Unexpected loss and default correlation · RiskAtoms Academy

By the end you will be able to

1Describe a credit loss distribution and measure unexpected loss as a quantile minus expected loss.
2Explain how default correlation widens the distribution, using the binomial case as the benchmark.
3Use the one-factor (Vasicek) model: conditional PDs, simulation, and the closed form at 99.9%.
4Read the corporate book's exact loss distribution and see why lumpiness and correlation both matter.

Why it matters

RiskAtoms Bank expects to lose £2.81m a year on its twelve corporate borrowers. If they defaulted independently, three or more defaults in one year would happen about once in 1,088 years. But borrowers share one economy: in a recession they weaken together. With a modest correlation the same event comes about once in 170 years. Capital exists for those years. Measuring them means modelling how defaults cluster.

Core explanation

Losses come in clusters

Illustrative dataRiskAtoms Bank plc's credit book

One made-up lending book on one made-up date (31 December, year 0), in pounds: a ten-grade master scale, a one-year rating transition matrix, twelve invented corporate borrowers (£587m of exposure, expected loss £2.81m), retail portfolios and twenty past workout cases. The firms are invented and the numbers are round illustrative values, not anyone's data. The same book runs through every lesson in Module 3.3.

Correlations here are illustrative: asset correlation 0.20 for the corporates and 0.10 for the personal loans.

Expected loss is the average (L3.3.10: Expected loss). Unexpected loss is how far a bad year can go beyond it. As with VaR (L2.5.06: Value at risk: the idea), we pick a high quantile of the one-year loss, a credit VaR, and subtract the expected loss. Banks typically use 99.9% or higher for capital. The shape of the distribution depends on one thing above all: whether defaults happen independently.

Independent defaults: the binomial benchmark

Take 100 equal personal loans from the bank's score band B, each with PD 3.5%. If they defaulted independently, the number of defaults would be binomial (M-C04: The binomial distribution): on average 3.5, standard deviation $\sqrt{100 \times 0.035 \times 0.965} = 1.84$ , and at most 10 defaults in 999 years out of 1,000. Independent defaults diversify fast: a big enough pool would lose almost exactly its EL every year.

Correlated defaults

Real defaults are not independent. A recession raises every borrower's PD at once. Two loans whose defaults tend to come together have a positive default correlation (M-C10: Covariance and correlation). In the model below, two band B loans have a joint default probability of 0.193% against 0.123% if independent: a default correlation of only 2.07%. Small as it is, it transforms the pool:

Independent (binomial)Correlated (asset correlation 0.10)

100 band B loans	Independent	Correlated
Average defaults	3.50	3.50
Standard deviation	1.84	3.21
P(no default)	2.8%	13.0%
P(10 or more)	0.27%	5.42%
99% quantile	8	15
99.9% quantile	10	21

The average does not move: correlation never changes expected loss. But good years get better (no default at all in 13% of years) and bad years much worse: the 99.9% quantile doubles from 10 to 21 defaults. The variance rises by a factor $1 + (n-1)\rho_D = 1 + 99 \times 0.0207 = 3.05$ .

The one-factor model

Where does such a correlation come from? Oldřich Vasicek's model gives each borrower a hidden, standard normal score (M-C06: The normal distribution), its asset value. The score is part systematic (the economy, $Z$ , shared by everyone) and part idiosyncratic ( $\varepsilon_i$ , the firm's own luck). The weight on the economy is set by the asset correlation $\rho$ . A borrower defaults when its score falls below $\Phi^{-1}(\text{PD})$ . Given the economy, defaults are independent, each with a conditional PD:

Kind of year (band B loan, PD 3.5%, ρ = 0.10)	Conditional PD
Boom (Z = +2)	0.50%
Average year (Z = 0)	2.81%
Recession (Z = −2)	10.69%
1-in-1,000 year (Z = −3.09)	18.95%

So a loan that defaults 3.5% of the time on average defaults 0.50% of the time in a boom and 10.7% in a recession. Correlated defaults are just independent defaults with a common, moving PD.

A very large pool: the closed form

In a very large pool the borrowers' own luck averages out and the default rate in a year is the conditional PD itself. The whole band B segment (14,000 loans) is close to that. Its default rate then has the Vasicek distribution: skewed, with most years below the average and a long right tail. The median year sees 2.81% default, below the 3.5% mean; the 1-in-1,000 year sees 18.9%.

Density of the yearly default rate, band B

Asset correlation	0	0.05	0.10	0.15	0.20
99.9% default rate, PD 3.5%	3.5%	12.5%	18.9%	25.2%	31.5%

The corporate book, exactly

Twelve borrowers are far from a large pool. With twelve names there are only $2^{12} = 4{,}096$ combinations of who defaults, so the bank can compute the loss distribution exactly, integrating over the economy for the correlated case:

Corporate book, one year	Independent	Asset correlation 0.20
Expected loss	£2.81m	£2.81m
Standard deviation of loss	£6.18m	£6.86m
P(no default)	77.6%	79.6%
P(two or more defaults)	2.01%	3.46%
P(three or more defaults)	0.092%	0.590%
99% loss quantile	£34.20m	£34.20m
99.9% loss quantile	£44.38m	£56.54m
Unexpected loss at 99.9%	£41.57m	£53.72m

IndependentAsset correlation 0.20

The steps are single names. At 99% both models land on £34.2m: one default of Bramwick Stores, the largest loss if default. Correlation shows further out, where it takes two or three defaults together: at 99.9% it adds £12.2m. Lumpiness matters as much. If each borrower sat in a very large pool of its own kind, the one-factor capital would add up to £24.4m; the book's actual 99.9% unexpected loss is £53.7m. The gap is concentration, the subject of L3.3.13: Concentration risk.

Exact or simulated?

Exact is possible here because the book is small. Real books have thousands of names and several factors, so banks simulate. A seeded run of 20,000 years (seed 33110) gives a 99.9% loss of £53.38m against the exact £56.54m: the 99.9% quantile rests on the worst 20 simulated years, so it is noisy, as in L3.2.11: Monte Carlo VaR.

The formal version

Variance of the number of defaults. Let $D = \sum_{i=1}^n \mathbf 1_i$ with $\Pr(\mathbf 1_i = 1) = p$ and pairwise default correlation $\rho_D$ . Each indicator has variance $p(1-p)$ , each pair covariance $\rho_D\,p(1-p)$ (M-C10: Covariance and correlation):

\begin{aligned} \operatorname{Var}(D) &= \sum_i \operatorname{Var}(\mathbf 1_i) + \sum_{i \ne j} \operatorname{Cov}(\mathbf 1_i, \mathbf 1_j) \\ &= n\,p(1-p) + n(n-1)\,\rho_D\,p(1-p) \\ &= n\,p(1-p)\,\bigl[1 + (n-1)\rho_D\bigr] \end{aligned}

With $\rho_D = 0$ it is the binomial variance. As $n$ grows, the default rate $D/n$ has variance tending to $\rho_D\,p(1-p)$ , not zero: correlated risk does not diversify away.

From asset to default correlation. Two borrowers default together when both scores fall below the threshold. Conditional on $Z$ they are independent, so

\begin{aligned} \Pr(\text{both}) &= \int_{-\infty}^{\infty} p(z)^2\,\varphi(z)\,dz = 0.001925 \\ \rho_D &= \frac{\Pr(\text{both}) - p^2}{p(1-p)} = \frac{0.001925 - 0.001225}{0.035 \times 0.965} = 0.0207 \end{aligned}

The large-pool quantile. The default rate tends to $p(Z) = \Phi\bigl((\Phi^{-1}(p) - \sqrt\rho\,Z)/\sqrt{1-\rho}\bigr)$ , which falls as $Z$ rises. So

\begin{aligned} \Pr\bigl(p(Z) \le x\bigr) &= \Pr\!\left(Z \ge \frac{\Phi^{-1}(p) - \sqrt{1-\rho}\,\Phi^{-1}(x)}{\sqrt\rho}\right) \\ &= \Phi\!\left(\frac{\sqrt{1-\rho}\,\Phi^{-1}(x) - \Phi^{-1}(p)}{\sqrt\rho}\right) \\ q_\alpha &= p\bigl(-\Phi^{-1}(\alpha)\bigr) = \Phi\!\left(\frac{\Phi^{-1}(p) + \sqrt\rho\,\Phi^{-1}(\alpha)}{\sqrt{1-\rho}}\right) \\ \text{UL}_\alpha &= \text{LGD} \times \text{EAD} \times (q_\alpha - p) \end{aligned}

Try it

Simulate the model yourself. Start with ρ = 0 and 100 loans: the bars match the binomial. Raise ρ and run again: the tail stretches. Switch to 1,000 loans: the simulated 99.9% moves towards the large-pool closed form. The years run in chunks between frames, so the page stays responsive.

Try itSimulate correlated defaults

Asset correlation ρ

Loans in the pool

Years to simulate

0 / 10,000 years

Exact 99.9% default rate21.0%

Simulated 99.9%–

Average default rate (exact)3.50%

Simulated average–

Standard deviation (exact)3.21%

Simulated standard deviation–

Very large pool, closed form at 99.9%18.9%

Worst simulated year–

Worked example

Unexpected loss on the band B personal loans

The segment: 14,000 loans, £130m outstanding, PD 3.5%, LGD 70%, asset correlation 0.10 (illustrative). Treat it as a very large pool.

The closed form at 99.9%

Step 1 of 5

1
Threshold and quantile
$\Phi^{-1}(0.035) = -1.8119, \qquad \Phi^{-1}(0.999) = 3.0902$

Regulatory lens · looking ahead

The Basel IRB risk-weight formula is this closed form. The Basel Committee's explanatory note (2005) sets the confidence level at 99.9% and builds on an adaptation of Merton's model to credit portfolios (the one-factor model of this lesson) and on Gordy's asymptotic single risk factor model, in which a bank's capital for a loan does not depend on the rest of its portfolio. It prescribes the asset correlations: between 12% and 24% for corporates, depending on PD; 15% for residential mortgages; 4% for qualifying revolving retail; and between 3% and 16% for other retail. It adds a maturity adjustment. The formula in full is in M4.2: Credit risk capital; multi-factor portfolio models, simulation with many names and copulas are in M6.5: Portfolio credit models.

Spark Atoms

Spark AtomA note that became a rule

Oldřich Vasicek first set out the limiting loss distribution in technical notes at KMV in 1987 and 1991. He published it in Risk in 2002, by which time a version of it was on its way into the Basel II capital formula used by banks around the world.

Spark AtomOne year in a thousand

The 99.9% confidence level means that, if the model were right, a bank holding exactly the capital the formula asks for would see its credit losses exceed that capital in about one year in a thousand. The Basel Committee's note admits the level “might seem rather high” and says it was chosen partly to protect against errors in banks' own PD, LGD and EAD estimates and other model uncertainties.

Key takeaways

Unexpected loss is a high quantile of the loss distribution minus expected loss; capital covers it.
Independent defaults are binomial and diversify away; correlated defaults do not.
A default correlation of only 2.1% triples the variance of defaults in a pool of 100: Var = n p(1 − p)[1 + (n − 1)ρ_D].
The one-factor model: defaults are independent given the economy, with conditional PD Φ((Φ⁻¹(PD) − √ρ Z)/√(1 − ρ)).
Large pool at 99.9%: q = Φ((Φ⁻¹(PD) + √ρ Φ⁻¹(0.999))/√(1 − ρ)); band B: 18.9%, UL £14.1m.
Small books are lumpy: the corporate book's 99.9% UL is £53.7m, more than the £24.4m the large-pool formula suggests.

Reaction

Check what stuck

Question 1 · objective 1

A bank has 200 equal loans with PD 5%, defaulting independently. What is the standard deviation of the number of defaults?

Question 2 · objective 2

The same 200 loans now have a default correlation of 0.02. By what factor does the variance of the number of defaults rise?

Question 3 · objective 3

In the one-factor model, what default rate does a very large pool with PD 1% and asset correlation 12% reach at the 99.9% quantile, in %?

%

Question 4 · objective 2

A default correlation of 2.1% sounds negligible. Why does it nearly double the spread of defaults in the 100-loan pool?

AEvery pair of loans adds a covariance term, and there are 99 × 100 ordered pairs against 100 variancesBBecause default correlation is always rounded upCBecause the PD rises with correlationDIt does not change the spread, only the mean

Question 5 · objective 4

The corporate book's 99% loss quantile is £34.2m with or without correlation. Why?

AThe book is lumpy: at 99% the loss is one default of Bramwick Stores, the largest loss-if-default, whichever model is usedBCorrelation never matters at 99%CThe simulation was not run long enoughDExpected loss is the same in both

Glossary

Unexpected loss (UL): Unexpected loss: how far the credit loss can exceed the expected loss, measured here as a high quantile of the loss minus EL (sometimes as the standard deviation of the loss).
Credit VaR: A high quantile of the credit loss distribution over a year, for example 99.9%. Capital is set to cover it minus expected loss.
Default correlation: The correlation between two borrowers' default indicators (1 if default, 0 if not).
Asset correlation: The correlation between two borrowers' asset values (or credit scores) in a threshold model of default.
Systematic factor: A risk factor shared by all borrowers, such as the state of the economy. It cannot be diversified away.
Idiosyncratic risk: The part of a borrower's risk that is its own: management, a lost contract, a fire. It diversifies away in a large book.
Conditional PD: The probability of default given the state of the economy: low in good years, high in bad ones.
Granular (very large) pool: A portfolio of so many small exposures that no single name matters: only the economy drives its default rate.

Deep Dives & Origins

Deep Dives

Basel Committee on Banking Supervision, An Explanatory Note on the Basel II IRB Risk Weight Functions, BIS, July 2005. Intermediate.
C. Bluhm, L. Overbeck and C. Wagner, Introduction to Credit Risk Modeling, 2nd ed., Chapman & Hall/CRC, 2010, chapter 2 on modelling correlated defaults (Bernoulli mixtures and one-factor models). Advanced.

Origins

O. A. Vasicek, “Loan portfolio value” (The distribution of loan portfolio value), Risk, December 2002.
O. A. Vasicek, Probability of Loss on Loan Portfolio, KMV Corporation, 1987 (as cited in Vasicek 2002).
M. B. Gordy, “A risk-factor model foundation for ratings-based bank capital rules”, Journal of Financial Intermediation 12(3), 2003.
Basel Committee on Banking Supervision, An Explanatory Note on the Basel II IRB Risk Weight Functions, BIS, 2005.

Watch this lesson

Video version

Prefer to watch? The same lesson as a 12-minute video will sit here, made from this page's script.