Credit Risk and Interest Rate Risk: Two Limbs of One Yield

Interest rate risk is the risk that a bond's price falls because the rate used to discount its payments moved. Credit risk is the risk that this borrower does not pay what was promised. One corporate yield carries both. Palash Cements Limited's 9.10 per cent a year is the 6.90 per cent five year government SPOT rate plus a 2.20 percentage point spread, and only the spread limb can end in a missing payment.

A single number does two completely different jobs here. Palash Cements Limited, an invented issuer used throughout this sequence, borrows for five years at 9.10 per cent a year. The 9.10 per cent is not one thing. Part of it is what anybody pays for the use of five years of somebody else's money, and the rest of it is what this particular borrower pays on top because it is this particular borrower. The two parts move on different days, for different reasons, and one of them can end in a payment that simply never arrives while the other never can.

The clearest starting point is away from bonds entirely: renting a shop on a market street. The rent has a floor set by the street itself: what any shop of that size on that street costs, whoever takes it. On top of the floor, a landlord charges a particular tenant a bit more or a bit less depending on how the last tenancy went, whether the deposit was paid on time, whether the shutter was ever pulled down mid month. If the rent rises next year, the tenant has learned that the total rose. The tenant has not learned whether the street repriced or whether the landlord repriced that particular tenant. A street repricing and a landlord repricing are two different pieces of news that arrive wearing the same clothes, and telling them apart requires knowing what a comparable shop on the same street now costs.

What is interest rate risk, in one sentence?

Interest rate risk is the risk that a bond's price falls because the rate used to discount its payments has moved. No borrower appears anywhere in that definition, and the absence is the point. Who promised the money does not matter. The mechanism sits in the arithmetic of discounting rather than in anybody's willingness to pay, so a bond issued by the most reliable borrower imaginable carries this risk in exactly the same way as a bond issued by the least reliable one.

The payments on a fixed rate bond are fixed, so when the discounting rate moves, the only thing left in the sum that can move is the price. Palash Cements Limited's bond promises Rs 91.00/- at the end of each of five years and Rs 1,000.00/- of face amountThe amount printed on the bond that is repaid at maturity, and the base on which the coupon rate is applied. at the end of the fifth. The six amounts are written into the contract, and a contract does not consult the market. If the rate a buyer applies to them changes, the only free variable left is what the buyer will hand over today.

The arithmetic is easier to feel than to read, so here is the household version. Suppose a relative agreed two years ago to pay a lender Rs 10,000/- in five years' time, and suppose everybody around that lender is now able to place money at a better rate than they could then. The claim has not changed. The alternative uses of the same rupee got better, so the value of the claim in today's money fell. Nobody broke a promise. The promise simply became worth less to hold, and anybody offered the claim today would pay less for it.

Why every rate carries the word SPOT or the word FORWARD

Every rate below carries the word SPOT or the word FORWARD, without exception, and the reason the labelling has to be that absolute is worth understanding. A SPOT rateThe rate for money placed today and returned at one stated future date, with nothing happening in between. is the rate for money placed today and returned at one stated future date. The five year government SPOT rate of 6.90 per cent a year is therefore the reference every spread below is subtracted from: money placed today, returned in five years, with an annual compounding convention. A FORWARD rateThe rate for money placed at one future date and returned at a later one, obtained by arithmetic from two SPOT rates rather than from an opinion about the future. is the rate for money placed at one future date and returned at a later one. A FORWARD rate is not a separate opinion about what will happen. The FORWARD rate is already sitting inside the SPOT curve and can be pulled out of it by arithmetic.

Here is the pull, worked out on the invented curve behind this sequence. The one year SPOT rate is 5.90 per cent a year and the two year SPOT rate is 6.25 per cent a year, both on annual compounding. Two years of growth at the two year SPOT rate is 1.0625 multiplied by itself. The product is 1.12890625. One year of growth at the one year SPOT rate is 1.0590. Dividing the first by the second and subtracting one gives the one year FORWARD rate one year from now: 6.601157 per cent a year, or 6.6012 per cent a year to four decimal places.

Now put the FORWARD rate beside the three year SPOT rate on the same invented curve, 6.55 per cent a year. The gap is 6.6012 less 6.55, or 0.0512 percentage points, measured on the unrounded figure as 5.1157 basis points. Two numbers, five basis points apart, and completely different objects: one is what money placed today and returned in three years earns, and the other is what money placed in a year and returned a year later earns. Nothing separates those two numbers in writing except the word attached to each of them, so the label carries the entire distinction.

The five year government SPOT rate of 6.90 per cent a year is the one used from here on. The rate comes from the same invented curve, is stated on annual compounding, and is the only rate below that belongs to a borrower other than Palash Cements Limited.

What is credit risk, in one sentence?

Credit risk is the risk that this particular borrower does not pay what was promised, in full and on time. No rate appears anywhere in that definition, and again the absence is the point. Paying or not paying has nothing to do with what the government SPOT curve is doing on any given morning. A borrower can fail to pay in a period when every rate on the curve sat perfectly still, and a borrower can pay every rupee on the day it was due in a period when the whole curve moved a long way.

Read the definition slowly and notice that it has two halves. The second half is the one readers drop. In full is one condition. On time is a separate condition. A borrower who pays nothing has failed. A borrower who pays sixty paise in the rupee has failed. A borrower who eventually pays every rupee but pays it three years after it was due has also failed to do what was promised. The lender could not use that money in the meantime and had to find it somewhere else, so the delay cost something real. The three outcomes are not the same size and not the same shape, but they belong to one category: the promise was not kept as written.

Everyday version again. A cousin borrows Rs 50,000/- and says she will return it in a year. Three things can happen that are not the thing she said. She returns nothing. She returns Rs 30,000/-. She returns all of it, in the third year, after the lender has already borrowed elsewhere to cover the gap. The difference between those three is felt by a household lender and by a lender holding a bond alike, and the arithmetic of the loss is different in each case. None of the three has anything to do with what banks are paying on deposits this month, so none of them is an interest rate problem.

Two definitions established earlier come in here without being rebuilt. The recovery rateThe share of the amount owed that a lender actually gets back after a borrower fails to pay, stated on the same base as the amount owed. is the share of what was owed that a lender eventually gets back, and loss given defaultOne hundred per cent less the recovery rate, measured on the same base, so a 40 per cent recovery is a 60 per cent loss given default. is one hundred per cent less the recovery rate on the same base. Both quantities matter only where the spread limb is read as a price for an expected loss.

Whether Palash Cements Limited can actually pay is a separate question. Working a ratio, reading a covenant and ranking a claim are how that question gets answered, and each is set out under credit analysis. What the two risks are and how they differ has to be settled first.

How does one corporate yield split into the two?

Palash Cements Limited's five year bond carries a 9.10 per cent annual coupon 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.. Issuing at par forces the yield to be the same 9.10 per cent a year. Every price below is struck on annual compounding, meaning one discounting period a year, so an amount five years away is divided by 1.0910 five times over at a 9.10 per cent annual rate. The compounding convention is not housekeeping. The same coupon, the same maturity and the same yield stated on a semi annual convention give a different price from figures that look identical in print, so the convention travels with the number.

The government SPOT rate for the same five years, on the same annual compounding convention, is 6.90 per cent a year. The split is now one subtraction.

Turn the relationship round and the spread is what falls out, and falling out of a subtraction is how a spread is actually obtained. Nobody announces a spread; two prices are observed and the difference between the yields is taken.

The split is a subtraction between two observable prices and not a judgement anybody made. Nobody sat down and decided that 6.90 points of the yield should be the price of time and 2.20 points should be the price of this borrower. Two instruments were priced, two yields were solved out of those prices, and the difference is what it is. The subtraction is therefore reliable in a way that an opinion is not, and it is also silent, on its own, about what the 2.20 is compensation for. Having the number and knowing what it means are two separate things.

The units rule, because this is where mixing them is commonest

A percentage point and 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. are the same quantity in two units. One basis point is one hundredth of a percentage point. So 2.20 percentage points is 220 basis points, and 0.30 percentage points is 30 basis points, and the two are never swapped mid sentence. A spread written in both units the first time it appears leaves a reader carrying the conversion rather than guessing at it.

And no spread is ever written as a bare number. A spread is always over something and always for a stated maturity. The figure here is 220 basis points over the five year government SPOT rate, for five years. Strip either qualifier off and the number stops meaning anything: 220 basis points over what, and for how long, are not decorative questions.

What is the split worth in price rather than in points?

Two decimal places in a rate can look like a rounding error to somebody who has never converted one into money. It is not. The spread is applied to every payment in the bond, once for each year those payments are away, and the effect compounds. The cleanest way to see the size of it is to price the same six promised amounts twice: once at Palash Cements Limited's own yield, and once as though those amounts were being discounted at the government SPOT rate instead.

Put 9.10 per cent a year into that sum and the answer is Rs 1,000.000000/-. An answer of exactly face is what at par means, and it is the check that the coupon and the yield really are the same number. Put 6.90 per cent a year into the same sum, with the identical six amounts, and the answer is Rs 1,090.446383/-. The gap between the two is Rs 90.4464/- on Rs 1,000.00/- of face, or 9.0446 per cent of the face amount. Ninety rupees of price is what the 2.20 percentage point spread is worth today.

The difference column, read downwards, shows the compounding at work. A 2.20 point spread applied for one year does not do much, so the first coupon is only Rs 1.716570/- cheaper. The same spread applied for five years to the largest amount in the bond does a great deal, and the last row is Rs 75.681595/- cheaper. Four fifths of the whole gap sits in that final row. A reader who has been told that a spread is worth two decimal places and then sees ninety rupees of price has learned something about how discounting scales that no rate on its own can show.

Which of the two can end with a payment that never arrives?

People reach for size when they compare these two, or for how jumpy each one is. Neither size nor jumpiness separates them, and no figure above settles either comparison. The separation is one of kind, and it survives however large or small either limb happens to be on a given day.

Interest rate risk moves a price. If the bond is then held to maturity and the borrower pays, every rupee that was promised arrives anyway. The holder of Palash Cements Limited's bond who sat through a rise in the government SPOT rate and did not sell still receives Rs 91.00/- at the end of each of five years and Rs 1,000.00/- at the end of the fifth. The price moved in the middle. The promise did not.

Credit risk can end with a payment that never arrives at all. If Palash Cements Limited stops paying, no move in any rate anywhere brings the missing amount back. The arithmetic was never the problem, so no path through the arithmetic recovers it. A fall in the government SPOT rate does not refill a coupon that was not paid.

Interest rate risk changes what a promise is worth today. Credit risk changes whether the promise is kept at all. An account that presents the two as flavours of one thing has hidden the only separation that survives. Everything else about them is a matter of degree. This is not.

The sentence about holding to maturity can be over read, so one honest qualifier attaches to it immediately. Saying that a holder to maturity is untouched by interest rate risk does not mean the price move cost nothing at all. A holder who has to sell early, or who has to show a carrying price to somebody, or who could have placed the same money at a better rate for those years, has met a real consequence. The true claim is narrower and still worth having: the promised amounts still arrive in full. The claim is about the payments, not about every consequence a price move can have for a holder.

How many borrowers does each one touch at once?

Move from one bond to a pile of them and the two risks stop resembling each other very quickly, even though on a single bond they can look like two similar shaped things sitting side by side in one yield.

A move in the government SPOT curve reaches every bond priced off it in the same instant. A curve move does not knock on doors one at a time. If the five year government SPOT rate moves, the five year point of every borrower's pricing moves with it, whatever those borrowers have been doing. A change in what lenders think of Palash Cements Limited reaches Palash Cements Limited. The news reaches nobody else, and no mechanism in the split would carry it further.

A holding of fifty different borrowers has fifty separate credit stories and one rate story. However alike the two risks look on a single bond, they stop behaving alike as soon as holdings are added together. Fifty independent things that can each go wrong on their own schedule behave completely differently in aggregate from one thing that moves everything at once.

The everyday version is a street of shops. If the municipality raises the ground rent for the whole lane, every shopkeeper in the lane pays more on the same morning, and no shopkeeper's own conduct had anything to do with it. If one shopkeeper's supplier stops delivering, that shop suffers and the one next door does not notice. A person who has lent small amounts to ten shopkeepers in that lane is carrying ten separate stories of the second kind and one shared story of the first kind, and they are not the same problem even though both can reduce what comes back.

Can a bond carry one and not the other?

Yes, in one direction, and no in the other, and the asymmetry is worth setting out precisely because it is easy to state carelessly.

The direction that works comes first. A government bond in its own currency is the reference that a spread is measured from. Its yield IS the government SPOT rate for that maturity, so subtracting the government SPOT rate from it leaves nothing over. The government bond therefore carries the rate limb and no spread limb. Treating one borrower as the reference is a convention that gives a spread something to be measured against, and it is not a claim that any government anywhere is certain to pay. How such an exposure is treated for capital is set by the Reserve Bank of India at rbi.org.in.

Now the direction that does not work. No bond escapes the rate limb. A borrower cannot be so reliable that time stops costing anything. Reliability is what removes a spread; it has no effect at all on the fact that money placed for five years has to be discounted for five years. The rate limb is a property of the calendar and the curve, not of the borrower, so no bond anywhere carries a spread limb without a rate limb under it.

The shop rent shows it again. A landlord might waive the extra he charges a tenant who has been trouble, so the tenant pays exactly the street rate. No landlord waives the street rate itself. The street rate is not about the tenant at all, and it stands whether the shop is occupied by a saint or a stranger.

Why does the same price fall have two possible causes?

Here is the situation almost every reader is actually in. One corporate bond is held. One number is looked at. The number is lower than it was. And what has the holder learned?

The holder has learned that the yield rose. A price fall and a yield rise are one event in the arithmetic of discounting, and no further evidence is needed for it. Which limb rose has NOT been learned. The government SPOT rate for the same maturity could have risen with the spread completely unchanged. The spread could have widened with the government SPOT rate completely unchanged. Both could have moved, in the same direction or in opposite directions with one dominating. The price is a single number and it cannot carry that information.

Worked through with the figures above, it stops being an abstraction. Suppose Palash Cements Limited's yield goes from 9.10 per cent a year to 9.40 per cent a year, an increase of 0.30 percentage points, or 30 basis points. Here are two completely different stories, both of which produce exactly that.

1. Cause A. The rate limb moved.The five year government SPOT rate rises from 6.90 per cent a year to 7.20 per cent a year, a rise of 30 basis points. The credit spread stays exactly where it was at 2.20 percentage points, which is 220 basis points. Add them: 7.20 plus 2.20 is 9.40 per cent a year. Nothing whatever happened to Palash Cements Limited.
   Nothing about this borrower changed, and every other five year borrower repriced too.
2. Cause B. The spread limb moved.The five year government SPOT rate stays exactly where it was at 6.90 per cent a year. The credit spread widens from 2.20 percentage points to 2.50 percentage points, which is from 220 basis points to 250 basis points. Add them: 6.90 plus 2.50 is 9.40 per cent a year. Something changed in what lenders think of this borrower and nothing changed on the curve.
   Only this borrower repriced, and the curve did not move at all.

Now price both. Put 9.40 per cent a year into the same discounting sum, with the identical Rs 91.00/- coupons and the identical Rs 1,000.00/- of face, annual compounding throughout, and the price is Rs 988.451156/- under Cause A and Rs 988.451156/- under Cause B. The yield is the same both times, so the sum is the same both times. The price fell by Rs 11.548844/- from Rs 1,000.000000/- in both stories, a fall of 1.1549 per cent of the Rs 1,000.00/- face amount.

The number underneath is identical to the last paisa, and the two sentences a reader would write above it are about completely different things. A price fall on its own therefore attributes nothing.

What is the spread limb actually paying for?

Reading the 2.20 percentage points takes the most careful language in the subject. The spread is not a fee that somebody charges for arranging things. The spread is a price, and like any price it can be read backwards to see what would have to be true for it to make sense.

The reading uses the two definitions carried in from earlier. Assume a recovery of 40 per cent of the amount owed. Then the loss given default is one hundred per cent less 40 per cent on the same base, or 60 per cent. The spread is treated as the annual price of an expected loss, and an expected loss is a default rate multiplied by what is lost when default happens.

Only one of the three quantities is unknown, so solve for it. Dividing both sides by the loss given default is allowed here because the loss given default is 0.60 and is not zero, and it leaves the implied default rate alone on one side.

A triangle that only works in one direction has not been understood, so run it back the other way as a check. Multiplying 3.6667 per cent a year by a loss given default of 0.60 gives 2.2000 percentage points a year, the spread the working started from. The arithmetic closes.

The three honest limits on that number, none of which is optional

First, the 40 per cent recovery is an assumption, and no measurement anywhere above supports it. Move the assumption and the answer moves with it, while the spread does not move at all. Holding the 220 basis point spread completely still, the table below shows a 30 per cent assumed recovery implying 3.1429 per cent a year, 40 per cent implying 3.6667 per cent a year, 50 per cent implying 4.4000 per cent a year and 70 per cent implying 7.3333 per cent a year. One price, four answers, and the assumption is doing that much of the work.

Second, the whole spread is being treated as compensation for credit, and in a real market it is not. Some part of a spread pays for not being able to sell the bond easily at the moment of wanting to. Every basis point of that read as credit makes the implied default rate too high. Splitting 0.40 percentage points off the 2.20 as payment for something other than default leaves 1.80 percentage points. Divided by the same 0.60 loss given default, 1.80 gives 3.0000 per cent a year rather than 3.6667 per cent a year. A small reallocation has made a large move, and the subtraction produces one spread with no way to separate the two parts.

Third, an implied default rate is what the price says and nothing more. It is not a forecast and it is not a measured frequency of anything. Nobody counted defaults to produce 3.6667 per cent a year. The 3.6667 was solved backwards out of one spread and one assumption, on one invented bond. Treating it as the probability that Palash Cements Limited fails misreads the arithmetic that produced it, and it is the single most common way a credit analysis does damage. The word implied travels with the figure at every use for exactly that reason.

How does a lender or an analyst use the split in practice?

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