Credit Spread: The Subtraction and What It Pays For

A credit spread is what a borrower pays above the government SPOT rate at the identical maturity, and it is stated as a rate a year. Palash Cements Limited issues five year debt at 9.10 per cent a year while the matching five year government SPOT rate reads 6.90 per cent a year. Subtract: 2.20 percentage points, or 220 basis points. Both inputs need one maturity and one compounding convention.

Every number the instrument opens with also stands as ordinary text here, so a reader who never touches a field still has the whole worked case. 9.10 less 6.90 gives 2.20 percentage points, or 220 basis points. The five payments discount to Rs 1,000.000000/- at 9.10 per cent a year and to Rs 1,090.446383/- at 6.90 per cent a year, a gap of Rs 90.4464/- and 9.0446 per cent of the Rs 1,000.00/- face amount. And 2.20 over 0.60 gives 3.6667 per cent a year, with 3.6667 times 0.60 giving 2.2000 percentage points back.

The subtraction is the whole computation. Two numbers in, one number out, and it fits on the back of a bus ticket. Which raises the honest question straight away: if the arithmetic is one line, what is the rest of this guide for?

The rest is for everything the subtraction quietly assumes. The difference between two rates only means something about a borrower when both rates are the price of the same thing over the same period, and almost every wrong answer on this material comes from breaking that condition rather than from botching the arithmetic. Take away the part the two rates share, the cost of five years of money to anybody, and what is left is what is different about the borrower. Take away a slightly different part, say ten years of money instead of five, and what is left is a mixture of two things wearing one label.

So the rest of this guide is the conditions, in the order the instrument applies them, and it ends where a credit spread stops being able to tell anybody anything.

Where do the two numbers come from, and in what form?

Input one is the borrower's yield a year, on annual compounding, for a stated maturity. For Palash Cements Limited, invented, the five year bond carries a 9.10 per cent annual couponThe payment a bond makes each period, fixed at issue as a rate on the face amount, and unmoved by anything afterwards. on Rs 1,000.00/- of face and was issued at parPriced at the face amount, which forces the coupon rate and the yield to be the same number., at Rs 1,000.00/-. A price equal to the face amount is what makes the coupon rate and the yield the same number here. So the yield is 9.10 per cent a year without anything being solved. Most bonds encountered in practice are not issued at par, so the yield has to be solved out of a price. The field note is the same either way: what the instrument wants is the yield, not the coupon.

Input two is the government SPOT rateThe rate for money placed today and returned at one stated future date. a year for the same maturity, on the same compounding convention. On the invented curve used throughout this sequence, the five year government SPOT rate is 6.90 per cent a year. The whole point of the subtraction is that a specific node was taken off a specific curve at a specific maturity, so the rate is written as the five year government SPOT rate every time, never as the benchmark on its own and never with any other name.

Consider a household comparing two loan offers for a scooter. One quote is for three years and the other for five. Part of the difference between the two quotes is simply the difference between three years of money and five years of money. Which lender is charging more for the risk cannot be settled until both are put on the same term. The instrument has exactly that problem. A reader who never sees the wrong answer never learns to recognise it, so the instrument does not hide the mismatch by refusing to run. Instead it computes what was asked for, marks it as measured against the wrong reference, and prints the two parts the answer is made of.

What exactly gets subtracted, and on which convention?

Now the arithmetic, all one line of it. Nine point one zero less six point nine zero is two point two zero. The subtraction is trivial and the conditions on it are not. A tool for this job therefore spends most of its effort checking the inputs and almost none of it computing.

The compounding convention belongs inside the arithmetic and not in a note underneath it. Every price in this guide is struck on annual compoundingOne discounting period a year, so an amount is divided once by one plus the rate for each year that passes.: one discounting period a year, so an amount due in three years is divided by 1.0910 three times over at a 9.10 per cent annual rate. The convention is not housekeeping. Palash Cements Limited's five payments discount to Rs 1,000.000000/- exactly on that convention, and being issued at par means exactly that. The same coupon, the same maturity and the same yield stated on a semi-annual convention would give a different price from figures that look identical in print. With the convention written beside the price a reader can reproduce the sum. Left out, the sum cannot be reproduced, whatever else has been stated.

Why does one spread have two numbers, 2.20 and 220?

Because it is one quantity written in two units, and this is where the errors on this material actually live. A basis pointOne hundredth of a percentage point, so 2.20 percentage points is 220 basis points and 0.50 percentage points is 50 basis points. is one hundredth of a percentage point. A percentage point figure multiplied by 100 gives basis points; a basis point figure divided by 100 gives percentage points. There is no third quantity hiding in there.

State both units the first time a spread appears and never let a percentage point figure sit in a sentence beside a basis point figure without both being labelled. A spread written as a bare 2.2 in one paragraph and a bare 220 in the next has produced two numbers where there is one quantity, and the reader who has to reconcile them will assume somebody made a mistake. There is a second discipline riding along with it: a spread is never written as a bare number at all. A spread is always over something and always for a stated maturity. Two hundred and twenty basis points here means 220 basis points over the five year government SPOT rate, for five years. Strip either of those away and the figure stops being a measurement.

What is 2.20 percentage points worth as a price?

A rate is an abstraction. A price is a number of rupees actually paid or not paid, and this is the form in which the size of a spread lands. A guess is worth making before the arithmetic.

The method is a comparison of two discountings of the same cash flows. Palash Cements Limited's bond pays Rs 91.00/- at the end of each of the first four years and Rs 1,091.00/- at the end of the fifth. The final payment is the last coupon plus the face amount. Discounting those five payments at the borrower's own 9.10 per cent a year gives Rs 1,000.000000/-, the price, and a price equal to the face amount is what at par means. Discounting the very same five payments at the 6.90 per cent five year government SPOT rate gives Rs 1,090.446383/-.

A spread of 2.20 in rate terms is applied to every payment for five years rather than to one of them, so it is worth just over nine per cent of the face amount in price terms. The difference is Rs 90.446383/- on Rs 1,000.00/- of face, printed as Rs 90.4464/-, and that is 9.0446 per cent of the face amount. Set beside a rate that reads 2.20, the price figure is the one that tends to land.

A reader meeting this for the first time often multiplies instead: 2.20 percentage points of Rs 1,000.00/- is Rs 22.00/- a year, five years of that is Rs 110.00/-, so why does the tool print Rs 90.4464/-? Because the multiplication treats every year's Rs 22.00/- as worth the same today, and the discounting does not. Year one contributes Rs 1.716570/- of the gap and year five contributes Rs 75.681595/-, and once the five contributions are added at today's value they come to Rs 90.446383/- rather than Rs 110.00/-. The ladder below is the whole sum, laid out so it can be added by hand.

What does the spread imply about a default rate?

A spread is not a fee. A spread is a price for an expected loss, and an expected loss is a rate of default multiplied by what is actually lost when a default happens. The loss and the default rate give a small triangle of three quantities, and any two of them solve for the third. The subtraction has already produced the spread, and the recovery rate arrives as an assumption supplied by the analyst. The third corner is the default rate the price implies.

The relationship runs forwards on the figures above. At an ASSUMED recovery of 40 per cent of the amount owed, the loss given defaultOne hundred per cent less the recovery rate, on the same base, so a 40 per cent recovery means a 60 per cent loss given default. is 100 less 40: 60 per cent of that same amount owed, or 0.60 as a decimal. Divide the spread by it: 2.20 over 0.60 is 3.6667 per cent a year, and it reads as 3.67 per cent a year.

A relationship shown one way is a formula and a relationship shown both ways is understanding, so now run it backwards. The backwards direction is not optional. Multiply the 3.6667 per cent a year by the same 0.60 loss given default and the answer is 2.2000 percentage points, or 220 basis points. The spread the subtraction began with has come back, and the triangle closes on itself.

One warning about carrying the decimals. A reader who checks this with a calculator will otherwise think a mistake has been made. Two point two zero divided by zero point six zero is exactly three and two thirds per cent. The tool carries all of it inside the multiplication and prints 3.67 per cent only as a reading. The printed 3.67 multiplied back by 0.60 lands on 2.2020 percentage points rather than 2.2000, and the two ten-thousandths are the rounding, not an error in the relationship.

What moves a spread when the borrower has not moved?

The curve movement slider on the calculator above changes which numbers move. The yield entered does not move. The build-up shows the government limb sliding and the spread absorbing every point of the slide, and the movement line underneath the build-up says so in as many words.

Move the curve down 0.40 percentage points and the five year node reads 6.50 per cent a year instead of 6.90. Nothing was typed into the yield field, so Palash Cements Limited is still yielding 9.10 per cent a year. The spread now reads 2.60 percentage points, or 260 basis points, and its price value climbs from Rs 90.4464/- to Rs 108.0477/- on Rs 1,000.00/- of face. The whole of the widening came out of the limb that has nothing to do with the borrower, so forty basis points arrived without one thing about the borrower changing.

Push the slider the other way and the effect reverses without becoming any more honest. At a five year node of 7.30 per cent a year the spread reads 1.80 percentage points, 180 basis points, and the borrower looks safer than it did an hour ago while having done nothing at all. A reader watching only the spread will report a tightening. A reader watching both limbs will report that the government rate rose.

The case that reads correctly is the last row below, where both limbs move together. If the borrower's yield falls to 8.70 per cent a year while the five year node falls to 6.50, the spread stays at 2.20 percentage points, and that unchanged figure is the true statement: five years of money got cheaper for everybody, and this borrower is charged exactly what it was charged before.

So the question to ask of a spread that has moved is not how far it moved. The question is which of the two limbs moved, and the only way to answer that is to have written both of them down the last time the spread was recorded. A spread recorded on its own, as a bare 220, cannot be taken apart afterwards.

Which spread measure is this one, and what is it not?

This tool computes a G-spreadThe spread measured against a single government SPOT rate at the matching maturity, rather than against the whole curve.: one yield less one government SPOT rate at one matching maturity, and nothing else. The subtraction reaches neither a Z-spread nor an option-adjusted spread. Both of those need the whole SPOT curve rather than a single node, and both are covered separately.

The difference is not academic. On this very bond, computed against the whole invented SPOT curve rather than against the five year node alone, the Z-spread comes to 227.1722 basis points against this tool's 220.00 basis points. The gap of 7.1722 basis points belongs to the two measures rather than to the bond. Nothing about the bond changed between the two figures. An option-adjusted spread does not appear at all here, and the cell for it is drawn empty on purpose: this record holds no bond carrying an option, so there is nothing to compute and no figure will be invented to fill the space.

The practical rule that follows is short. An answer from here should never be compared against a spread built another way. A spread whose measure is unknown is not yet a number that can be set beside this one.

What happens when the two maturities do not match?

The arithmetic of the mistake shows exactly what the wrong answer is made of, and it is worth seeing. Two hundred and twenty basis points less 175 basis points is 45 basis points. And 45 basis points is exactly 7.35 less 6.90, the rise in the government SPOT curve between five years and ten years. The mismatched answer is not noise: it is the true spread less the slope of the government curve over the stretch accidentally borrowed, and nothing on the face of it says so.

Who actually runs this subtraction, and what for?

The household version comes first. Everyone has met it. A neighbour asks to borrow Rs 50,000/- for two years and offers to pay more than a two year deposit would pay. The extra is not a fee for the paperwork and it is not generosity. The extra is the price of the chance that the money does not all come back, and the size of it depends on how much would be expected back if things went badly. A spread is that same idea at kitchen table scale, and everything in this guide is the same idea with the arithmetic tightened.

A lender runs the subtraction because it is the only way to see what it is charging for risk as distinct from what it is charging for time. A five year loan at 9.10 per cent a year sounds like one decision. Split into a 6.90 per cent limb that is the cost of five years of money and a 2.20 percentage point limb that is everything else, it becomes two decisions, and only the second one is about the borrower. When the government SPOT curve shifts, the first limb moves and the lender's judgement of the borrower has not changed at all. Anyone reading a yield without splitting it will keep mistaking one for the other.

An analyst runs it to compare the same borrower with itself across dates, or two borrowers with each other on a matched maturity. Comparison needs something to compare against, and the second use is exactly where a single worked bond stops: no second issuer, no rating and no run of past spreads stands beside this one. Naming that absence is the honest move. A tool that cannot compare should say so rather than produce a lonely number and let the reader supply the comparison from memory.

A holder of the bond runs it because the price form is what actually shows up in a valuation. Rs 90.4464/- on Rs 1,000.00/- of face is 9.0446 per cent of the amount at stake, and at wedding scale, on Rs 20,00,000/- of face, the same spread is worth about Rs 1,80,893/- of price. The price figure is the number that makes people concentrate. The rate form is easier to say and the price form is easier to feel, and a good tool prints both.

What can this calculator not tell anybody?

A calculator that prints a number invites the reader to treat the number as a finding. So the refusals are printed beside the answer rather than under it, and there are four of them.

Whether 220 basis points is enough is a judgement, and a subtraction makes no judgements. How likely Palash Cements Limited is to fail stays out of reach as well. The impliedSolved backwards out of a price under a stated assumption, rather than measured or forecast. default rate came out of the price rather than out of any count of defaults. How much of the 220 basis points pays for default rather than for something else cannot be recovered either. One subtraction produces one number and never a split of it. And what the spread will do next would need a series of past spreads set beside this single day's figure.

The first limit: the recovery figure is an assumption

No document supports the 40 per cent. No recovery study was read for it and none exists in this guide, so the figure is written with the word ASSUMED attached every time it appears. Move it and the answer moves with it. The spread does not move at all. The whole of the point sits there: the assumption is doing as much work as the price is, and the price never changed.

Same price, four answers, and the spread column never budges. An implied default rate quoted without the recovery assumption beside it hands the reader one column of that table and throws the rest away.

The second limit: the whole spread is being read as payment for default

In a real market some part of a spread pays for something other than the chance of not being paid back, and the most familiar of those is the difficulty of selling the bond on demand. Every basis point of that read as credit makes the implied default rate too high. Suppose 0.40 percentage points of the 2.20 were payment for something other than default. Then 1.80 over 0.60 is 3.0000 per cent a year rather than 3.6667 per cent a year, a difference of two thirds of a percentage point on the implied rate from a split of 40 basis points on the spread. One subtraction cannot split the spread into those two parts.

The third limit: an implied default rate is what the price says

An implied default rate is not a forecast and not a measured frequency of anything. Nobody counted a default to produce 3.6667 per cent a year. The figure was solved backwards out of one spread and one assumption, on the invented figures in this guide. Presenting it as the probability that Palash Cements Limited fails is a misreading of the arithmetic that produced it, and it is a misreading that happens because the number looks so much like a probability when it is printed cleanly beside a company name.

Palash Cements Limited and the SPOT curve used throughout are invented.
Educational material. Not advice on any investment, tax, budget or market position.