[PPT] - An Intensity Based Non-Parametric Default Model for Residential PowerPoint Presentation

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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

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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

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Introduction

In April 2001 Swiss banks held over CHF 500 billion of debt in form
f 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.

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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.

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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 Xt

def

= 1{T≤t} =

1

default prior to time t, else. The observation of default is censored, it is

bserved only if the payment fails over a pe-

riod of fixed length (usually 90 days) after it was due.

✲

tl−1 tl tl+1

fix

T

bs.

Common reasons for default

unemployment;
divorce;
significantly interest rate increase.

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Conditional intensity process

Let Yi = (Yi,1, . . . , Yi,p), Yi,q = (Yi,q(t))t≥0 be a collection of predictors for obligor i, i = 1, . . . , n. Fi,t = σ(Yi,s : s ≤ t) is the σ-algebra generated by

Yi,t = (Yi,s)0≤s≤t.

Di,t = σ(Xi,s : s ≤ t) is the σ-algebra generated by the default indicator Xi = (Xi,s)t≥0 of obligor i. We define the enlarged filtration Gi = (Gi,t)t≥0 by Gi,t = Fi,t ∨ Di,t.

Definition. Let Fi = (Fi,t)t≥0 be the flow of information from the pre-

dictors for obligor i. The conditional intensity process of the time to default Ti given Fi, is the nonnegative, Fi-predictable process λFi

i

such that the process Mi = (Mi,t)t≥0 defined by Mi,t = Xi,t −

t∧Ti

λFi

i,u du

is a Gi-martingale.

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Properties of the conditional intensity process λFi

i Let Si(t|Fi,t) = P

Ti > t|Fi,t
be the conditional survival probability and

fi(t|Fi,t) = limsց0 1

sP

Ti ∈ (t, t+s]|Fi,t
be the conditional density func-

tion of Ti. . Under technical conditions, λFi

i

and fi exist, and we have

limsց0 1

sP

Ti ∈ (t, t + s]|Gi,t
= 1{Ti>t} λFi

i,t.

λFi

i,t = fi(t|Fi,t) Si(t|Fi,t).

Si(t|Fi,t) = exp
−

t∨di

di

λFi

i,u du

where di = time of issue.

On the set {Ti > t} and for ∆t ≪ 1, λFi

i,t ∆t approximates the conditional

probability that a default occurs during (t, t + ∆t].

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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
ccur independently.
Fit the model to a mortgage portfolio for determining the form of

the linking functions.

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The model

Let Yi = (Yi,1, . . . , Yi,p), Yi,q = (Yi,q(t))t≥0 be a collection of predictors for the intensity process λFi

i

for obligor i. We model λFi

i

as a function

f Yi.

We suppose that λFi

i,t = λi,0 hi,0(t − di) p

q=1

hi,q(Yi,q(t)). We write λFi

i,t = λFi i,t(θi; Yi,t) where θi = (log λi,0, log hi,0, . . . , log hi,p).

Here hi,0, hi,1, . . . , hi,p are the link functions to be estimated later. Let ηFi

i,t(θi; Yi,t) = log λFi i,t(θi; Yi,t). Then we obtain

ηFi

i,t(θi; Yi,t) = log λi,0 + log hi,0(t − di) + p

q=1

log hi,q(Yi,q(t)). We suppose that E

log hi,q(Yi,q(t))
= 0 for i = 1, . . . , n, q = 1, . . . , p.

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Assumptions

The θi’s are the same for all obligors in the same rating class.

⇒ Functional form depends only on the rating class.

Given Ti > 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.

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Estimation of the model for one rating class

n . . . 7 6 5 4 3 2 1 t0 t1 t2 · · · tm = T X X X RP RP RP RP RP: repayment. X: default.

θ = (log λ0, log h0, log h1, . . . , log hp) are the

same for every obligor.

Group obligors such that their predictors

Yi and their time of issue di are identical in

every group (J groups).

Let 0 = t0 < t1 < · · · < tm = T.
Oj,l = number of outstanding mortgages

during (tl, tl+1] in group j.

Dj,l = number of mortgages defaulted dur-

ing (tl, tl+1] in group j.

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Conditional likelihood function

f the discretized model

Assuming that λ and the predictors are constant on [tl, tl+1), then on the set {Ti > tl} for obligor i in group j and Yj = yj we have

P

Ti ∈ (tl, tl+1]|Gi,tl
=

P

Ti ∈ (tl, tl+1]|Fj,tl
P
Ti > tl |Fj,tl
=

S(tl |Fj,tl) − P

Ti > tl+1|Fj,tl
S(tl |Fj,tl)

= 1 − exp

−(tl+1 − tl)λtl(θ; yj,tl)
def

= uj,l(θ). The likelihood function for the observation is thus given by L(θ) =

m−1

l=0

J

j=1

Oj,l

Dj,l

uj,l(θ)Dj,l
1 − uj,l(θ)

Oj,l−Dj,l

binomial distribution

.

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Generalized additive model (GAM)

Let V be a real random variable. Let Y = (Y1, . . . , Yp) be a set of predictors. Given Y, V has the conditional distribution function FY with µ(Y) = E

V |Y
. We assume that for functions f1, . . . , fp, we have

G(µ(Y)) = η(Y) = α +

p

q=1

fq(Yq) where G is the link function, E

fq(Yq)
= 0 for q = 1, . . . , p.

η is called an additive form, θ = (α, f1, . . . , fp) are the unknown param- eters to be estimated. The triple (η, G, FY) is called a GAM. Remarks

If all the fq’s are linear functions, then (η, G, FY) is called a gener-

alized linear model (GLM).

For observations (Vi)i=1,...,M we need Vi|Yi ∼ FYi, independently.
The GAM serves as a diagnostic tool for suggesting transformations
f the predictors.

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GAM estimation

If V |Y ∼ FY has an exponential family density fY(v; ξ, φ) = exp

vξ − b(ξ)

a(φ) + c(v, φ)

,

v ∈ support(FY) 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

FY = 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.

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Backfitting algorithm

Let G = id and (η, id, FY) the simple additive model, with FY an expo- nential family density. We have for i = 1, . . . , M Vi = α +

p

q=1

fq(Yi,q) + ǫi, where ǫi = Vi − E

Vi|Y
. The backfitting algorithm proceeds as follows:
Initialization r = 0:
f0

q ≡ 0 for q = 1, . . . , p,

α0 = 1

M

i=1 Vi.

Iteration r → r + 1: cycle over q = 1, . . . , p
fr+1

q

= Sλ

q

  Vi −

αr −

q−1

q′=1
fr+1

q′

(Yi,q′) −

p

q′=q+1
fr

q′(Yi,q′)

Yq

  

i=1,...,M

until maxi=1,...,M

fr+1

q

(Yi,q) − fr

q (Yi,q)

is small enough.

Sλ

q denotes a smoothing operator (linear) with smoothing factor λ.

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Stepwise selection technique

Choice of the smoothing method (smoothing spline, local regres-

sion, kernel regression,. . . ).

For each function fq define a set Θq of alternatives of increasing

complexity for the corresponding smoothing operator Sλ

q , in terms

f the number of degrees of freedom d

f (d f = 0 for one Sλ

q means

that fq ≡ 0, d f = 1 means fq linear).

Let

θ1 ∈ Θ = R×Θ1×· · ·×Θp. Define θ2 by increasing the complexity

ne step forward in Θq′ for exactly one q′ = 1, . . . , p in

θ1 ( θ1, θ2 are nested models).

Compare the two models by testing the null hypothesis H0 : θ =

θ1 against the alternative HA : θ = θ2 using a χ2-test.

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Akaike information criterion

Let (η, G, FY) be a GAM and let FY be an exponential family density with dispersion parameter φ. We define the Akaike information criterion for the model θ ∈ Θ by AIC

θ = D(

θ; v) + 2 φ d f

θ,

where d f

θ is the number of degrees of freedom of the model.

AIC is a penalized version of the deviance D.
AIC accounts for the number of degrees of freedom used by the

smoothers.

Usually a lower AIC implies that the model fits better then another.
AIC offers a criterion for comparing two models

θ1, θ2 ∈ Θ, nested

r non-nested.
No specific statistical test is associated with comparing AIC’s.

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Reformulation of the default model as GAM

Let Vj,l = Dj,l Oj,l uj,l(θ) = 1 − exp

−(tl+1 − tl) λ(tl, θ|yj,tl)
then

Vj,l ∼ 1 Oj,l binomial(Oj,l, uj,l(θ)) G(uj,l(θ)) = log λ0 + log h0(t − dj) +

p

q=1

log hq(yj,q(tl)) where uj,l(θ) = Eθ

Vj,l |yj,tl
and G : (0, 1) −

→ R, µ → log(− log(1 − µ)) is the link function (the complementary log log-function). ⇒ Generalized additive model.

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Data set

Sub-portfolio P with 73683 Swiss residential mortgages.
t0 = 1st quarter 1994, tm = 4th quarter 2000.
Observation of P follows at the end of each quarter (March 31,

June 30, September 30, December 31).

The mortgage product and the mortgage interest rate ri,tl applied

during the quarter [tl−1, tl) are available for i = 1, . . . , 73683 and l = 1, . . . , m.

Obligors belongs to 26 different economic and political regions

(26 cantons).

Two rating classes are considered:

A=higher rating and B=lower rating.

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Predictors

For obligor i (i = 1, . . . , 73683) we use the following predictors.

Quarter of the year Yi,0: Yi,0(tl) = k, if tl is the k-th quarter.
Quarterly regional unemployment rate Yw,1, if obligor i lives in region

w = 1, . . . , 26.

Lags of 1, . . . , 16 quarters for the regional unemployment rate are

considered (notation: Y (r)

w,1, w = 1, . . . , 26, r = 1, . . . , 16).

Indicator variable Yi,3 for mortgage product:

adjusted-rate (Yi,3 = 1), fixed-rate mortgage (Yi,3 = 2).

Levels Yi,4 for the relative interest rate change over last quarter:

Yi,4(tl) =

          

1 if xi,tl < 0, 2 if xi,tl = 0, k + 1 if xi,tl ∈ (ak−1, ak], k = 2, 3, 5 if xi,t > 0.5, where xi,tl =

ri,tl ri,tl−1

− 1, a1 = 0, a2 = 0.25 and a3 = 0.5.

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Selected models

We have J = 260 groups of obligors characterized by the predictor

realizations (6500 observations of Oj,l and Dj,l for each rating class).

3265 non-zero observations of Oj,l for rating A, and 2713 non-zero
bservations of Oj,l for rating B.

The following models has been selected by our criterion:

Rating A

G(uj,l( θA)) = αA +

f(11)

1,A (y(11) j,1 (tl)) +

+ β3,A 1{yj,3(tl)=1} + γ3,A

+

f4,A(yj,4(tl)).

Rating B

G(uj,l( θB)) = αB +

f(q)

0,B(y0(tl)) +

f(8)

1,B(y(8) j,1 (tl)) +

+ β3,B 1{yj,3(tl)=1} + γ3,B

+

f4,B(yj,4(tl)).

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Parametric estimates

Rating

α
β3
γ3

Estimate

9.9108
1.3568

0.6740 A Standard error 0.7752 0.4443 0.2207

Approx. 95% CI
11.4612
2.2454

0.2326

8.3604
0.4682

1.1154 Estimate

6.8644
1.7893

0.8462 B Standard error 0.3636 0.1690 0.0799

Approx. 95% CI
6.1372
2.1273

0.6864

7.5916
1.4513

1.006

Parametric estimates for the two models (Rating A and Rating B), with standard errors and approximated 95% confidence intervals. c

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Non-parametric estimates: rating A

2 4 6 8 unemployment rate (lagged 11 quarters)

2
1

1 2 3 f(unemployment)

Spline estimation f (11)

1,A

with 1.2 degrees of free- dom. Dotted lines give the approximated 95% confidence interval.

1 2 3 4 5 interest rate change (intervals)

2

2 4 6 f(interest rate change)

Spline estimation f4,A with 2 degrees of freedom. Dotted lines give the approximated 95% confi- dence interval. c

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Non-parametric estimates: rating B

1 2 3 4 quarter of year

0.8
0.6
0.4
0.2

0.0 0.2 0.4 f(quarter)

Spline estimation f (q)

0,B with 2 degrees of freedom.

Dotted lines give the approximated 95% confi- dence interval.

2 4 6 8 unemployment rate (lagged 8 quarters)

1.0
0.5

0.0 0.5 f(unemployment)

Spline estimation f (8)

1,B with 1.1 degrees of free-

dom. Dotted lines give the approximated 95% confidence interval. c

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Non-parametric estimates: rating B (2)

1 2 3 4 5 interest rate change (intervals)

4
2

2 4 f(interest rate change)

Spline estimation f4,B with 1.9 degrees of free- dom. Dotted lines give the approximated 95% confidence interval. c

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Simulation under different scenarios

25 30 35 40 45 50 55 60 0.00 0.02 0.04 0.06 Number defaults Probability 5 10 15 20 25 30 0.00 0.02 0.04 0.06 0.08 0.10 0.12 Number defaults Probability

1000 simulations of the total number of defaults during the first quarter 2001 in a portfolio P′ with 100000 obligors. Obligors in P′ are distributed among the 26 regions, the 2 mortgage products and the 2 rating classes as in portfolio P at the end of the last quarter 2000. Two scenario for the interest rate are considered: increase of 0.75% (left histogram), decrease of 0.5% (right histogram). c

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