Default Rate, Loss Given Default and Exposure Compared

Three questions hide inside one word. How often does a borrower fail, and that is a rate a year. How much of the amount owed disappears when it does, and that is a share. How much was owed at that moment, and that is rupees. Multiply the three and rupees a year fall out. An expected loss has always been exactly that.

What are the three questions hiding inside one word?

Ask somebody why they will not lend money to a person they know, and listen to the answer they give. The answer is almost never one reason. The answer is a tangle. The person is unreliable, the amount is large, there is nothing to fall back on, they have been slow before. All of that arrives as a single feeling with a single name. The name is risk, and a feeling cannot be divided by anything.

Three separate questions, asked one at a time, produce something quite different. How often does this person fail to pay? Quite often, comes the answer, maybe one time in ten. When they do fail, how much of what is owed never comes back? Most of it comes back eventually, they say, it just takes a year. And how much is owed right now? Rs 4,000/-, they say, without hesitating.

The same person who could not explain their own reluctance has just supplied three usable numbers, and the reason is that each question has its own unit and none of them can be answered in another one. One time in ten is a frequency. Most of it comes back is a share of something. Rs 4,000/- is an amount. A reader who holds the three apart can say which one they are least sure about, and that is the whole difference between having a view and having a mood.

Credit arithmetic does exactly this, formally, and gives the three questions three names. The first is a probability of defaultThe chance that a borrower fails to pay what was promised, stated as a rate a year unless some other period is written beside it.. The second is a loss given defaultThe share of the amount owed that never comes back once a borrower has failed, written as a percentage of that same amount owed.. The third is an exposure at defaultHow much is owed at the moment failure actually happens, written in rupees rather than as a rate or a share.. Separating the three means stating the base and the period of each one in the same sentence as its figure, and the operation that joins them is a multiplication rather than an addition.

The worked case is the one this sequence has been carrying throughout. Palash Cements Limited, invented for this sequence and holding no credit rating anywhere, has issued a five year bond carrying a 9.10 per cent annual coupon on Rs 1,000.00/- of face, priced at par, annual compounding, one discounting period a year. The five year government SPOT rate on the invented curve behind this sequence is 6.90 per cent a year, on the same annual convention. One taken from the other leaves a gap of 2.20 percentage points, or 220 basis points, and that gap is a credit spreadThe difference between what a borrower promises and what the government pays for the same length of time, written in percentage points and in basis points..

Everything that follows is an attempt to see what is inside that 2.20 points. The spread turns out to contain two of the three components tangled together and none of them separately, the third has to be supplied from outside the price entirely, and the answer the three produce closes back onto the spread it came from. None of that is a disappointment. Knowing the shape of an object is the point of taking it apart.

How often does the borrower fail, and over what stretch of time?

A probability of default is the chance that a borrower does not deliver what the contract promised, read as widely as the contract itself reads. Paying nothing counts. Paying half counts. Paying every rupee eleven months after it was due counts too, and the three are not the same event and do not cost the same money. Which of them the word covers in any particular rule set is decided by the Securities and Exchange Board of India (SEBI) at sebi.gov.in, named among the sources below.

Two things must be attached to a default rate before it means anything at all, and a figure quoted without either of them is not a quantity, it is a decoration. The first is the period. The second is the source.

The period comes first. Readers drop the period more often than anything else. A chance of failing over the next twelve months and a chance of failing at some point across five years are different quantities that happen to be written with the same per cent sign. A shop that closes one January in every twenty is not the same object as a shop with a one in twenty chance of closing this month. Here every default rate is written per year, with the words a year attached every single time it appears, figures included. The moment those two words come loose the number starts drifting into a five year statement that nobody wrote.

Now the source. The source is the harder half. There are two completely different ways a default rate can come into existence and they share nothing except the unit they are printed in.

The first way is to count. Take a defined population of borrowers, watch it across a defined stretch of years, record what fraction failed, and divide. Counting produces a counted rate, and its meaning is exactly as wide as the population it was counted over and no wider. The second way is to solve backwards. Take a price, take an assumption about how much is lost when failure happens, and ask what default rate would make the price consistent with the assumption. Solving backwards produces an impliedSolved backwards out of a price under a stated assumption, rather than measured from what happened or forecast from anything. rate.

Every default rate used here is the second kind. No population of borrowers was ever assembled and no year was ever counted. The working figure of 3.6667 per cent a year was divided out of one spread under one assumption, and it describes the price rather than the company.

The distinction is worth insisting on because the two look identical on paper and are used for opposite purposes. A counted rate is evidence about borrowers. An implied rate is a restatement of a price. A reader handed 3.67 per cent a year beside a company name with no label will file it as evidence, every time, and no amount of careful arithmetic afterwards will get it back out of that drawer. So the word implied travels with the figure at every single use here, and the assumption that produced it is named in the same breath.

How much is lost, and lost of what exactly?

The second question is not how often but how much, and it is asked about a completely different thing. A loss given default is the share of the amount owedThe sum the contract says must be handed over, which is not the same thing as the price somebody paid to acquire the right to receive it. that never comes back after failure. The share is written as a percentage of that amount owed, or as a decimal share of the same amount, and the two are the same statement.

The risk sequence settled the recovery rate earlier, so it is inherited here rather than redefined. A recovery rate is the share of the amount owed that does come back. The loss given default is one hundred per cent less that, taken on the identical base, so naming either one has already named the other.

The working figure goes through it. The assumed recovery is 40 per cent of the amount owed, an assumption examined properly further on. One hundred less forty is sixty, so the loss given default is 60 per cent of the amount owed, written 0.60 of the amount owed, and both of those readings carry the base out loud.

The base is not decoration attached to the figure, it is half of the figure, and a loss given default of 0.60 applied to the wrong base is a different quantity wearing the same name. This is the single most expensive habit in credit arithmetic, so it is worth a sentence of its own: the base is the amount owed, never the price somebody paid, and never the coupon.

The reason shows in a household example that has nothing to do with bonds. A cousin borrows Rs 50,000/- from a lender. Later, a relative offers to take over that claim for Rs 60,000/-, thinking the cousin is good for it and wanting the interest. If the cousin then fails and 40 per cent of the amount owed comes back, the amount owed is still Rs 50,000/- and 60 per cent of it is Rs 30,000/-. The loss is not 60 per cent of the Rs 60,000/- somebody paid. The price paid changes who carries the loss and how large that person feels it to be, but the contract owes Rs 50,000/- and the recovery is measured on the contract.

On a bond bought at par the two bases coincide exactly. The coincidence is precisely why the error is so hard to see. Palash Cements Limited issued at par, so the price and the amount owed are both Rs 1,000.00/-, and a reader who learned the arithmetic here can apply the share to either figure and get the right answer by accident. The failure block below shows what it costs the first time a bond is bought at any other price, and the answer there is Rs 54.2678/- on a single bond.

How much is owed at a moment nobody can date?

The third question is the plainest to ask and the most awkward to answer. How much is actually owed at the instant failure happens? The figure is the exposure at default, a rupee amount rather than a rate or a share.

For a bond that pays a coupon each year and returns the face amount once at the end, the exposure is the face amount still outstanding plus anything accrued and unpaidInterest already earned by the lender because time has passed, but not yet handed over, so it sits inside the amount owed at the moment of failure. at that moment. A coupon that was earned by the calendar but never arrived is part of what is owed, and leaving it out understates the exposure by however much had built up.

Two features make the exposure genuinely different in kind from the other two components, and both of them matter more than they first look. The first is the unit. A default rate and a loss given default are both pure numbers that can be quoted about a borrower in the abstract, and the exposure cannot: it is rupees, and rupees belong to a particular lender holding a particular claim. Two lenders can agree completely about how often this borrower fails and how much comes back and still carry wildly different exposures. One of them holds ten times as much of the bond as the other.

The second is the timing. The exposure is measured at the moment of failure, and nobody can date that moment in advance. The timing is not a technicality that quietly resolves itself. Think of a caterer who takes a wedding booking. Their exposure to that household is small the week the booking is made, grows steadily as they buy vegetables and hire staff and put down deposits with a tent supplier, peaks the morning of the wedding, and collapses the moment the final payment lands. If the household fails to pay, what the caterer loses depends entirely on which day it happened. The rate at which weddings go wrong and the share of an unpaid bill that is eventually recovered are both unchanged by the calendar. The exposure is nothing but calendar.

Here the arithmetic runs into an honest gap. There is no issue size anywhere in this sequence. Palash Cements Limited has an invented bond with an invented coupon and no stated quantity of it in existence, and inventing one now would put a fabricated total rupee figure into a teaching record where every other number is derived in front of the reader. So the exposure is left unfixed and everything here is worked per Rs 100/- of face amount instead. Every rupee answer below carries those five words with it.

How does a default rate differ from a loss given default?

The default rate and the loss given default are swapped for one another more often than any other pair in credit, and the swap is silent. Both are written as percentages. Both attach to the same borrower. Both get larger when things get worse. A reader who has met them once and not held them apart deliberately will merge them within a week, and the merged object cannot be used for anything.

Probability of Default vs Loss Given Default

The two questions do not sound alike even when the figures do, so keep the pair apart with the questions.

A probability of default answers how often. Its subject is the event. A default rate says nothing whatsoever about size, and a borrower with a very high one might cost a lender almost nothing each time. A loss given default answers how much of the amount owed. Its subject is the severity once the event has happened. A loss given default says nothing whatsoever about frequency. The figure is a conditional statement about a day that has not arrived, so a borrower that has never missed a payment in its life can still carry a loss given default of 0.95 of the amount owed.

Two households on the same street make the difference physical. The first borrows small amounts constantly and is late roughly one time in three, and always, eventually, pays in full because they are simply disorganised rather than short. High how often, very low how much. The second has borrowed twice in twenty years and repaid both on the day, but everything they hold is pledged elsewhere, so if they ever did fail there would be nothing at all left for the lender. Very low how often, very high how much. Nobody would describe those two households with the same word, and yet a single figure called risk does exactly that.

The arithmetic consequence is the part worth carrying away: because the two are multiplied, the same answer arises from an unlimited number of pairs, and being handed the answer tells a reader nothing about either factor. Hold the product still at 2.20 per cent of the amount owed a year and the pairs that produce it are not a short list. The pairs trace a curve.

Look at the shape rather than the values for a moment. The curve never touches either axis, and that is not a drawing accident. A loss given default of zero would mean everything comes back, and then no default rate however large could produce a loss at all. A default rate of zero would mean failure never happens, and then no loss given default however severe could produce one either. Both factors have to be present for the product to exist, and the units say the same thing in a different language.

Why does the arithmetic multiply instead of add?

Because addition is not available. The sentence sounds like a joke and is a literal statement about units.

There are three quantities. One is a rate per year. One is a decimal share of the amount owed. One is a number of rupees. Add them and nothing says what the answer is measured in. Three point six six six seven per cent a year plus zero point six zero plus one hundred rupees. The three ingredients do not live on the same scale, and writing them in a row does not make them comparable. No unit exists for that sentence to land in. The mistake is the same category as adding a speed to a distance to a colour.

Multiplication behaves completely differently, and the difference is the whole reason the formula looks the way it does.

The units are what prove the setup is right, so watch them travel through that multiplication. A rate per year carries the unit per year. A share of the amount owed is a pure number: rupees divided by rupees, with the rupees cancelling. A share therefore has no unit of its own, and its base has to be stated in words instead. The exposure carries the unit rupees. Put them together and per year times nothing times rupees gives rupees per year. Rupees per year is precisely the unit an expected credit lossThe three components multiplied together, which lands in rupees a year on a stated exposure rather than as a rate or a share. is supposed to arrive in.

The units checkConfirming that the units on the left of a calculation actually produce the units on the right, so a rate times a share times an amount really does give an amount a year. is not pedantry, it is the fastest test available that an expected loss has been set up correctly, and it catches most of the ways it can be set up wrongly. An answer that comes out in per cent has left out the exposure. An answer in rupees with no period attached has quietly dropped the year off the default rate. An answer larger than the exposure itself has been multiplied by a percentage where a decimal was meant. None of those need any knowledge of credit to spot. Each one needs a single glance at the unit.

Run through with the working figures, the answer is small enough to say out loud. A default rate of 3.6667 per cent a year, as a decimal, is 0.036667. Multiplied by 0.60 of the amount owed it gives 0.022 of the amount owed a year. Multiplied by Rs 100.00/- of face amount, the expected credit loss is Rs 2.20/- a year, per Rs 100/- of face amount, on this exposure, under this assumption, at annual compounding.

Notice how many qualifiers that sentence carries and resist the urge to trim them. Each one is naming a base or a period that somebody could otherwise substitute a different value into without realising they had changed the question.

Which of the three can a price actually supply?

Now to the question a reader arrives with. If the market has priced this bond, and a price contains information, which of the three components does it give up?

The honest answer is none of them on its own. The price gives a spread, and a spread is two of the three components already multiplied together, with the third absent entirely.

The observation itself is the only quantity here that nobody had to supply, so start there. Palash Cements Limited must promise 9.10 per cent a year for five years, annual compounding. Over that identical stretch, the five year government SPOT rate stands at 6.90 per cent a year on the same convention. The difference between what the two must promise is 2.20 percentage points, and the same quantity in the other unit is 220 basis points. The 220 basis points are the spread, and the spread is the whole of what the price handed over.

Two things follow immediately and both are worth pausing on. First, the exposure is not in there at all. A spread is a rate, quoted on whatever amount somebody happens to hold, so it says nothing about how much is owed and could not: the same 220 basis points applies whether a lender holds Rs 100/- of face amount or a hundred times that. Second, the two components that are in there are stuck together. One equation, two unknowns, and no amount of staring at the price will separate them.

Nothing whatever inside the price prefers either, so one of the two has to be supplied from outside it, and which one is chosen is a decision with consequences.

Assume the recovery and the price yields the default rate. Assume the default rate and the price yields the recovery. Both are legitimate. Both use identical arithmetic. Neither is more true than the other. An account that shows only the first direction has quietly implied that the first direction is the natural one, and it is not. The first direction is taken here because the risk sequence already settled what a recovery rate is, and that is a choice rather than a necessity.

One rounding caution, given before it can trip anybody rather than after. Carry four decimals inside the multiplication. Two point two zero divided by zero point six zero is exactly three and two thirds per cent a year, so 3.6667 per cent a year is a reading of the figure rather than the figure itself. A reader who takes the printed 3.67 per cent a year and multiplies it back by 0.60 lands on 2.2020 percentage points instead of 2.2000, misses by two ten thousandths of a point, and concludes that they made an error. The reader did not. The rounding did.

What happens when one of the three turns out to be wrong?

Here is where holding the three apart stops being tidiness and starts paying. An error in one component does not behave like an error in another. The three fail in genuinely different ways, and the difference is large enough to change what an analyst would go and check.

The exposure is the simple one, so take it first. The exposure multiplies the whole answer and nothing else touches it. Double the exposure and the expected credit loss doubles, exactly, with the rate a year and the share of the amount owed left completely alone. Per Rs 100/- of face amount the answer is Rs 2.20/- a year; per Rs 200/- of face amount it is Rs 4.40/- a year. The relationship is a straight line through the origin, with nothing subtle in it at all.

Now take the recovery assumption, and the behaviour is so different that most readers refuse to believe it the first time. When the default rate has been implied from a spread, getting the recovery assumption wrong does not change the expected loss by a single paisa.

The reason repays slow reading. The sentence is worth more than the arithmetic. The spread is fixed at 2.20 percentage points a year: that is an observation and no assumption can move it. The default rate was not observed, it was calculated as the spread divided by the loss given default. The default rate was defined as a quotient with that denominator, so a change in the loss given default changes the rate by exactly the offsetting amount. Multiplying the two back together multiplies the spread by a number and then divides by the same number.

Four assumptions, four wildly different default rates, and one expected credit loss that never moves. The recovery assumption therefore controls not the answer but the story about why the answer is what it is. At an assumed recovery of 30 per cent of the amount owed the story reads: failure is fairly rare and ruinous. At 70 per cent it reads: failure is twice as frequent and each one is survivable. Both stories cost a lender the same Rs 2.20/- a year per Rs 100/- of face amount, and the price cannot say which of them is true.

Why does the answer close back onto the spread it came from?

One step remains, and it is the step that decides what the whole calculation is worth. Take the two components that came out of the price and multiply them back together.

Expressed on Rs 100/- of face amount the same thing reads Rs 2.20/- a year, and Rs 2.20/- on Rs 100/- is 2.20 per cent of the amount owed a year, which is 2.20 percentage points, which is 220 basis points. Four ways of writing one quantity.

The closure is easy to mistake for a successful check, so say what it actually means. The loop is a successful check, in the narrow sense that it confirms no arithmetic slipped. The loop is not confirmation of anything about Palash Cements Limited. The default rate was solved out of that spread. Putting it back cannot produce anything except that spread. An expected credit loss computed this way is not independent evidence about a borrower, it is the spread written in different units.

None of that is a criticism of the method, and none of it makes the method useless. Converting a rate into rupees a year is genuinely useful: a lending desk cannot add 220 basis points across a set of holdings, and it can add rupees. The conversion adds no information. Whatever the price knew, the answer knows, and not one thing more.

What are the three limits that travel with the number?

Three limits travel with the number. None is optional and none can be dropped when the figure is quoted somewhere else.

First, the recovery of 40 per cent of the amount owed is an assumption, and no counted evidence stands behind it. The figure was chosen for being round and legible in the arithmetic, and moving it moves the implied rate with it: 30 per cent gives 3.1429 per cent a year, 40 per cent gives 3.6667, 50 per cent gives 4.4000 and 70 per cent gives 7.3333. Same price, four answers, and the assumption is doing that much of the work.

Second, the whole of the spread has been treated as payment for credit and nothing else. In a real market some part of a spread compensates a holder for the difficulty of selling a bond when they want to, and every basis point of that read as credit pushes the implied default rate above where it belongs. Carve 0.40 percentage points out of the 2.20 and treat them as buying something other than protection against failure, and 1.80 points of credit are left. Divided by the same 0.60, the 1.80 gives 3.0000 per cent a year rather than 3.6667. Nothing available separates the two portions, and the whole 2.20 has been treated as credit for that reason.

Third, an implied default rate is what a price says. The rate is not a forecast of anything and not a measured frequency of anything. Nobody counted a failure to produce 3.6667 per cent a year. The 3.6667 was divided backwards out of one spread under one assumption, and reading it as the chance that Palash Cements Limited fails misreads the arithmetic that made it.

So what are the three components honestly good for?

Less than the neat formula suggests, and more than nothing. The three components are good for locating a disagreement. Two people who both say a bond looks expensive have said nothing to each other until they say which component they disagree about, and once they do, the argument becomes finite.

The components keep units straight, and keeping units straight sounds small but is not. A rate a year, a share of the amount owed, and a rupee amount at a moment are three things that cannot be swapped, and a reader who writes the base and the period beside every one of them has closed off most of the ways a credit calculation goes quietly wrong.

And they convert a rate into money. A spread can then be set beside a cost, a cushion or a fee that is also quoted in money. The three are not good for telling anybody how likely a particular borrower is to fail. No failure was counted anywhere here, and no arithmetic performed on a price can turn a price into a count.