Probability of Default: Estimating the Chance of Failure to Pay

A probability of default estimates how likely one named borrower is to fail to pay over a stated period, almost always one year. The estimate is a property of the borrower and never of the facility. A probability turns into money only when it is multiplied by how much is at risk and how much of that is lost. On its own it ranks names and prices nothing.

Everything below leans on one separation. A ranking and a quantity are not the same object. A ranking puts things in order and says nothing about the distance between them. A quantity says how far apart they are and what each one is worth. A probability of defaultAn estimate of how likely one borrower is to fail to pay over a stated period, almost always one year. is a quantity bolted on to a place in a ranking, and the moment it stops being treated as a quantity a loan book gets read exactly backwards.

The same idea appears in ordinary life. A landlord with ten tenants can name the one most likely to miss the rent next month. Naming that tenant is a ranking, and a ranking is genuinely useful. The second question is different. Which tenant hurts most if the rent stops? The likeliest misser has a single room at Rs 6,000/- a month. The steadiest tenant has the whole ground floor at Rs 60,000/-. The order of likelihood and the order of money are two different lists, and only the second one decides whether the landlord can service the loan on the building. A lender that reads its own risk grades as a ranking of loss has confused the order of the names with the size of the money sitting behind them.

Everything worked below belongs to Vindhya Commercial Bank Limited, an invented mid-sized Indian commercial bank, and every amount attached to it is stated in Rs crore. The ten probabilities are that bank's own one year estimates, produced from its own default history on its own book. Not one of them is a market figure or an agency figure, and none of them can be lifted onto another lender's book.

What is a probability of default a probability of?

Almost every muddle on this subject begins by attaching the estimate to the wrong thing. Start with the noun it attaches to. A probability of default is a probability that one named borrower fails to pay. Not that a loan goes bad. Not that a facility is written off. Not that a sector has a hard year. The borrower is the unit. One entity either keeps paying or does not, so a borrower with four facilities across three branches still has one probability of default.

The rule sounds obvious until a lender attaches the estimate to the facility instead. Counterparty C1, Nirjhar Industries Limited, has a working capital line, a term loan and two derivative trades with Vindhya Commercial Bank Limited. Attach a probability to each and the bank holds three or four different numbers for one company, with no honest way to reconcile them. The company does not fail three times in different amounts. The company fails once. How much is at risk when it fails and how much of that is never recovered are the other two numbers in the product, and pushing them into the probability is what destroys the product.

The second noun is harder and matters more. A probability is a probability of an event, and an event that has not been written down cannot be counted. So before a bank can estimate anything it has to publish a definition of defaultThe event the probability is a probability of, defined in writing before it is measured, because an undefined event cannot be counted. and then live with it. Is a borrower in default the day a payment is late? After a stated number of days? When the bank concludes the borrower is unlikely to pay in full without the bank enforcing its security? Different definitions produce different histories, different histories produce different rates, and two banks quoting a probability of default for the same borrower may simply be counting two different events.

Think of a school reporting how many pupils failed an examination. The number is meaningless until somebody states the pass mark, and it changes the moment the pass mark moves, without a single pupil answering a single question differently. A default definition is that pass mark. Vindhya Commercial Bank Limited applies one definition across the whole book, and one definition is what makes its ten grade probabilities comparable with one another. A single definition applied everywhere is not what makes them comparable with anybody else's.

Why one year, and what does a one year number leave out?

Almost every probability of default in use is a one year figure, and the reason is comparability rather than truth. Loans have wildly different lives. C1 carries a weighted average remaining life of 3.4 years, C5 Tapti Agro Processing Limited carries 1.8 years, and C2 Sahyadri Power Transmission Limited carries 8.2 years on a power transmission asset. If each name were quoted over its own life, every number would be answering a different question. The figures could not then be added, ranked or held against a single limit. Fixing the window at one year makes ten answers to one question, and a book that can actually be summed.

The cost of that convention is that a one year number is silent about every year after the first. C2 at internal grade 3 carries a 0.20 per cent one year probability. A figure that small sounds like a name barely worth watching. The tenor sits beside it. The bank is exposed to C2 for 8.2 years, so it faces that first year again, and again, and again, on a borrower whose circumstances will not stay still. The one year estimate never claimed to cover years two to eight. The estimate is a reading taken at a point, and the exposure outlives the reading many times over.

A health check makes the same point. A doctor telling a forty year old that the chance of a serious event in the next twelve months is low has said something true and useful, and has said nothing at all about the next twenty years. Nobody hearing it thinks the twenty year answer is the same. In credit the number arrives with no window printed on it, so people do think exactly that. State the horizon every time the figure is quoted. Left unstated, the figure will be read as a lifetime chance.

There is a second thing hiding in the horizon, and it decides how much the whole scale moves in a bad year. A probability can be estimated through the cycleA probability estimated to hold across good and bad years rather than to describe this year., meaning it is built to hold as a long run average across good years and bad ones, or point in timeA probability estimated to describe conditions as they are now, which moves much more than a through the cycle one., meaning it is built to describe conditions as they are right now. The first hardly moves when the economy turns and lets borrowers slide down the scale instead. The second moves a great deal and keeps borrowers roughly where they are. Both are legitimate. The illegitimate move is holding one kind of estimate and reading it as though it were the other kind.

Where does the number come from, if it is not an agency figure?

Where the number comes from is the most common mistake on this subject, and the mistake runs in one specific direction. The assumption is that the probability arrives from outside, published by somebody with authority, the way a temperature arrives from the weather office. It does not. The probability of default against each grade in this case is Vindhya Commercial Bank Limited's own estimate, produced from its own observed default history on its own borrowers, and it is the property of that one bank.

The mechanics of producing it are simple to state and hard to do well. The bank sorts its past borrowers into grades, watches what share of each grade actually defaulted over the following twelve months, does that across enough years to have something worth averaging, and lands on a rate for each grade. Feed the same exercise a different book and it produces different rates. A bank lending mostly to established manufacturers and a bank lending mostly to first-generation traders will observe different default experience, and both sets of numbers can be honest.

The consequence is a limit on how far the ten figures travel. The ten probabilities below are not a scale for Indian corporate credit. The ten figures do not state what a grade 5 borrower's chance of failure is. The ten figures are what one invented bank has observed on its own book, written down. Lifting them out and applying them to another lender's borrowers takes one institution's history and dresses it as a general fact. A probability of default travels with the book it was measured on, and it does not travel further than that.

How any particular borrower arrived in grade 4 rather than grade 5, what the assessment criteria are, and how a scale is checked against what actually happened are separate questions with their own machinery. Here the grade arrives as an input and the estimate that sits against it is the object under the microscope.

How does an internal grade scale carry a probability of default?

A bank does not estimate a separate probability for every borrower it lends to. The bank builds a scale, puts each borrower on a rung of it, and estimates one probability per rung. Vindhya Commercial Bank Limited runs a ten rung scale. Grades 1 to 9 are borrowers that are still paying, and grade 10 is the rung for borrowers that have already stopped. An internal gradeA place on a lender's own ordered scale, carrying a probability of default the lender estimated from its own history. is therefore two things at once: a place in an order, and a carrier for a number.

The picture below is built to keep those two apart, so look at them separately. As an order, the ten rungs are evenly spaced, one step at a time, and the step size means nothing at all. As numbers, the steps are anything but even. Grade 1 carries 0.03 per cent and grade 5 carries 0.90 per cent, so five rungs down the ladder is thirty times the estimated chance of failure. Grade 9 carries 15.00 per cent, five hundred times grade 1. The rungs are a ranking and the probabilities are a quantity, and the only reason a scale carries numbers at all is that the ranking cannot say how far apart the rungs are.

Two things are worth taking from that picture. First, the middle of this scale roughly doubles at each step: 0.45, 0.90, 1.80, 3.60 and 7.20 are each exactly twice the one above. A doubling at every step is why the plotted points fall on a near straight line once the axis is logarithmic. Second, look at the sliver at the foot. On a plain linear axis grades 1, 2, 3 and 4 all live inside the first 3.0 per cent of the width, so a chart drawn that way silently tells the reader that the four best grades are the same thing. The four grades are not the same thing. Grade 4 carries fifteen times grade 1.

Why is grade 10 never added to the grades above it?

Grade 10 is the odd rung and it is worth stopping on. Arithmetic goes wrong on that rung more often than anywhere else on the scale. Grade 10 does not carry a probability of default. Grade 10 carries the borrowers that have already defaulted. Asking for the chance that a grade 10 borrower fails inside the next twelve months is asking for the chance of something that has happened, and there is no useful answer to that question.

At Vindhya Commercial Bank Limited, grade 10 holds Rs 1,764 crore of gross advances. The Rs 1,764 crore in grade 10 is the same Rs 1,764 crore as the bank's gross non-performing book, one number wearing two labels rather than two amounts of money. The non-performing book is not a separate pile of loans sitting somewhere else. The non-performing book is grade 10 seen through the reporting lens instead of the risk grading lens. Adding the two counts the same money twice.

Everything that follows therefore runs across grades 1 to 9 and stops. Grades 1 to 9 hold Rs 57,036 crore. With grade 10 added, gross advances come to Rs 58,800 crore, the whole book. Both figures are correct and they answer different questions. Any expected loss computation is a statement about a loss that has not happened yet, so a rung holding losses that have already happened cannot appear in it. The defaulted exposures are dealt with by provisions taken against them, and provisioning is a separate machine.

What is the sub-investment grade share actually measured on?

The bank runs a limit, L4, on how much of the book may sit in its weakest still-paying grades. The measure L4 runs on is sub-investment gradeAt this bank, grades 7, 8 and 9 and nothing else, because grade 10 has already defaulted., and at Vindhya Commercial Bank Limited it means grades 7, 8 and 9 and nothing else. Grade 7 holds Rs 4,704 crore, grade 8 holds Rs 2,352 crore and grade 9 holds Rs 1,176 crore, giving Rs 8,232 crore. The sum of the three is 14.0 per cent of gross advances of Rs 58,800 crore. Limit L4 caps that at Rs 8,820 crore, so utilisation sits at 93.3 per cent and the limit is within.

Now watch what happens if somebody adds grade 10, on the entirely reasonable-sounding argument that a defaulted borrower is surely the weakest of all. The measure becomes Rs 8,232 crore plus Rs 1,764 crore, being Rs 9,996 crore. Against the same Rs 8,820 crore cap that reports 113.3 per cent. A limit that is genuinely within is now reported as breached by thirteen per cent, and the reported breach is made entirely of money the bank has already written down against. Nobody has lent anything new. One line of arithmetic has manufactured a breach, and a breach report goes to a committee.

What happens when the probability meets an exposure?

A probability on its own is an opinion about a name. Expected lossProbability of default times loss given default times exposure at default, being three numbers multiplied and never added. is money, and it is a product of three numbers: how likely the borrower is to fail, how much is outstanding when it does, and what share of that is genuinely gone. Vindhya Commercial Bank Limited assumes a loss given defaultThe share of an exposure that is actually lost when a borrower fails, after everything recovered. A severity, not a likelihood. of 40.0 per cent on every grade that has not defaulted. The 40.0 per cent is that bank's own assumption and not anybody's requirement.

The multiplication runs grade by grade. The exposure sitting in the grade, multiplied by the grade's probability and by 40.0 per cent, gives the expected loss the model asks that grade to carry. Done nine times, it gives the whole book's expected loss for the year.

The nine grades worked end to end

The total hides the shape, so the nine expected loss figures matter more than the total. Every figure in this table belongs to the invented bank, and the loss given default of 40.0 per cent is that bank's own assumption applied unchanged to every grade.

Each expected loss row above is rounded to two decimals, so adding the printed column gives Rs 343.61 crore while the unrounded total is Rs 343.60 crore. The rounding difference is one paisa of arithmetic and no more.

Now go back to the prediction. Grades 1, 2 and 3 hold 27.0 per cent of the book and carry Rs 8.44 crore, being 2.5 per cent of the expected loss. Grades 4 and 5 hold 42.0 per cent and carry Rs 69.85 crore, being 20.3 per cent. Grades 6, 7, 8 and 9 hold 28.0 per cent and carry Rs 265.31 crore, being 77.2 per cent. Roughly a quarter of the book carries a fortieth of the expected loss, and roughly another quarter carries more than three quarters of it. Grade 9 alone holds 2.0 per cent of the book and carries 20.5 per cent of the expected loss, Rs 70.56 crore against Rs 0.35 crore from grade 1, a ratio of two hundred to one.

Sit with the middle panel and the bottom panel together for a moment. The bank has put most of its money in the middle of its own scale, and a sensibly run book looks exactly like that. But expected loss does not care where the money is, it cares where the money multiplied by the estimate is, and that quantity keeps climbing right to the bottom rung. Grade 9 is the smallest grade on the book and the largest single contributor to the expected loss, and no chart of exposures alone would ever have shown that.

Take one last look at that picture with two of the bank's real names in mind. C1, Nirjhar Industries Limited, is the largest single-name exposure on the book at Rs 1,680 crore funded, and it sits in grade 4. C9, Lohit Valley Tea Estates Limited, is one of the smallest of the top ten at Rs 600 crore funded, and it sits in grade 7. On funded exposure alone, C1 carries Rs 3.02 crore of expected loss and C9 carries Rs 8.64 crore. The smaller name carries almost three times the expected loss on thirty six per cent of the money, and any report ranked by exposure puts it six places lower.

Why do two grades that look nothing alike carry the same expected loss?

Look back at grade 7 and grade 8 in the table. Grade 7 carries a 3.60 per cent estimate on Rs 4,704 crore. Grade 8 carries 7.20 per cent on Rs 2,352 crore. Every visible property of those two rungs differs. One is twice as risky as the other by the bank's own estimate, and it holds half the money. And they contribute exactly the same expected loss: Rs 67,73,76,000 each, to the rupee, not approximately and not by luck.

The reason is that expected loss is a multiplication, and multiplication does not care which factor is which. Double one factor, halve the other, and the product stands still. Draw it as a rectangle with the exposure as the width and the estimate as the height, and the two grades are the same rectangle with the sides swapped around. Two grades three hundred basis points apart on the scale land on one number the moment the exposure is put beside them. The equality is the cleanest available proof that a probability of default on its own is a ranking and never a quantity of money.

What does the model ask for, and what does the bank actually hold?

The nine grades produce Rs 343.60 crore of expected loss for the year. Set that beside what Vindhya Commercial Bank Limited has actually put aside against the same exposures, namely Rs 494.4 crore of standard asset provisionA provision held against exposures that have not defaulted, which sits separately from provisions against those that have.. The bank is holding Rs 150.8 crore more than its own model asks for, and a reader meeting those two numbers for the first time will read that as prudence.

The reason for the difference is a rule rather than a judgement. The bank provides the higher of its own model output and the standard asset provision the applicable rules require of it, and on this book the second number is the higher one. The size of that requirement comes from the Reserve Bank of India. So the Rs 494.4 crore is not the bank disagreeing with its own model. The higher figure is the bank obeying a floor that sits above the model on these particular exposures.

What does a downturn do that the multiplier cannot show?

The control above is honest arithmetic and a poor description of a bad year. Scaling every estimate by one number says that every borrower on the book got worse by the same proportion on the same day. No downturn works like that. A downturn produces grade migrationA borrower moving from one grade to another, which is what a downturn actually does to a book.: names slide down the scale. The probability attached to grade 5 can sit exactly where it always was while the amount of money standing in grade 5 walks out and reappears in grade 6.

Work one instance. Move Rs 2,000 crore of exposure from grade 5 to grade 6 and change nothing else. Grade 5 falls to Rs 12,112 crore and grade 6 rises to Rs 10,232 crore. Both estimates stay at 0.90 and 1.80 per cent. The expected loss rises by Rs 2,000 crore times the 0.90 percentage point difference times 40.0 per cent, being Rs 7.20 crore, so the total moves from Rs 343.60 crore to Rs 350.80 crore. Expressed on the dial, that entire migration is a multiplier of 1.021.

Now push it as far as it will go on that one rung. Move the whole of grade 5, all Rs 14,112 crore of it, down to grade 6. The move adds Rs 50.80 crore and reaches Rs 394.41 crore, still Rs 99.99 crore short of the Rs 494.4 crore the bank holds, and a multiplier of only 1.148. Emptying the largest grade on the book one rung downward does not get two thirds of the way to the crossing the dial reaches at 1.44. The multiplier is a way of feeling the arithmetic and never a scenario.

How does a lender actually use the number once it has one?

Four uses, and they are worth separating because they make different demands on the same figure. The first is ranking, and it is the only one the probability can do alone. When two proposals arrive and the bank can fund one, the grade decides which name is the better credit. Nothing has to be multiplied out for that comparison to work. The ranking use therefore feels natural and the other three feel like extra arithmetic.

The second is pricing. A lender charging the same rate to a grade 3 name and a grade 8 name is being paid the same for two very different chances of not being paid at all. Grade 8's estimated chance of failure is thirty six times grade 3's on this bank's own scale. The probability enters the pricing conversation as one input among several, and what any particular rate should be is a separate question.

The third is sizing the provision, worked at length above: the estimate meets an exposure and a severity, and money comes out. The fourth is monitoring, and it is the one practitioners rate most highly. A book quietly sliding from grade 5 into grade 6 is losing money before a single borrower has missed anything. The interesting number is rarely today's grade, it is the movement of names between grades over the last two quarters.

An analyst standing outside the bank has none of this. The analyst cannot see grade 5. An outside analyst can see the disclosed split of the book, the provisions held, the credit cost charged for the year and the movement in the non-performing book. Reading those four together is an attempt to reconstruct from the outside what the ten grade table says on the inside. The gap between what a model asks and what is held is worth attention wherever it is visible.

The household version is short. Most people know which of their relatives they would lend Rs 10,000/- to and which they would not, and could put five of them in order without hesitating. None of them could say how much more likely the fifth is to not repay than the first, or how much would actually be lost in each case after everything recovered. The order comes easily and the quantity does not, and a lender that never does the second half of the work is running on the same instinct with a larger number of zeros.

Where do the approaches to estimating it come from?

The idea that a bank might use its own estimates rather than a set of externally supplied weights did not arrive from nowhere, and knowing where it came from is part of knowing what kind of number it is. The reasoning was written by the Basel Committee on Banking Supervision at the Bank for International Settlements. The Committee published both a standardised approach, where the weights come from outside the bank, and internal ratings based approaches, where a bank's own estimates of the probability of default may enter the computation under conditions.

The Basel Committee is where the idea originated. The Basel text is not the rule an Indian bank is held to. The Reserve Bank of India sets what Vindhya Commercial Bank Limited may actually use for capital purposes and what standard asset provision it must actually hold, and naming only the global standard is the confident and common error on this subject. The two sit in a fixed order: the Basel Committee is where the reasoning was published, and the Indian requirement is what binds an Indian balance sheet.