Expected Credit Loss: Three Numbers, One Multiplication
An expected credit loss is a rupee amount a year, and it is one multiplication rather than a judgement. Take the annual default rate implied by Palash Cements Limited's 220 basis point spread, 3.6667 per cent a year at an assumed 40 per cent recovery, multiply by a loss given default of 0.60, multiply by a Rs 10,00,00,000/- holding, and Rs 22,00,000/- a year falls out.
The expected credit loss on a holding of a size set at the controls
Entering figures recomputes the whole build-up, down to the check that the answer still agrees with the extra coupon this price pays. Educational illustration on invented figures, derived here, and not a market level or a forecast.
| The step | What is done at this setting | Reading |
|---|---|---|
| The credit spread | 9.10 per cent a year less 6.90 per cent a year | 220 basis points |
| Less the part paid for something else | nothing taken off, so the whole spread is read as payment for default | less 0 basis points |
| The credit spread read as payment for default | what is left of the spread after that | 220 basis points |
| Loss given default | one less an ASSUMED recovery of 40 per cent of the amount owed | 0.60 |
| The annual default rate | implied: 2.2000 divided by 0.60 | 3.6667 per cent a year |
| How often times how much | 3.6667 per cent a year times 0.60 of the amount owed | 2.2000 per cent of the amount owed a year |
| Times the exposure | 2.2000 per cent of Rs 10,00,00,000/- | Rs 22,00,000/- a year |
| The same amount discounted one year | divided once by 1.0910, taking off Rs 1,83,501/- | Rs 20,16,499/- |
The opening position works out the same way on paper. A coupon of 9.10 per cent a year against a government SPOT rate of 6.90 per cent is a credit spread of 220 basis points. At an ASSUMED recovery of 40 per cent of the amount owed the loss given default is 0.60, so the implied annual default rate is 2.20 divided by 0.60, or 3.6667 per cent a year. Multiply by 0.60 and by a holding of Rs 10,00,00,000/- of face and the expected credit loss is Rs 22,00,000/- a year, or Rs 20,16,499/- discounted one year at 9.10 per cent. The year the borrower actually fails costs Rs 6,00,00,000/-.
Drag the recovery slider from 40 to 90 per cent of the amount owed and change nothing else. Which reading moves?
Three inputs, two multiplication signs, one answer. Anyone who can work out fifteen per cent of a restaurant bill can already do the arithmetic. The interesting part of a multiplication like this is never the sum. Everything the sum quietly assumes, and everything the answer does not entitle anybody to say afterwards, is where the difficulty sits.
An expected credit lossA rupee amount a year, being the annual default rate multiplied by the loss given default multiplied by the exposure. is a rupee amount attached to one holding for one year, and every wrong answer on this material comes from mislabelling one of the three inputs rather than from botching the multiplication. A rate gets entered as though it were a share. A share of the amount owed gets applied to the price paid instead. A rate for one year gets read as a chance across five. The arithmetic never complains. So the labelling has to be done first and out loud.
What is actually being multiplied, and what does each number measure?
Input one is the annual default rate, and its period is one year. It reads 3.6667 per cent a year, and the word attached to it every single time is impliedSolved backwards out of a price under a stated assumption, rather than counted in any record or forecast from anything.. The rate is not found anywhere: it is solved out of Palash Cements Limited's 220 basis point spread by dividing that spread by the loss given default. The opening of this sequence ran that same arithmetic in both directions. Nobody counted a failure to produce it. Its period is one year and not five, so it is not the chance that this borrower fails at some point before the bond matures.
Input two is the loss given defaultThe share of the amount owed that is not recovered, being one hundred per cent less the recovery rate on that same base., and its base is the amount owed. It reads 0.60, and it is a share rather than a rate, so it carries no period at all. It is one hundred per cent less the assumed recovery rate on the same base, and the assumed recovery here is 40 per cent of the amount owed. The base is the most commonly slipped label on the subject. A recovery of 40 per cent means 40 per cent of what the borrower owed, not 40 per cent of the price paid for the bond and not 40 per cent of the coupon expected.
Input three is the exposure at defaultThe rupee amount owed at the moment failure happens: face still outstanding plus anything accrued and unpaid., and it is a rupee amount at one named moment. Here it is a holding of Rs 10,00,00,000/- of face, found nowhere on this platform. It is an assumption, labelled as one beside every rupee figure it produces, and what that costs is taken up next.
Four quantities are now on the table and no two of them are the same kind of thing: a rate per year, a share of the amount owed, a rupee amount at a moment, and, once they are multiplied, a rupee amount per year on a stated holding. Think of a household budgeting for a scooter that breaks down. How often it breaks down in a year is one number, what share of its value a breakdown ruins is a second, what it is worth is a third, and nobody would add those three. The units fit together only one way, so everybody multiplies them without being told to.
| ECL | the expected credit loss, in rupees a year on the stated holding |
| d | the implied annual default rate, as a decimal per year, from the spread divided by L |
| L | the loss given default, as a decimal share of the amount owed |
| E | the exposure at default, in rupees of face at the moment failure happens |
What is the base of the loss given default, and what is the period of the annual default rate?
Before the next section runs it. A rate a year is about to be multiplied by a share of an amount and by a rupee amount. What unit must the answer be in?
What unit does the answer come out in, and how is it checked?
Rs 10,00,00,000/- times 0.0366667 times 0.60 gives Rs 22,00,000/- a year. The reason to walk that slowly is not the arithmetic but the units. The units are the only error check available.
Take the first two inputs on their own. A rate of 3.6667 per cent a year multiplied by a share of 0.60 of the amount owed gives 2.20 per cent of the amount owed a year: still a rate, still per year, but its base is now the amount owed rather than nothing in particular. Bring in the third input and 2.20 per cent of Rs 10,00,00,000/- is Rs 22,00,000/-, with the per year riding straight through onto the answer. If the number that comes out is not in rupees a year, one of the three inputs went in wearing the wrong unit, and that is the only check this tool offers. The middle line of that descent is worth a second look. 2.20 per cent of the amount owed a year is the spread in a different costume, and the reason for that is taken up shortly.
The compounding convention belongs inside the arithmetic rather than in a note beneath it. Every price here is struck on annual compoundingOne discounting period a year, so an amount is divided once by one plus the annual rate for each year that passes., so at a 9.10 per cent annual rate a payment five years out is divided by 1.0910 five times over. Palash Cements Limited's five payments discount to Rs 1,000.000000/- exactly on that convention. Pricing at exactly the face amount is what at parPriced at the face amount, which forces the coupon rate and the yield to be the same number. means and why the yield reads straight off the coupon. The instrument uses the same convention to discount the expected credit loss one year: Rs 22,00,000/- divided once by 1.0910 is Rs 20,16,499/-, so a year of discounting takes Rs 1,83,501/- off it. On a semi-annual convention the same coupon, maturity and yield give a different price out of figures that look identical in print.
A reader multiplies 3.6667 by 0.60 by Rs 10,00,00,000/- without dividing the rate by a hundred first. Roughly how wrong is the answer?
Palash Cements Limited's bond is described as issued at par with a 9.10 per cent annual coupon on Rs 1,000.00/- of face. What has to be stated before anybody can reproduce that price?
Where did the Rs 10,00,00,000/- holding come from?
Straight answer: from here, and nowhere else. There is no issue size for Palash Cements Limited's bond anywhere on this platform, no register of who holds what, and no position record of any kind. The Rs 10,00,00,000/- of face is an assumption made so the multiplication has a third input, and it is stamped as one beside every rupee figure derived from it.
The exposure is the only one of the three inputs that scales the answer straight through without touching either of the other two, so every rupee figure here is proportional to a number invented for the purpose. At Rs 20,00,00,000/- in the exposure field the expected credit loss doubles to Rs 44,00,000/- a year, with the implied annual default rate and the loss given default sitting exactly where they were. At one bond, Rs 1,000.00/- of face, the answer is Rs 22.00/- a year, the same 2.20 per cent at a scale small enough to hold in the hand.
The other two are not free that way. The spread of 220 basis points fixes their product, so pulling one moves the other in the opposite direction by exactly as much. The exposure is the only input that can change the answer on its own, so when a rupee answer changes, look there first.
The holding doubles to Rs 20,00,00,000/- of face and nothing else changes. What happens to the expected credit loss?
Before the arithmetic. How does the expected credit loss compare with the extra coupon the spread pays on this same Rs 10,00,00,000/- holding?
Why does the answer land exactly on the extra coupon?
Forget the three inputs and just count the money coming in. A year of coupon on Rs 10,00,00,000/- of face at 9.10 per cent a year is Rs 91,00,000/-. Lent for the same five years at the five year government SPOT rateThe price of money handed over now and returned at one named date in the future, quoted as a rate a year. of 6.90 per cent instead, one year of income would have been Rs 69,00,000/-. Subtract: Rs 22,00,000/- a year of extra coupon, the whole of what the spread pays on this holding.
The three inputs multiplied gave Rs 22,00,000/- a year, and the extra coupon gives Rs 22,00,000/- a year. Not close. Identical, to the rupee, at any holding size whatever, and that is the check the instrument prints under its own build-up at every setting.
The two figures agree because the default rate was solved backwards out of that same spread before it was ever multiplied by anything, so the multiplication hands back the number it was built from and produces no independent finding about the borrower at all. The circle closes in two steps. The implied annual default rate is the spread divided by the loss given default, so multiplying it back by the loss given default cancels that factor and leaves the spread; and the spread multiplied by the exposure is the extra coupon. There was never room for a different answer.
| S | the credit spread, as a decimal per year, here 0.0220 |
| L | the loss given default, as a decimal share of the amount owed, here 0.60 |
| E | the exposure at default, in rupees of face, here Rs 10,00,00,000/- |
| yc | the borrower's yield a year, here 9.10 per cent on annual compounding |
| rg | the government SPOT rate a year at the matching five year maturity, here 6.90 per cent |
What that buys is a sanity check and a description. A rupee expected credit loss that does not agree with the extra coupon on the same holding means an input has been typed wrong, and the agreement itself says cleanly what a spread is for: the extra coupon is exactly consumed, on average, by the loss it is compensating for. What it does not buy is any information about Palash Cements. A rate counted independently is not the rate this price implies. Set the instrument's rate source to supplied and the check stops holding, and the tool prints the gap rather than pretending the two agree.
The household version lands quickly. A neighbour who has been late with money before asks to borrow Rs 1,00,000/-, and the lender charges Rs 4,000/- of interest a year where a reliable cousin would have been charged Rs 2,000/-. That is an extra Rs 2,000/- a year, set by the lender. Asked what chance there is that the neighbour will not pay, the lender can divide that Rs 2,000/- by whatever share of the money a bad outcome would cost and produce a percentage. That percentage restates the lender's own pricing decision. It is not a discovery about the neighbour, and reporting it as one would be silly.
| The route | What is worked | On this holding |
|---|---|---|
| The coupon route | Rs 10,00,00,000/- at 9.10 per cent a year | Rs 91,00,000/- |
| The coupon route | Rs 10,00,00,000/- at the 6.90 per cent government SPOT rate | Rs 69,00,000/- |
| The coupon route | The difference, which is what the spread pays | Rs 22,00,000/- |
| The three input route | 3.6667 per cent a year, times 0.60, times Rs 10,00,00,000/- | Rs 22,00,000/- |
| The gap between them | Nothing, and there is a reason nothing is left over | Rs 0/- |
What happens to the answer when the recovery assumption moves?
The 220 basis point spread stays completely still, because it is a fact about the price and no assumption can change it. Moving the assumed recovery rate then moves the two inputs it touches. At a recovery of 30 per cent of the amount owed the loss given default is 0.70 and the implied annual default rate is 2.20 divided by 0.70, or 3.1429 per cent a year; at 70 per cent the pair reads 0.30 and 7.3333 per cent a year. Across that range the default rate has more than doubled while the loss given default has fallen to well under half, and the table below sets out the two settings in between. The expected credit loss, in rupees, is Rs 22,00,000/- a year at every one of them.
Their product was fixed by the price of the bond before either factor had been named, so the two trade against each other exactly, and what a recovery assumption changes is the story about why the loss is what it is and never the loss itself. That is the sentence to carry away from this section, and the control below shows it happening rather than asking for it to be taken on trust.
| R | the ASSUMED recovery rate, as a decimal share of the amount owed |
| L | the loss given default, one less the assumed recovery on the same base |
| d | the implied annual default rate, as a decimal per year |
| S | the credit spread, as a decimal per year, fixed at 0.0220 by the price |
| Assumed recovery | Loss given default | The spread | Implied default rate | Expected credit loss |
|---|---|---|---|---|
| 30 per cent | 0.70 | 220 basis points | 3.1429 per cent a year | Rs 22,00,000/- |
| 40 per cent | 0.60 | 220 basis points | 3.6667 per cent a year | Rs 22,00,000/- |
| 50 per cent | 0.50 | 220 basis points | 4.4000 per cent a year | Rs 22,00,000/- |
| 70 per cent | 0.30 | 220 basis points | 7.3333 per cent a year | Rs 22,00,000/- |
Move the recovery assumption and watch the answer refuse
One control. The spread is a fact about the price of Palash Cements Limited's bond, so it stays at 220 basis points at every position of the control and no assumption can move it. Educational illustration: every figure is derived inside this calculator from the settings shown.
- The rate it used was solved out of a price, so it cannot say how likely Palash Cements Limited is to fail.
- Nor can it say what a reporting entity must record. The accounting basis for that belongs to the Institute of Chartered Accountants of India at icai.org.
- Whether Rs 22,00,000/- a year is adequate compensation is a view, and the arithmetic carries none.
- Nor does it reach across more than one year: arithmetic of a different shape, covered separately.
Both ends of that control are worth reading off in full. At the far left, a recovery of nothing at all, the loss given default is 1.00, the implied annual default rate is 2.20 per cent a year and a default costs the whole Rs 10,00,00,000/-. At the far right, a recovery of 80 per cent of the amount owed, the loss given default is 0.20, the implied rate is 11.0000 per cent a year and a default costs Rs 2,00,00,000/-. The expected credit loss is Rs 22,00,000/- a year at both.
That pair of endpoints makes the point better than any explanation. At one end the tool tells a story about a borrower that almost never fails but wipes the holder out when it does; at the other, about a borrower that fails five times as often and costs a fifth as much each time. Two different pictures of the same company, one rupee answer, and nothing about the world changed between them. Nothing about the world was ever an input.
The assumed recovery moves from 40 to 70 per cent of the amount owed. What happens to the expected credit loss in rupees?
Does a loss of Rs 22,00,000/- ever actually happen?
No. Not in any year, not on this holding, not once. And this is the property most worth carrying away from the whole subject.
In any single year, one of exactly two things happens to Palash Cements Limited's five year bond. Either the borrower pays what was promised and the loss on the holding is Rs 0/-, or it does not and the loss is the loss given default of 0.60 applied to the whole Rs 10,00,00,000/-, or Rs 6,00,00,000/-. There is no third outcome and nothing in between. The bond does not partly fail in a way that costs the holder twenty two lakh rupees.
So where does Rs 22,00,000/- come from? It is an expected valueA weighted average of outcomes, which need not equal, and usually does not equal, any outcome that can actually occur.: an outcome of Rs 0/- carrying a weight of 96.3333 per cent, and an outcome of Rs 6,00,00,000/- carrying a weight of 3.6667 per cent, averaged together. Check it exactly rather than against a rounded decimal. The failing outcome is 0.60 of the holding, and its weight is a rate that is itself 2.20 divided by 0.60, so weighting one by the other puts the 0.60 straight back and leaves 2.20 per cent of Rs 10,00,00,000/-, or Rs 22,00,000/-. The expected credit loss is an average of two amounts and is equal to neither of them, so it is the one figure of the three that cannot be observed in any year.
Everybody has met this shape outside finance. A shopkeeper whose awning blows off once every eight monsoons does not spend an eighth of an awning a year; he spends nothing for seven years and then buys a whole awning. The average is a good number for planning across many shopkeepers or many years and a terrible description of any actual year.
| p | the weight on the failing outcome, here 3.6667 per cent for one year |
| 1 - p | the weight on the paying outcome, here 96.3333 per cent for the same year |
| L × E | the rupee loss in a year of failure, here 0.60 of Rs 10,00,00,000/-, which is Rs 6,00,00,000/- |
| ECL | the expected credit loss, here Rs 22,00,000/- a year |
The picture above draws two points and nothing between them, deliberately. Drawing the shape between Rs 0/- and Rs 6,00,00,000/- would need a loss distribution, a correlation assumption and an ordering of cash flows by period, and none of the three has been stated. A smooth curve across that gap would look far more knowing than the arithmetic behind it.
In a year when Palash Cements Limited fails, what does this Rs 10,00,00,000/- holding actually lose?
The error that gets made, and what it costs
Here is the artefact. Somebody works the arithmetic correctly, gets Rs 22,00,000/- a year, and writes it into a two line note that reads: expected credit loss on the Palash Cements Limited holding, Rs 22,00,000/-. Position covered. Then they move on, and the holding is treated as handled.
| The note as written | Amount |
|---|---|
| Expected credit loss on the holding, one year | Rs 22,00,000/- |
| Position covered | yes |
Nothing is wrong with the arithmetic. Rs 22,00,000/- is the right answer to the question the multiplication asks. What is wrong is the reading. That figure is the average of Rs 0/- with a weight of 96.3333 per cent and Rs 6,00,00,000/- with a weight of 3.6667 per cent, and it is an amount that occurs in no year at all. Writing position covered against it treats an average as though it were the worst case.
Who makes it: anybody who meets an average and reads it as the typical case. Most readers do that most of the time, and it has nothing to do with how good they are at sums. The pull is very strong when the number is printed cleanly beside a company name.
What it costs: a holding sized as though a bad year costs Rs 22,00,000/-, when the year it actually goes wrong costs Rs 6,00,00,000/-. That is a little over twenty seven times as much, and it is the whole of the difference between the two readings.
The repair, in one line: write both numbers or neither. Rs 22,00,000/- a year expected, Rs 6,00,00,000/- in a year of failure, and no claim about which year that would be. The arithmetic supports no such claim.
Who runs this multiplication, and what for?
The household version is the same shape. A cousin who repairs two-wheelers asks for Rs 2,00,000/- to go into his workshop for a year and offers Rs 30,000/- of interest, where the bank would have paid Rs 12,000/-. That is an offer of Rs 18,000/- of extra income, and the multiplication above asks what that Rs 18,000/- is buying. If a bad year would cost 60 per cent of the money, the lender's own pricing has implied a failure rate of 15 per cent a year. Writing it down puts the figure where it can be looked at and believed or not, and the discipline does not get more sophisticated when the amounts get larger.
A spread stated as 220 basis points is easy to nod at, and Rs 22,00,000/- a year on a Rs 10,00,00,000/- book is not, so a lender runs the multiplication to translate a rate decision into a rupee amount it can actually carry. The rate is an abstraction; the rupees appear in a plan. The exposure is the one input the lender fully controls, so the multiplication also makes the size of the holding visible as a decision.
An analyst runs it in the other direction, as a consistency check on somebody else's pricing: take the rupee amount and the holding, and back out what default rate and recovery pair would have to be true for that price to make sense. If the pair that emerges is one nobody would state out loud, the price has said something the words around it did not. Backing the pair out is a reading of a price rather than a forecast, and a reading is a much more defensible claim.
An investor uses it to keep two amounts separate: what the position costs on average, Rs 22,00,000/-, and what it costs in the year it goes wrong, Rs 6,00,00,000/-. A position sized against the first and one sized against the second are not the same position, and the multiplication makes the gap explicit rather than a matter of temperament.
A reporting entity has a fourth use, and it is covered separately. Measuring and reporting an expected credit loss in a set of accounts follows an accounting basis set by the Institute of Chartered Accountants of India at icai.org, and it is a different exercise from the arithmetic here. The accounting basis has to be confirmed at its source before any of this arithmetic is carried to a reporting question.
What does this tool refuse to say?
A tool that prints a rupee figure beside a company name is inviting the reader to carry it away as a finding, so the four refusals sit in the output panel rather than in a footnote where they would arrive after the damage.
The first limit: the recovery figure is an assumption
Nothing on this platform supports 40 per cent. No recovery study was read for it and no authority has been asked. It is a number chosen so the arithmetic has something to work on, and it is doing a great deal of the work. Hold the 220 basis point spread still and the implied annual default rate reads 3.1429 per cent a year at a recovery of 30 per cent of the amount owed and 7.3333 per cent at 70 per cent. Same price, and the larger answer is more than twice the smaller. Push the slider to 100 per cent and the instrument prints no rate at all. At that assumption a default costs nothing, and the division has nothing to divide by.
An implied default rate quoted without the recovery assumption that produced it is not a shortened statement but a different and much stronger claim than the arithmetic supports. The word ASSUMED travels with the recovery figure every time the implied rate is written down, and the two are quoted together or not at all.
The second limit: the whole spread is being read as payment for default
The multiplication takes all 220 basis points and treats every one as compensation for the borrower failing. In a real market that is not the only thing a spread pays for: some part of it compensates a holder for not being able to sell the bond easily when they want to, and every basis point of that read as credit makes the implied default rate too high.
Work the size of it on the instrument. Set the second slider to 40 basis points and 180 remain as payment for default. Divide 1.80 by the loss given default of 0.60 and the implied annual default rate falls from 3.6667 to 3.0000 per cent a year, roughly a fifth off it, and the expected credit loss falls in step from Rs 22,00,000/- a year to Rs 18,00,000/- on the same holding. The extra coupon falls with it. Both are the same spread. Nothing on this platform says whether 40 basis points is the right split or whether the right split is nothing at all. The control therefore starts at nothing.
The third limit: an implied default rate is what the price says
It is not a forecast and it is not a measured frequency. Nobody counted failures to produce 3.6667 per cent a year: it was solved backwards out of one spread and one assumption, and the reconciliation with the extra coupon showed how completely it hands that spread back when it is multiplied out again. Treating it as the probability that Palash Cements Limited fails misreads the arithmetic that produced it, and doing so beside a company name is how arithmetic turns into an accusation.
This tool produced an expected credit loss of Rs 22,00,000/- a year on a real looking holding. What has it established about Palash Cements Limited?
Where the rule set lives, and why not one row here is filled in
Everything above is arithmetic on invented figures, and the only convention stated inside it is the compounding convention: annual throughout, and stated because a price cannot be reproduced without it. Every item below moves and sits with an authority, so each is named here rather than written out. Confirm each at its source before relying on it.
| The item | Where it is settled |
|---|---|
| The capital treatment that applies to holding a credit exposure | The Reserve Bank of India, rbi.org.in |
| The valuation norm that decides the price at which a credit holding is carried | The Reserve Bank of India, rbi.org.in |
| What counts as a default for reporting purposes, and who decides it has happened | The Securities and Exchange Board of India (SEBI), sebi.gov.in |
| The treatment that applies to a holding once it has stopped paying | The Reserve Bank of India, rbi.org.in |
| What recovery assumption a regulated holder must apply, if any | The Reserve Bank of India, rbi.org.in |
| The process by which an unpaid claim is resolved, and in what order claims are met | The insolvency authority, ibbi.gov.in |
| The accounting basis on which an expected credit loss is measured and reported | The Institute of Chartered Accountants of India, icai.org |
Because the arithmetic above is written free of any rule set, a second market becomes an addition to this table rather than a rewrite of the arithmetic.
References
| Source | Named for | Where |
|---|---|---|
| The Institute of Chartered Accountants of India | The accounting basis on which an expected credit loss is measured and reported, a different exercise from the arithmetic above. | icai.org |
| The Reserve Bank of India | The capital treatment of a credit exposure, the valuation norm for a credit holding, the treatment that applies once a holding has stopped paying, and any recovery assumption a regulated holder must apply. | rbi.org.in |
| SEBI | What counts as a default for reporting purposes and who decides it has happened, and the rating agencies and their published methods. | sebi.gov.in |
| The insolvency authority | The process by which an unpaid claim is resolved and the order in which claims are met. | ibbi.gov.in |
Palash Cements Limited and the government SPOT curve are invented.
Educational material. Not advice on any investment, tax, budget or market position.