Probability of default · RiskAtoms Academy

By the end you will be able to

1Tell a through-the-cycle PD from a point-in-time PD, and say which uses need which.
2Estimate a grade's PD from default history, and from ratings mapped to agency data.
3Put an upper bound on the PD of a low-default portfolio.
4Update a PD on new information with Bayes' rule, in probability and odds form.

Why it matters

In year −4 of RiskAtoms Bank's history, 6.8% of its 2,280 rated companies defaulted. In year −8 only 1.6% did. Over ten years the rate was 2.6%. Which number is the PD? All three, depending on the question being asked. And when Larkhollow Freight's early-warning flag goes up, the right PD for it changes again.

Core explanation

Where a PD comes from

L2.2.03: Credit and counterparty risk introduced the probability of default; L3.3.04: Internal ratings and master scales attached one to each grade of the master scale. This lesson asks how the bank knows those numbers are right, and when to move them.

Illustrative dataRiskAtoms Bank plc's credit book

One made-up lending book, in pounds, at one made-up year end: twelve invented corporate borrowers (exposure at default £587m, expected loss £2.81m a year), 152,100 retail accounts in mortgages, personal loans and cards (exposure £4.485bn), a ten-grade master scale and an illustrative rating transition matrix. No firm or person in it is real. The same book runs through every lesson in Module 3.3, so the figures agree from lesson to lesson.

This lesson also uses ten years of the bank's default history for its whole rated book of 2,280 companies, simulated from the master-scale PDs and an economic cycle with a recession in years −5 to −3. Seeded, so the counts never change.

Through the cycle or point in time

A point-in-time (PIT) PD answers “how likely is default in the next twelve months, given everything we know today?” It rises in a recession. A through-the-cycle (TTC) PD answers “how likely is default in an average year?” It is steadier. Real rating systems sit somewhere between the two. Pricing, early-warning and accounting provisions (L3.3.18: Provisioning basics) want PIT figures; regulatory capital and limits want TTC ones, so that they do not swing with the economy.

Take Larkhollow Freight, grade 7, TTC PD 2.26%. A PIT system that scaled its PD with the cycle would have moved it between grades 6 and 8 over the ten years, and its expected loss between £138k and £506k a year. The TTC grade stays at 7 throughout.

Sort itTTC or PIT?

Which kind of PD fits each use best?

Setting IRB regulatory capital for the corporate book.
Deciding which borrowers to put on the watchlist this month.
Setting a counterparty limit meant to last through the cycle.
Estimating the next twelve months' expected credit losses for the accounts.

From default history

The direct way to check a grade's PD is to count. Watch the grade's borrowers each year, count who defaults under the bank's definition of default, and pool the years. Each year's rate is a PIT snapshot; the pooled rate over a full cycle is a TTC estimate.

ExploreTen years of defaults, grade by grade

Each bar is one year's default rate among the bank's 380 grade-7 borrowers.

Grade7

one year (point in time) pooled ten-year rate 2.32% (88 defaults in 3,800 borrower-years) master-scale PD 2.26%

Worst year 4.74%, best year 0.79%. The recession in years −5 to −3 lifts every grade at once.

Grade	Borrower-years	Defaults	Pooled rate	Master PD	95% upper bound
1	400	0	0.00%	0.03%	0.75%
2	900	0	0.00%	0.07%	0.33%
3	1,600	2	0.13%	0.14%	0.39%
4	3,000	9	0.30%	0.28%	0.52%
5	4,200	25	0.60%	0.57%	0.83%
6	4,500	50	1.11%	1.13%	1.40%
7	3,800	88	2.32%	2.26%	2.76%
8	2,500	111	4.44%	4.53%	5.18%
9	1,400	161	11.50%	11.31%	13.00%
10	500	152	30.40%	30.00%	33.96%

From grade 4 down, the pooled rates sit close to the master scale: the scale is well calibrated there. The other route is the one L3.3.05: External ratings in practice built: map each grade to agency ratings and borrow the agencies' decades of default data, which helps most where the bank's own history is thin.

When there are almost no defaults

Grades 1 and 2 show no defaults at all in 1,300 borrower-years. That does not mean their PD is zero. In a low-default portfolio the question becomes: how high could the PD be and still make zero defaults plausible? The answer is an upper confidence bound. For grade 1 it is 0.75% at 95%, against a master-scale PD of 0.03%: the data cannot tell grade 1 from grade 5.

ExploreHow high could the PD be?

Set the borrower-years observed and the defaults seen among them.

Borrower-years observed400

Defaults seen0

Confidence

Observed rate0.000%

Upper bound at 95%0.746%

Grade of the bound5

With no defaults the bound is 1 − (1 − confidence)^1/n, close to 3/n at 95%. It assumes defaults are independent; in practice they cluster in recessions, which widens the bound further.

The formal version

Three estimators

With $n_{k,t}$ borrowers in grade $k$ at the start of year $t$ and $d_{k,t}$ defaults during it, the pooled long-run estimate is

\hat p_k = \frac{\sum_t d_{k,t}}{\sum_t n_{k,t}}, \qquad \hat p_{7} = \frac{88}{3{,}800} = 2.32\%

With $d$ defaults in $n$ independent borrower-years, the upper bound at confidence $\gamma$ is the largest $p$ under which seeing $d$ or fewer is still at least $1-\gamma$ likely. With no defaults it has a closed form:

\begin{aligned} P(D \le d \mid n, p) &\ge 1 - \gamma \\ d = 0:\quad (1-p)^n = 1-\gamma \;&\Rightarrow\; p^{*} = 1 - (1-\gamma)^{1/n} \approx \frac{-\ln(1-\gamma)}{n} \\ n = 400,\ \gamma = 95\%:\quad p^{*} &= 0.75\% \end{aligned}

To push the grade-1 bound down to its 0.03% PD would take 9,984 borrower-years without a default: about 250 years at the bank's current 40 grade-1 names. For grade 3, with 2 defaults in 1,600 borrower-years, the bound is 0.39%, found by solving the binomial condition numerically (M-C04: The binomial distribution).

Finally, Bayes' rule (M-C12: Conditional probability and Bayes' rule) updates a prior PD on a signal $S$ to a posterior. Step through it for Larkhollow Freight:

\frac{P(D \mid S)}{P(\bar D \mid S)} = \underbrace{\frac{P(S \mid D)}{P(S \mid \bar D)}}_{\text{likelihood ratio}} \times \frac{P(D)}{P(\bar D)}

Try it

Pick a borrower and tell the bank what its early-warning flag says. Notice how far the same flag moves a grade-3 PD and a grade-9 one, and what happens when the flag stays quiet.

Try itUpdate a PD on new information

Borrower

Early-warning flag

Covenant breach this quarter

Flag's hit rate, P(flag | default)60%

Flag's false-alarm rate, P(flag | no default)5%

Of 10,000 borrowers like this one, about 226 default in a year. The flag catches 136 of them, and also fires for 489 of the 9,774 that survive. So a raised flag means default about 136 times in 624: 21.7%.

PD before (grade PD)2.26%

Combined likelihood ratio12.00×

PD after21.72%

Grade7 → 10

EL before£230,068

EL at the new grade£3,054,000

The covenant breach uses its own rates (50% hit, 4% false alarm), and the two signals are treated as independent given the outcome, an assumption to test, since both feed on the same weak trading. Rates are illustrative and match M-C12.

Worked example

Larkhollow Freight raises the flag

The bank's early-warning model looks at account behaviour, overdraft use and late filings. Over past years, 60% of borrowers that defaulted within a year had raised its flag, and 5% of those that did not default raised it too (illustrative rates, the same as in M-C12: Conditional probability and Bayes' rule). This quarter Larkhollow Freight's flag goes up.

Of 10,000 grade-7 borrowers	Flag raised	Flag quiet	Total
Default within a year	136	90	226
Survive	489	9,285	9,774
Total	624	9,376	10,000

P(D \mid S) = \frac{0.60 \times 2.26\%}{0.60 \times 2.26\% + 0.05 \times 97.74\%} = 21.72\%

The PD rises from 2.26% to 21.7%: past the 20% edge of grade 10, the watchlist. At the posterior PD, with LGD 35.1% on EAD £29m, EL is £2.21m a year; at grade10's central PD of 30% it is £3.05m, against £230k before.
The flag is not a verdict. Of every 624 grade-7 borrowers that flag, 489 survive the year. A credit officer reviews the file before the grade moves, as L3.3.04: Internal ratings and master scales required of overrides.
Had the flag stayed quiet, Bayes' rule would have lowered the PD to 0.96%, grade 6 (EL £115k). Silence is information too, though most banks act only on the flags.
If the quarterly accounts also show a covenant breach (50% hit, 4% false alarm), the odds multiply again by 12.5: from 0.0231 to 3.47, a PD of 77.6%. That assumes the two signals are independent given the outcome. Both feed on the same weak trading, so the true figure is lower.

Prior matters

The same flag on a grade-3 borrower (PD 0.14%) gives a posterior of only 1.65%. Rare events stay fairly rare even after a strong signal: most flags on good borrowers are false alarms.

Regulatory lens

The Basel IRB standard sets the rules for PDs used in capital. Default means more than 90 days past due on a material obligation, or unlikely to pay in full (CRE36.69). A grade's PD must be a long-run average of one-year default rates (CRE36.64), from an observation period of at least five years for at least one source (CRE36.80). Banks may estimate it from internal default experience, by mapping to an agency's scale, or as the average of a statistical model's PDs, and may pool data across institutions (CRE36.78–79). Every estimate carries a margin of conservatism, larger where data are limited (CRE36.68 and 36.79). And no corporate PD may fall below a floor of 0.05% (CRE32.4), which settles the grade-1 question in this lesson by rule rather than by data. Capital itself comes in M4.2: Credit risk capital.

Spark Atoms

Spark AtomPublished after his death

Thomas Bayes never published his rule. He died in 1761; his friend Richard Price found the essay among his papers and had it read to the Royal Society, which printed it in its Philosophical Transactions in 1763, as “An Essay towards solving a Problem in the Doctrine of Chances”.

Spark AtomThe most prudent estimate

In 2004 Katja Pluto and Dirk Tasche proposed a way to set PDs for low-default portfolios: assume the grades are ordered correctly and take the upper confidence bound for each, working from the best grade down. Their “most prudent estimation” is now widely used for sovereign, bank and top-rated corporate books.

Key takeaways

A PIT PD moves with the economy; a TTC PD averages over the cycle. Capital wants TTC; early warning and provisions want PIT.
Pool defaults over a full cycle to estimate a grade's TTC PD: defaults over borrower-years. The book's 10 years include one recession.
With few or no defaults, report an upper confidence bound; with none it is about 3/n at 95%.
Bayes' rule updates a PD on a signal; in odds form each signal multiplies the odds by its likelihood ratio.
A low prior keeps the posterior modest: most flags on good borrowers are false alarms.

Reaction

Check what stuck

Question 1 · objective 1

In a recession, which kind of PD rises sharply?

AThrough-the-cycleBPoint-in-timeCNeitherDBoth by the same amount

Question 2 · objective 2

A grade had 150 borrowers in each of four years and 12 defaults in total. What is its pooled default rate, in %?

%

Question 3 · objective 3

A portfolio shows no defaults in 1,000 borrower-years. What is the 95% upper bound for its PD, in % (two decimals)?

%

Question 4 · objective 4

A borrower's PD is 1%. A signal fires for 50% of future defaulters and 5% of survivors. It fires. What is the PD now, in % (one decimal)?

%

Question 5 · objective 2

Under the Basel IRB rules, the PD a bank assigns to a grade must be…

Athis year's default rate for the gradeBa long-run average of one-year default rates, from at least five years of dataCthe agency default rate for the matching letter, with no adjustmentDthe highest default rate seen in any year

Glossary

Probability of default (PD): Probability of default: the chance that a borrower defaults within a stated horizon, usually one year.
Through the cycle (TTC): Through the cycle: a PD meant to hold on average over a full economic cycle, so it moves little in booms and recessions.
Point in time (PIT): Point in time: a PD for the next twelve months given conditions now, so it rises in recessions and falls in booms.
Default (regulatory definition): For capital purposes: the borrower is more than 90 days past due on a material obligation, or the bank judges it unlikely to pay in full.
Pooled default rate: The total number of defaults over all years divided by the total number of borrower-years observed.
Borrower-year: One borrower observed for one year. 400 borrowers watched for ten years give 4,000 borrower-years.
Low-default portfolio: Low-default portfolio: a book with so few defaults (sovereigns, banks, top-rated companies) that default rates alone cannot pin down the PD.
Upper confidence bound: The highest PD that is still reasonably consistent with the defaults observed, at a chosen confidence level.
Prior: The probability you hold before new information arrives: here, the grade's PD.
Posterior: The probability after updating on new information, by Bayes' rule.
Likelihood ratio: Likelihood ratio: how much more often a signal fires for borrowers that default than for those that do not, P(S | D) / P(S | not D).
Margin of conservatism: An add-on to a PD estimate to allow for weak data or methods, required of IRB banks.

Deep Dives & Origins

Deep Dives

K. Pluto and D. Tasche, “Estimating probabilities of default for low default portfolios”, arXiv:cond-mat/0411699, 2004. Advanced.
B. Engelmann and R. Rauhmeier (eds), The Basel II Risk Parameters, 2nd ed., Springer, 2011. Intermediate.

Origins

Basel Committee on Banking Supervision, Basel Framework, CRE36: IRB approach: minimum requirements to use IRB approach.
Basel Committee on Banking Supervision, Basel Framework, CRE32: IRB approach: risk components.
T. Bayes, “An Essay towards solving a Problem in the Doctrine of Chances”, Philosophical Transactions of the Royal Society of London 53, 1763, pp. 370–418.

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