The Credit Curve: One Point, and What a Shape Would Say

A credit curve puts maturity along the bottom and one borrower's credit spread up the side, so every mark on it reads as a spread in basis points at a single moment. Palash Cements Limited has one bond on this platform. The bond runs five years, and it prices 220 basis points over the matching government SPOT rate. One bond at one maturity gives one mark. The rest of both axes stays blank on purpose.

What sits on each axis of a credit curve?

Start with the picture. The picture is where the confusion begins. A credit curveThe credit spread of one borrower plotted against maturity at one moment in time. One borrower, one moment, several maturities. is a chart with two axes, and it looks almost exactly like a government yield curve. Same sweep, same left to right reading, same habit of bending near the short end. The resemblance between the two charts is the whole trouble. Two pictures that share a shape get read as one object, and they are not one object.

Take the axes one at a time. Across the bottom runs maturityThe date the last payment on a bond falls due. Measured here in years from today, so a five year bond sits at the mark labelled five., measured in years from today. Up the side runs the credit spreadA borrower yield less the government SPOT rate for the identical maturity, stated as a rate a year and written in percentage points and in basis points., measured in basis points. And the quiet word doing the heavy lifting is the word one. One borrower. One moment in time. A different borrower gives a different chart. The next day gives a different chart again. A credit curve is a photograph of a single borrower taken at a single instant, and every mark on it belongs to that borrower and that instant and to nothing else.

Now set the government yield curve beside it and note what changes. A government yield curve puts a yield on the vertical axis: at each maturity it gives the rate the government pays for money of that length. On the invented curve used throughout this sequence, that is 6.90 per cent a year at five years and 7.35 per cent a year at ten years, on annual compounding throughout. A credit curve puts something else entirely on that vertical axis. The credit curve puts the gap there. At each maturity it gives the borrower's yield less the government SPOT rateThe price of money handed over now and returned at one named date in the future. One rate for each date, and no reinvestment assumed in between. of the same length, and nothing else survives the subtraction.

Here is the consequence, and it catches almost everybody the first time. The government SPOT rate at each maturity has already been taken out of every mark on the credit curve before it was plotted, so a credit curve can lie perfectly flat while the government curve behind it climbs steeply. Compute the climb so it is not a claim on trust: on the invented curve, ten years reads 7.35 per cent a year and two years reads 6.25 per cent a year, and 7.35 less 6.25 is 1.10 percentage points, which is 110 basis points of rise across that stretch. A borrower charging the identical spread at every one of those maturities would draw a flat credit curve straight through a government curve that rose by 110 basis points underneath it. The two pictures disagree because they are not measuring the same thing.

Why does one issuer at one maturity not give a curve?

Because a curve is a shape, and a shape needs more than one mark to exist. The rule sounds obvious written down, and it is routinely ignored in practice.

Palash Cements Limited, invented for this sequence and the only issuer anywhere on this platform, has issued exactly one bond: five years, a 9.10 per cent annual coupon on Rs 1,000.00/- of face, annual compounding. The five year government SPOT rate on the invented curve reads 6.90 per cent a year. The subtraction runs: 9.10 less 6.90 leaves 2.20 percentage points, and the same quantity in the other unit is 220 basis points. The subtraction gives a location, and a location is a pair of coordinates. Five years across. Two hundred and twenty basis points up. One mark on a chart.

One mark is not a curve, and the honest treatment of the rest of the chart is to leave it alone. There is no second maturity for this borrower to plot. There is no second borrower to plot beside it. So the axes stretch away in both directions carrying nothing, and the reason for the emptiness is printed inside the picture rather than left as blank space a reader might quietly fill in.

The same evidence anywhere else in life would be treated with more care. A shopkeeper names the price of a five kilogram sack of rice today. The shopkeeper has named one price for one size on one day. The price of the five kilogram sack says nothing about what a one kilogram packet costs, nothing about what a fifty kilogram sack costs, and certainly nothing about whether the price per kilogram rises or falls with the size of the sack. Somebody who draws a line through that single price and announces a pricing policy has invented the other end of that line. A single observation fixes a position and says nothing about a direction, and drawing a line through one mark publishes an opinion wearing the costume of a measurement.

There is a second absence on this chart. Palash Cements Limited carries no credit rating anywhere on this platform. A rating scale, and the definition attached to every step of it, belongs to the rating agencies and to the Securities and Exchange Board of India (SEBI) at sebi.gov.in, and those definitions are revised. The row where a rating would sit is therefore left empty with its reason printed inside it, and every sentence here holds without one.

The same discipline applies to the evidence behind the numbers. There is no measured default frequency on this platform, no recovery study and no series of past spreads. Arithmetic performed openly on invented inputs is not evidence about how often anything has actually happened.

What would the shape say if one existed?

Everything in this part is written in the conditional, from the first word to the last, and the conditional is not politeness. No shape exists on this platform for this borrower. The three readings below are worth knowing because they turn up in other people's charts, and knowing them is exactly what stops a reader attaching one of them to a borrower that has produced only a single mark.

If a borrower's spread were wider at ten years than at five, the extra would be payment for the five additional years of exposure to whatever might go wrong inside them. That is the reading most people expect, and it has a plain intuition behind it. A neighbour who asks to borrow for six months and a neighbour who asks to borrow for six years are asking for two different things, and the longer request carries more of everything that can change: their job, their household, the trade they work in, their health. Charging more for the longer request is charging for a longer stretch of not knowing.

If a borrower's spread were narrower at ten years than at five, the price would be saying that the near years are the dangerous ones and that a borrower who survives them is through the worst of it. This reading surprises people until they meet the everyday version. A shop that has just opened is fragile in a way that a shop of ten years standing is not. Anybody asked to lend to a stall that opened last month would want the most protection over the first stretch, not the last. A borrower with a heavy repayment falling due soon, or a business waiting on one contract to be renewed, sits in that shape: get past the near danger and what remains looks calmer.

If a borrower's spread were flat across maturities, the price would be saying each additional year carries the same charge as the year before it. No accumulation of extra worry with length, no relief from surviving the early stretch, just the same charge repeated.

All three readings are legitimate and all three appear in real charts. None of the three belongs to this borrower, and none of them appears on this platform. The picture below draws all three faintly through the single real mark for one reason only: to show how comfortably a lone mark accommodates every one of them.

Why is a spread a rate a year rather than a total?

One word decides how a spread is read, and the word is year.

Two hundred and twenty basis points is not a charge for the five years taken together. The spread is a charge for one year, applied in each of five years. Palash Cements Limited pays a 9.10 per cent annual coupon on Rs 1,000.00/- of face and the reference borrower pays 6.90 per cent a year across the identical stretch, and both of those are rates a year. Subtract two rates a year and a rate a year comes out. Nothing in the subtraction converted the answer into a total, and nothing about the bond running five years divides the answer by five.

The same quantity is handled correctly in any rental agreement. A shop taken on rent at Rs 40,000/- a month for three years costs Rs 40,000/- in each of thirty six months. Nobody reads the agreement and concludes the shop costs Rs 40,000/- for the whole three years. But a rate wearing a percentage sign loses its period the moment somebody writes it at speed, and the sentence "the spread on this bond is 220 basis points over five years" is exactly that loss. A rate stripped of its period turns into a different quantity without anybody noticing the substitution.

The same period travels through everything the spread implies. The relationship a spread claims to satisfy was settled earlier in this sequence. Run it and watch the period travel.

One quantity is known and two are not, so an assumption has to be supplied before the arithmetic can move. Assume a recovery of 40 per cent of the amount owed. The recovery assumption rests on nothing measured, and every figure after it moves when the assumption moves.

At an assumed recovery of 40 per cent of the amount owed, the loss given default is 100 less 40, so 60 per cent of that same amount owed, or 0.60 as a decimal share. Two of the three quantities are now in hand, so the third falls out by division.

A relationship shown in one direction is only a recipe. Run it back the other way as a check. Take 3.6667 per cent a year and multiply by 0.60. Back comes 2.2000 percentage points, the 220 basis points the whole thing began with. The arithmetic closes on its own starting figure, and every number inside it wears the words a year.

Each of the four quantities carries a different base and a different period, and the multiplication joining them means nothing until each one is labelled. Hold them apart before going further.

How does an annual default rate accumulate across five years?

By multiplying what remains, not by adding what happens.

One assumption travels with every figure from here onwards, so state it out loud. Suppose the implied annual default rate of 3.6667 per cent a year held in each of the five years. Nothing on this platform supports that supposition. Holding the rate constant is itself an assumption, and the accumulation cannot be shown without one.

Now think about what survives a year rather than what fails in it. If the chance of failing inside any one year is 3.6667 per cent, then the chance of survivalReaching the end of a stated period without having failed to pay. Survival is the complement of failing inside that period, and the two sum to one hundred per cent of the same base. through that year is 100 less 3.6667, which is 96.3333 per cent. To reach the end of two years the borrower must survive the first year and then survive the second, and two things that both have to happen are joined by multiplication rather than by addition.

Repeated multiplication of this kind is compoundingApplying a rate repeatedly to whatever is left after the previous application, rather than to the original amount every time. working on a survival share instead of on money, and it behaves exactly the way compounding always behaves: each step operates on what is left after the step before it, never on the original. Apply it five times and 0.963333 recurring raised to the fifth power gives 82.962712 per cent as the five year survival share.

The quantity a reader actually asked for is the other one, and it is now a subtraction rather than a sum.

Put the five year survival share of 82.962712 per cent into that and the accumulated default chanceThe chance of failing at some point across a stated number of years, as opposed to the chance of failing inside any single one of them. across five years is 100 less 82.962712, and the answer is 17.037288 per cent. Survival multiplies down the chain, and the accumulated chance of failing is whatever that chain leaves behind. The second figure therefore has to be computed from the first rather than assembled on its own.

Why does adding the annual rate five times give the wrong answer?

Because the addition charges the borrower for years the borrower could not reach.

The natural move comes first, along with what it produces. Multiplying 3.6667 per cent a year by five years gives 18.3333 per cent. Set beside the compounded 17.037288 per cent, the addition overstates by 1.2960 percentage points. A little over a point does not sound like much, and the size of it is not the interesting part. The reason is.

Here is the reason in one sentence. A borrower that failed in year two cannot fail again in year three, and the addition charges year three regardless. Adding the annual rate five times quietly assumes that a borrower is available to fail in every one of the five years, which is a statement about the world that nobody would put their name to if it were written out in full. Each step in the chain operates only on the share that is still standing, so compounding survival never makes that assumption.

The everyday version is blunter and it lands faster. A stall that shuts down in March cannot also shut down in August. Counting both closures counts a stall shutting twice, and there was only ever one stall. Anybody would spot the error told that way; almost nobody spots it when it is dressed as a multiplication.

Compute that at every year and the gap does not sit still. The gap runs 0.0000 percentage points at one year, then 0.1344, then 0.3984, then 0.7871, and 1.2960 at five years. The gap widens at every single step, so the same habit applied to a longer bond produces a larger error rather than the same one. That is what makes the mistake structural rather than a rounding annoyance: it is not a fixed slip that can be learned and allowed for, it is a slip that grows with exactly the quantity being measured.

How honest is the number the accumulation was built on?

Three limits sit on the implied annual default rate of 3.6667 per cent a year, and every one of them travels wherever that figure travels. A limit kept in a note is a limit somebody will quote around.

First, the 40 per cent recovery is an assumption and nothing anywhere on this platform supports it. No recovery study was read to produce it. Watch what happens when the price is held perfectly still and only the assumption is moved.

Same price, four answers, and no price moved by so much as a paisa between the rows. The assumption is doing that much of the work, and it follows that the accumulation built on top of it moves too. Every accumulated figure in this guide is the accumulation of the middle row and of nothing else.

Second, the whole spread has been treated as compensation for credit, and in a real market some part of a spread pays for the difficulty of selling a bond at the moment a holder wants to sell. Every basis point of that read as credit makes the implied default rate too high. Split 0.40 percentage points off the 2.20 and treat them as the price of something that is not default: 1.80 points of credit remain, and 1.80 divided by 0.60 gives 3.0000 per cent a year rather than 3.6667. There is no way on this platform to separate the two, and saying so is better than working around it. The error runs one way only. The entire spread was divided as though every basis point of it were credit, so the answer can only come out too high.

The two limits stack rather than cancel. Change the recovery assumption and the answer moves; carve a slice off the spread and it moves again; do both and it moves twice. The accumulated five year figure inherits every one of those movements, and stands as a consequence of two stated assumptions rather than as a property of a company.

Third, an impliedSolved backwards out of a price under a stated assumption, rather than counted from what has happened or projected forward into what will. default rate is what the price says. It is not a forecast and it is not a measured frequency of anything. Nobody counted defaults to produce 3.67 per cent a year. The rate was solved backwards out of one spread and one assumption, and a reader who carries it away as the probability that Palash Cements Limited fails has misread the arithmetic that produced it.

What may a reader honestly say about the empty axes?

Nothing, and the carefulness is the teaching rather than an apology for it.

There is no bond for this borrower at any maturity other than five years, so there is no second mark. A single mark admits a line through it running in any direction at all, and drawing one would be publishing an opinion and calling it a curve. The list of what a reader may take away instead is longer than it first looks, and it is worth stating precisely.

The axes can be taken away: what a credit curve puts on each one and why that differs from what a government yield curve puts there. The single mark and the subtraction that produced it can be taken: 9.10 less 6.90 gives 2.20 percentage points, or 220 basis points, at five years. The three readings a shape would carry if a shape existed can be taken, held firmly in the conditional. And the accumulation arithmetic can be taken whole. One mark is all it needs, and it would work identically on any other borrower whose price was available.

The emptiness of these axes is the finding. A record that contains one issuer at one maturity supports exactly one mark, and the discipline that stops a writer filling in the rest is the same discipline that stops an analyst filling in a curve between two prices somebody actually dealt at. The habit is identical; only the scale differs.