A Brief Note on Implied Historical LGD

A Brief Note on Implied Historical LGD Rogério F. Porto Bank of Brazil SBS Q.1 Bl. C Lt.32, Ed. Sede III, 10.andar, 70073-901, Brasília, Brazil rogerio@bb.com.br +55-61-3310-3753 March 14, 2011 1

Executive Summary We discuss, in a formal way, the implied historical method used to evaluate loss given default (LGD), mainly for Basel II requirements. For an ex post evaluation, we show that, under some mild assumptions, this method gives results equivalent to the workout LGD method when the average expositions at default for defaulted and non-defaulted assets are equal. We also show that when these exposures are not equal, the difference between the two methods is more pronounced for a bigger probability of default. Finally, we discuss how the ex post and the ex ante evaluations are asymptotically equal. keywords: credit risk, loss given default, recovery rates, implied historical method, Basel II, New Accord. 2

1 Introduction Financial institutions (FIs) that provide credit to their clients are subject to default risk. To enforce that the FIs will positively contribute to the financial system s wealth, regulators, at several countries, usually ask for capital allocations in order to maintain a buffer for unexpected and non mitigated defaults. This requirement is, among others, a subject in the Basel s New Accord, named Basel II (Basel Committee on Banking Supervision, 2004, p. 2). A typical default, of interest to credit risk management of retail portfolios, occurswhentheclientdoesnotpaypartoforalltheduesinacontract. More generally, any predicted cash inflow that is not observed due to a contractual default, in a financial instrument, can be considered as an event of interest. This event is called default. For credit risk measurement, following Basel II, we can attach, to any financial asset and at each moment in time, a monetary value or exposure (Basel Committee on Banking Supervision, 2006, p.53, 218), a maturity (M), a probability of default (PD) and an economic loss related to a possible default. Generally, it is reasonable to assume that, if there is no default, then there is no economic loss as well. After a default event, the FIs try to minimize their economic loss through asset recovering actions like direct debt collection, guarantees selling, renewals or asset selling. Most of the regulators under Basel II, permit that these informations can be used in an internal ratings-based approach (IRB) for credit risk management (Basel Committee on Banking Supervision, 2006, p.52, 211). These informations originate the so called credit risk parameters. Measuring these parameters require definitions, evaluations and data bases. In this paper, we discuss some aspects related to the evaluation of one of these parameters, the loss given default (LGD). There are basically four methods to evaluate the LGD. Two of them use financial market public data and will not be discussed here. The other two use FIs internal data and are discussed in this paper. In fact, we give sufficient and necessary conditions for their equivalence. Some practical implications are discussed through the paper but statistical aspects are not considered. One of these two methods can be applied to virtually any asset, but the other can only be applied to retail assets, according to the actual Basel II requirements. Thus, during the paper, buckets or cohorts of retail assets are our main focus. The paper is organized in the following way. In Section 2 we review the main definitions. In Section 3, LGD is discussed for one unique financial asset. Its extension for a portfolio of assets is done in the following Section 4. A second method to evaluate LGD is discussed in Sections 5 and 6, for two 3

moments in time: ex post and ex ante facto. Finally, in Section 7 we collect the conclusions of our study. Mathematical proofs are in Section 8. 2 Definitions In this section we roughly review some definitions that will be useful along the paper. Details can be consulted on the cited bibliography. The asset maturity (M) will not be considered. The definition of default includes various aspects (Basel Committee on Banking Supervision, 2006, 452) but usually considers whether the obligor is past due on any material credit obligation. The idea is to achieve a definition that does not generate lots of defaults while keeping FIs credit risk comparable. At the time of default, the monetary value of the exposure is called exposure at default (EAD). For an asset portfolio, we can evaluate ex post the probability of randomly choosing a defaulted asset, called probability of default, by PD = D/N, where D denotes the number of defaulted assets among N assets, in a fixed time frame H (Basel Committee on Banking Supervision, 2005a, p. 18). If the definition of default is adequate, it is reasonable to suppose that there is neither economic loss nor profit bigger than the EAD. Then we consider LGD as the ratio of economic loss to EAD for each asset (Basel Committee on Banking Supervision, 2006, 297), (Basel Committee on Banking Supervision, 2005a, p. 61). Ex post values for PD and LGD are also called default rate and loss given default rate (or realized LGD), respectively (Basel Committee on Banking Supervision, 2005b, p. 2). The EAD, M, PD and LGD are called credit risk parameters. 3 Individual LGD Consider the individual evaluation of LGD. For this, let LGD i,t0 the LGD for an asset i, i = 1,...,D, evaluated at the time t 0 when default occurs. Up to the time when the asset i is totally recovered, consider that the FI receives J i values R i,j > 0 at times t R i,j, j = 1,...,J i. That is, cash inflows like debts collected, sold guarantees etc. Also, consider that the FI pays K i values P i,k > 0 at times t P i,k, k = 1,...,K i. That is, cash outflows like costs, fees etc. 4

For adequate comparison with the of asset i at t 0, the values R i,j and P i,k need to be discounted by rates r i,j r ( t 0,ti,j) R and si,k s ( t 0,ti,k) P, respectively, for j = 1,...,J i and k = 1,...,K i. Thus, LGD i,t0 = 1 Ji R i,j j=1 r i,j K i P i,k k=1 s i,k = 1 Ji j=1r i,j K i k=1 P i,k, is called realized LGD (Basel Committee on Banking Supervision, 2005a, p. 66). Note that the LGD can be evaluated at any time t and not only at the time t 0 of default. For a fixed time of default t 0, the values of, R i,j and P i,j simply need to be discounted from t 0 to t, to reflect the time value of money. Note that this is not a redefinition of the time of default from t 0 to t that, to be done right, should take care of its impact on the values of, R i,j and P i,j. The important point is that the economic loss must be evaluated at the same time as the EAD, in order to keep monetary comparability and to guarantee that the LGD lies between 0 and 1. If LGD evaluation is done before the asset is completely recovered, the realized LGD is called workout LGD (Basel Committee on Banking Supervision, 2005a, p. 65). In practice, we want the recovery process to be almost close to the end. 4 Portfolio LGD Consider now the LGD evaluation for a portfolio with D defaulted assets. We need the following assumption: S1 the values R i,j, P i,k and, for all j = 1,...,J i, k = 1,...,K i, i = 1,...,D, are evaluated at the same time t = t 0. Note that t 0 is not necessarily the time of default, since each asset in the portfolio can default ina different time. Thus, the usual way to evaluate the portfolio LGD is to treat it as a synthetic asset, such that ( D Ji j=1r i,j K i k=1 LGD t0 = 1 P ) i,k D (1) has the same form of the realized LGD and still guarantees that LGD t0 [0,1]. 5

The following proposition shows how to obtain LGD t0 for a portfolio from the individual LGD i,t0, i = 1,...,D previously evaluated. Proposition 1 Under assumption S1, where w i = / D l=1 EAD l. D LGD t0 = w i LGD i,t0, That is, the portfolio LGD is equal to the EAD-weighted average of the individual LGD. LetLGD t0 = D 1 D LGD i,t0, betheaverageportfoliolgd.then, note that, LGD t0 LGD t0, unless = C for all i = 1,...,D, where C > 0 denotes some fixed constant. Thus a, statistical model for the average LGD will not be the portfolio LGD unless we have the EAD be approximately constant. 5 Ex post implied historical LGD For portfolios made of retail assets, it is permitted that the LGD be evaluated by a implied historical method (Basel Committee on Banking Supervision, 2005a, p. 62), (Basel Committee on Banking Supervision, 2006, 465). This method seems to be derived from the expected loss (EL) formula (Schuermann, 2004), (Basel Committee on Banking Supervision, 2004, p. 4): EL = EAD PD LGD, i.e., EL LGD = EAD PD. ThisLGDisalwayspositiveforpositivevaluesofEL,EADandPD.However, if EL/s bigger than PD, then LGD will be bigger than 1. This, for instance, may happen for some specific discount rates. Note that now the portfolio includes non-defaulted assets. Then, in what follows, consider the evaluation of LGD for a portfolio with N assets, where the assets i = 1,...,D are defaulted while the assets i = D + 1,...,N are not defaulted. Thus, ex post, D J i K i EL = R i,j P i,k j=1 k=1 and, additionally to S1, we need the following assumption: 6

S2 it is possible to obtain values evaluated at the time t = t 0, for all i = D+1,...,N. Again, t 0 is not necessarily the time of default. Since the assets considered in the assumption S2 are not defaulted, in practice their EAD values are based in possible defaults (in accordance with our definition of default) at the time closest to (but not before) t 0. Generally this is done by the use of credit conversion factors (CCF) (Basel Committee on Banking Supervision, 2005a, p. 94). Then, it only makes sense to consider t 0 as the time when there is non zero exposures for all N assets in the portfolio. The EAD values, like LGD, need to be evaluated at the same time t 0, in order to correctly reflect the time value of money of the LGD components. Finally, the implied historical LGD (ILGD) is defined as ILGD t0 = [ D EADi ( Ji j=1r i,j K i N D N k=1 P i,k )]. (2) That is, under assumptions S1 and S2, the numerators of the implied LGD ILGD t0 andoftherealizedlgd(1)lgd t0 arethesame, whilethedenominator of ILGD t0 uses the portfolio s total exposure, including the non-defaulted assets, multiplied by PD. The following result gives the necessary and sufficient condition for equality between the realized portfolio LGD (1) and the implied historical LGD (2). Proposition 2 Under the assumptions S1 e S2, D LGD t0 = βilgd t0 αd 1 = (N D) 1 N i=d+1, where LGD t0 and ILGD t0 are respectively given by (1) and (2) e β = α + PD(1 α). The immediate consequence of Proposition 2 is that the realized portfolio and implied historical LGD values are the same if and only if the defaulted and non-defaulted average exposures are equal, in which case, α = 1 = β. By Proposition 2, if the LGD values evaluated by the two methods are different, then the average exposures are not equal or the formulas used for them are different related to the exposures, losses, costs or any combination of these. In a sense, one can consider this result as a justification for the homogeneity requirement in credit risk evaluation (Basel Committee on Banking Supervision, 2006, p. 91, 401, and p. 107, 492). 7

PD 0.0 0.2 0.4 0.6 0.8 1.0 0.8 0.6 0.4 0.2 1 1.6 1.2 1.4 1.8 0.0 0.5 1.0 1.5 2.0 α Figure 1: Contour plot with iso-β curves, where β = α + PD(1 α), for values β = 0.2,0.4,...,1.8. It is interesting to analyze the impact of differences in the average exposures for the two LGD evaluation methods: if the defaulted assets average exposure is greater than the non-defaulted assets average exposure, then the implied historical LGD will be greater than the realized LGD and viceversa. Besides that, the impact of the difference will be bigger for portfolios with bigger PD. (Figure 1). In fact, by writing (1 β) = (1 α)(1 PD), and noticing that PD < 1 (since the portfolio includes defaulted and nondefaulted assets), one can not only see that α = 1 β = 1, but also that α < 1 β < 1 and α > 1 β > 1. In cases where the values N, N R i,j and N P i,j are recorded by date for a bucket of N similar assets, as happen in some accounting databases, then it may be easier to evaluate the implied LGD than the portfolio LGD, mainly when D is large and it is hard to obtain individual values for the defaulted assets. In this case, using simple averages and the default rate, one can decide what method to use taking into account the regulator s and the institution s tolerance to approximations. Likewise, this can be useful when evaluating LGD for a cohort of similar contracts, like loans from a lending campaign, for instance. 8

6 Ex ante implied historical LGD Banks that have received supervisory approval to use the IRB approach may rely on their own internal estimates of risk parameter in determining the capital required for a given exposure (Basel Committee on Banking Supervision, 2006, p.52, 211). Thus, to be effective, these estimates need to be evaluated ex ante the defaults, for the non-defaulted assets. In this section we relate the ILGD t0 to a formula for LGD estimate that is well known in the literature. Consider a portfolio with N non-defaulted assets. In this case, using random variables, one can postulate the model L i = D i LGD i, (3) for i = 1,...,N, where L i denotes the value of the losses with the asset i, D i is the indicator function of default of asset i and and LGD i denote the same as in Section 3. Under various independence assumptions, there are correspondent formulas using (3), where the most interesting for this paper are the following. If LGD i is independent of D i, then E( LGD i ) = E(L i) E(D i ) = E(L i) P (D i = 1), (4) where the loss given default is a monetary value (see, e.g., Li, 2010). On the other side, if, LGD i and D i are mutually independent, then E(LGD i ) = E(L i ) E( D i ) = E(L i ) E( )P (D i = 1), (5) where the loss given default is a value between zero and one (see, e.g., Barco, 2007; Guo et al., 2007; Seidler and Jakubík, 2009). In the literature (see, e.g., Hillebrand, 2006)), sometimes it is assumed that for all i = 1,...,N, E(L i ) = el, P (D i = 1) = pd and E( ) = ead, then, from (4), E( LGD i ) = el/pd and, from (5), E(LGD i ) = el/(ead pd). Additionally, if the assets are independent random variables, by the Weak Law of Large Numbers and by the Slutsky Theorem, the ex post LGD value (Section 5) converges in distribution to E(LGD i ), in (5), as N increases. In this case, for sufficiently large values of D and N, LGD t0 ILGD t0 and E(LGD t0 ) LGD t0. Equation (5) seems to be the most used by regulators (see, e.g., Hungarian Financial Supervisory Authority, 2008), together with the associated assumptions. However, they suggest an estimation procedure using historical time series. In this case, additional assumptions are necessary to obtain results similar to ours, shown for cross-section data, but this will not be treated here. 9

7 Conclusion Inthispaper, wehavepresentedsomeaspectsofthelossgiven default(lgd) evaluation. We have shown formulas to obtain LGD for a portfolio, from the individual assets LGD. Also we have given a necessary and sufficient condition for equivalence of two generally suggested ways of LGD evaluation, realized and implied historical LGD. The condition is that the defaulted and non-defaulted assets average exposure be the same. Some implications when this condition is not met were discussed, highlighting the attention to the evaluation of other credit risk parameters and their impact in the LGD evaluation by the two ways. Finally, we have discussed the case of ex ante evaluation of LGD in crosssection data, giving explicit assumptions for comparison between this and the ex post implied historical LGD evaluation. 8 Proofs 8.1 Proof of Proposition 1 D w i LGD i,t0 = 1 D J i Dl=1 R i,j EAD l j=1 K i P i,k k=1 = LGD t0. 8.2 Proof of Proposition 2 LGD t0 = βilgd t0 D N D = β N N = α N D D D + D N = α N D D i=d+1 D Ni=D+1 N D D = α. D 10

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