Common mistake 1 with Credit risk & Basel capital

Treating EL and capital as additive buffers for the same risk. EL is priced; capital is for the unexpected part.

Common mistake 2 with Credit risk & Basel capital

Assuming defaults are independent. That makes a large portfolio look almost riskless.

Common mistake 3 with Credit risk & Basel capital

Using a through-the-cycle PD in a stress scenario, or the reverse.

Common mistake 4 with Credit risk & Basel capital

Applying the corporate correlation formula to retail or mortgage exposures, which have their own.

EconStatsCraft › Credit risk & Basel capital

Credit risk & Basel capital

Expected and unexpected loss from PD, LGD and EAD, then the Vasicek one-factor portfolio: the worst-case default rate, credit VaR, Basel IRB capital with the maturity adjustment, risk-weighted assets and the portfolio loss distribution.

The intuition

Lending to one borrower, you either get paid or you lose LGD of the loan. Lending to thousands, the loss rate becomes predictable, except that borrowers tend to default together when the economy turns. Expected loss is the average year, priced into the loan. Capital is for the bad year, and how bad that year gets depends on how strongly defaults move together.

When to use it

Pricing a loan, setting a credit reserve, computing regulatory capital under Basel IRB, or answering the credit risk questions in FRM Part II and CFA Level II.

Where the estimator comes from

In the Vasicek model each borrower's asset return is A_i=\sqrt{\rho}\,Z+\sqrt{1-\rho}\,\varepsilon_i, with a common factor Z, and default occurs when A_i<\Phi^{-1}(PD). Conditional on Z, defaults are independent, with probability \Phi\!\left(\frac{\Phi^{-1}(PD)-\sqrt{\rho}Z}{\sqrt{1-\rho}}\right). In a large portfolio this is the default rate. Setting Z at its 0.1% quantile gives the worst-case default rate, and Basel capital is LGD times the excess of that over PD, scaled for maturity.

The model

EL = PD\cdot LGD\cdot EAD,\qquad \text{WCDR} = \Phi\!\left(\frac{\Phi^{-1}(PD)+\sqrt{\rho}\,\Phi^{-1}(X)}{\sqrt{1-\rho}}\right),\qquad K = LGD\,(\text{WCDR}-PD)\cdot MA

What it assumes

A large, fine-grained portfolio of similar loans, so idiosyncratic risk diversifies away and only the common factor remains.
One systematic factor drives every borrower, with the same correlation ρ.
LGD and EAD are fixed numbers, not random. In reality LGD rises in recessions, exactly when defaults do.
Basel sets ρ from PD for corporate exposures; you can override it.

Statistical properties

With ρ = 0 a large portfolio's default rate would equal PD with near certainty; correlation is what makes credit risky.
The loss distribution is strongly right-skewed: most years are quiet, a few are very bad.
Capital rises with PD, but less than proportionally, because Basel lowers ρ as PD rises.
The maturity adjustment raises capital for longer loans, which have more time to be downgraded.

How to read the output

Expected loss is a cost, covered by the interest margin. Capital covers the loss above expected at 99.9% confidence.
The worst-case default rate can be many times PD. With PD 2% and ρ 0.16 it is about 19%.
Higher correlation fattens the tail: defaults cluster, and capital must rise even though the average is unchanged.
The break-even spread, about PD × LGD, is the extra yield needed just to cover expected loss, before any return on capital.

A worked reading

A portfolio of 100 loans of 1,000,000 each, with a 2% one-year PD and 45% LGD. Expected loss is 2% × 45% × 100,000,000 = 900,000 a year, a cost covered by charging about 90 basis points of spread. Basel sets the correlation at 0.164 for this PD, and at 99.9% the worst-case default rate is 19.0%, nearly ten times the average. The credit VaR beyond expected loss is about 7.66 million, and IRB capital with the maturity adjustment is 9.19% of exposure.

Symptom → diagnosis → what to do

If you see this	It means	Do this
Capital lower than expected loss	Capital is set beyond EL, by design	Correct; EL is provisioned separately
Worst-case default rate near 100%	Correlation set very high	Check ρ; Basel corporates use 0.12 to 0.24
Results do not change with portfolio size	Correct for the asymptotic model	Size only scales the cash amounts
Break-even spread looks small	It covers EL only	Add a return on capital and costs for a full price

Common mistakes

Treating EL and capital as additive buffers for the same risk. EL is priced; capital is for the unexpected part.
Assuming defaults are independent. That makes a large portfolio look almost riskless.
Using a through-the-cycle PD in a stress scenario, or the reverse.
Applying the corporate correlation formula to retail or mortgage exposures, which have their own.

Where to read this properly

Vasicek (2002), The Distribution of Loan Portfolio Value. Risk, 15(12).
Basel Committee on Banking Supervision (2005), An Explanatory Note on the Basel II IRB Risk Weight Functions.
GARP, FRM Part II, Credit Risk Measurement and Management.

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Credit risk & Basel capital

The intuition

When to use it

Where the estimator comes from

The model

What it assumes

Statistical properties

How to read the output

A worked reading

Symptom → diagnosis → what to do

Common mistakes

Where to read this properly

See also

Related Finance & risk

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