the basel ii rwa formula
TRANSCRIPT
The Basel II Risk-Weighted Assets FormulaMichael Walker
[email protected]/02/02
Vasicek1 considers the fractional number of defaults in a portfolio of a large number of riskycorrelated loans. He finds that the probability that the fractional number of defaults is less that θ is
W (θ) = N
[√1−RN−1(θ)−N−1(PD)√
R
].
where PD is the probability of default on a single loan and R is the correlation parameter, assumedthe same for all pairs of loans.
Define the “fractional defaults at risk,” FDaR, to be the fractional number of defaults that willnot be exceeded within a 99.9% confidence level. Then
W (FDaR) = 0.999
Solving for FDaR gives
FDaR = N
[N−1(PD) +
√RN−1(0.999)√
(1−R)
].
As stated above, this calculation is valid in the limit that the number of loans in the portfoliobecomes very large. In this limit, and in the absence of correlation (i.e. R → 0) the binomialdistribution for the fractional number of defaults θ is accurately described by a one-point densitywith its weight concentrated at θ = PD. Thus FDaR = PD. Also, for very strong correlation(i.e. R → 1), the distribution of fractional defaults is described by a two-point density with weight1− PD at θ = 0 and weight PD at θ = 1. Thus FDaR = 1 for PD > 0.001, while FDaR = 0 forPD < 0.001. These intuitive results are in agreement with those of the above formula.
The Basel II formula for (corporate, etc.) risk-weighted assets is now
Risk-Weighted Assets = 12.50× FDaR× LGD × EAD ×MatAd,
where LGD is the loss given default, EAD is the exposure at default, and MatAd is a maturityadjustment. Taking 8% of the risk-weighted assets gives the maximum loss at the 99.9% confidencelevel, and the required buffer capital is set equal to this loss.
The formula for the correlation parameter given in Basel II, which has an empirical basis,2 canbe simplified to
R = 0.12[1 + exp(−50PD)].
The maturity adjustment given in Basel II has the form
MatAd =1 + (M − 2.5)× b× PD
1− 1.5× b× PD
whereb = [0.08451− 0.05898× log(PD)]2.
Here M is an effective maturity, constrained to lie between one year and 5 years (IRB approach).In the calculation of FDaR, losses are deemed to occur only in the event of default. However, it isclear that when, say, a two-year AA-rated loan downgrades after one year to grade BB, there arealso credit losses which should be taken into account.3 There is of course no such adjustment, i.e.MatAd = 1, when the effective maturity M equals the time horizon of one year. This intuition isin agreement with the above formula for MatAd.
1 O. Vasicek, Probability of Loss on a Loan Portfolio, KMV (1987,1991).2Jose A. Lopez, The Empirical Relationship between Average Asset Correlation, Firm Probability ofDefault and Asset Size, (find using Google).3Michael B Gordy, A Risk-Factor Model Foundation for Ratings-Based Bank Capital Rules, J. Fi-nancial Intermediation 12, 199 (2003).