Myung-Jig Kim / Journal of Economic Research 11 (2006) 203–223
203
Downturn LGD, Best Estimate
of Expected Loss, and Potential
LGD under Basel II
: Korean Experience
Myung-Jig Kim1
Hanyang University
Received 14 September 2006 ; Accepted 3 November 2006
Abstract
The Advanced-IRB banks should be able to demonstrate to the
regulatory supervisors that the long-run LGDs and downturn
LGDs are validated with their own data on historical recovery
rates. For defaulted exposures, the CRD (Capital Requirements
Directive) requires the use of the best estimate of expected loss
(BEEL) and of the potential LGD (PLGD) that reflects possible
additional unexpected losses during the recovery period. This paper attempts to provide a concrete illustration of the method that
allows one to compute various LGD estimates, particularly the
BEEL and PLGD, the latter of which have been often overlooked
in the previous studies. The proposed LGD model is essentially
the same as the single-factor model developed by such authors as
Frye (2000) and Dullmann and Trapp (2004), among others, and is
applicable to banks which experience the data scarcity in preparing the adoption of the Advanced-IRB approach. By considering
the extreme quantile value of the latent factor implied from the
model, this model can also be used to generate stress LGD for a
stress testing purpose dictated in Pillar II of the New Accord.
Keywords : Advanced-IRB Approach; Best Estimate of Expected
Loss; Downturn LGD; Potential LGD; Single-Factor Model.
JEL classification : G21, G33, C13
1
Correspondence : Myung-Jig Kim, College of Economics and Finance, Hanyang
University, (Phone) +8211-1716-6134 (E-mail) [email protected]. The author
wishes to thank two anonymous JER referees for their helpful comments. The guidance on choosing this topic along with helpful comments by the BIS Team of the
Korea Development Bank is acknowledged with thanks, but responsibility for any
errors is entirely the author’s.
204
1
Downturn LGD, BEEL and PLGD under Basel II
Introduction
The data scarcity is one of the major challenges faced by the northeast Asian banks under the New Capital Accord, or better known as
Basel II. The measurement of the default probability (PD), hence the
loss given default (LGD) as well, for low default portfolios (LDP), for
instance, might belong to this category, but recently some progress has
been made to this problem by the advancement of a quantitative approach that can produce conservative PD estimates (see, for example,
Pluto and Tasche (2005) and Benjamin, Cathcart and Ryan (2006)).
Nevertheless, not many efforts have been made to the development of
the framework that would produce conservative LGD estimates for, say,
different types of collateral, when the data are scarce. Even large banks
in the region have been paying relatively little attention to the LGD
rating system as the improvement of the PD rating system of the comprehensive credit rating system was perceived as its more imminent task
down the road.
Those banks which plan to adopt the advanced internal ratings based
(IRB) approach in Basel II will be allowed to use their own estimates
of LGD.2 Since defaults tend to be clustered during times of economic
distress, the Basel Committee requires that Advanced-IRB banks LGD
estimates be long-run estimates that account for the possible correlation
between recovery rates and default rates.3 The long-run LGD estimate
is typically computed as a default-weighted average of the annual LGDs.
The Basel Committee also requires that, if banks anticipate that
an economic downturn is approaching, the LGD estimates that are appropriate for an economic downturn should be used because the longrun default-weighted average might not be conservative enough to represent expected loss rate during the recession period.4 Alternatively,
banks may build sophisticated LGD models that consider explicitly the
macroeconomic or systematic risk factors driving the recovery rates.
2
See, for example, Resti and Sironi (2004) for detailed explanation on three different approaches to LGD measurement designed by the Basel Committee, namely,
the standardized approach, the foundation approach and the advanced approach.
3
Para. 13, BCBS, 2006
4
Para. 218 and 239, CEBS, 2006
Myung-Jig Kim / Journal of Economic Research 11 (2006) 203–223
205
These models are then used to produce the expected LGDs under various economic scenarios. In any case, the IRB banks should be able to
demonstrate to the regulatory supervisors that the long-run LGDs and
downturn LGDs are indeed validated with their own data on historical
recovery rates. A minimum collection period for LGD archives should
ideally cover at least one complete economic cycle. For instance, in the
case of corporate loans, it must never be shorter than seven years.
The Basel II paradigm treats the LGD for defaulted exposures different from normal or non-defaulted exposures. The BCBS (2006), CEBS
(2006), and academics provide lengthy description of the measurement
and validation of downturn LGD for non-defaulted exposures, both theoretically and empirically, but give little guidance on the measurement
and validation for the estimation of expected losses applicable to defaulted exposures. The CEBS guideline (2006), for instance, interprets
that
“...for defaulted exposures, the CRD (Capital Requirements
Directive) requires the use of an estimate of expected loss
(ELBE) that should be the best estimate of expected loss,
given economic circumstances. In such cases, LGD is defined as the sum of ELBE and a measure reflecting possible
additional unexpected losses during the recovery period. ...”
(Para. 239e, CEBS (2006).)
The tentative guidelines on the IRB approach issued in 2005 by
the Financial Supervisory Service (FSS) of Korea resemble the ones
above: Advanced-IRB banks require the use of the best estimate of expected loss (BEEL) and potential LGD (PLGD) that reflects possible
additional unexpected loss associated with the measurement of BEEL.
Question is how one can implement it.
Since the development of the LGD rating system is at the early stage
for many commercial banks in the northeast Asia, such important parameters of the credit risk as downturn LGD, BEEL, and PLGD cannot
be readily estimated based on the historical data. As such, one alternative is to consider the theoretical model that admits the limited data as
an input, but still is capable of producing these risk parameters during
the transition period.
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Downturn LGD, BEEL and PLGD under Basel II
The purpose of this paper is to propose the simple theoretical model
that may be capable of producing the various LGD estimates, particularly the BEEL and PLGD, which have been often overlooked in the
previous studies. The proposed LGD model is essentially the same as
the single-factor model developed by such authors as Frye (2000) and
Dullman and Trapp (2004), among others. The advantages of the singlefactor LGD model include the simplicity of the implementation in spite
of the data scarcity and of the way that banks set the level of conservatism levied to their LGD estimates reflecting the incompleteness of
the historical LGD data. By considering the extreme quantile value of
the latent factor in the model, this model can also be used to generate
stress LGD for a stress testing purpose.
This paper is organized as follows: Section 2 briefly reviews the onefactor LGD model and shows how to apply this model to compute the
BEEL and PLGD. Section 3 explains the maximum likelihood estimation, followed by the empirical results for an illustration in Section 4.
Section 5 summarizes and concludes the paper.
2
Theoretical Framework of the Estimation of
Downturn LGD, Best Estimate of Expected
Loss, and Potential LGD
A theoretical framework that estimates downturn LGD, BEEL, and
potential LGD when the data are sparse is based upon the single-factor
approach that underlies the derivation of the Basel II capital requirement formula. A single-factor model has been developed and applied to
problems of portfolio credit risks and the capital adequacy by numerous
authors such as Frye (2000), Vasicek (2002), and Gordy (2003). A particular model this paper focuses on is the framework that is proposed by
Frye (2000), Pykhtin (2003), and Dullmann and Trap (2004) in an effort to explain the relationship between default rates and recovery rates.
These authors begin by assuming that the asset return process of firm
follows a one-factor process:
Aj =
√
ρX +
p
1 − ρZj
(1)
Myung-Jig Kim / Journal of Economic Research 11 (2006) 203–223
207
Aj denotes the firm j’s asset return, X denotes the systematic risk
factor which is assumed to be a standard normally distributed random
variable, i.e., X ∼ N (0, 1), Zj denotes the idiosyncratic risk factor that
is uncorrelated with X and is assumed to be a standard normally distributed random variable, i.e., Zj ∼ N (0, 1), and ρ is a parameter that
represents the coefficient of correlation between asset returns and the
systematic risk and hence the asset return process is also assumed to be
a standard normally distributed random variable.
The last assumption enables the threshold of the firm j’s default
to be expressed as Aj = Φ−1 (P D) and thus the firm j’s default frequency (DF) conditional on the systematic risk in a particular year can
be written
DFj
= P r Aj < Φ−1 (P D) | X = x
√
p
= Pr
ρX + 1 − ρZj < Φ−1 (P D)
!
√
Φ−1 (P D) − ρX
√
= Φ
1−ρ
(2)
The CDF of the standard normal distribution is denoted by Φ(·) and
Φ−1 (·) hence denotes the inverse CDF. Recall that the generic benchmark risk weight (BRW) formula provided in the Basel II proposals is
of the form
BW R = LGD × F actor(M aturity) × F actor(P D, ρ)
(3)
Bluhm and Overbeck (2003, 2004) showed that the last line of equation
(2) corresponds to F actor(P D, ρ) in the BRW formula.
In modeling the recovery rate (R) and linking it to the one-factor
model of equation (2), Frye (2000) and Dullmann and Trapp’s (2004) approaches are essentially the same except for the distributional assumptions. Namely, the former assumes the normal distribution whereas the
latter, drawn from earlier works of Schonbucher (2003) and Pykhtin
(2003), considers more flexible assumptions of logit-normally and lognormally distributed LGDs:
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Downturn LGD, BEEL and PLGD under Basel II
Dullmann and Trapp (2004):
Logit normal case:
√
√
Yj = µ + σ ωX + σ 1 − ωWj ,
R(Yj ) =
X ∼ N (0, 1),
Wj ∼ N (0, 1)
exp(Yj )
1 + exp(Yj )
(4)
(5)
Or
R(Yj )
ln
1 − R(Yj )
!
√
√
= µ + σ ωX + σ 1 − ωWj
(6)
Log-normal case:
√
√
R(Yj ) = exp µ + σ ωX + σ 1 − ωWj
(7)
X is the systematic risk factor introduced in equation (1), ω is the
correlation parameter, and Wj denotes the idiosyncratic risk factor that
is specific to the firm j’s recovery rate. The Fryes (2000) representation
is of the simple form:
Frye (2000):
Nomal case:
√
√
R(Yj ) = µ + σ ωX + σ 1 − ωWj
(8)
One advantage of specifications in equations (6) and (7) is that recovery rates are always located within zero and unity, whereas equation (8) provides no such warranty in practice. Using the relation that
LGD = 1−R, letting t denote a particular year, and using the change-ofvariable technique, the conditional distribution of LGD corresponding
to equations (6)-(8) can be derived as follows:
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Myung-Jig Kim / Journal of Economic Research 11 (2006) 203–223
PDF of LGD in case of logit-normal asumption:
f (LGDt ; µ, σ, ω | Xt )
r
=
1
exp
2πσ 2 (1 − ω)LGDt2 (1 − LGDt )2
−
ln
1−LGDt
LGDt
2 !
√
ωσXt
−µ−
2σ 2 (1 − ω)
(9)
PDF of LGD in case of log-normal assumption:
f (LGDt ; µ, σ, ω | Xt )
s
=
1
[ln (1 − LGDt ) − µ −
exp −
2
2
2πσ (1 − ω)(1 − LGDt )
2σ 2 (1 − ω)
√
2
ωσXt ]
!
(10)
PDF of LGD in case of normal assumption:
f (LGDt ; µ, σ, ω | Xt )
s
=
!
√
2
1
[LGDt − 1 + µ + ωσXt ]
exp −
2πσ 2 (1 − ω)
2σ 2 (1 − ω)
(11)
When model parameters are available, the conditional expectation
of LGD can be easily computed. For instance, in case of the log-normal
specification of the recovery rates, it is computed as:
σ 2 (1 − ω)
E(LGDt | Xt = xt ) = 1 − exp µ + σ ωXt +
2
√
!
(12)
The expressions for the conditional mean and variance of the logitnormal distribution are not known, but can be computed numerically
as follows:
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Downturn LGD, BEEL and PLGD under Basel II
E(LGDt | Xt = xt ) =
Z
1
LGDs f (LGDt | Xt = xt )ds
(13)
0
V ar(LGDt | Xt = xt ) =
1
Z
( LGDs − E(LGDt | Xt = xt ))2
0
× f (LGDt | Xt = xt )ds
SD(LGDt | Xt = xt ) =
q
V ar(LGDt | Xt = xt )
(14)
(15)
If the past history of the systematic risk factor during the economic
downturn Xdownturn is known, the downturn LGD is simply the expectation of LGD conditional on Xdownturn . If the value cannot be
readily assigned due to the data scarcity, one can rely on the probability instead. Namely, one would assume an appropriate quantile for
X such that Xt = −3.09 = −Φ(0.999) if the downturns tend to be extremely severe. The choice of the confidence level will therefore depend
upon how one would perceive the severity of the economic downturn.
Likewise, the best estimate of economic loss for defaulted facilities
can be defined as the expectation of LGD conditional on the expectation
of the economic state until one year from now. Hence, expressions such
as equations (12) and (13) provide a consistent framework to estimate
both downturn LGD and BEEL.
It is often difficult to forecast the economic conditions one year forward and so does predicting the Xt ’s. Therefore, it is natural to adjust
it to reckon the uncertainty in the estimation of BEEL. The potential
LGD that is intended to incorporate such uncertainty could be computed in the FDT model framework by first noting that estimated Xt
can be obtained from equation (2)
X̂ =
Φ−1 (Pd
D) −
!
√
1 − ρ̂Φ−1 (DF )
√
≡ g −1 (DF )
ρ̂
(16)
There are two sources of uncertainty in this expression, one from
the coefficient of correlation ρ̂ and the other from Pd
D. Therefore, if
211
Myung-Jig Kim / Journal of Economic Research 11 (2006) 203–223
parameter estimates along with its covariance matrix are available, the
uncertainty of X̂ can be computed as
V ar(X̂t ) =

"
∂g −1 (·) ∂g −1 (·)
,
∂ρ
∂P D
#"
V ar(ρ)
Cov(ρ, P D)
Cov(ρ, P D) V ar(P D)
#




∂g −1 (·)
∂ρ
∂g −1 (·)
∂P D



 (17)

Therefore, X̂ that is to be used in the computation of the potential
LGD can be calculated under the relevant level of confidence, say 97.5%,
as follows:
Xt,P LGD = X̂t − 1.96 · SD(X̂t )
(18)
The potential LGD is then simply the expectation of LGD conditional on the value of Xt,P LGD . In this way, downturn LGD, BEEL and
potential LGD may be computed in the single FDT model framework
when the data are scare. Next section discusses the MLE of the model
parameters for such computations.
3
Maximum Likelihood Estimation
Both Frye (2000) and Dullmann and Trapp (2004) proposed twostep MLE procedure for estimating the model parameters. In the first
step, the distribution of default frequency is derived and is maximized
with respect to ρ and P D . Since the systematic risk factor is a standard normal variate, the distribution of DF , f (DFt ; P D, ρ) is derived
by applying the change-of-variable technique as follows:
dg −1 (DF ) h
i
f (DFt ; P D, ρ) = f g −1 (DF )
dDF
(19)
Letting, γ = Φ−1 (P D), δt (DFt ) and recalling that dδt /dDF =
t ), equation (19) can be rewritten
φ−1 (δ
212
Downturn LGD, BEEL and PLGD under Basel II
f (DFt ; P D, ρ) = −
= −
= −
s
√
[g −1 (DF )]2
1
2
· φ (δt ) · √ e−
2π
!
!−1 √
2
δ
[g −1 (DF )]2
1−ρ
1
t
· √1 e−
2
· √ e− 2
√
ρ
2π
2π
√
2
!
√
γ− 1−ρδt
δt2 √
− 12
1−ρ
ρ
·e 2 ·e
√
ρ
1−ρ
√
ρ
!
1 − ρ −γ
·e
ρ
=
−1
2 +(1−2ρ)δ 2 −2√1−ργδ
t
t
2ρ
(20)
The ML estimates of P D and ρ are then obtained by maximizing
the sample log-likelihood function.5
max
P D, ρ
`(P D, ρ; DF1 , · · · , DFT ) =
T
X
ln(f (P D, ρ; DFt ))
(21)
t=1
Given ML estimates of P D and ρ, the latent variable Xt can be
imputed using equation (16) above.
The second step of the estimation is to estimate parameters of recovery rates (Rt ), namely, µ ,ω ,σ. and Denoting Dt the number of
5
Alternate estimation procedure for asset correlation ρ based on the volatility of
default rates time-series has been suggested by Bluhm and Overbeck (2003, 2004).
They showed that the variance of the conditional default rates given in equation (2)
can be expressed
V ar(DF (X)) = JDP (ρ, P D) − P D × P D
The formula for the joint default probability (JDP) of two obligors in a uniform
portfolio with parameters ρ and P D is defined
1
JDP =
2π
p
1 − ρ2
Z
Φ−1 (P D)
Z
Φ−1 (P D)
exp −
−∞
−∞
u2 − 2ρuν + ν 2
2(1 − ρ2 )
dudν.
If one computes the sample mean and variance of the observed annual default frequency data for P D and V ar(DF (X)), respectively, only unknown parameter in the
variance expression above is the correlation coefficient ρ.
Myung-Jig Kim / Journal of Economic Research 11 (2006) 203–223
213
defaulted facilities in year t, the likelihood function of recovery rates
conditional on Xt is:
Logit-normal recovery rates:
L(µ, σ, ω; R1 , · · · RT , D1 , · · · DT )
s
=
h i2 
√
Rt
D
ln
−
µ
−
ωσX
t
t
1−Rt
Dt


exp −
 (22)
2
2
2
2
2πσ (1 − ω)Rt (1 − Rt )
2σ (1 − ω)

Log-normal recovery rates:
L(µ, σ, ω; R1 , · · · RT , D1 , · · · DT )
s
=
!
√
2
Dt [ln (Rt ) − µ − ωσXt ]
Dt
exp −
2σ 2 (1 − ω)
2πσ 2 (1 − ω)Rt2
Normal recovery rates:
(23)
6
L(µ, σ, ω; R1 , · · · RT , D1 , · · · DT )
s
=
6
!
√
2
Dt
Dt [Rt − µ − ωσXt ]
exp −
2πσ 2 (1 − ω)
2σ 2 (1 − ω)
(24)
To see this, let Dj,t denote the j-th defaulted facility in year t, hence, Dt =
Likewise, the recovery rate in year t is computed as Rt = ΣJj=1 Rj,t /Dt .
Since the normal recovery process
for the j-th defaulted facility in year is defined
p
√
√
as Rj,t = µ + σ ωXt + σ (1 − ω)Wj,t , it follows that Rt = Σµ + Σσ ωXt +
p
Σσ (1 − ω)Wj,t /Dt . Therefore, it can be seen that Rt is normally distributed with
√
√
the mean µ + σ ωXt and the variance Σσ 2 1 − ω/Dt2 = σ 2 (1 − ω)/Dt , which yields
the likelihood function in equation (24). The likelihood functions in equations (22)(23) can also be derived in the similar manner using the change-of-variable technique.
ΣJj=1 Dj,t .
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Downturn LGD, BEEL and PLGD under Basel II
The sample log-likelihood function, Tt=1 ln(L(·)), can be maximized
with respect to unknown parameters, µ, σ, and ω.
Technically, σ and in equations (22)-(23) are not identified separately. Dullmann and Trapp (2004) suggested estimating σ using historical data first. For instance, the sample standard deviation is computed using time-series ln(Rt /(1 − Rt )) for logit-normal specification,
for log-normal specification, ln(Rt ) and for normal specification, Rt respectively.
P
4
Empirical Results
This section provides empirical illustration of the proposed approach
using the particular data set from a bank operating in Korea. Table 1
shows annual default rates for corporate exposures and recovery rates
and the number of defaulted obligors for particular type of collateral.
From reading of the cyclical component of the coincident index published by the National Statistical Office, year 2004 might be regarded
as the economic downturn period. Such years as 2001 and 2003 have
experienced a partial downturn lasting about six months, followed by a
brief recovery. Year 2002 might be regarded as the expansionary period
Annual default rates in Table 1 reveal the pro-cyclical movement
resulting in the lowest in year 2002 and the highest in year 2004. Recovery rates, however, seem to show no evidence of the negative correlation with default rates, nor cyclical effects, as is documented in the
literature (see, Allen and Saunders (2003), for detailed survey on the
empirical evidence of cyclical effects in recovery and the negative correlation between PD and LGD). Hence, the application of the proposed
model to this data is tantamount to making a strong assumption that
the substantial portion of the variation of recovery rates is attributable
to that of the idiosyncratic component.
Myung-Jig Kim / Journal of Economic Research 11 (2006) 203–223
215
Table 1. Historical Data
Year
Annual
Default Rate
Recovery
Rate
2001
2002
2003
2004
0.03671
0.02052
0.03451
0.04375
0.324224
0.361381
0.400248
0.453286
Number of
Defaulted
Obligors
11
7
13
16
Estimated
Systematic
Risk Factor1
-0.4026
1.63836
-0.17621
-1.05954
Note: 1 Estimated systematic risk factor Xt is computed using the ML estimate ρ
and P D reported in the upper panel of Table 2 and the expression (16) in the text.
Table 2 reports ML estimates of the coefficient of correlation (ρ),
long-run probability of default (P D ), and the rest of parameters of the
recovery process assuming different distributional assumptions.
Table 2. ML Estimates
Parameter
ρ
PD
Estimate Standard Error
0.01514
0.01054
0.03392
0.00469
Logit-normal recovery rates
µ
-0.37981
0.0243
ω
0.32303
0.05000
σ(historical) 0.020242
Log-normal recovery rates
µ
-0.9288
0.01571
ω
0.26296
0.05888
σ(historical)
0.12392
Normal recovery rates
µ
0.39472
0.006975
ω
9.39E-19
3.23E-15
σ(historical)
0.04781
t-value
1.4360*
7.2304
-15.629
6.4610
-59.124
4.466
56.589
0.000
Note: * indicates the significance at 7.6% level from one-tailed test. T, as opposed
to (T-1), was used as a divisor in computing the historical standard deviation. The
usual convergence criteria could not be applied due to small observations. It was set
equal to 1e-00. The program is written in GAUSS, along with the OPTMUM library.
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Downturn LGD, BEEL and PLGD under Basel II
The magnitude of estimated coefficient of correlation is 0.015, which
is somewhat small compared to ones reported in the previous literature.
The corresponding p-value was 7.6%. When the method was applied
to longer time-series of default rates from the Korean rating agencies,
the magnitude was much larger and the significance level was higher.
Hence, small magnitude of estimated coefficient of correlation might be
attributable to a small sample problem. As for comparison, the value
of ρ computed from using the method in footnote 4 was 0.013, which is
similar to that of the MLE above.7
The last column in Table 1 shows the implied systematic risk factor
based upon estimates in Table 2, computed from using the formula in
equation (16). The correlation between implied Xt and GDP growth
rates was about 75%. Hence, the substantial portion of the variation in
Xt may be attributable to business cycle effects.
Also reported in Table 2 are estimated parameters of the recovery
rates processes. The long-run recovery rates µ’s that correspond to the
neutral economic state, i.e., Xt = 0, are all significant. The correlation
parameters ω’s are also significant, except for the Fyres (2000) normal
specification. Hence, log-normal and logit-normal specifications appear
to be more relevant for this data. Figure 1 depicts the density functions of LGD based upon parameter estimates in Table 2 and assuming
Xt = 0. As expected, both log-normal and logit-normal distributions
are slightly skewed.
7
Input parameters and computed JDP were Var(DF)=7.1131e-05, PD=0.03387,
and JDP =0.0012186, respectively.
Myung-Jig Kim / Journal of Economic Research 11 (2006) 203–223
217
Figure 1. Density Functions of LGD Conditional on Neutral Economic
State
Note: The neutral economic state means setting the value of Xt being equal to zero.
Figure 2 depicts the density function of LGD under the assumption
of economic downturn. The economic downturn is defined as the state
that the systematic risk factor takes on value at the 99.9% quantile, i.e.,
Xt = −3.09 = −Φ−1 (0.999). It shows that both log-normal and logit
normal density functions shifts to the right as the conditioning variable
Xt takes on a large value. On the other hand, the LGD density with
the normal assumption does not shift as it is not state-dependent. It is
expected that expected value of downturn LGD from either log-normal
or logit-normal assumption would not yield a material difference.
218
Downturn LGD, BEEL and PLGD under Basel II
Figure 2. Density Functions of LGD Conditional on Economic
Downturn
Note: Economic downturn is defined at the 99.9% level, i.e., Xt = −3.09 =
−Φ−1 (0.999).
In order to estimate BEEL, one has to predict the economic state
that would prevail for one year. Figure 3 depicts the density assuming
that value of Xt for upcoming year is -1.095, i.e., the value of the systematic risk in year 2004. Assuming the log-normal specification, for example, the BEEL is then computed as BEEL = E(LGDt | Xt = −1.0595)
= 62.86%, which is about 4.5% smaller than downturn LGD estimate.
219
Myung-Jig Kim / Journal of Economic Research 11 (2006) 203–223
Figure 3. Density Functions of LGD Conditional on Predicted
Economic State
Note: Predicted economic state is assumed to be equivalent to setting Xt = −1.0595.
It is straightforward to compute the potential LGD (PLGD). Assume that the confidence level is 97.5% and the recovery is log-normally
distributed. Assume further that the predicted economic state is equivalent to setting Xt = −1.0595. Using the covariance matrix of ρ and
P D estimates
"
V ar(ρ)
Cov(ρ, P D)
Cov(ρ, P D) V ar(P D)
#
"
=
0.00011
−
7.75010e-06 2.20045e-05
#
the standard deviation of the systematic risk factor is computed as
the square-root of expression (17), which yielded SD(X̂t ) = 0.62476.
Then, predicted value of the systematic factor that reflects the uncer-
220
Downturn LGD, BEEL and PLGD under Basel II
tainty surrounding the upcoming economic state can be expressed as
Xt,P LGD = X̂t − 1.96(X̂t ) = −2.2840. The potential LGD is, therefore,
the conditional expectation of LGD on Xt,P LGD : P LGD = E(LGDt |
Xt = −2.2840) = 65.64%.
Table 3 summarizes estimates of various LGDs assuming the lognormal assumption.8 Since predicted economic state used in the computation of the BEEL and PLGD was not as severe as the assumed
economic downturn, estimates of the BEEL and PLGD are smaller
than that of downturn LGD. Long-run LGD, which is computed as the
default-weighted LGD, was similar in magnitude to one that assumed
the neutral economic state. Hence, it is apparent that the long-run
LGD of this particular data set does not seem to provide an adequate
additional conservatism to LGD estimates in the spirit of Basel II.
Table 3. Estimates of Various LGDs: Log-Normal Assumption
Neutral
State1
BEEL2
PLGD3
Downturn
LGD4
60.27%
62.86%
65.64%
67.36%
Default
weighted
LGD5
60.53%
Historical
Maximum6
67.58%
Note:
1. Assumed Xt = 0
2. Assumed Xt = −1.0595
3. Assumed Xt = −1.0595, the covariance matrix of ρ and P D obtained from the
estimation in upper panel of Table 2, and 97.5% significance level.
4. Assumed Xt = −3.09 = −Φ−1 (0.999).
5. Computed as (1/T )ΣTt=1 LGDt (DFt /ΣTt=1 DFt ).
6. Maximum LGD during the 2001-2004 periods.
As the data are accumulated over time, banks would understand
better the nature of the systematic risk factor process and perhaps be
able to link it to macroeconomic variables of its interest such as interest rate, credit spread, GDP growth rate, inflation rate, and etc. Under
8
Estimates of various LGDs for different types of collateral, not reported in the
paper due to data confidentiality, have revealed the similar behavior as ones reported
in Table 3. Results from backtest using the 2001-2003 period as in-sample for the
estimation and year 2004 as out-of-sample, also not reported here, as part of the LGD
validation suggested that these estimates provide, on average, an added conservatism
of about 9%.
Myung-Jig Kim / Journal of Economic Research 11 (2006) 203–223
221
this circumstance, the stress test that is described in Pillar II of Basel II
might be implemented in the same theoretical framework that is used to
estimate downturn LGD, BEEL and PLGD above. Since the relationship between the systematic factor and other macroeconomic variables
may not be confirmed at present, the values of Xt that correspond to the
significance level exceeding 99.9% might be considered as a compromise
during the transition period.
5
Summary and Conclusions
Banks planning to adopt the Advanced-IRB approach should be able
to demonstrate to the regulatory supervisors that the long-run LGDs
and downturn LGDs are validated with their own data on historical
recovery rates. For defaulted exposures, the CRD requires the use of
the best estimate of expected loss (BEEL) and of the potential LGD
(PLGD) that reflects possible additional unexpected losses during the
recovery period. These are not a trivial task to many banks in the
northeast Asia as the development of the LGD rating system is often at
the early stage.
For instance, the interim assessment of the validity of AIRB-seeking
banks estimates of the LGDs by the Financial Supervisory Services
(FSS) of Korea performed in early 2006 suggests that they vary across
banks for similar products and their variances are too large to be acceptable for the approval purpose and that it might amplify the variability
of the regulatory capital requirement. One way to overcome this inaccuracy of LGD estimates is to create the pooled public LGD benchmark
database and to narrowly define the product types. Also the FSS plans
to extend the public LGD database to cover the severe recessionary
period experienced during 1997-1998 in cooperation with large Korean
banks. It would, however, take some time.
This paper proposes, as a short-term resolution, the use of the theoretical LGD models that has been popular in the literature and are
consistent with the derivation of the Pillar I formulae of the regulatory
capital requirements. Namely, this paper attempts to provide a concrete
illustration of the method that allows one to compute various LGD estimates, particularly the BEEL and PLGD, which has not been explicitly
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Downturn LGD, BEEL and PLGD under Basel II
addressed in the previous studies and supervisory guidelines. The proposed LGD-BEEL model draws upon the well-known Frye (2000) and
Dullmann and Trapp (2004) single factor LGD models. It is simple to
implement, while the imposition of additional conservatism during the
data build-up period can be readily calibrated. Although the model
is applied to the extremely small sample, it still showed the meaningful parameter estimates. Using the particular Korean data, this paper
also finds that the original Frye model that assumes the normality of
the recovery process was not able to produce the cyclical LGD estimate,
whereas both the lognormal and logit normal recovery assumptions were
supporting the presence of the cyclicality of the LGDs.
The proposed method would be valuable to northeast Asian banks
which commonly experience the data scarcity in the preparation of the
advanced-IRB approach, the appropriateness of which might be further
discussed in the Subgroups, like the CEBS, of the cooperative supervisory framework that has been established in 2006 among Korea (FSS),
Japan (FSA) and China (CBRC).
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