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Inflated mixture models: Applications to multimodality in loss given default. Mauro Ribeiro de O. Júnior @MauroRdeOJr
Caixa Econômica Federal (Brazilian bank) Department of Statistics Federal University of Sao Carlos, Brazil Thursday 27 August 2015
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Holyrood Park It may give us some idea about what is modality...
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Agenda
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Illustrative examples of LGD distributions I Tong, Edward NC, Christophe Mues, and Lyn Thomas. "A zero-adjusted gamma model for mortgage loan loss given default."International Journal of Forecasting 29.4 (2013): 548-562.
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Why zero-adjusted?
Because the support of the gamma distribution does not include the zero point, i.e., the support is (0, +∞).
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How zero-adjusted?
figamma (y ; ϑ) =
δ0 , (1 − δ0 )fgamma ,
where fgamma follows a gamma distributions.
if if
y =0 0 < y < +∞,
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We’ve got more?
Inverse Gaussian distribution, with support (0, +∞). Log normal distribution, with support (0, +∞). (truncated) Normal Gaussian distribution, with support [0, +∞).
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Illustrative examples of LGD distributions II Calabrese, Raffaella. "Downturn loss given default: Mixture distribution estimation."European Journal of Operational Research 237.1 (2014): 271-277.
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Who is beta? and Mixture? and Inflated Mixture?
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The beta distribution The well-known beta distribution has 2 parameters, mean µ ∈ (0, 1), and precision φ > 0. Its density function is defined for y ∈ (0, 1), where Γ(.) is the gamma function: f (y ; µ, φ) =
Γ(φ) y (µφ−1) (1 − y )(1−µ)φ−1 . Γ(µφ)Γ((1 − µ)φ)
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The mixture of beta distributions Given f1 (y ; µ1 , φ1 ) and f2 (y ; µ2 , φ2 ), two beta distributions, we set a mixture of two beta distributions, now with 5 parameters!: fm2 b (y ; π, µ1 , φ1 , µ2 , φ2 ) = πf1 (y ; µ1 , φ1 ) + (1 − π)f2 (y ; µ2 , φ2 ).
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The inflated mixture of beta distributions
The Y distribution is said to be an inflated (in zeros and ones) mixture of two beta distributions, with a 7-parameter ϑ = (δ0 , δ1 , π, µ1 , φ1 , µ2 , φ2 ), if its density function is given by: δ0 , (1 − δ0 − δ1 )fm2 b , fim2 b (y ; ϑ) = δ1 ,
if if if
y =0 0 β x> β 1+e 1i 1 +e 2i 2 x> β 3 e 3i x> β 1+e 3i 3 x> β 4 e 4i x> β 1+e 4i 4
,
if if if
y =0 0 β x> β 1+e 1i 1 +e 2i 2
,
Note that the inflated mixture of beta regression model can be viewed as an extension of the inflated beta regression model introduced by Ospina, R. & Ferrari, S. L. (2012). A general class of zero-or-one inflated beta regression models. Computational Statistics & Data Analysis, 56(6), 1609-1623.
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Application data
We consider a sample of the portfolio 1, containing 5000 retail loans. We consider two covariates made available by the bank. Let (x1 , x2 , x3 ) , where x1 is the interceptor parameter, i.e., x1 = 1, and two others are real covariates. x2 represents two group of clients according to the behavioural risk presented. The bank has its behaviour score model and segregated their customers into two groups, roughly, x2 = 0 to customers with poor credit risk and x2 = 1 with better credit risk. x3 is related to the loan characteristics. The loan classified as x3 = 0, represents a group of loans with term relatively shorter than the group with x3 = 1.
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Application data
Thus, we use the following setting of the link functions: e(β11 x1i +β12 x2i +β13 x3i ) δ0i = (β11 x1i +β12 x2i +β13 x3i ) 1+e +e(β21 x1i +β22 x2i +β23 x3i ) e(β21 x1i +β22 x2i +β23 x3i ) δ1i = 1+e(β11 x1i +β12 x2i +β13 x3i ) +e(β21 x1i +β22 x2i +β23 x3i ) µ1i µ 2i
=
e(β31 x1i +β32 x2i +β33 x3i ) 1+e(β31 x1i +β32 x2i +β33 x3i )
=
e(β41 x1i +β42 x2i +β43 x3i ) 1+e(β41 x1i +β42 x2i +β43 x3i )
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The results
The results corroborate the finding that longer term loans held by lower credit risk clients have a much lower loss given default that the remaining group. Tabela: Summary of average LGD estimated by the inflated mixture of two beta regression model Portfolio 1 mean lgd = 0.5216
Subgroups x2 = 0 x3 = 0 x3 = 1 x2 = 1 x3 = 0 x3 = 1
Qtd 1,266 1,269 1,234 1,231
Observed 0.6325 0.6324 0.4262 0.3890
Estimated 0.6383 0.6255 0.4265 0.4101
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The results: estimated parameters Tabela: MLE results for the inflated mixture of two beta regression models Parameter β11 β12 β13 β21 β22 β23 π β31 β32 β33 φ1 β41 β42 β43 φ2
Estimative (est) 0.1890 0.8122 0.0851 0.9171 -0.2448 0.0013 0.5784 -0.7320 -0.2577 0.0728 1.1757 1.1076 -0.1125 -0.0508 62.8578
Standard error (se) 0.0707 0.0809 0.0802 0.0641 0.0786 0.0777 0.0828 0.1005 0.1086 0.1079 0.0538 0.0293 0.0349 0.0351 0.1110
|est|/ se 2.6731 10.0333 1.0599 14.3059 3.1132 0.0173 6.9789 7.2790 2.3727 0.6750 21.8377 37.7084 3.2177 1.4478 566.0437
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Selected Reference
Calabrese, R. Downturn loss given default: Mixture distribution estimation. European Journal of Operational Research, 237(1), 271–277, 2014. Ospina, R. & Ferrari, S. L. A general class of zero-or-one inflated beta regression models. Computational Statistics & Data Analysis, 56(6), 1609–1623, 2012. Tong, Edward NC, Christophe Mues, and Lyn Thomas. A zero-adjusted gamma model for mortgage loan loss given default. International Journal of Forecasting, 29(4), 548–562, 2013.
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Contact
[email protected] The research was sponsored by CAPES - Process number: BEX 10583/14-9, Brazil. Oliveira, M.R.1 , Louzada, F.2 , Pereira, G.H.A.3 , Moreira, F.4 , Calabrese, R.5 , Inflated Mixture Models: Applications to Multimodality in Loss Given Default (July, 2015). Available at SSRN: http://ssrn.com/abstract=2634919
Thank you! 1 Department of Credit Modelling at Caixa Econômica Federal Bank and Department of Statistics at Federal University of São Carlos, Brazil 2 Department of Statistics at University of São Paulo, Brazil 3 Department of Statistics at Federal University of São Carlos, Brazil 4 University of Edinburgh Business School, UK 5 University of Essex Business School, UK