How to Analyse a Change in Interest-Rate Conditions

Analyse a rate move in eight steps: name which rate moved, record the size in basis points, establish whether the move was parallel, estimate the price effect from the MODIFIED duration, reprice in full, record the gap between estimate and repricing as an output in its own right, separate the whole exposure from the active part, and write down every question the run could not answer.

A rate move arrives as one piece of news and immediately turns into five or six separate questions, each with its own answer and its own reliability. The value of a fixed order is not tidiness. A fixed order stops the questions being answered in the wrong sequence, and the wrong sequence is where every expensive error on this kind of work actually lives. Estimating before establishing which rate moved. Reporting a whole position as though somebody had chosen it. Handing over a figure with no way for the reader to tell how far it can be trusted.

Everything the eight steps use has been settled already: what a MODIFIED durationThe sensitivity figure: how much a price moves, in per cent, for a change of one percentage point in the yield. It is not a length of time and is never written in years. is, what convexityThe curve that a straight line misses. Two bonds with the same sensitivity figure can still move by different amounts, and this is the name for that difference. is, what a basis pointOne hundredth of one percentage point. A hundred of them make one percentage point, which is why a rate move is written in them rather than in fractions. is, how a curve gets split into separate sensitivities, and what a borrower's extra yield pays for. All of that machinery is covered separately and is used throughout the eight steps. The order, the recording format at each step, and two complete runs that can be checked line by line are what this guide adds.

Why does the order matter more than any single calculation?

Think about a household that hears on the evening news that interest rates have gone up. The immediate reaction in most homes is to reach for the loan statement and work out the new instalment. The instalment is a perfectly good calculation performed on a question nobody has established. Which rate went up? The one the home loan is priced off, the one the recurring deposit earns, or a policy rate that neither of them tracks in the same week? The arithmetic is fine. The multiplication has simply been done before the thing being measured was named, so the answer is precise about a quantity nobody can identify.

A desk analysing a bond position makes exactly the same mistake in a more expensive currency. The screen shows a price that has fallen. Somebody reaches for the sensitivity figure and multiplies. Nine minutes later a note goes out saying the position lost a certain number of rupees because rates rose. Nobody in the chain has checked whether the government rate for that date moved at all, or whether the entire move sat in the borrower's own extra yield. A move in the extra yield is a different exposure with a different remedy.

Each step in the order below closes a question that the step after it depends on, and nothing else fixes the order. A single sensitivity figure only speaks for a move that hit every date on the curve by the same amount, so step four cannot be run honestly until step three has produced a verdict. Until step five has been done there is nothing to subtract from, so step six cannot exist. A run that splits the position from the decision at the beginning ends up doing the estimate and the repricing twice, so step seven has to come last among the arithmetic steps.

Step one: which rate actually moved?

A corporate bond's yield has two parts stacked on top of each other. There is the government rate for the same date, and there is the spreadThe part of a borrower's yield that sits above the government rate for the same date. It is compensation for something, and what it compensates for is settled separately. the borrower pays above it. A price fall is consistent with the government part moving, with the spread part moving, or with both moving by amounts that partly cancel. The observed price move on its own picks none of the three, so a run that starts by computing an effect has already committed to a cause nobody checked.

The recording format for this step is one line: the rate that moved, the date or maturity it applies to, and the word SPOT or the word FORWARD attached to it. A SPOT rateThe rate on money handed over today and repaid on one future date, with nothing in between. and a FORWARD rateThe rate on money handed over on a future date and repaid later still, with the terms agreed now. are different objects that can sit within a few basis points of each other on any ordinary curve, so the word is not decoration. Leave it off and the next person to read the line has to guess.

If the government rate for the same date was not observed separately, the step does not fill the gap with an assumption. The step writes the gap down as an unanswered question and carries it to the closing step. Leaving the gap open feels unsatisfying and is the correct behaviour. The alternative is a run that reports a spread move it never measured. A refusal is visible and a guess is not, so a cause that is guessed at is worse than a cause the run declines to name.

Step two: how big was it, and in what unit?

The second step writes the size of the move as a number of basis points, and never as a share of the rate. The distinction sounds like pedantry until the two quantities separate. On the invented ten year bullet bond this material runs on, the yield to maturity starts at 8.50 per cent a year. Half a percentage pointThe plain difference between two percentages. Moving from 8.50 to 10.50 is two of them, whatever share of the starting figure that happens to be. is 50 basis points. Half a per cent of that rate is 8.50 multiplied by 0.005. The answer is 0.0425 percentage points, or 4.25 basis points. The same six words describe two quantities that differ by just under twelve times, and only the unit tells them apart.

So the recording format for step two has four fields: the rate before, the rate after, the difference in basis points, and the period each rate applies to. For the run below that reads: 8.50 per cent a year before, 10.50 per cent a year after, a difference of 200 basis points, annual compounding on both sides. There is one arithmetic check that catches most of the errors made at this step. The difference in basis points divided by one hundred must equal the difference in percentage points. Here 200 divided by 100 is 2.00, and 10.50 less 8.50 is 2.00 as well. A percentage with its base somewhere else is a number waiting to be misread. If a percentage has been written anywhere in the line, the base of that percentage has to be written on the same line.

Step three: was the move parallel, and what happens when that cannot be told?

A curve carries a rate at every date, and there is no rule saying they all move together. The two year point can move by 40 basis points while the ten year point moves by 5. Step three asks whether the move that just happened landed on every point by the same amount, and it demands a written verdict rather than a silence that a reader will interpret as a yes.

The recording format is the move observed at each point on the curve that was actually looked at, followed by one of exactly three verdicts. PARALLEL, meaning the observed points moved by the same amount. NOT PARALLEL, meaning they did not. NOT ESTABLISHED, meaning the run did not observe enough points to say. Each verdict sends the rest of the run down a different route, and the third route is the one that has to be carried in writing rather than in somebody's memory.

One MODIFIED duration is built to answer exactly one question: what happens when every date moves by the same amount. A PARALLEL verdict is therefore the verdict that lets a single figure carry the whole run. A NOT PARALLEL verdict means one figure cannot answer it, and what is needed instead is a key-rate mapA set of separate sensitivity figures, one for each point on the curve, used when the points do not all move by the same amount. built for that purpose. A NOT ESTABLISHED verdict means the run continues on a stated assumption of parallel, and the word assumed then travels with every figure underneath it, in the working, in the note, and in whatever gets pasted into an email three days later.

Step four: what does the estimate say, and what is it worth?

Now the arithmetic starts, and it is deliberately the cheapest arithmetic available. Write the move out as percentage points, multiply it against the MODIFIED duration, and record what comes back both as a percentage price change and as a rupee amount, with the base of that percentage sitting on the same line.

On the invented ten year bullet bond, the MODIFIED duration is 6.5613 and the move is 2.00 percentage points. The product is 13.1226 per cent. The base is the bond's price of Rs 1,000.00/-, so the rupee figure is Rs 131.2260/-, and the estimated price after the move is Rs 868.7740/-. Four things go into the record: the duration used, the move, the percentage with its base, and the rupee amount.

The fifth thing that goes into the record is a label, and it is the part most often left off. The step four figure is an estimate taken from the slope alone, and it is known before it is computed to overstate a loss on a rise in the yield and to understate a gain on a fall. The overstatement is not a caveat somebody added out of caution. Laying a straight line against a curved relationship does that, a property settled elsewhere, and the label exists so that the reader of the note does not have to rediscover it. A figure carrying that label can be used with confidence about what kind of figure it is. The same figure without the label is a prediction, and it is a prediction that is wrong in a direction anybody could have named in advance.

Step five: why reprice when an estimate is already in hand?

Step five discounts every dated amount again at the new rate and records what comes out. On the ten year bullet bond that means ten amounts: nine coupons of Rs 85.00/- and a final year carrying a coupon of Rs 85.00/- alongside the Rs 1,000.00/- of face amount, so Rs 1,085.00/- in the tenth year. Discount all ten at 10.50 per cent a year, annual compounding, and the price is Rs 879.7045/-. Against the base of Rs 1,000.00/- that is a fall of Rs 120.2955/-, or 12.0295 per cent.

Step five is a calculation in its own right and not a check on step four, and treating it as a check is the single change that makes the whole procedure unreliable. A run that reprices only when the estimate looks surprising has made the repricing conditional on a hunch, and hunches are exactly what a procedure exists to remove. The estimate does not decide whether the repricing happens. The repricing happens.

Adding the right hand column exactly as it is printed gives Rs 879.7046/-, one ten thousandth of a rupee above the Rs 879.7045/- struck from the unrounded amounts. Ten roundings, each worth up to half of one hundredth of a paisa, do that between them. A reader who adds the column and lands on the higher figure has not made a mistake and should not spend twenty minutes believing they have. Both figures are printed above for that reason.

Step six: what is the gap between the two, and why does it get its own step?

Subtract the estimate from the repricing and write down what is left, with its sign and its size in both percentage points and rupees. The estimate said the price would land at Rs 868.7740/-. The repricing says Rs 879.7045/-. The difference is Rs 10.9305/-, or 1.0931 percentage points of the same Rs 1,000.00/- base.

The gap between the estimate and the repricing is a finding of the run, not a correction applied to an earlier step, and giving that gap a step of its own is what turns a calculation into something another person can audit. If the error were buried inside step four as an adjustment, the output would be a single better number and nobody downstream would know how good it was. Kept as its own line, it tells the reader two separate things.

The first is size. At a move of 200 basis points, the cheap estimate is out by 1.0931 percentage points of the base. On any position worth having that is real money. The gap widens more sharply than the move that produced it, so at a move of 25 basis points the same estimate would be out by a small fraction of 1.0931 percentage points. A reader who has the error figure knows which regime the run was in. A reader who does not has to treat a small move and a large one with identical confidence, and one of those two confidences is misplaced.

The second is the sign, and this is the cheapest input check available anywhere in the procedure. Where an instrument pays amounts that cannot change, the estimate from the slope has to land short of the repricing on a rise in the yield and past it on a fall. Here it lands below: Rs 868.7740/- against Rs 879.7045/-. If it had landed the other way, the correct conclusion is not that something interesting has been discovered about the bond. An input is wrong and has just announced itself.

Step seven: whose exposure is it, the position's or the decision's?

Everything so far has been run on one bond. Step seven only bites when there is a holding and something it is measured against, and it is the step that decides whether a report is honest about what anybody actually chose.

The invented holding here is Rs 5,000 crore of fixed income with a MODIFIED duration of 5.20, measured against a benchmarkThe reference holding a portfolio is measured against, so that the difference between the two is the part somebody chose rather than the part they inherited. whose MODIFIED duration is 4.80. The difference is 0.40. Read the unit with care: a MODIFIED duration answers how far a price moves for one percentage point of yield, so it is a responsiveness and never a stretch of calendar, and subtracting one from another leaves that same unit behind.

Apply a parallel rise of 100 basis points, or 1.00 percentage point. Take 5.20 per cent on a base of Rs 5,000 crore and the whole exposure comes to Rs 260.00 crore. Take 4.80 per cent on that same base and the benchmark part comes to Rs 240.00 crore. Take the 0.40 per cent that is left and the active part comes to Rs 20.00 crore. Set those last two beside each other and they land back on the Rs 260.00 crore, so the split closes exactly.

The whole exposure and the active part differ by thirteen times on this holding, and a rupee figure without the word WHOLE or the word ACTIVE beside it is one of those two numbers being read as the other. Reported as a decision, Rs 260.00 crore says somebody made a very large call. Reported as a position, it says almost the opposite: Rs 240.00 crore of it was never a choice at all, and the only thing anybody decided was worth Rs 20.00 crore. Both sentences are true of the same holding on the same day.

The recording format is therefore two rupee figures, two percentages, the base of each, and the word WHOLE or the word ACTIVE against each. And the step comes last among the arithmetic rather than first for a plain practical reason: a run that splits at the beginning ends up estimating twice and repricing twice, once for each half, when one pass through the arithmetic would have done.

Step eight: what did the run fail to answer?

The closing step writes down, as outputs, every question the run met and could not answer. Not as an apology at the foot of the note. As numbered lines sitting alongside the findings, in the same typeface, with the same weight.

The list this particular procedure produces is fairly predictable, and predictability is a good sign rather than a boring one. Whether the move was parallel, if step three could not establish it. What a move that was not parallel would have done, if no map of separate sensitivities exists. Which of the two parts of a corporate yield moved, if the government rate for the same date was never observed on its own. And anything at all about what happens next. The procedure does not ask that question and cannot answer it.

A run that returns seven findings when its inputs supported four looks more complete than one that returns four findings and three refusals, and it is the one nobody can audit. The reader of the fuller note has no way to tell which of the seven was worked and which was filled in, so the three weak ones quietly contaminate the four good ones. The reader of the shorter note knows exactly what they have. Auditability is the whole argument for the closing step, and it is why the refusals are written as outputs rather than left as omissions.

What do the two runs look like when they are put side by side?

The procedure has now been described. Here it is run twice, end to end, on the two invented cases this material holds, and the second run is deliberately shorter because its inputs are thinner.

Run one, the ten year bullet bond

Step one. The rate that moved is the yield to maturity on a bond carrying no credit element at all, so there is no spread sitting on top of a government rate and nothing to separate. An empty field and a field that says nothing applies read very differently a month later, so the run records the absence explicitly.

Step two. The yield starts the day at 8.50 per cent a year and finishes it at 10.50, a difference of 200 basis points, or 2.00 percentage points, annual compounding on both sides. The check runs: 200 divided by 100 is 2.00, and 10.50 less 8.50 is 2.00.

Step three. The ten year bullet bond is a single instrument valued at one yield to maturityThe single rate that makes a bond's dated amounts add back to its price. One number standing in for a whole schedule., so whether a curve moved in parallel does not arise. The verdict recorded is not applicable, a third thing again from parallel and from not established.

Step four. The MODIFIED duration is 6.5613, so 6.5613 multiplied by 2.00 is 13.1226 per cent of the Rs 1,000.00/- price, or Rs 131.2260/-, giving an estimated price of Rs 868.7740/-. Labelled an estimate from the slope alone.

Step five. Repricing the ten dated amounts one by one, at a new rate of 10.50 and on annual compounding, gives Rs 879.7045/-, a fall of 12.0295 per cent measured on the same Rs 1,000.00/- base.

Step six. The estimate said Rs 868.7740/- and the repricing says Rs 879.7045/-, so the error is Rs 10.9305/-, or 1.0931 percentage points of the Rs 1,000.00/- base. The estimate reached further than the repricing on a rise in the yield, so the sign is the expected one.

Step seven. Not applicable. One bond with nothing to measure it against has no active part, and the run says so instead of skipping the line.

Step eight. Two refusals. The run has said nothing about what happens next, and nothing about what a move that was not parallel would have done.

Run two, the Rs 5,000 crore holding

Step one. The rate that moved is the government curve.

Step two. 100 basis points, or 1.00 percentage point, annual compounding.

Step three. Recorded as an assumption. No observation of the shape of the curve on the day exists in this material, and no scenario for a move that was not parallel, so every figure below carries the word PARALLEL and the word ASSUMED beside it.

Step four. The holding's MODIFIED duration is 5.20, so 5.20 multiplied by 1.00 gives 5.20 per cent, and on a base of Rs 5,000 crore that comes to Rs 260.00 crore.

Step five. Not available. No schedule of dated amounts exists holding by holding, and a full repricing cannot be performed on a sensitivity figure alone. The cell is left empty with that reason inside it.

Step six. Therefore also not available. With no repricing there is nothing to subtract the estimate from, so the Rs 260.00 crore stands as an unchecked estimate from the slope alone, known to overstate a loss on a rise in the yield and never verified here.

Step seven. The active part is the 0.40 by which the holding's 5.20 exceeds the benchmark's 4.80, or 0.40 per cent worked on that same base of Rs 5,000 crore, coming to Rs 20.00 crore. The whole is Rs 260.00 crore. The two figures differ by thirteen times.

Step eight. Four refusals. The parallel verdict was assumed rather than observed. No repricing was possible. No error figure could be produced. Nothing has been said about a curve that twists rather than shifts.

Put the two runs beside each other and the finding they produce together is the one worth carrying away. Run one carries figures at four of its eight steps and refuses nothing; run two carries figures at three, refuses two outright, and closes with four recorded refusals, and the second output is what an honest run looks like when the inputs are thinner. Nobody was lazier on the second run. The material simply held less, and the run said so in the places where it held less rather than in a footnote.

Who actually runs this, and what do they write down?

Three kinds of reader use this order, and they stop at different steps.

A treasury analyst inside a lender runs all eight and stops nowhere. Their whole holding of government paper sits on a balance sheet that has to report, so the estimate at step four is the morning number, the repricing at step five is the afternoon number, the error at step six is what tells them whether the morning number was good enough to keep using next week, and the split at step seven is what separates the exposure the balance sheet was always carrying from whatever the desk added to it. The refusals at step eight are what goes into the covering note, and they are read.

A research analyst covering a borrower usually stops at step one, and that is the correct place to stop. A government rate move says nothing about the borrower and a spread move says a great deal, so their question is which of the two parts of the yield moved. If step one cannot separate the two, the honest output is a refusal, and a refusal that arrives on the day is worth considerably more than an attribution that arrives on the day and turns out to be guessed.

A household runs a version of this without any of the vocabulary. Somebody with a fixed deposit ladder and a home loan hears that rates have moved and reaches, correctly, for step one before step four: which rate, the deposit rate or the loan rate? The deposit rate and the loan rate move a household's position in opposite directions. Then step two, in plain terms, is asking by how much and from what, rather than accepting the word higher. The steps are not a professional ritual; they are what anybody does who wants their answer to survive being questioned. The professional version only adds the recording format, and that format exists so a second person can check the first person's work without redoing it.