Modelling and estimating loss given default rates
Dirk Tasche Deutsche Bundesbank Banking and Financial Supervision Department dirk.tasche@gmx.net
Basel II and Credit Risk Modelling in Consumer Lending Southampton, September 2006
opinions expressed in this presentation are those of the author and do not necessarily reect views of the Deutsche Bundesbank.

Outline

Introduction (slide 1)
The Basel II model in brief (Slide 4)
Statistical issues with Basel II (Slide 7)
Conditionally independent LGDs and defaults (Slide 11)
Co-monotonic LGDs and defaults (Slide 16)

Conclusions (Slide 22)

Introduction (1)
Basel II internal ratings based approach (IRBA): Banks calculate regulatory capital requirements for exposures based on own estimations of risk parameters.
Retail portfolio: PD, LGD, EAD.
Sovereign, bank, and corporate portfolios: Foundation approach: PD. Advanced approach: PD, LGD, EAD, eective maturity.

Introduction (2)

PD: probability of default. EAD: (expected) exposure at default (in currency units). LGD: (expected) loss given default (as percentage of realized EAD). Basel II Framework provides guidance on how to estimate these parameters (446 to 479). LGD to be estimated as downturn LGD.

Basel

Committee on Banking Supervision: International Convergence of Capital Measurement and Capital Standards. A Revised Framework. Updated November 2005. Committee on Banking Supervision: Guidance on Paragraph 468 of the Framework Document. July 2005.

Introduction (3)

Goal with this presentation:
Provide background information on model assumptions that underly the requirements for parameter estimation.
Discuss some specic issues with (downturn) LGD estimation. The rationale behind the downturn concept. Which LGD concepts would be consistent with the Basel II credit portfolio model?
The Basel II model in brief (1)

EADA LGDA 1DA

(4.1) Basic model. L portfolio-wide loss, A obligors. DA default event, 1DA {0, 1} default indicator. EADA, LGDA random EAD and LGD of obligor A. EADA, LGDA independent and independent of all other random variables.
The Basel II model in brief (2)

DA = {

1 A A 1(P DA)}
P DA probability of default of A, standard normal distribution function.
S systematic risk factor, A idiosyncratic risk factors. S, (A)A independent and identically standard normally distributed (4.1) is probit-model if S is considered explanatory variable.
B asset correlation of A and B.
The Basel II model in brief (3)
L = 1(P DA) AS E[EADA] E[LGDA] 1 A A (6.1)
L is derived from (4.1) and (5.1) by taking expectation conditional on S, i.e. L = E[L | S]. If the number of obligors is large, L can be understood as asymptotic approximation to L. Independence assumption is crucial for the appearance of unconditional expectations E[EADA] and E[LGDA] in (6.1). Advantage of using (6.1): L is strictly decreasing dierentiable transformation of standard normal distribution.

e.g., Gordy, M. (2003) A Risk-Factor Model Foundation for Ratings-Based Bank Capital Rules. Journal of Financial Intermediation 12(3), 199-232.
Statistical issues with Basel II (1)
Is it reasonable to assume that both EAD and LGD are random variables?
Relevance, in particular, for credit lines.
In general, nite limits are assigned to credit lines.
Suggestion: Replace random EAD by deterministic limit and summarize uncertain exposure at default and LGD in one (random) percentage of loss.
Drawback: Extra LGDs for credit lines to be estimated.
Conditionally independent LGDs and defaults (2)
In practice, recovery rates are observed instead of LGD. Example: exposure EAD secured with two collaterals, with values Vi and recovery rates RRi, i = 1, 2. Recovery rate RRun for unsecured part. Expected loss at default: E[LAD] = E (EAD V1 RR1 V2 RR2) (1 RRun)
?? V1 V2 = E[EAD] 1 E[EAD] E[RR1] E[EAD] E[RR2] (1 E[RRun])
(6.1) to be modied accordingly. May EAD, RRun, RR1, RR2 and the default event be assumed independent?
Statistical issues with Basel II (3)
By assumption for (4.1), default events DA and loss-given-default rates LGDA are independent. Cannot be veried at single obligor level. At portfolio level: assume homogeneity with respect to E[LGDA], i.e. E[LGDA] = const = for all A. n number of obligors.
1 Realized default rate (within one period): DR = n A 1DA.

A LGDA 1DA A 1DA 9

Average realized loss-given-default rate: ALGD =
Statistical issues with Basel II (4)
Under assumptions of Slide 9: cov[DR, ALGD] = E[DR ALGD] E[DR] E[ALGD]

1 = nE A 1 =n A

LGDA 1DA ] E[DR] E[1DA ] E[DR]
=0 Hence, with independent default events and LGDs, realized default rates and average realized LGDs would be uncorrelated. In contrast to empirical evidence.
e.g., Altman, E.I., Brady, B., Resti, A. and A. Sironi (2004) The Link between Default and Recovery Rates. Working paper S-CDM-04-07, Salomon Center for the Study of Financial Institutions.
Conditionally independent LGDs and defaults (1)
Assumption: LGDA and DA independent conditional on S, EADA constant. (6.1) then has to be modied (E[LGDA | S] for E[LGDA]): 1(P DA) AS = L EADA E[LGDA | S] (11.1) 1 A A Example: LGDA =
with Y = rA S + 1 rA A, A standard normal and independent of S and all other variables.

Approach

A Y A , 1 A

(11.2)

presented here is similar to Pykhtin, M. (2003) Unexpected recovery risk, RISK 16(8), 7478.
Consequence of (11.2): E[LGDA | S] = A rA S A 1 A rA (12.1)
LGDA is only observed when obligor A defaults. Hence the observable distribution of LGDA is the distribution of LGDA conditional on DA. In particular:
2 ( A ,1 (P DA); ( A) = E[LGDA] < E[LGDA | DA] = P DA

A A rA )

(12.2) P[LGDA t] > P[LGDA t | DA]
denotes the bivariate standard normal distribution function.
Conditionally independent LGDs and defaults (3)

LGD densities

unconditional conditional on default

Density

Loss fraction

Parameter values

= 0.6, = 0.05, r = 0.75, P D = 0.1,
Conditionally independent LGDs and defaults (4)
Consequent extension of the Basel II risk weight formulae to the correlated LGD and default case as modelled by (11.1) would require evaluation of (12.1) at S = 1(0.001). Table shows bias encountered when E[LGDA | S = q(S)] is erroneously calculated with observable (i.e. conditional on default) distribution of LGD (dotted line in graph on Slide 13). Level LGD distribution conditional unconditional

5% 87.0% 82.5%

2.5% 1% 0.5% 0.1% E[LGDA | S = q(S)] 88.4% 89.8% 90.8% 92.5% 84.1% 85.8% 86.9% 88.9%

0.05% 93.1% 89.6%

= s quantile of S at level , i.e. s solution of P[S s] =.
Conditionally independent LGDs and defaults (5)
As indicated by (12.2), the conditional on default distribution of LGDA is not simple Estimating parameters for (11.2) is numerically demanding.
Additionally, databases seem insucient for reliable estimation of the parameters.
As a consequence, the Basel Committee decided to require direct estimation of downturn LGDs instead of evaluating (12.1).
Downturn LGD: roughly described as average LGD observed during economic downturns.
Co-monotonic LGDs and defaults (1)
Counter-intuitive features of conditional independence approach: By construction, LGD assumes a realization even if there is no default (cf. (4.1)). Consequence: Distributions of observable LGD and built-in-model (unconditional) LGD dier (cf. Slide 13).

Recourse to actuarial approach to aggregated claim modelling, by assuming co-monotonicity of LGD and default indicator.

Tasche,

D. (2004) The single risk factor approach to capital charges in case of correlated loss given default rates. Working paper.
Co-monotonic LGDs and defaults (2)
Basic idea: Replace product of variables LGDA 1DA in (4.1) by single variable XA 0 such that (i) {XA > 0} = DA = {

1 A A 1(P DA)}.

(ii) Distribution of XA conditional on DA reects observed LGD distribution FA of obligor A, i.e. P[XA x | XA > 0] = FA(x). Assume (for sake of simplicity) that FA is continuous and strictly increasing. By (ii), FA(0) = 0. EADA assumed constant.
Co-monotonic LGDs and defaults (3)
if ZA > P DA; 0, XA = F ZA , if ZA P DA; A P DA

(18.1)

ZA = ( {ZA P DA} = DA.

1 A A )

ZA is uniformly distributed on [0, 1]. Conditional on DA, 1 PZA is uniformly distributed on [0, 1]. D
Hence, conditional on DA, XA has distribution FA. Small realizations of ZA (obligor A deep in default) imply large realizations of XA (counter-monotonicity!).
Co-monotonic LGDs and defaults (4)
When applying the co-monotonic approach, (11.1) has to be modied to L =

EADA E[XA | S]

(19.1)
Calculating E[XA | S = s] for realization s of S: E[XA | S = s]
( A s+ 1 A A) 1 = E 1{( s+1 )P D } FA 1 P DA A A A A

1 (P DA ) A s 1 A

( A s+ 1 A z) 1 (z) FA 1 P DA

(19.2)

denotes the standard normal density.
Co-monotonic LGDs and defaults (5)
1 (P DA ) A S Comparing E[LGDA | S] 1 A

(cond. independence ap-

proach, (11.1)) and E[XA | S] (co-monotonicity approach, (19.1)).
LGDA from (11.2). Parameters as on Slide 13. FA(t) = P[LGDA t | DA]
1 = 1 P DA 1A 1 (t) , 1(P DA); rA A A

(20.1)

With identical observable LGD distributions, the co-monotonicity approach yields lower capital charges. Level Approach cond. independence co-monotonicity independence 2.5% 1% 0.5% 0.1% Capital charge per of exposure 22.4% 27.4% 33.8% 38.4% 48.4% 21.3% 25.7% 31.3% 35.4% 44.0% 21.0% 25.2% 30.5% 34.3% 42.2% 5% 0.05% 52.4% 47.4% 45.3%
Co-monotonic LGDs and defaults (6)
Co-monotonicity approach: Parameters A, P DA and LGD distribution FA to be estimated. Estimation of FA straightforward as, theoretically, built-in-model and observable LGD distribution are identical. Conditional independence approach: Parameters A, P DA, rA, A, A to be estimated. Estimation of rA, A, A non-trivial as built-in-model and observable LGD distribution are dierent. (19.2) numerically more demanding than (11.1) and (12.1).

Conclusions

The Basel Committee decided to take account of correlation between loss rates and default rates for regulatory Basel II capital requirements. Having considered the conditional independence approach, as a consequence of the inherent estimation problems the committee did not adopt this approach. Instead, the committee adopted a direct estimation (downturn LGD) approach which is not fully compatible with the underlying credit portfolio risk model. Although there is empirical evidence for correlation between loss rates and default rates, industry so far does not take account of it in their internal models.