Parallel Shift vs Steepening vs Flattening vs Twist

Four words, four claims, and every one of them is about two named maturities. A move is parallel when each recorded rate shifts by one identical amount. A move steepens when the gap between two maturities gets wider and flattens when that gap gets narrower. A move twists when one end sees a rise in the yield while the other sees a fall. Drop the maturities and none of the four says anything.

Somebody says the curve steepened today. The sentence tells a listener less than it sounds. The sentence has not said whether rates went anywhere in particular, whether anybody lost money, or even which two maturities the speaker was looking at. One distance got bigger. The whole content of the word is that much, and the same is true of the other three. The four words are compressions, and a reader who does not know what has been compressed out of them will hear things that were never said.

All four are unpacked below against one made up curve. The curve, an invented one, records 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. Three of those six do the work below, the two year, the five year and the ten year. Each is a SPOT rate: money put down today against one single dated repayment, nothing paid out along the way, one date to attach the rate to. Every sum here runs on ANNUAL compounding: the rate lands once in the year, and whatever it earned stays in to earn again next time round. Naming that basis is not a formality. Running the identical figures twice a year instead prices the very same promise differently, so a sum quoted with its basis left off cannot be checked by anybody. Two units run through everything below and swapping them is a real error rather than a slip of style. One whole unit of a rate is a percentage point. Cut into a hundred slices, each slice is a basis point. The figure 1.10 in the first unit and the figure 110 in the second are therefore one distance measured twice.

Discounting a dated payment is settled elsewhere and simply assumed below. A curve fixes a rate at particular maturities and says nothing between them, so every move here lands on a recorded node and nowhere else. A curve move gets reported as three separate readings, a change in the level, a change in the slope and a change in the curvature, and how the slope and the curvature are worked out is covered separately. So is what a single rate move does to a single price. Rate sensitivityHow far the price of a bond moves when the rate used to value it moves. Covered separately; here it is only assumed. is therefore assumed here rather than built again. A curve move is a set of rate moves. The permission carries.

The naming has been settled nowhere, and the naming is the entire difficulty. Five scenarios get declared below. Every one of them is an input written down here and applied to the recorded nodes; not one is observed anywhere, in this record or outside it, and each carries the word declared at the point it is used.

Why does a move that three numbers already describe need four separate names?

The four names are built on top of the three numbers and make no sense underneath. The three numbers come first. Any move of the curve at all, however scrambled, can be reported as a change in the level, a change in the slope and a change in the curvature. The three readings are covered separately and are only applied here.

The level is the height of the whole thing. The slope is one distance: a longer rate less a shorter one, with both maturities named. The curvature is the three point calculation, sometimes called a butterflyThe market's name for the three point calculation that reads the bend in a curve, worked from a short maturity, a long one and one in between. Named for the shape of a position built around it. Building that position is covered separately., which asks whether the middle of the curve sits above or below a straight line drawn between the two ends.

Here is the thing worth noticing about those three. The three readings were deliberately built so that each one can move while the other two sit perfectly still. That is not an accident of arithmetic, it is the design. The level is a common amount, the slope is a difference, the curvature is a difference of differences, and each strips out what the one before it measured. The same trick is what a shopkeeper uses when she asks first what her total takings were, then whether the morning outran the evening, then whether the middle of the day behaved like the average of the two. Three questions, three answers, and knowing one of them says nothing about the other two.

So why four names rather than three? Because the names are shorthand for particular patterns across those three readings, and shorthand is what people actually speak. Parallel is the level moving on its own. Steepening and flattening are the slope moving, one word for each direction. Two of the four names therefore read the same number and differ only in which way it went. Twist is a fifth thing again, a fact about direction at the two ends that no single reading reports at all.

Four names, three readings, and no clean mapping between them. The words get muddled for exactly that reason. The rest of this guide defines each one in full against the same three recorded nodes, and then shows a move that none of the four covers.

What makes a move parallel, and which two readings does that leave untouched?

A move is parallel when each node shifts by one identical amount, and all of them the same way. Parallel has no more definition than that, and notice it is a condition on the moves rather than a description of the result. Check the moves, and if they are all equal the word applies.

Take a household example first. The shape of it matters more than the arithmetic. A vegetable seller raises the price of every single item on her cart by two rupees. Onions, tomatoes, coriander, all of it, two rupees more. The cart is dearer, plainly. But which item is dearest is exactly what it was this morning, and the distance between the cheapest and the dearest has not shifted by a paisa. Somebody looking only at the gaps would report that nothing had happened at all, and they would be right about the gaps and wrong about the cart.

The move is now declared and applied. A rise of 50 basis points at every recorded node, declared here as an input and observed nowhere. The two year SPOT rate goes from 6.25 per cent to 6.75 per cent. The five year goes from 6.90 per cent to 7.40 per cent. The ten year goes from 7.35 per cent to 7.85 per cent. Every node saw a rise in the yield, and every rise was the same size.

Take all three readings in turn. The level has risen by 50 basis points, and because the moves were equal there is a single number to report. The slope, worked as the ten year node less the two year node, stood at 1.10 percentage points and stands at 1.10 percentage points now. 7.85 less 6.75 gives back exactly what 7.35 less 6.25 gave. The curvature counts the five year node twice over and then takes away each outer node once. The curvature reading was 0.20 percentage points and is 0.20 percentage points still. 14.80 less 6.75 less 7.85 lands where 13.80 less 6.25 less 7.35 landed. One reading moved, two did not, and that is not a result of these particular numbers but the definition itself doing its work.

The point is worth saying once more in plain words, and it is the reason parallel is the easy case. The slope adds one rate and subtracts one rate. The curvature adds two and subtracts two. Any amount added to all three nodes therefore cancels itself out of both. Only the level does no subtracting at all, and only the level is left holding it.

One last thing before moving on, and it matters twice more below. A parallel move is the only one of the four where a single number describes what happened to every rate at once. The single number is what makes the parallel case costable later, and its absence is what makes the other three uncostable. The arithmetic depends on it.

What has to happen before a move earns the word steepening?

One thing only. The distance between two named maturities has to get wider. Nothing else is required, nothing else is implied, and the moment anything else is added the description has stopped being a steepening and started being a guess.

The declared move: the two year SPOT rate holds at 6.25 per cent and does not budge; the five year rises 25 basis points to 7.15 per cent; the ten year rises 50 basis points to 7.85 per cent. Declared here, applied to the recorded nodes, observed nowhere.

Read it. The short endThe part of a curve covering the nearest maturities, where an authority's own rate has most of the say. Which maturities count as short is a matter of convention rather than arithmetic, so the maturity is named every time instead. did not move at all, so there is no single level change available to report; one node stayed put while another rose, and a level reading only exists when the nodes move together. The slope went from 1.10 percentage points to 1.60, a widening of 50 basis points. 7.85 less 6.25 is 1.60. The curvature is twice 7.15 less 6.25 less 7.85. The arithmetic gives 14.30 less 14.10, or 0.20 percentage points, precisely where it stood.

So a steepening happened here with the short end absolutely still. A steepening is not a rise in the yield everywhere and never was. That sentence is worth reading twice, because the everyday picture people carry is a curve tilting like a see-saw, and a see-saw has one end going each way. The declared curve did nothing of the sort. Every node either rose or stayed exactly where the record fixed it, and the word steepening still applies. The word is about one distance and one distance alone.

There is a household version of this that lands immediately. A tailor charges Rs 300/- for a same week order and Rs 300/- for one he can take a month over. No gap. Now he leaves the same week price alone and raises the month one to Rs 340/-. The gap between his two services has widened from nothing to Rs 40/-, and he has not raised a single price that a same week customer pays. Somebody reporting only that his prices went higher has said something true about one service and false about the other. The long endThe part of a curve covering the most distant maturities. As with the short end, where it starts is convention rather than calculation. Every claim here names the actual maturity instead. of a curve behaves the same way, and the word for it names the gap rather than the prices.

Can a curve flatten while not one recorded rate falls anywhere?

A curve can. The declared move below does exactly that, and it is worth slowing down over. A flattening is a move in which the distance between two named maturities narrows. The word claims nothing more. A flattening is the mirror of a steepening, one reading moving the other way, and it says as little about the level as its mirror does.

Declare it: the two year SPOT rate rises 50 basis points to 6.75 per cent, the five year rises 25 basis points to 7.15 per cent, and the ten year holds at 7.35 per cent. Now go along the row and look for a fall. There is not one. Two nodes saw a rise in the yield and the third stayed precisely where the record fixed it. Nothing on this curve went lower by any amount at all.

And yet the slope, worked the same way it was worked twice above, was 1.10 percentage points and is now 0.60. 7.35 less 6.75 is 0.60. The gap narrowed by 50 basis points. A flattening is a claim about the distance between two rates and carries no information whatever about the level. Reading a narrowed gap as a fall in rates is the single most common confusion of the four.

The vegetable seller again, running the other direction. Coriander was cheap and out of season onions were dear. She raises coriander by two rupees and leaves onions alone. Every price on her cart either rose or stayed put, and the gap between the cheapest thing and the dearest thing got smaller. Anybody who insists that a narrowing gap requires something getting cheaper has confused a distance with a direction, and there is no arithmetic that will rescue the confusion, only a habit of reading the two numbers separately every single time.

The curvature, for completeness, is twice 7.15 less 6.75 less 7.35. The arithmetic gives 14.30 less 14.10, or 0.20 percentage points once again. Three declared scenarios in a row have now left the curvature exactly where it began, and the run stops being a coincidence two blocks from here.

What does a twist report that neither of the other readings carries?

A twist is a move in which the short end and the long end go in opposite directions around some maturity in between. One end sees a rise in the yield while the other sees a fall. The opposition is the whole of the claim, and it is a different kind of claim from the other three. A twist is about direction at two places rather than about a number anywhere.

Declare one: the two year SPOT rate rises 50 basis points to 6.75 per cent, the five year holds at 6.90 per cent, and the ten year falls 50 basis points to 6.85 per cent.

Read it and something awkward turns up straight away. The slope was 1.10 percentage points. The slope now is 6.85 less 6.75, or 0.10. The slope narrowed, and narrowing is exactly what the word flattening means. So this declared move is a twist and a flattening at the same time, and the two names are not alternatives to pick between. A reader who has been treating the four words as four boxes, one of which a move must fall into, has just watched a move land in two of them.

None of this is a defect in the vocabulary. The overlap is what happens when four shorthand names are laid over three readings that were never designed to line up with them. Flattening reports the slope. Twist reports something the slope cannot report. The two ends went opposite ways. Work through it: the slope subtracted 6.75 from 6.85 and got 0.10, and that 0.10 would have arrived just as happily from a curve where both ends rose, or both fell, or neither moved and only one did. The single figure 0.10 does not know, and cannot be made to know, that one end went one way and the other went the other way.

Which gives the practical rule this block exists for. A twist is the one move of the four that has to be written out node by node rather than summarised. Summarising it destroys the fact that made it a twist. Every other name survives compression. A twist does not.

Which move walks straight past the slope without disturbing it?

Here is a fifth declared move, and it is the one that shows why the four names in the title are not a complete list of anything. The two year SPOT rate holds at 6.25 per cent. The ten year holds at 7.35 per cent. Only the five year moves, rising 20 basis points to 7.10 per cent.

The slope is the ten year node less the two year node. 7.35 less 6.25 is 1.10 percentage points. The slope was 1.10 percentage points before as well. Neither of the two rates the slope consults changed by a single basis point, so the slope has nothing whatever to report, and no amount of staring will make it report anything.

Now read the curvature. Before, the curvature was twice 6.90 less 6.25 less 7.35. The arithmetic gives 13.80 less 13.60, or 0.20 percentage points. After, the curvature is twice 7.10 less 6.25 less 7.35. The arithmetic there gives 14.20 less 13.60, or 0.60 percentage points. The reading has trebled, a rise of 0.40 percentage points, or 40 basis points.

The curve moved, visibly and by a measurable amount, and one of the three readings reported absolutely nothing. That is the reason a curve move is read with three numbers rather than two, and it is not a theoretical worry. A middle of the curve lifted while both ends stood still. In shape terms the curve became more humpedA curve whose middle stands above a straight line drawn between its two ends, so it bulges rather than running straight. Which shapes a curve can take is covered separately and is only borrowed here as a word. than it was. None of the four names in the title fits it. The node moves were unequal, ruling out parallel. The slope did not move, ruling out both steepening and flattening. No two ends went opposite ways, and indeed neither end went anywhere. A twist is ruled out as well.

The tailor, one last time. He leaves his same week price alone and leaves his month price alone, and raises only the price of the two week job in the middle. The gap between his fastest and his slowest service has not changed by one rupee. A customer who only ever compares those two would swear nothing happened, and a customer who wanted the two week job would tell them otherwise.

What decides whether the curvature holds still or does not?

Something has been happening quietly across four scenarios and it is time to name it. The curvature read 0.20 percentage points before the parallel rise and 0.20 after. Same before and after the steepening. Same before and after the flattening. Same before and after the twist. Then the fifth move arrived and it went to 0.60. Four in a row holding still is too many to shrug at, and it turns out not to be luck at all.

Work the change in the curvature rather than the curvature itself. If the short node moves by some amount, the middle node by another and the long node by a third, then the change in the three point calculation is twice the middle node's move, less the short node's move, less the long node's move. Set that to nothing and rearrange, and a condition drops out that is startlingly simple.

Now run the four named moves through it. The parallel rise moved the middle 50 basis points, and the average of 50 and 50 is 50. The steepening moved the middle 25, and the average of nothing and 50 is 25. The flattening moved the middle 25, and the average of 50 and nothing is 25. The twist moved the middle not at all, and the average of a rise of 50 and a fall of 50 is nothing. All four named moves were built with the middle node obeying that condition exactly. The curvature sat still through every one of them for that reason and no other.

And the fifth move breaks it in the plainest possible way. Both ends stayed put, so their average is nothing, and the middle went 20 basis points anyway. The condition fails by the full 20, the curvature moves by twice that, and 40 basis points is what the reading shows.

The condition converts a vague warning into a test that can be run in the head. Somebody describes a curve move and supplies three node figures. The two ends get averaged and the average is compared with the middle. If they match, the curvature has not moved and the four names might be enough. If they do not, one of the four names is about to describe the move badly, and the third reading is the one carrying the part the name will miss.

How do the five declared scenarios read when they are set out in one table?

Everything above, in six rows. The recorded row first, then the five declared moves applied to it. Down the slope column the four named moves separate cleanly; down the curvature column four of them are identical while the fifth stands out on its own. Every rate here is the invented curve's, every move is declared, and every slope and curvature figure was subtracted here on ANNUAL compounding.

Two columns are worth reading as columns rather than as rows. The slope column runs 1.10, 1.10, 1.60, 0.60, 0.10, 1.10. One reading is doing everything the four names describe: unchanged, wider, narrower, nearly gone, and unchanged again. The curvature column runs 0.20 five times and then 0.60 once. Between them those two columns tell the four moves apart and expose the one move the four names cannot reach. Rates in per cent a year, slope and curvature in percentage points; the one rupee cost these figures can support comes two blocks below, and only for one of the rows.

Two things about that control are worth noticing. First, carried slowly through the middle, the slope bar crosses its red rule: on one side of that crossing the word is a flattening and on the other it is a steepening, and there is no third thing happening. The word changes; the picture does not lurch. Second, and this is the part that catches careful readers, the curvature bar moves throughout. Moving one node alone never gives a clean example of any single named move. The middle node is not obeying the average condition, and the curvature is dragged along on every step. Carry the control to a rise of 20 basis points and the curvature bar disappears entirely, having reached nothing; go past that and it grows on the far side of the line as a negative reading.

Which of these four can be given a rupee cost here, and which cannot be given one at all?

One of the four, and the reason is worth more than the figure.

The record behind this guide carries an invented holdingA set of bonds owned together and measured as one thing. Building one, and deciding what should be in it, is covered separately; here it is only a quantity to attach arithmetic to. of Rs 5,000 crore with a modified durationOne number saying how far the value of a holding moves for a given move in the rate used to value it. Where it comes from, and why it is only an approximation, are covered separately. of 5.20. Declare a parallel rise of 100 basis points. Multiply: 5.20 times one percentage point is 5.20 per cent of the holding, and 5.20 per cent of Rs 5,000 crore is Rs 260 crore. Both figures come from the record and the second reproduces from the first in whole rupees rather than from any rounded percentage.

The record carries a second figure as well, and the two are easy to mix. Which number this one is has to be said clearly. Rs 260 crore is the whole exposure of the holding to that declared move. The figure is not a comparison against anything, not a decision anybody took, and not a measure of whether the holding was well built. Comparisons of that kind are covered separately.

The record behind this guide holds no non-parallel scenario, and inventing one would produce a number indistinguishable from a worked one. For the steepening, the flattening and the twist, no rupee figure appears at all. There is a second reason underneath the first, and it survives even where the missing scenario is supplied: a single duration figure describes how a holding answers when every rate moves together, and it carries no information whatever about where along the curve that holding actually sits. Two holdings with the same 5.20 can be built quite differently, one packed into short maturities and one into long, and a non-parallel move is exactly the question those two would answer differently. The single figure cannot tell them apart, and the failure is a limit of the measure rather than a gap in this record.

What are these four words actually for, once they leave a textbook?

Three places, and not one of them involves inventing a curve move.

The first is reading what somebody else wrote. A note arrives saying the curve steepened, or that the market saw a bull flattening, or that there was a twist at the front end. The single most useful move a reader can make is to look for the two maturities, and where they are absent, treat the sentence as an atmosphere rather than a statement. That is not pedantry. A steepening between two years and ten years and a steepening between ten years and thirty years are different events with different causes and different consequences, and a note that mentions neither pair has left out the part that would let anybody check it. Once the pair is found, the rest follows in seconds: which node moved, by how much, and did anything at all happen to the middle.

The second is writing a risk line that somebody else will act on. Here the four words carry one specific and unglamorous job. The four words say whether one number may be used. If the move was parallel, one duration figure describes the whole thing and the sentence is short. If it was anything else, the honest line names the nodes and their moves and stops, and it is longer and less satisfying and cannot be misread. A line that gives a rupee figure for a non-parallel move has quietly asserted something about how the holding is spread along the curve that the figure itself does not contain.

The third is a household, and this is the one nearly everyone meets without noticing. Somebody hears on the news that the curve steepened, and they have money to place. How much has that sentence told them? Almost nothing usable. A steepening is perfectly consistent with the short end sitting still, as the third block above showed. The sentence has not said whether the rate on offer for a two year deposit changed. The sentence has not said whether anything got dearer or cheaper. One gap got wider, somewhere, between two maturities the newsreader did not name. Knowing that much, and knowing precisely how little it is, is worth more than a confident misreading, and it is the correct place to stop. Deciding what to do about that money afterwards is a separate question.

One thing runs through all three. None of them calls for a number to be produced. Each calls for working out which claim was actually made, and then checking whether the maturities behind it were supplied. Four words that sound like they carry much more have no more practical content than that.

How is a curve move named from the numbers rather than from the picture?

Two curves drawn at slightly different vertical scales look like completely different events and are not. A picture is therefore the worst thing to name a move from. Work from the node figures instead, in a fixed order, and the order matters because two of these steps can each report nothing while the curve has plainly moved.

Two of those steps deserve a sentence each. Step four exists because of the fifth scenario, where both ends held and the middle rose 20 basis points: steps two and three each reported nothing and the curve had moved by a measurable 40 basis points of curvature. Skipping step four turns the four names in the title into a set of boxes a move gets forced into, and forcing is how a description becomes wrong.

Step six is the one that actually gets skipped, and it is skipped because it feels like housekeeping rather than content. It is not housekeeping. A steepening between two years and ten years and a steepening between five years and thirty years are separate events. A sentence naming neither pair cannot be checked, cannot be reproduced, and cannot be disagreed with. Being impossible to disagree with sounds like a strength and is the opposite. The maturities are what make the sentence a claim.

One thing about that list is worth saying plainly, and it explains why the list is short. Every sum in this guide is a subtraction between rates on one made up curve, and a subtraction needs no convention at all except the basis those rates are quoted on. The basis is stated inside the arithmetic as ANNUAL. No line above could be checked without it. Everything else in the block above sits outside the arithmetic rather than underneath it. A second market with different conventions would add rows to that block and change not one figure in the tables.