An Intensity Based Non-Parametric Default Model for Residential Mortgage Portfolios Enrico De Giorgi, RiskLab, ETH Zurich joint work with Vlatka Komaric, Credit Suisse Group Risk Day 2001, Zurich October 19, 2001 E-mail: degiorgi@math.ethz.ch Homepage: http://www.math.ethz.ch/ ∼ degiorgi/ RiskLab: http://www.risklab.ch
An Intensity Based Non-Parametric Default Model for Residential Mortgage Portfolios Part 1 I Introduction II Definitions Part 2 III The model IV Estimation methodology V Data set and results VI Conclusion � 2001 (E. De Giorgi, RiskLab) c 1
Introduction • In April 2001 Swiss banks held over CHF 500 billion of debt in form of mortgages (about 63% of the total loan portfolios value held by Swiss banks). • Estimated Swiss real estate market value is between 2300 and 2800 billion CHF (more than twice the market capitalization of all stocks included in Swiss Performance Index). • About 86% of Swiss real estate are in the hands of private individ- uals. • Not much research was done in this area so far: Lack of information, confidentiality, old and insufficient data, mort- gages are regarded as a ”low risk” product in Switzerland. � 2001 (E. De Giorgi, RiskLab) c 2
Mortgage characteristics A mortgage is a loan secured by a real estate property. The purpose of the private clients is to finance the property. We have the following traditional products: • Adjusted-rate and term (ARM): − no fixed maturity, interest rate follows market, but with a lag and is subject to politics; − prepayment is free for the clients (embedded option). • Fixed-rate and term: − maturity and interest rate are fixed by the issue of the mortgage; − maturity usually of 2-5 years; − prepayment costs are charged to clients. � 2001 (E. De Giorgi, RiskLab) c 3
Default event Definition. An obligor is said to default at time T if he loses the ability to make the next interest payment. Define the default indicator process for t ≥ 0 by � 1 default prior to time t , def = 1 { T ≤ t } = X t 0 else. The observation of default is censored, it is t l − 1 t l t l +1 observed only if the payment fails over a pe- fix ✲ riod of fixed length (usually 90 days) after it obs. T was due. Common reasons for default • unemployment; • divorce; • significantly interest rate increase. � 2001 (E. De Giorgi, RiskLab) c 4
Conditional intensity process Let Y i = ( Y i, 1 , . . . , Y i,p ), Y i,q = ( Y i,q ( t )) t ≥ 0 be a collection of predictors for obligor i , i = 1 , . . . , n . F i,t = σ ( Y i,s : s ≤ t ) is the σ -algebra generated by � Y i,t = ( Y i,s ) 0 ≤ s ≤ t . D i,t = σ ( X i,s : s ≤ t ) is the σ -algebra generated by the default indicator X i = ( X i,s ) t ≥ 0 of obligor i . We define the enlarged filtration G i = ( G i,t ) t ≥ 0 by G i,t = F i,t ∨ D i,t . Definition. Let F i = ( F i,t ) t ≥ 0 be the flow of information from the pre- dictors for obligor i . The conditional intensity process of the time to default T i given F i , is the nonnegative, F i -predictable process λ F i such i that the process M i = ( M i,t ) t ≥ 0 defined by � t ∧ T i λ F i M i,t = X i,t − i,u du 0 is a G i -martingale. � 2001 (E. De Giorgi, RiskLab) c 5
Properties of the conditional intensity process λ F i i � � Let S i ( t |F i,t ) = P T i > t |F i,t be the conditional survival probability and � � f i ( t |F i,t ) = lim s ց 0 1 T i ∈ ( t, t + s ] |F i,t be the conditional density func- s P tion of T i . . Under technical conditions, λ F i and f i exist, and we have i � � = 1 { T i >t } λ F i • lim s ց 0 1 T i ∈ ( t, t + s ] |G i,t i,t . s P i,t = f i ( t |F i,t ) • λ F i S i ( t |F i,t ) . � � � t ∨ d i λ F i • S i ( t |F i,t ) = exp − where d i = time of issue. i,u du d i On the set { T i > t } and for ∆ t ≪ 1, λ F i i,t ∆ t approximates the conditional probability that a default occurs during ( t, t + ∆ t ]. � 2001 (E. De Giorgi, RiskLab) c 6
Overview of the model • Form homogeneous groups characterized by their credit rating. • Model time-to-default as the first jump-time of an inhomogeneous Poisson process with stochastic intensity (doubly stochastic Poisson process or Cox process). • Link the intensity to explaining factors (economic environment, mortgage characteristics, obligor characteristics). • Given a realization of explaining factors, suppose individual defaults occur independently. • Fit the model to a mortgage portfolio for determining the form of the linking functions. � 2001 (E. De Giorgi, RiskLab) c 7
The model Let Y i = ( Y i, 1 , . . . , Y i,p ), Y i,q = ( Y i,q ( t )) t ≥ 0 be a collection of predictors for the intensity process λ F i for obligor i . We model λ F i as a function i i of Y i . We suppose that p � λ F i i,t = λ i, 0 h i, 0 ( t − d i ) h i,q ( Y i,q ( t )) . q =1 We write λ F i i,t = λ F i i,t ( θ i ; Y i,t ) where θ i = (log λ i, 0 , log h i, 0 , . . . , log h i,p ). Here h i, 0 , h i, 1 , . . . , h i,p are the link functions to be estimated later. Let η F i i,t ( θ i ; Y i,t ) = log λ F i i,t ( θ i ; Y i,t ). Then we obtain p � η F i i,t ( θ i ; Y i,t ) = log λ i, 0 + log h i, 0 ( t − d i ) + log h i,q ( Y i,q ( t )) . q =1 � � We suppose that E log h i,q ( Y i,q ( t )) = 0 for i = 1 , . . . , n , q = 1 , . . . , p . � 2001 (E. De Giorgi, RiskLab) c 8
Assumptions • The θ i ’s are the same for all obligors in the same rating class. ⇒ Functional form depends only on the rating class. • Given T i > t , the conditional probability that obligor i will survive time t + s for s > 0 depends on the history only through the predictors at time t . ⇒ Treats all the outstanding mortgages at time t in the same way. • Given the predictors up to time t , defaults of obligors up to time t are conditionally independent. ⇒ Dependence structure is totally described by the predictors. � 2001 (E. De Giorgi, RiskLab) c 9
Estimation of the model for one rating class • θ = (log λ 0 , log h 0 , log h 1 , . . . , log h p ) are the same for every obligor. · · · t m = T t 0 t 1 t 2 1 X 2 • Group obligors such that their predictors Y i and their time of issue d i are identical in 3 every group ( J groups). RP RP 4 5 X • Let 0 = t 0 < t 1 < · · · < t m = T . 6 7 . . • O j,l = number of outstanding mortgages . X during ( t l , t l +1 ] in group j . RP RP n RP : repayment. • D j,l = number of mortgages defaulted dur- X : default. ing ( t l , t l +1 ] in group j . � 2001 (E. De Giorgi, RiskLab) c 10
Conditional likelihood function of the discretized model Assuming that λ and the predictors are constant on [ t l , t l +1 ), then on the set { T i > t l } for obligor i in group j and Y j = y j we have � � T i ∈ ( t l , t l +1 ] |F j,t l � � P T i ∈ ( t l , t l +1 ] |G i,t l = P � � P T i > t l |F j,t l � � S ( t l |F j,t l ) − P T i > t l +1 |F j,t l = S ( t l |F j,t l ) � � = 1 − exp − ( t l +1 − t l ) λ t l ( θ ; y j,t l ) def = u j,l ( θ ) . The likelihood function for the observation is thus given by m − 1 J � O j,l � � � O j,l − D j,l � � u j,l ( θ ) D j,l L ( θ ) = 1 − u j,l ( θ ) . D j,l l =0 j =1 � �� � binomial distribution � 2001 (E. De Giorgi, RiskLab) c 11
Generalized additive model (GAM) Let V be a real random variable. Let Y = ( Y 1 , . . . , Y p ) be a set of predictors. Given Y , V has the conditional distribution function F Y � � with µ ( Y ) = E V | Y . We assume that for functions f 1 , . . . , f p , we have p � G ( µ ( Y )) = η ( Y ) = α + f q ( Y q ) q =1 � � where G is the link function, E f q ( Y q ) = 0 for q = 1 , . . . , p . η is called an additive form, θ = ( α, f 1 , . . . , f p ) are the unknown param- eters to be estimated. The triple ( η, G, F Y ) is called a GAM. Remarks • If all the f q ’s are linear functions, then ( η, G, F Y ) is called a gener- alized linear model (GLM). • For observations ( V i ) i =1 ,...,M we need V i | Y i ∼ F Y i , independently. • The GAM serves as a diagnostic tool for suggesting transformations of the predictors. � 2001 (E. De Giorgi, RiskLab) c 12
GAM estimation If V | Y ∼ F Y has an exponential family density � � vξ − b ( ξ ) f Y ( v ; ξ, φ ) = exp + c ( v, φ ) v ∈ support( F Y ) , a ( φ ) where ξ is the natural parameter ( b ′ ( ξ ) = µ ) depending on Y , and φ is the dispersion parameter, then the local scoring algorithm with backfit- ting can be applied to solve the GAM (Hastie and Tibshirani, 1990). Remarks • F Y = binomial( n, p ( Y )) is an exponential family density with φ = 1. • The local scoring algorithm maximizes the likelihood function by a modified Newton-Raphson procedure. • The local scoring algorithm converges for cubic smoothing splines. � 2001 (E. De Giorgi, RiskLab) c 13
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