How to analyse a Yield-Curve Scenario: The Eight Step Method

A yield-curve scenario is a complete set of rate values that somebody DECLARES. Analysing one means writing every node down, marking each as recorded or declared, recomputing every shape reading and every FORWARD rate from those node values alone, and then naming what the scenario cannot answer. Any figure in the finished analysis that traces to none of those three sources was invented.

The panel opens on the scenario this guide works through. Its rates belong to an invented SPOT curve built for teaching, and the move laid over them is a declared input rather than an observation. At that opening setting the two year rate falls 10 basis points to 6.15 per cent, the five year rate is held at 6.90, and the ten year rate falls 40 basis points to 6.95. The slope reading narrows from 1.10 percentage points to 0.80, the butterfly reading moves from 0.20 to 0.70, and the declared move splits into a level fall of 25 basis points, a slope component of minus 15 and a curvature component of 25, worth a gain of 1.20 percentage points of the declared position value. Each of those figures is worked out below rather than asserted here.

What is every reading on a curve actually made of?

Every reading anybody takes off a yield curve is arithmetic performed on the rates at particular maturities: a subtraction for the shape readings, and a division followed by a root for a FORWARD rate. No extra ingredient, and no judgement applied at the end.

An analysis is therefore only ever as complete as the set of rates it started from. The commonest failure in this work is not bad arithmetic at all but a rate nobody wrote down, quietly filled in, and then reported as though it had been given. That failure is invisible in the finished document. The sums are right, the columns add, and one input was supplied by the person doing the work rather than by the person who set the question.

So the procedure spends most of its effort on bookkeeping rather than computing: which figure came from where, written down before anything is done to it. The bookkeeping sounds like paperwork, and it carries the whole procedure.

Where does a yield-curve scenario come from?

A yield-curve scenario is a set of rates, one for each maturity the analysis needs, that somebody has DECLARED. Not observed, not forecast, not derived from anything else. A scenario whose origin is not written down gets read a week later, by somebody nowhere near the discussion, as a description of something that actually happened, and every conclusion drawn from it inherits an authority it never had.

A neighbour mentions that the flat downstairs went for Rs 2,00,000/-. Two months later the same figure comes back from three different people, and by then nobody can say whether it was a completed sale, an asking price, or something the neighbour guessed. The number did not change. The label on it was lost, and the label was the only thing that said what the number could be used for.

So the first step of the analysis is to write down who declared the scenario, and to record on the face of the work that it is a declaration rather than an observation. Where the scenario belongs to somebody else, their name goes beside it. A scenario with no author is a set of numbers nobody has agreed to. Where the author cannot be established, that absence is itself a finding worth reporting.

Why write the scenario as a table before writing it as a sentence?

Because a sentence hides gaps and a table shows them.

A caterer quotes for a wedding: dinner for three hundred at Rs 20/- a plate. The sentence reads complete. Write the same quote as a table, one row per item, and three rows have nothing in them: the tent, the service staff and the dessert counter. Every reader of the sentence filled those blanks differently in their own head. The table added no information. The table made an absence visible.

A curve scenario behaves exactly the same way. Written as a sentence: the curve flattened. Written as a table, on ANNUAL compounding:

The table is three lines long, and it settles the two things the sentence left completely open: which maturities moved, and by how much each one moved. Write the table first, every single time, and only then write the sentence that describes it.

Notice the middle row. The five year SPOT rate is in the table even though nothing happens to it. A row saying held is a decision that was taken and recorded. A row that is simply absent is a decision taken silently by whoever computes first, and it will not appear anywhere in the finished work.

Which cells are recorded and which are declared?

In the declared flattening, 6.90 per cent at the five year SPOT rate is a RECORDED node being held still. The 6.15 per cent at the two year SPOT rate and the 6.95 at the ten year SPOT rate are both DECLARED. Marking them apart does two separate jobs. The first is the one already made: a table whose cells are marked cannot quietly turn into a claim about the world some months on.

The recorded column has nothing to put against a maturity the curve does not fix, so the second job of marking is to make such a scenario impossible to run. This invented SPOT curve fixes six maturities and nothing 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. A scenario that moves a four year or a seven year SPOT rate cannot be analysed here. There is no starting value to move, and any figure reported from it would have been read off a drawn line rather than taken from the record. Name the gap and stop: that is the finished answer, not a shortfall in one.

How are the shape readings recomputed once a scenario is in place?

From the node values. Not from the changes. Getting the order the wrong way round usually gives an answer close enough to look right. The mistake therefore survives review.

The slope here always means one subtraction, with both maturities named every time: the ten year SPOT rate with the two year SPOT rate taken away from it. On the recorded nodes that leaves 1.10 percentage points, and under the declared flattening 0.80. The slope narrowed by 30 basis points, and those 30 basis points came out of the recomputation rather than going into it.

The butterfly reading always means the same three-term sum, with all three maturities named every time: double the five year SPOT rate, remove the two year SPOT rate, remove the ten year SPOT rate. Double 6.90 gives 13.80 on either set of nodes, and removing 6.25 and 7.35 leaves 0.20 percentage points while removing 6.15 and 6.95 leaves 0.70. The butterfly reading changed by 50 basis points even though the five year SPOT rate never moved. Both of the maturities standing either side of it did move.

Notice one sentence this scenario will not support. No rate in it saw a rise in the yield: the two ends both fell and the middle was held, and the narrowing that the slope reports says nothing whatever about that. The level and the shape are two separate questions, and this one scenario moved both. An analysis that reports the flattening and stops has described roughly half of what it was handed.

Move the control across its whole range in the first mode and watch what refuses to happen. The two year SPOT rate runs from 5.25 per cent to 7.25 and the ten year from 6.35 to 8.35, and at all forty one settings the slope reads 1.10 percentage points and the butterfly reading 0.20. Both readings are subtractions, and the same amount added to every term in a subtraction cancels out. A move that is identical at every maturity is therefore invisible to both of them. That is worth seeing rather than being told, because a reader who is merely told it assumes the readouts were rounded.

Switch to the second mode, where only the ten year node moves and every other node is pinned, and the readings come alive at once. This scenario said nothing about the far end, so the thirty year SPOT rate is held at its recorded 7.60 per cent. Once the declared move passes 25 basis points the ten year SPOT rate reads above that held figure, and the far end of the drawing turns over. The crossover is not a fault in the picture and not a slip in the arithmetic. Any one-variable scenario eventually reaches it once everything else is pinned, producing a curve nobody would have described in words, and the analysis reports that curve rather than tidying it away.

What does a scenario do to the FORWARD rates inside the curve?

The FORWARD rates are the step most analyses skip, and skipping them is understandable: the scenario named three SPOT rates, so three SPOT rates get reported. But a curve carries more than the rates written on it.

A FORWARD rate is arithmetic performed on two SPOT rates: the rate for a stretch of time between two future dates, coming out of the relationship between the rate to the near date and the rate to the far one. So a scenario that moves the SPOT rates moves every FORWARD rate that touches them, by amounts that look nothing like the moves declared.

Work one. On the recorded nodes the five year SPOT rate reads 6.90 per cent and the ten year 7.35. Raise 1.0735 to the tenth power, divide by 1.0690 raised to the fifth power, take the fifth rootThe number which, multiplied by itself five times over, gives back the number it was taken from. Taking it is the reverse of raising something to the fifth power. of the result and subtract one: the five year rate five years FORWARD is 7.801894 per cent, on ANNUAL compounding. The same sum under the declared flattening, where 6.90 is held and the ten year sits at 6.95, gives 7.000023 per cent. The declared move at the ten year node was 40 basis points, and the five year rate five years FORWARD moved 80.1871 basis points, slightly more than twice as far.

The doubling is not peculiar to these particular numbers. The FORWARD rate covers the stretch from year five to year ten, half of the ten year period, and the front half was pinned at 6.90 per cent. Half the length carries all of the declared move, so the move it carries is about twice as large.

Now work one in the other direction. The three year rate two years FORWARD covers years two to five, and on the recorded nodes it comes from 6.25 per cent at two years and 6.90 at five: raise 1.0690 to the fifth power, divide by 1.0625 squared, take the cube root, subtract one, giving 7.335541 per cent. Under the declared flattening the two year SPOT rate has come to 6.15 while the five year stays put, and the same sum gives 7.402942.

The three year rate two years FORWARD rose by 6.7401 basis points inside a scenario in which no SPOT rate rose at all. The reason is arithmetic rather than any story about expectations: the near end of the stretch fell while the far end was held, so the stretch between them carries more of the total and the rate covering it reads higher. A FORWARD rate is arithmetic that today's SPOT rates already contain, and it carries nobody's opinion about where a short rateThe interest rate on borrowing for a very brief period, such as overnight or a few weeks, as against the rate for lending across several years. will actually be.

One note on precision. The five year rate five years FORWARD is usually quoted to two places, as 7.80 per cent. Two places cannot show an 80.1871 basis point change against a 40 basis point declaration, so the 7.801894 above is the same number carried further. Wherever a FORWARD rate is written here, the two SPOT rates behind it are written in the same sentence, so a reader can rebuild it rather than take it.

What does the declared move do to a position?

A scenario is three declared moves. A position is three declared exposures. Multiplied together they give one total, but that total says nothing about which part of the move produced it. Splitting the move first does, and the split is arithmetic rather than judgement.

Any move at three nodes comes apart into exactly three components. The LEVEL component is the average of what the two end nodes did. Lifting one end by a given amount while lowering the other by the same opens the distance between them by twice that amount. The SLOPE component is therefore half the change in the slope reading. The CURVATURE component is half the change in the butterfly reading, and it measures how far the middle node sat off the straight line joining the two ends.

The declared flattening comes apart into a level fall of 25 basis points, a slope component of minus 15 and a curvature component of 25, and those three rebuild all three declared moves exactly. A split that cannot reproduce its own inputs is a description rather than a decomposition. The panel prints the rebuild at every setting of its controls.

The declared position is a flattener held in equal size at each end: 4.0 years of key rate exposureHow much a position moves in value for a one point change in the rate at one particular maturity, with the rates at every other maturity left where they are. Measured in years. at the ten year node against minus 4.0 years at the two year node, and nothing at the five year node. Under the declared flattening it gains 1.20 percentage points of its value, all of it on the slope line: 15 basis points of slope component against 8.0 years of slope exposure. The level line reads nothing because the two exposures cancel, the curvature line because no exposure sits at the middle node.

Which is exactly where a scenario that points the right way still loses value. Set the panel to the third worked setting. The curve flattens by the same 30 basis points, from a slope reading of 1.10 percentage points to 0.80, but every rate is higher rather than lower: 7.05 per cent at two years, 7.50 at five, 7.85 at ten. The position now carries minus 2.0 years at the two year node against 4.0 at the ten, so it is no longer balanced. The slope line pays 0.90 percentage points; the level line takes 1.30 away; the net is a loss of 0.40 percentage points on a scenario the position was put on to catch. A shape reading does not know where the curve sits. No shape reading could have warned of that loss.

Which questions can the analysis answer, and which can it never answer?

The analysis can answer anything that is arithmetic on the declared nodes: what each shape reading became, what each FORWARD rate became, how the declared move splits into its three components, and how far each component moved, in basis points and in percentage points. All of it is computable, and all of it is checkable by anybody holding the same table.

Three questions sit outside that, each left blank on purpose with its reason written inside the blank.

The first: what did a holding gain or lose. The scenario says nothing about what anybody held, so from the scenario alone the cell stays empty. Once the exposure of a position at each node is declared, the question becomes arithmetic, and the panel above runs it. No holding size is entered anywhere, so the answer is a percentage of a declared position rather than a rupee figure.

The second: how likely is this scenario. A likelihood needs a measured seriesThe same quantity recorded again and again over a stretch of time, so that how often it did various things can actually be counted. of past curve movements, counted rather than remembered. A declared scenario has no such count behind it, so a likelihood put on one is a feeling with a decimal point attached. The third: is the scenario reasonable. Reasonableness is the same question in a more comfortable voice, and it fails for the same reason: a judgement with nothing to judge against is a preference.

So the finished analysis carries three blank cells, each with its reason written inside it, and a reader who reaches one has arrived at the edge of the arithmetic rather than at something somebody forgot to fill in. A blank cell with nothing in it looks like an oversight and invites the next person to fill it.

How is a figure that came from nowhere caught?

The finished analysis takes one of four marks against every single number in it, with no fifth mark and no exemption for a number that looks obvious. RECORDED, meaning a node this curve fixes. DECLARED, meaning a value this scenario or this position set. DERIVED, meaning arithmetic on those two, written out beside the figure rather than promised. Or UNTRACEABLE, and anything carrying the fourth mark either gets its derivation written next to it or comes out of the analysis.

The four-mark pass is dull, it takes twenty minutes, and it is the single highest-value step in the whole procedure. A plausible figure with no source behind it survives every other kind of review there is. It is internally consistent, it sits in a sensible range, and it agrees with the figure beside it, so nothing about it invites recomputation. One question catches it, asked of every number in turn: where exactly did this come from, with the answer written down.

Here is the check run over the analysis worked in this guide, in full, so the count at the bottom can be verified rather than believed. The last seven rows are the position layer, and the position layer is in the check for the same reason as everything else: it sits in the analysis.

What does the procedure look like as a list?

Everything above collapses into a list that can be worked through with a scenario in one hand and a blank sheet in the other. The order is arranged so that a missing node becomes visible before any arithmetic gets done on top of it.

What does a finished analysis actually look like?

One declared scenario, taken through the procedure in order, so the shape of finished work is visible rather than described.

The eight rows above are the deliverable. The finished analysis is short, every line can be rebuilt by the person receiving it, and what it declines to say is stated as clearly as what it says.

The six SPOT rate levels, the two scenarios laid over them and the key rate exposures of the position are invented.
Educational material. Not advice on any investment, tax, budget or market position.