Charta is on Google Play: practise on the go. Get the app
Charta

Estimate a bank's default rate and its credit risk capital with the Vasicek model, and explain how that model is set up.

FRM® Part I · Valuation and Risk Models · Measuring Credit Risk · FRM1.VR.6.g

The idea

The Vasicek model estimates the default rate and capital for a large, homogeneous loan portfolio. It uses a one-factor Gaussian copula. Every borrower has the same one-year default probability PD and the same copula correlation ρ with one common factor. The portfolio is large enough that idiosyncratic risk diversifies away, so given the factor the default rate is fixed. The worst-case default rate over one year at confidence X is WCDR = N[(N⁻¹(PD) + √ρ × N⁻¹(X)) ÷ √(1 − ρ)]. Here N is the standard normal cumulative distribution. Credit risk capital is the loss at the confidence level minus expected loss. In symbols, Capital = EAD × LGD × (WCDR − PD). The Basel internal-ratings formula uses a confidence level of 99.9 percent. Watch the signs: N⁻¹(X) is positive for X above 50 percent. A higher ρ raises WCDR sharply, and with zero correlation WCDR equals PD.

Formulas

Vasicek worst-case default rate

WCDR = N[(N⁻¹(PD) + √ρ × N⁻¹(X)) ÷ √(1 − ρ)]

N
the standard normal cumulative distribution function, with N⁻¹ its inverse
PD
the one-year default probability of each loan
ρ
the copula correlation
X
the confidence level

For a large homogeneous portfolio. N⁻¹(X) is positive above 50 percent and N⁻¹(PD) negative for small PD; with ρ = 0 the WCDR equals PD.

Vasicek credit risk capital

capital = EAD × LGD × (WCDR − PD)

EAD
exposure at default
LGD
loss given default
WCDR
the worst-case default rate at the confidence level
PD
the one-year default probability

The loss at the confidence level less expected loss. Basel's internal ratings formula uses 99.9 percent; forgetting to subtract PD gives the full tail loss instead.

A practice question

From the question bank, with the reason each wrong option is wrong.

Which description fits the setup of the Vasicek model of portfolio default rates?

  1. A multi-factor model in which each pair of borrowers has its own correlation and default probability.Why not: Vasicek uses one common factor with identical PD and correlation across loans.
  2. A binomial model in which defaults are independent with probability equal to the average PD.Why not: Vasicek is built to introduce dependence through the common factor; it is not an independent binomial model.
  3. A one-factor Gaussian copula with identical PD and correlation for all loans, applied to a large portfolio so idiosyncratic risk diversifies away. - correct
  4. A rating-migration model that revalues each loan at its new rating at the horizon.Why not: That describes CreditMetrics; Vasicek models only default and no-default.

Given the factor, defaults are independent with a known conditional probability, and with a large portfolio the realised default rate equals that conditional probability. The only uncertainty is the factor.

A harder one

Using the Vasicek model, a portfolio has a one-year PD of 1% for each loan, a copula correlation of 0.12 and a confidence level of 99.9% (N^-1(0.999) = 3.090; N^-1(0.01) = -2.326). The worst-case one-year default rate is closest to:

  1. 0.01%Why not: Uses a minus sign on the factor term, which would give a best-case rather than worst-case default rate.
  2. 9.0% - correct
  3. 1.3%Why not: Uses rho and 1 - rho instead of their square roots.
  4. 1.0%Why not: Reports the average PD; with a positive correlation the worst case is far above it.

WCDR = N[(-2.326 + sqrt(0.12) x 3.090) / sqrt(0.88)] = N[(-2.326 + 1.070) / 0.938] = N[-1.339] = 9.0%.

The hardest tier

In a Vasicek model with PD 1% and a 99.9% confidence level, the copula correlation is raised from 0.10 to 0.20. Capital per unit of EAD x LGD is WCDR - PD. By what factor does it change, approximately?

  1. It rises by a factor of about 2.01 - correct
  2. It rises by a factor of about 1.41, the square root of the correlation ratioWhy not: Assumes capital scales with the square root of the correlation; the dependence is through the normal CDF and is much stronger.
  3. It rises by a factor of about 1.88Why not: Compares the WCDRs and forgets to deduct the 1% PD (expected loss) from each before taking the ratio.
  4. It stays unchanged, because PD and the confidence level are unchangedWhy not: Ignores that a higher correlation raises the worst-case default rate.

At rho = 0.10: WCDR = 7.75%, capital 6.75%. At rho = 0.20: WCDR = 14.55%, capital 13.55%. Ratio 2.01: capital is very sensitive to the correlation assumption.

In the app: 6 practice questions on this outcome, spaced review that brings it back before you forget it.

Start free →

More in Measuring Credit Risk