Estimating Credit Portfolio Loss with Dependent Loss Given Default
Estimating Credit Portfolio Loss with Dependent Loss Given Default Martin Hillebrand [email protected]
Technical University of Munich
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.1
Introduction Consider a portfolio of m credits.
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.2
Introduction Consider a portfolio of m credits. Portfolio loss within one year: m X X= 11i ·EADi ·LGDi i=1
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.2
Introduction Consider a portfolio of m credits. Portfolio loss within one year: m X X= 11i ·EADi ·LGDi i=1
11i ∈ {0, 1} is a Bernoulli(PDi (Y )) random variable; 11i = 1 if credit i defaults
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.2
Introduction Consider a portfolio of m credits. Portfolio loss within one year: m X X= 11i ·EADi ·LGDi i=1
11i ∈ {0, 1} is a Bernoulli(PDi (Y )) random variable; 11i = 1 if credit i defaults
Probability of default PDi (Y ) = Φ (c − eY ) Dependent on economic cycle (Y )
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.2
Introduction Consider a portfolio of m credits. Portfolio loss within one year: m X X= 11i ·EADi ·LGDi i=1
11i ∈ {0, 1} is a Bernoulli(PDi (Y )) random variable; 11i = 1 if credit i defaults
Probability of default PDi (Y ) = Φ (c − eY ) Dependent on economic cycle (Y ) EADi :
Martin Hillebrand
the exposure at default – constant: EADi ≡ 1/m
Estimating Portfolio Loss with Dependent LGD – p.2
Introduction Consider a portfolio of m credits. Portfolio loss within one year: m X X= 11i ·EADi ·LGDi i=1
11i ∈ {0, 1} is a Bernoulli(PDi (Y )) random variable; 11i = 1 if credit i defaults
Probability of default PDi (Y ) = Φ (c − eY ) Dependent on economic cycle (Y ) EADi : LGDi :
the exposure at default – constant: EADi ≡ 1/m
the loss given default Usually assumed to be constant
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.2
Introduction Consider a portfolio of m credits. Portfolio loss within one year: m X X= 11i ·EADi ·LGDi i=1
11i ∈ {0, 1} is a Bernoulli(PDi (Y )) random variable; 11i = 1 if credit i defaults
Probability of default PDi (Y ) = Φ (c − eY ) Dependent on economic cycle (Y ) EADi : LGDi :
the exposure at default – constant: EADi ≡ 1/m
the loss given default Usually assumed to be constant Our model: dependent on economic cycle (Z )
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.2
Merton type structural model Company i defaults if logreturn of asset value Ai falls 11i = 11{Ai <si } below some threshold value si :
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.3
Merton type structural model Company i defaults if logreturn of asset value Ai falls 11i = 11{Ai <si } below some threshold value si : d
Ai = N (0, 1) =⇒ si = Φ−1 (PDi )
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.3
Merton type structural model Company i defaults if logreturn of asset value Ai falls 11i = 11{Ai <si } below some threshold value si : d
Ai = N (0, 1) =⇒ si = Φ−1 (PDi ) p Ai = β i Y + 1 − β 2 εi
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.3
Merton type structural model Company i defaults if logreturn of asset value Ai falls 11i = 11{Ai <si } below some threshold value si : d
Ai = N (0, 1) =⇒ si = Φ−1 (PDi ) p Ai = β i Y + 1 − β 2 εi =⇒ PDi (Y ) = P(11i = 1|Y ) = Φ
Φ−1 (PDi ) − βi Y p 1 − β2
!
= Φ(c − eY )
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.3
PSfrag replacements
Dependence in the Data
0.6 0.5 0.4
E(LGD|Z)
Altman et al. (2003): Annual average LGD and default frequencies of American corporate bonds from 1982 to 2001 mean(LGD)=0.58 mean(PD)=0.035
0.7
Fig. 1: Dependence of E(LGD|Z) and PD(Y )
0.02
0.04
0.06
0.08
0.10
PD(Y )
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.4
PSfrag replacements
Dependence in the Data
0.6 0.4
0.5
E(LGD|Z)
Altman et al. (2003): Annual average LGD and default frequencies of American corporate bonds from 1982 to 2001 mean(LGD)=0.58 mean(PD)=0.035
0.7
Fig. 1: Dependence of E(LGD|Z) and PD(Y )
0.02
Recall the linear relation
0.04
0.06
0.08
0.10
PD(Y )
Φ−1 (PD(Y )) = c − eY. Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.4
PSfrag replacements
Transformed Data: Linear Dependence
0.4 0.2 −0.2
0.0
Φ−1 (E(LGD|Z))
0.6
Fig. 2: Dependence of transformed E(LGD|Z) and PD(Y )
−2.4
−2.2
−2.0
−1.8
−1.6
−1.4
Φ−1 (PD(Y ))
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.5
PSfrag replacements
Transformed Data: Linear Dependence
0.4 0.2 −0.2
0.0
Φ−1 (E(LGD|Z))
0.6
Fig. 2: Dependence of transformed E(LGD|Z) and PD(Y )
−2.4
−2.2
Martin Hillebrand
−1.8
−1.6
−1.4
Φ−1 (PD(Y ))
Linear ansatz:
with Z = dY +
−2.0
√
Φ−1 (E(LGD|Z)) = a + bZ 1 − d2 X
Estimating Portfolio Loss with Dependent LGD – p.5
Linear Regression Model From Φ−1 (PD(Y )) = b − eY
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.6
Linear Regression Model From Φ−1 (PD(Y )) = b − eY
and with Z = dY +
Martin Hillebrand
√
Φ−1 (E(LGD|Z)) = a + bZ 1 − d2 X
Estimating Portfolio Loss with Dependent LGD – p.6
Linear Regression Model From Φ−1 (PD(Y )) = b − eY
and with Z = dY +
√
Φ−1 (E(LGD|Z)) = a + bZ 1 − d2 X
we obtain the regression model
p Φ (E(LGD|Z = d · y + 1 − d2 X)) p c bd −1 = a + bd − Φ (PD(y)) + b 1 − d2 X e e −1
providing estimates for a, b, and d. Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.6
Downturn LGD
0.6 0.5 0.3
0.4
E(LGD|Z = zα )
0.7
0.8
PSfrag replacements
With the estimates, downturn LGD can easily be calculated by E(LGD|Z = zα ) = Φ(ˆ a + ˆb · zα ).
0.0
0.2
0.4
0.6
0.8
1.0
α
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.7
Downturn LGD
0.6 0.5 0.3
0.4
E(LGD|Z = zα )
0.7
0.8
PSfrag replacements
With the estimates, downturn LGD can easily be calculated by E(LGD|Z = zα ) = Φ(ˆ a + ˆb · zα ).
0.0
0.2
0.4
0.6
0.8
1.0
α
E.g. for α = 0.001, we obtain E(LGD|Z = z0.001 ) = 0.874. Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.7
Portfolio Loss Distribution For m → ∞, we obtain a formula for the portfolio loss distribution: Z A(l) P (L ≤ l) = 1 − Φ(B(l, y))ϕ(y) dy. −∞
and
Martin Hillebrand
c − Φ−1 (l) with A(l) = e l 1 −1 B(l, y) = √ a − dby − Φ . 2 Φ(c − ey) b 1−d
Estimating Portfolio Loss with Dependent LGD – p.8
Portfolio Loss Distribution For m → ∞, we obtain a formula for the portfolio loss distribution: Z A(l) P (L ≤ l) = 1 − Φ(B(l, y))ϕ(y) dy. −∞
and
c − Φ−1 (l) with A(l) = e l 1 −1 B(l, y) = √ a − dby − Φ . 2 Φ(c − ey) b 1−d
Observe that, if LGD ≡ 1,
P (L ≤ l) = 1 − Φ(A(l)). Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.8
PSfrag replacements
3: Credit Portfolio Loss Density Portfolio Loss Distribution Tail
PSfrag replacements
3: Credit Portfolio Loss Density Portfolio Loss Distribution Tail
Comparison with constant LGD
0.05
dependent LGD LGD=0.65 LGD=1
0
0.00
0.01
10
0.02
20
0.03
30
dependent LGD LGD=0.65 LGD=1
Credit Portfolio Loss Distribution Tail
0.04
40
Credit Portfolio Loss Distribution Histogram
0.00
0.05
0.10
Relative Loss
Martin Hillebrand
0.15
0.10
0.11
0.12
0.13
0.14
0.15
0.16
0.17
Relative Loss
Estimating Portfolio Loss with Dependent LGD – p.9
Impact on VaR The impact of dependent LGD modeling on portfolio VaR grows with increasing α: VaRα Dependent LGD LGD ≡ 0.65 LGD ≡ 1 ELGD(qα ) ∗ p(qα )
Martin Hillebrand
α = 99%
α = 99.5%
α = 99.9%
0.107 0.089 0.137 0.112
0.126 0.102 0.157 0.132
0.171 0.134 0.206 0.179
Estimating Portfolio Loss with Dependent LGD – p.10
PSfrag replacements
3: Credit Portfolio Loss Density Portfolio Loss Distribution Tail
PSfrag replacements
3: Credit Portfolio Loss Density Portfolio Loss Distribution Tail
0.08
Relative Loss
Relative Loss
Goodness of Fit: qq-plots
0.00
0.020 0.015 0.000
0.005
0.010
Estimated Quantiles
0.04 0.02
Estimated Quantiles
0.06
0.025
Hillebrand Giese Tasche
0.00
0.02
0.04 Empirical Quantiles
Martin Hillebrand
0.06
0.08
0.000
0.005
0.010
0.015
0.020
0.025
Empirical Quantiles
Estimating Portfolio Loss with Dependent LGD – p.11
Nonhomogeneous portfolios Data:
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.12
Nonhomogeneous portfolios Data: As in the standard Merton type model:
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.12
Nonhomogeneous portfolios Data: As in the standard Merton type model: Default probabilities: PDi
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.12
Nonhomogeneous portfolios Data: As in the standard Merton type model: Default probabilities: PDi Factor weights (“asset correlations”):βik
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.12
Nonhomogeneous portfolios Data: As in the standard Merton type model: Default probabilities: PDi Factor weights (“asset correlations”):βik Expected loss given default: E(LGDi )
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.12
Nonhomogeneous portfolios Data: As in the standard Merton type model: Default probabilities: PDi Factor weights (“asset correlations”):βik Expected loss given default: E(LGDi ) Additionally: LGD variance: σi
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.12
Nonhomogeneous portfolios Data: As in the standard Merton type model: Default probabilities: PDi Factor weights (“asset correlations”):βik Expected loss given default: E(LGDi ) Additionally: LGD variance: σi parameters b, d describing systematic dependence of √ LGD on systematic factors (a = 1 + b2 Φ−1 (E(LGD)))
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.12
Calculation Algorithm 1. Simulate factors Y = (Y1 , ..., Yk ) and X
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.13
Calculation Algorithm 1. Simulate factors Y = (Y1 , ..., Yk ) and X 2. For each obligor i: calculate conditional default probability PDi (Y )
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.13
Calculation Algorithm 1. Simulate factors Y = (Y1 , ..., Yk ) and X 2. For each obligor i: calculate conditional default probability PDi (Y ) d
3. Simulate default events Di = Binomial(PDi (Y ))
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.13
Calculation Algorithm 1. Simulate factors Y = (Y1 , ..., Yk ) and X 2. For each obligor i: calculate conditional default probability PDi (Y ) d
3. Simulate default events Di = Binomial(PDi (Y )) 4. Calculate expected conditional loss given default √ T β i 2 E(LGDi |Zi = zi ) where Zi = dX + 1 − d √ TY β i βi
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.13
Calculation Algorithm 1. Simulate factors Y = (Y1 , ..., Yk ) and X 2. For each obligor i: calculate conditional default probability PDi (Y ) d
3. Simulate default events Di = Binomial(PDi (Y )) 4. Calculate expected conditional loss given default √ T β i 2 E(LGDi |Zi = zi ) where Zi = dX + 1 − d √ TY β i βi
d
5. Simulate LGDi = Beta(E(LGDi |Zi = z), σi )
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.13
Calculation Algorithm 1. Simulate factors Y = (Y1 , ..., Yk ) and X 2. For each obligor i: calculate conditional default probability PDi (Y ) d
3. Simulate default events Di = Binomial(PDi (Y )) 4. Calculate expected conditional loss given default √ T β i 2 E(LGDi |Zi = zi ) where Zi = dX + 1 − d √ TY β i βi
d
5. Simulate LGDi = Beta(E(LGDi |Zi = z), σi ) Pm 6. Calculate X = i=1 Di · LGDi · EADi
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.13
Calculation Algorithm 1. Simulate factors Y = (Y1 , ..., Yk ) and X 2. For each obligor i: calculate conditional default probability PDi (Y ) d
3. Simulate default events Di = Binomial(PDi (Y )) 4. Calculate expected conditional loss given default √ T β i 2 E(LGDi |Zi = zi ) where Zi = dX + 1 − d √ TY β i βi
d
5. Simulate LGDi = Beta(E(LGDi |Zi = z), σi ) Pm 6. Calculate X = i=1 Di · LGDi · EADi 7. Repeat steps 1-6 10000 times
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.13
Summary We extend the standard Merton type model by including dependence of LGD and PD via systematic factors.
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.14
Summary We extend the standard Merton type model by including dependence of LGD and PD via systematic factors. We obtain a formula for the loss distribution of large homogeneous portfolios.
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.14
Summary We extend the standard Merton type model by including dependence of LGD and PD via systematic factors. We obtain a formula for the loss distribution of large homogeneous portfolios. Our model addresses the observation that the probability of default and the expected loss do not necessarily vary comonotonically over the economic cycle, but require at least two factors for an appropriate fit
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.14
Summary We extend the standard Merton type model by including dependence of LGD and PD via systematic factors. We obtain a formula for the loss distribution of large homogeneous portfolios. Our model addresses the observation that the probability of default and the expected loss do not necessarily vary comonotonically over the economic cycle, but require at least two factors for an appropriate fit The Basel Committee (2004) suggests to estimate “downturn LGD” and “downturn PD” separately. Based on the insight of the investigation, we cannot support this sugggestion, because it leads to an overestimation of risk. Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.14
Technical University of Munich
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.1
Introduction Consider a portfolio of m credits.
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.2
Introduction Consider a portfolio of m credits. Portfolio loss within one year: m X X= 11i ·EADi ·LGDi i=1
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.2
Introduction Consider a portfolio of m credits. Portfolio loss within one year: m X X= 11i ·EADi ·LGDi i=1
11i ∈ {0, 1} is a Bernoulli(PDi (Y )) random variable; 11i = 1 if credit i defaults
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.2
Introduction Consider a portfolio of m credits. Portfolio loss within one year: m X X= 11i ·EADi ·LGDi i=1
11i ∈ {0, 1} is a Bernoulli(PDi (Y )) random variable; 11i = 1 if credit i defaults
Probability of default PDi (Y ) = Φ (c − eY ) Dependent on economic cycle (Y )
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.2
Introduction Consider a portfolio of m credits. Portfolio loss within one year: m X X= 11i ·EADi ·LGDi i=1
11i ∈ {0, 1} is a Bernoulli(PDi (Y )) random variable; 11i = 1 if credit i defaults
Probability of default PDi (Y ) = Φ (c − eY ) Dependent on economic cycle (Y ) EADi :
Martin Hillebrand
the exposure at default – constant: EADi ≡ 1/m
Estimating Portfolio Loss with Dependent LGD – p.2
Introduction Consider a portfolio of m credits. Portfolio loss within one year: m X X= 11i ·EADi ·LGDi i=1
11i ∈ {0, 1} is a Bernoulli(PDi (Y )) random variable; 11i = 1 if credit i defaults
Probability of default PDi (Y ) = Φ (c − eY ) Dependent on economic cycle (Y ) EADi : LGDi :
the exposure at default – constant: EADi ≡ 1/m
the loss given default Usually assumed to be constant
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.2
Introduction Consider a portfolio of m credits. Portfolio loss within one year: m X X= 11i ·EADi ·LGDi i=1
11i ∈ {0, 1} is a Bernoulli(PDi (Y )) random variable; 11i = 1 if credit i defaults
Probability of default PDi (Y ) = Φ (c − eY ) Dependent on economic cycle (Y ) EADi : LGDi :
the exposure at default – constant: EADi ≡ 1/m
the loss given default Usually assumed to be constant Our model: dependent on economic cycle (Z )
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.2
Merton type structural model Company i defaults if logreturn of asset value Ai falls 11i = 11{Ai <si } below some threshold value si :
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.3
Merton type structural model Company i defaults if logreturn of asset value Ai falls 11i = 11{Ai <si } below some threshold value si : d
Ai = N (0, 1) =⇒ si = Φ−1 (PDi )
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.3
Merton type structural model Company i defaults if logreturn of asset value Ai falls 11i = 11{Ai <si } below some threshold value si : d
Ai = N (0, 1) =⇒ si = Φ−1 (PDi ) p Ai = β i Y + 1 − β 2 εi
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.3
Merton type structural model Company i defaults if logreturn of asset value Ai falls 11i = 11{Ai <si } below some threshold value si : d
Ai = N (0, 1) =⇒ si = Φ−1 (PDi ) p Ai = β i Y + 1 − β 2 εi =⇒ PDi (Y ) = P(11i = 1|Y ) = Φ
Φ−1 (PDi ) − βi Y p 1 − β2
!
= Φ(c − eY )
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.3
PSfrag replacements
Dependence in the Data
0.6 0.5 0.4
E(LGD|Z)
Altman et al. (2003): Annual average LGD and default frequencies of American corporate bonds from 1982 to 2001 mean(LGD)=0.58 mean(PD)=0.035
0.7
Fig. 1: Dependence of E(LGD|Z) and PD(Y )
0.02
0.04
0.06
0.08
0.10
PD(Y )
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.4
PSfrag replacements
Dependence in the Data
0.6 0.4
0.5
E(LGD|Z)
Altman et al. (2003): Annual average LGD and default frequencies of American corporate bonds from 1982 to 2001 mean(LGD)=0.58 mean(PD)=0.035
0.7
Fig. 1: Dependence of E(LGD|Z) and PD(Y )
0.02
Recall the linear relation
0.04
0.06
0.08
0.10
PD(Y )
Φ−1 (PD(Y )) = c − eY. Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.4
PSfrag replacements
Transformed Data: Linear Dependence
0.4 0.2 −0.2
0.0
Φ−1 (E(LGD|Z))
0.6
Fig. 2: Dependence of transformed E(LGD|Z) and PD(Y )
−2.4
−2.2
−2.0
−1.8
−1.6
−1.4
Φ−1 (PD(Y ))
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.5
PSfrag replacements
Transformed Data: Linear Dependence
0.4 0.2 −0.2
0.0
Φ−1 (E(LGD|Z))
0.6
Fig. 2: Dependence of transformed E(LGD|Z) and PD(Y )
−2.4
−2.2
Martin Hillebrand
−1.8
−1.6
−1.4
Φ−1 (PD(Y ))
Linear ansatz:
with Z = dY +
−2.0
√
Φ−1 (E(LGD|Z)) = a + bZ 1 − d2 X
Estimating Portfolio Loss with Dependent LGD – p.5
Linear Regression Model From Φ−1 (PD(Y )) = b − eY
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.6
Linear Regression Model From Φ−1 (PD(Y )) = b − eY
and with Z = dY +
Martin Hillebrand
√
Φ−1 (E(LGD|Z)) = a + bZ 1 − d2 X
Estimating Portfolio Loss with Dependent LGD – p.6
Linear Regression Model From Φ−1 (PD(Y )) = b − eY
and with Z = dY +
√
Φ−1 (E(LGD|Z)) = a + bZ 1 − d2 X
we obtain the regression model
p Φ (E(LGD|Z = d · y + 1 − d2 X)) p c bd −1 = a + bd − Φ (PD(y)) + b 1 − d2 X e e −1
providing estimates for a, b, and d. Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.6
Downturn LGD
0.6 0.5 0.3
0.4
E(LGD|Z = zα )
0.7
0.8
PSfrag replacements
With the estimates, downturn LGD can easily be calculated by E(LGD|Z = zα ) = Φ(ˆ a + ˆb · zα ).
0.0
0.2
0.4
0.6
0.8
1.0
α
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.7
Downturn LGD
0.6 0.5 0.3
0.4
E(LGD|Z = zα )
0.7
0.8
PSfrag replacements
With the estimates, downturn LGD can easily be calculated by E(LGD|Z = zα ) = Φ(ˆ a + ˆb · zα ).
0.0
0.2
0.4
0.6
0.8
1.0
α
E.g. for α = 0.001, we obtain E(LGD|Z = z0.001 ) = 0.874. Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.7
Portfolio Loss Distribution For m → ∞, we obtain a formula for the portfolio loss distribution: Z A(l) P (L ≤ l) = 1 − Φ(B(l, y))ϕ(y) dy. −∞
and
Martin Hillebrand
c − Φ−1 (l) with A(l) = e l 1 −1 B(l, y) = √ a − dby − Φ . 2 Φ(c − ey) b 1−d
Estimating Portfolio Loss with Dependent LGD – p.8
Portfolio Loss Distribution For m → ∞, we obtain a formula for the portfolio loss distribution: Z A(l) P (L ≤ l) = 1 − Φ(B(l, y))ϕ(y) dy. −∞
and
c − Φ−1 (l) with A(l) = e l 1 −1 B(l, y) = √ a − dby − Φ . 2 Φ(c − ey) b 1−d
Observe that, if LGD ≡ 1,
P (L ≤ l) = 1 − Φ(A(l)). Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.8
PSfrag replacements
3: Credit Portfolio Loss Density Portfolio Loss Distribution Tail
PSfrag replacements
3: Credit Portfolio Loss Density Portfolio Loss Distribution Tail
Comparison with constant LGD
0.05
dependent LGD LGD=0.65 LGD=1
0
0.00
0.01
10
0.02
20
0.03
30
dependent LGD LGD=0.65 LGD=1
Credit Portfolio Loss Distribution Tail
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40
Credit Portfolio Loss Distribution Histogram
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Relative Loss
Martin Hillebrand
0.15
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0.11
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0.13
0.14
0.15
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Relative Loss
Estimating Portfolio Loss with Dependent LGD – p.9
Impact on VaR The impact of dependent LGD modeling on portfolio VaR grows with increasing α: VaRα Dependent LGD LGD ≡ 0.65 LGD ≡ 1 ELGD(qα ) ∗ p(qα )
Martin Hillebrand
α = 99%
α = 99.5%
α = 99.9%
0.107 0.089 0.137 0.112
0.126 0.102 0.157 0.132
0.171 0.134 0.206 0.179
Estimating Portfolio Loss with Dependent LGD – p.10
PSfrag replacements
3: Credit Portfolio Loss Density Portfolio Loss Distribution Tail
PSfrag replacements
3: Credit Portfolio Loss Density Portfolio Loss Distribution Tail
0.08
Relative Loss
Relative Loss
Goodness of Fit: qq-plots
0.00
0.020 0.015 0.000
0.005
0.010
Estimated Quantiles
0.04 0.02
Estimated Quantiles
0.06
0.025
Hillebrand Giese Tasche
0.00
0.02
0.04 Empirical Quantiles
Martin Hillebrand
0.06
0.08
0.000
0.005
0.010
0.015
0.020
0.025
Empirical Quantiles
Estimating Portfolio Loss with Dependent LGD – p.11
Nonhomogeneous portfolios Data:
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.12
Nonhomogeneous portfolios Data: As in the standard Merton type model:
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.12
Nonhomogeneous portfolios Data: As in the standard Merton type model: Default probabilities: PDi
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.12
Nonhomogeneous portfolios Data: As in the standard Merton type model: Default probabilities: PDi Factor weights (“asset correlations”):βik
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.12
Nonhomogeneous portfolios Data: As in the standard Merton type model: Default probabilities: PDi Factor weights (“asset correlations”):βik Expected loss given default: E(LGDi )
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.12
Nonhomogeneous portfolios Data: As in the standard Merton type model: Default probabilities: PDi Factor weights (“asset correlations”):βik Expected loss given default: E(LGDi ) Additionally: LGD variance: σi
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.12
Nonhomogeneous portfolios Data: As in the standard Merton type model: Default probabilities: PDi Factor weights (“asset correlations”):βik Expected loss given default: E(LGDi ) Additionally: LGD variance: σi parameters b, d describing systematic dependence of √ LGD on systematic factors (a = 1 + b2 Φ−1 (E(LGD)))
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.12
Calculation Algorithm 1. Simulate factors Y = (Y1 , ..., Yk ) and X
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.13
Calculation Algorithm 1. Simulate factors Y = (Y1 , ..., Yk ) and X 2. For each obligor i: calculate conditional default probability PDi (Y )
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.13
Calculation Algorithm 1. Simulate factors Y = (Y1 , ..., Yk ) and X 2. For each obligor i: calculate conditional default probability PDi (Y ) d
3. Simulate default events Di = Binomial(PDi (Y ))
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.13
Calculation Algorithm 1. Simulate factors Y = (Y1 , ..., Yk ) and X 2. For each obligor i: calculate conditional default probability PDi (Y ) d
3. Simulate default events Di = Binomial(PDi (Y )) 4. Calculate expected conditional loss given default √ T β i 2 E(LGDi |Zi = zi ) where Zi = dX + 1 − d √ TY β i βi
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.13
Calculation Algorithm 1. Simulate factors Y = (Y1 , ..., Yk ) and X 2. For each obligor i: calculate conditional default probability PDi (Y ) d
3. Simulate default events Di = Binomial(PDi (Y )) 4. Calculate expected conditional loss given default √ T β i 2 E(LGDi |Zi = zi ) where Zi = dX + 1 − d √ TY β i βi
d
5. Simulate LGDi = Beta(E(LGDi |Zi = z), σi )
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.13
Calculation Algorithm 1. Simulate factors Y = (Y1 , ..., Yk ) and X 2. For each obligor i: calculate conditional default probability PDi (Y ) d
3. Simulate default events Di = Binomial(PDi (Y )) 4. Calculate expected conditional loss given default √ T β i 2 E(LGDi |Zi = zi ) where Zi = dX + 1 − d √ TY β i βi
d
5. Simulate LGDi = Beta(E(LGDi |Zi = z), σi ) Pm 6. Calculate X = i=1 Di · LGDi · EADi
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.13
Calculation Algorithm 1. Simulate factors Y = (Y1 , ..., Yk ) and X 2. For each obligor i: calculate conditional default probability PDi (Y ) d
3. Simulate default events Di = Binomial(PDi (Y )) 4. Calculate expected conditional loss given default √ T β i 2 E(LGDi |Zi = zi ) where Zi = dX + 1 − d √ TY β i βi
d
5. Simulate LGDi = Beta(E(LGDi |Zi = z), σi ) Pm 6. Calculate X = i=1 Di · LGDi · EADi 7. Repeat steps 1-6 10000 times
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.13
Summary We extend the standard Merton type model by including dependence of LGD and PD via systematic factors.
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.14
Summary We extend the standard Merton type model by including dependence of LGD and PD via systematic factors. We obtain a formula for the loss distribution of large homogeneous portfolios.
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.14
Summary We extend the standard Merton type model by including dependence of LGD and PD via systematic factors. We obtain a formula for the loss distribution of large homogeneous portfolios. Our model addresses the observation that the probability of default and the expected loss do not necessarily vary comonotonically over the economic cycle, but require at least two factors for an appropriate fit
Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.14
Summary We extend the standard Merton type model by including dependence of LGD and PD via systematic factors. We obtain a formula for the loss distribution of large homogeneous portfolios. Our model addresses the observation that the probability of default and the expected loss do not necessarily vary comonotonically over the economic cycle, but require at least two factors for an appropriate fit The Basel Committee (2004) suggests to estimate “downturn LGD” and “downturn PD” separately. Based on the insight of the investigation, we cannot support this sugggestion, because it leads to an overestimation of risk. Martin Hillebrand
Estimating Portfolio Loss with Dependent LGD – p.14
