Credit Risk: What Is Lost When a Borrower Stops Paying

Credit risk is the chance that a promise to repay is kept late, kept partly, or broken. A lender prices that chance into the rate it charges, so a company borrowing over five years pays more than the government does across an identical stretch of the SPOT curve. The gap between the two rates is a credit spread, and it buys one expected loss: how often a promise breaks, scaled by how much money breaks with it.

Why does one borrower pay more than another for the same five years?

The place to begin is something visible rather than something that has to be accepted on trust. Two borrowers want money for the same five years. Palash Cements Limited, an invented cement maker, issues a five year bond carrying a 9.10 per cent annual coupon on Rs 1,000.00/- of face. The government, borrowing over the same five years on the invented SPOT curveA set of rates for money placed today and returned at one stated future date, one rate for each date. The curve is the reference every other borrower is measured against. , pays 6.90 per cent a year.

Subtract, and look at the answer before anybody names it. Nine point one zero less six point nine zero is 2.20 percentage points. The same quantity in the other unit is 220 basis pointsOne hundredth of a percentage point. So 2.20 percentage points is 220 basis points, and 0.50 percentage points is 50 basis points., because one basis point is one hundredth of a percentage point. Two borrowers, one length of time, one difference, and that difference is the only quantity in this guide that was observed rather than supplied.

The same shape appears on any street. Two stalls outside the same office sell the same lunch, one at Rs 90/- and one at Rs 110/-. The Rs 20/- gap is not the price of lunch. The gap is the price of whatever is different about the two stalls. Until somebody says what that difference is, the Rs 20/- is a number without a meaning attached. The same is true of 220 basis points.

The 220 basis point difference has a name, and the name is a credit spreadThe difference between a borrower's yield and the government SPOT rate for the same maturity, written in percentage points and in basis points.. Two rules travel with it from here on, and both of them have cost somebody a correction. First, a spread is written in both of its units the first time it appears, and a figure in percentage points is never left standing beside a neighbour quoted in basis points. Second, no spread is ever a bare number. A spread is always over something and always for a stated length of time, so 220 basis points here means 220 basis points over the five year government SPOT rate, for five years, and it means nothing at all detached from those two facts.

What is credit risk, and what exactly is at risk?

Credit risk is the risk that a borrower does not pay what was promised, in full and on time. Read both halves of that sentence, because most readers hold only the first. A borrower who pays nothing has defaultedFailed to pay what was promised, in full and on time. A late payment and a partial payment both count; it is not only the case where nothing arrives.. A borrower who pays half has also defaulted. So has one who pays everything eleven months late. The three are not the same event and they do not cost a lender the same money.

Credit risk therefore cannot be carried in one number. A household asked about lending Rs 20,000/- to a cousin names two separate things without being prompted: how likely it is that the money does not come back on time, and how much of it would still be recovered if things went badly. The two questions have different answers, and a lender who has only thought about the first has thought about half the problem.

The finance version keeps those two halves apart. How often is the promise broken, and how much is lost each time it is broken. Everything that follows is those two quantities multiplied together, read in one direction and then the other.

What is the extra 2.20 percentage points actually paying for?

A lender pricing Palash Cements Limited's bond is not being paid 2.20 percentage points a year to feel nervous about it. The extra rate is meant to cover, on average and over a long run of similar lending, the money that does not come back. Pricing in the money that does not come back is an ordinary commercial idea. A caterer who has been left unpaid once in fifty weddings prices the fifty first wedding with that one in it, and does not call the extra a fee for courage.

So write the spread as what it claims to be. The spread is a rate per year, and it equals a default rate per year multiplied by the share of the exposure lost when a default happens. The second quantity is the loss given defaultOne hundred per cent less the recovery rate, measured on the same base, which is the amount owed. If forty per cent of the amount owed comes back, sixty per cent of it is lost., and the product of the two is an expected lossA default rate multiplied by a loss given default, stated per year. An expected loss is an average over many similar exposures, not a prediction about any single one. stated per year.

The relationship has three quantities in it, and exactly one of them has been handed over. The 2.20 percentage points was observed by subtraction. The default rate is not written on any bond. The loss given default is not written on any bond either. One equation with two unknowns has no single answer, so before anything can be solved, one of the two has to be supplied from outside.

How does a spread become a default rate?

Supply the second quantity. Assume that when a borrower stops paying, 40 per cent of the amount owed is eventually recovered. The 40 per cent is a recovery rateThe share of the amount owed that a lender gets back after a borrower stops paying, stated as a percentage of the amount owed rather than of the price paid., and nothing about the bond fixes where the figure came from.

At an assumed recovery of 40 per cent of the amount owed, the loss given default is 100 less 40, or 60 per cent of that same amount owed, written as 0.60. Since 0.60 is not zero, the relationship from step one can now be divided through by it, and that division is the whole trick.

The 3.6667 per cent a year is the annual default rate the price implies, under a 40 per cent assumed recovery and under nothing else. Read as a reading rather than as an exact value it is 3.67 per cent a year, and the four decimals are kept wherever the figure is used again, for a reason that appears in the next part.

Four quantities have now appeared and each one carries a different base and a different period. Naming them in the same sentence as the figure is not pedantry; it is what stops the arithmetic being applied to the wrong number. A recovery rate is a percentage of the amount owed, never of the price paid and never of the coupon. A loss given default is a percentage of that same amount owed. A default rate is a rate per year, not a chance across the whole life of the bond and not a count of events. An exposure is a rupee amount at a stated moment, and an expected credit loss is a rupee amount per year on a stated exposure.

Does the arithmetic close when it is run backwards?

Run it the other way and see. Take the 3.6667 per cent a year and multiply it by the 0.60 loss given default. The answer is 2.2000 percentage points, or 220 basis points, the spread the whole thing started from. The loop shuts.

A relationship shown in one direction is a recipe, and a relationship that closes on its own starting figure has been checked. That is the entire reason for insisting on the second direction. A reader who has only watched the division has learned which button to press. A reader who has watched the multiplication land back on 220 basis points has watched the claim hold, and can now spot the day it does not.

One caution about rounding, given in advance rather than after the reader has been tripped. Four decimals must be carried inside the multiplication. Two point two zero divided by zero point six zero is exactly three and two thirds per cent a year, so the printed 3.67 per cent is a reading and not the value. Multiplying the reading back gives 2.2020 percentage points, not 2.2000. A reader who lands there without warning concludes that they made the error. Two ten thousandths of a percentage point is nothing at all in money and everything in confidence.

Where did the 40 per cent recovery come from?

From nowhere. The honest answer is worth more than a comfortable one. No recovery figure is printed on any bond, and no study of how much has historically come back stands behind the 40 per cent. The 40 per cent is an assumption supplied by whoever is doing the arithmetic, and it is doing an enormous share of the work.

Watch what happens when the spread is held perfectly still and only the assumption moves. The price does not change. The subtraction does not change. The 220 basis points does not change. And the answer changes anyway.

One price, one spread, four answers, and none of the four is more true than the other three. Notice also that the answer does not move in even steps. Going from 30 to 40 per cent assumed recovery adds about half a point to the implied rate; going from 50 to 70 adds nearly three. The division has a shrinking denominator, so the answer runs away as the assumption gets generous, and that is worth feeling rather than reading.

What else could be sitting inside the same 2.20 points?

Everything above has treated every single basis point of the spread as payment for default. Treating every basis point that way was a choice, and it was made silently. In a real market a lender also wants paying for other things, and the most familiar of them is not being able to sell the bond quickly when they want to. Nothing in the number itself says which basis point is which. The spread arrives as one lump.

So split it and see what happens. Take 0.40 percentage points off the 2.20 as payment for something other than default. The credit part is then 1.80 percentage points, or 180 basis points. Dividing 1.80 by the same 0.60 loss given default gives 3.0000 per cent a year rather than 3.6667. Whatever inside the spread is not payment for default has been counted as payment for default, so the error runs in one direction only and always makes the implied default rate too high.

The direction of the error follows from the arithmetic and can be shown. The size cannot be shown, because no market quote separates one part of a spread from another. The 0.40 percentage points is an assumed split, not a measured one.

Why is the word implied carrying so much weight?

An impliedSolved backwards out of a price under a stated assumption, rather than measured from what has happened or forecast about what will happen. default rate is the rate that would make the price make sense under the stated assumption. Nobody counted a single default anywhere to produce 3.6667 per cent a year. Nobody surveyed anything. One subtraction and one assumed recovery were pushed through a division, and out came a figure with a per cent sign on it.

The implied default rate is a property of the price, read backwards through one assumption, and not a property of the borrower. The price implies a number. The number does not produce the price, and it does not describe Palash Cements Limited, whose circumstances are nowhere examined here.

There is a second absence worth naming. Palash Cements Limited carries no credit rating anywhere in this arithmetic. A rating scale, and the definition attached to each step of that scale, are set by the rating agencies and by the Securities and Exchange Board of India (SEBI) at sebi.gov.in, and both are revised. Every statement above holds without needing a rating.

How much is the spread worth as a price today?

A spread of 2.20 percentage points sounds like a rounding error next to a 9.10 per cent coupon. Price it and it stops sounding small. Palash Cements Limited's five year bond pays Rs 91/- at the end of each of five years and Rs 1,000.00/- more at the end of the fifth, on Rs 1,000.00/- of face. Every price here is struck on annual compoundingOne discounting period a year, so an amount is divided by one plus the annual rate once for each year. A sum cannot be reproduced unless the convention is stated., one discounting period a year, and a sum like this one cannot be reproduced by anybody who does not know which convention was used.

Discount those six amounts at 9.10 per cent a year and they total Rs 1,000.000000/- exactly. Pricing at the face amount is what at parPriced at the face amount, so the coupon rate and the yield come out as the same number. means, and it is why the coupon rate and the yield are the same number. Now discount the very same six amounts at the five year government SPOT rate of 6.90 per cent a year, used as a single rate, and they total Rs 1,090.446383/-.

The spread is worth Rs 90.4464/- of price today on Rs 1,000.00/- of face, or 9.0446 per cent of the face amount. That is the same 2.20 percentage points as before, seen from the price side rather than the rate side, and it is large because it is applied to every payment for five years rather than to one of them.

What are the three limits that travel with the answer?

The three limits are collected in one place, so nothing depends on a reader having held each one from where it first appeared. All three are compulsory and none of them is a footnote.

First, the 40 per cent recovery is an assumption, and no bond document, no study and no market quote fixes it. Hold the 220 basis point spread completely still and a 30 per cent assumed recovery implies 3.1429 per cent a year, 40 per cent implies 3.6667 per cent a year, 50 per cent implies 4.4000 per cent a year and 70 per cent implies 7.3333 per cent a year. Same price, four answers, and the assumption is doing that much of the work.

Second, the whole spread has been treated as compensation for credit. In a real market some part of a spread pays for other things, and every basis point of that read as credit pushes the implied default rate too high. Splitting 0.40 points off the 2.20 drops the implied rate from 3.6667 to 3.0000 per cent a year. No quoted spread arrives with the two parts separated.

Third, an implied 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, and treating it as the probability that this issuer fails has misread the arithmetic that produced it.

So what can a spread honestly be used for?

Less than most readers expect, and still enough to be worth having. A spread is a compact statement of what a market is charging a borrower over a reference for a stated length of time. A spread can be re-read under a different assumption and the two readings compared. A spread can be split into parts and argued about part by part, and splitting it that way is covered separately. A subtraction is something anybody can repeat, so a spread travels well between people.

A spread cannot tell anybody how likely a particular borrower is to fail, and holding both of those at once is the whole skill this guide teaches. No second issuer is priced alongside Palash Cements Limited, no default frequency is measured and no recovery study is available, so the only honest comparison is between two readings of the same price under two different assumptions. The comparison between two readings is real, and it is just not the one most readers came for.