared input, set so that exactly one node moves while the other two hold still.

The slope now reads 7.35 less 6.25, or 1.10 percentage points. Neither number in the subtraction moved, so the slope is the same as before, to the last decimal. The butterfly reading now runs twice 6.75, or 13.50, less 6.25, less 7.35, and lands on minus 0.10 percentage points. The reading was 0.20, so it has travelled 0.30 percentage points. The curve changed shape, the reading that measures shape moved 30 basis points and crossed to the other side of nothing, and the slope reported that absolutely nothing had happened.

The gap between those two verdicts is the entire case for bringing in a third maturity, and it is also why the four standard names for how a curve moves are not a complete list of what a curve can do. A move that lifts the middle relative to both ends, or drops it, is neither a shift of the whole curve nor a change in the distance between two ends. A move like that has to be measured somewhere else, and the somewhere else is a triple.

Which changes in the slope does a butterfly reading fail to register at all?

The reverse question is the other half, and it is the half almost nobody gets shown. Set out without what it misses in return, a reading that catches something the slope misses gets treated as a better slope rather than as a different measurement.

Put two declared moves side by side. Both of them open the curve out between two years and ten years, so both are steepenings in the ordinary sense, and both are written down here rather than observed anywhere.

Read the bottom two rows against each other. Both give a slope of 1.50 percentage points, identical to the last decimal, and their butterfly readings are 0.40 percentage points apart and on opposite sides of nothing. A reader handed only the slope has no way whatever to tell those two apart. A reader handed only the butterfly reading has no way to tell either of them from the recorded row. One of the two reads 0.20 percentage points, just as the recorded row does.

The reason the pivot leaves the reading untouched is worth working out rather than accepting. One wing fell 20 basis points and the other rose 20. The average of 6.05 and 7.55 is 6.80 per cent, exactly what 6.25 and 7.35 averaged to. The point the body is being measured against did not move. The body did not move. So the distance between them is what it was, and twice that distance is what it was. The two wings have swung 40 basis points apart from each other all the same.

A pivot is where a steepenerA structure whose result turns on the distance between two chosen maturities widening. and a butterfly stop being interchangeable ideas in a reader’s head. A structure built around the slope and a structure built around a triple respond to genuinely different features of the same curve, and a single curve movement can feed one and starve the other. How the two compare as structures is worked through separately; what matters here is only that the two readings are measuring different things and that neither is a substitute for the other. The same holds for a flattenerA structure whose result turns on that same distance closing up instead., the slope reading running the other way.

Look at the two short green brackets in that drawing. The two brackets are the same length. The body is the same distance from the average of its wings in both panels; what differs is which side of it the body is on, and that flip is the whole of the 0.40 percentage point gap between the two readings.

The second panel deserves its own drawing. Its mechanism is easy to say and hard to believe until it is seen: the point the body is being measured against never moved.

What happens to both readings when one wing moves?

Everything so far has been a still picture: a curve, a declared move, two numbers before and two after. A still picture cannot show that the slope and the butterfly reading are not independent of each other. Moving a wing moves a term that sits inside both subtractions, so both readings have to go somewhere, and the interesting part is exactly how tightly they are tied to each other.

The control below moves the two year SPOT rate and nothing else. The five year SPOT rate stays at 6.90 per cent. The ten year stays at 7.35 per cent. Watch three things at once: the travelling node, the straight line joining the two wings, and the hollow marker sitting at the middle of that line. The body never moves in this drawing, and yet its distance from the hollow marker changes at every setting. A butterfly reading measures exactly that distance.

Three things in the calculator above are worth pausing on. The first is that the hollow marker moves. The hollow marker is the average of the two wings, so half of every move at the two year node passes straight into it, and the body sits still while the average slides underneath. The second is that the reading passes through exactly nothing once the two year SPOT rate is set here at 6.45 per cent, where the average of the wings comes to 6.90 and lands precisely on the body, and the readout there is 0.00 percentage points. The bar does not shrink towards nothing and stop; it collapses to nothing inside its track and then grows out the other side. At the low end of the control, declared at 5.25 per cent, the reading comes to 1.20 percentage points against a slope of 2.10; at the high end of 7.25 the reading comes to minus 0.80 against a slope of 0.10.

The third is the speed, and it gives up the sharpest thing in this panel. A move of 10 basis points in the two year SPOT rate moves the slope 10 basis points; the butterfly reading moves 10 as well, the same way. Set against each other at any setting whatever, the slope stands ahead of the butterfly reading by exactly 0.90 percentage points, from one end of the control to the other. The 0.90 percentage point gap is forced rather than lucky: the two year SPOT rate enters both subtractions once and with the same sign, so it drops out of the difference between them altogether. On this one control the two readings carry nothing independent of each other. Moving a wing is therefore the wrong experiment for telling a butterfly and a slope apart, and moving the body is the right one.

What does it mean when the reading crosses over to the other side of nothing?

A butterfly reading carries a sign, and the sign is not decoration attached to a size. The sign is half of what the number reports, and it is the half that goes missing when a reading gets written down at speed.

On the recorded nodes the reading is 0.20 percentage points, and the plain-English content of that is: the body stands above the average of its two wings. Under the declared fall of 15 basis points in the yield at the body, it is minus 0.10 percentage points, and the plain-English content is: the body has gone under. Take instead the third case, in which a rise of 40 basis points was declared at the ten year node’s yield. The reading there is minus 0.20 percentage points, so the body is under again, and for a completely different reason. Nothing happened to the body at all in that third case. A wing climbed past it and dragged the average of the wings up with it.

So two readings that are both negative, on the same curve, describe two situations with almost nothing in common. One had a body that dropped. The other had a body that stood perfectly still. The sign says which side of the average the body is on, and that is the only thing the reading was ever measuring, so a reading quoted without its sign, or described as a size, has thrown its content away and kept its packaging.

There is a second thing the sign does. The sign shows that the reading crosses rather than merely shrinking. A distance that gets smaller and smaller and then stops at nothing behaves like a gap closing. The butterfly reading does not stop: it arrives at nothing, passes through, and opens out again with the body on the other side. Between minus 0.20 percentage points and 0.20 percentage points there is a position where the body sits exactly on the average of its wings, and there is nothing special about that position except that it is where the sign changes hands.

The four declared moves, gathered in one place

Every move declared so far is collected below, with the recorded row at the top. Each row is a set of values written down here and observed nowhere, and every reading in it was recomputed from the values in its own row rather than adjusted from the row above.

Three things fall out of reading that table down its columns rather than across its rows. Everything that happened in the top three rows happened at the body, and the slope column cannot see any of it, so those three rows all carry a slope of 1.10 percentage points and butterfly readings of 0.20, minus 0.10 and 0.80. The bottom two rows share a slope of 1.50 percentage points and carry readings 0.40 percentage points apart. And the average-of-the-wings column shifts exactly once in five rows, at 7.00 per cent, in the one row where a single wing moved on its own.

The reading on the first row and the reading on the last row are both 0.20 percentage points, on two curves whose wings stand 20 basis points apart at each end. Nothing demonstrates more plainly that a butterfly reading describes a shape rather than a curve.

Does one curve have one butterfly reading, or several?

Several. The count of readings on one curve is the part of the subject that quietly ruins the most conversations, and the arithmetic that settles it takes about a minute.

The record fixes six maturities. Any three of them in increasing order give a triple, and every triple has its own reading. Here are five of them, worked from the same six numbers given at the start.

Five readings. Two signs. One curve. Not one of them is wrong, not one is a better description than another, and a sentence claiming this curve is curved in some particular direction, with no three maturities attached, is not a statement about anything at all. The reading is a measurement taken between three named points, and it belongs to those points rather than to the curve they were taken off.

There is a second reason a reading cannot travel from one triple to another, and it is easy to miss. Look at the years. The one, two and three year triple has gaps of one year and one year. Every other triple on this list has unequal gaps: one and two, two and five, three and five, five and twenty. Four of these five triples are unevenly spaced in years, and the triple used throughout is one of the four. Uneven spacing is fine. The arithmetic never mentions years and simply averages two rates. But any drawing of a butterfly has to place the body at the middle of its wings whether or not it sits at the middle of them in time. Every three-node drawing above is therefore on an ordinal axis and says so on its face.

What does a butterfly reading not settle about how the three parts are sized?

Everything above has been about a reading, and a reading is arithmetic on three rates. A structure is a different animal. If somebody builds something at three maturities, they have to decide how much sits at each of them, and that decision is made in money rather than in rates.

A reading and a structure are genuinely different questions, and the gap between them is where most confusion about butterflies lives. The weights of one, two and one belong to the reading and they operate on rates. The weights say nothing whatever about how much money goes where. Two legsThe separate holdings inside a structure. Something built at three maturities has three of them, each carrying its own amount. carrying the same amount at different maturities do not respond alike to the same move in their own rates. A longer claim has more waiting time inside it and so more price sensitivityHow much a price moves for a given move in its own rate. It is measured under interest rate risk, and no figure for it appears here.. A sizing decision worked out from those sensitivities is what makes a three-part structure indifferent to a change in the slope and responsive only to curvature. A sizing decision is not the reading, and it does not fall out of the reading.

Sizing is covered separately, and the six SPOT rates above do not hold what a sizing needs. Six SPOT rates are what the record fixes. The record holds no price sensitivity at any of those maturities, and no holding of any size. And it says nothing at all about what somebody would end up with once the curve has changed shape rather than merely shifted. Three quantities are required to size three legs, and none of the three exists here. So the block below carries empty cells with the reason written next to them. The reading is arithmetic anybody can recheck in thirty seconds. The sizing comes from price sensitivities, covered separately.

Who takes a reading like this on an ordinary working day, and what for?

Start with an everyday case that has nothing to do with markets. Walk into a bank and look at the board of deposit rates. One year pays one rate, three years pays another, five years pays a third. Almost everybody reads that board as a list, and the only comparison they make is between the top and the bottom of it. Now do this instead: average the one year rate and the five year rate, and ask whether the three year rate sits above or below that average. If it sits above, the middle of the board is being paid more generously than a straight run between its ends would give. If it sits below, the middle is the thin part. The comparison just made is a butterfly reading, done with three numbers off a noticeboard and no arithmetic beyond an average and a subtraction.

Now the working versions, and there are three worth naming. The first is someone reading a note that says a curve steepened. A steepening, on its own, is compatible with the middle of the curve having gone anywhere at all, and the reader who asks what the reading on a named triple did is asking the question the note left out. The second is somebody reconciling two descriptions of the same day. Two desks report the same curve, one calls it more curved and one calls it less, and the useful move is to stop arguing about the adjective and ask each of them which three maturities they were on, at which point the disagreement usually turns out not to exist.

The third is the one that keeps a person honest about their own work. Anyone describing what a position responds to has to be able to say which readings would move and which would not, and that is a much harder test than it sounds. Consider the pivot from earlier: two wings swinging 20 basis points apart from each other, a real change in the curve by anybody's account, and a butterfly reading that does not shift by a single basis point. A person who cannot predict that in advance does not yet know what their own description is measuring. None of this settles what anybody should hold. The test is being able to say precisely what a quoted number is and is not sensitive to.

How is a curvature reading taken, step by step?

Six steps, in order, and the first one is the only one that gets skipped.

Step five looks like busywork, so it is worth one more sentence. Under the symmetric pivot the two wings moved by 20 basis points each and the reading moved by nothing; under the rise at one wing a single node moved by 40 basis points and the reading moved by 40. The change in the reading depends entirely on which node moved, so no rule of thumb connects it to the size of a declared move. Recomputing from the values takes ten seconds and is always right.

Where does the arithmetic stop, and who keeps the wording beyond it?

Everything above this line came out of subtraction. Six invented rates, four moves declared here rather than watched, and a rule of one, two and one. Not a single figure needed permission from anybody. The run of pure subtraction ends here. Take a reading like this on a curve somebody actually publishes, and six separate rules of wording start to matter, each kept elsewhere and revised on somebody else's timetable.

A card that can be filled in at its own source beats a sentence typed from memory two years ago and left to rot, so the six cards below carry no contents. Each carries a different reason for standing empty.

The yield curve used here, and the four moves run against it, are invented.
Educational material. Not advice on any investment, tax, budget or market position.