Universidad Complutense de Madrid Máster en Ingeniería Matemática Curso 2007-2008 Modelling Week...

Universidad Complutense de Madrid

Máster en Ingeniería MatemáticaCurso 2007-2008

Modelling Week Second Edition

June 16 – June 24, 2008

Credit Scoring Modelling for Retail Banking Sector

Problem raised by Accenture.

Coordinators:

Ignacio Villanueva (UCM).

Estela Luna (Accenture).

Team members:

Elena Bartolozzi (Universitá di Firenze) Matthew Cornford (University of Oxford) Leticia García-Ergüín (UCM) Cristina Pascual Deocón (UCM) Oscar Iván Pascual (UCM) Francisco Javier Plaza (UCM)



Index Introduction Methodology and Data Univariate and Multivariate Analysis Model Creation Validation Calibration

Our problem is concerned with who a bank should loan its money to.

When a client applies for a loan, the bank would like to be sure that the client will pay back the full amount of the loan.

We need effective models that allow us to predict if a client will pay back the loan.

What we have is historical data for several variables. We are trying to fit a model to this historical data so we

can estimate a probability of default.


Our data is provided by Accenture and include details of completed loan agreements

The variables included are: Age Income Wealth Marital Status Length as a Client Amount of Loan Maturity Default


Sample Selection

We split the sample into two parts

The modelling sample

The validation sample


Modelling Sample A random sample from the data is selected. The size of the modelling sample is about 2/3 of the

original data This new sample is used to create the model.

Validation Sample The remaining data is used to validate the model We test how many defaults the model predicted and

which of them really did default.


We have a dependent variable, which is default, and some independent variables (age, income,…)

First of all, we do univariate analysis. For each variable, we calculate some statistics like

mean, standard deviation, skewness… We plot some histograms… This information can be use as a first check before

applying the model. It would be better if the data were homogeneous.


Univariate Analysis

We’ve used SAS software to generate these statistics:

output.htm


SAS Graph v9


This kind of analysis is very useful to detect outliers or transcription mistakes.

Multivariate Analysis

Correlations


Chi-squared test We try to calculate which of the variables are explanatory

variables, i.e. which variables does default depend on. We use the chi-squared test for that:

To begin with, we must discretize the continuous variables using percentiles.

After doing Chi-squared test, we look at the p-value. If p-value<0.05, we reject independency If p-value>0.05 we do not reject independency.


22

1

( )ni i

i i

O E

E

According to the results of the Univariate and Multivariate Analysis, the variables we include in our model are:

Age Income Wealth Marital Status Maturity


We apply a logit model using proc logistic in SAS and glmfit in MATLAB as well, obtaining the same results.


kk

jj

k

j

jj

k

j

jj

k

j

k

xxx

xxLogit

x

x

xxx

x

x

x

...)(1

)(log))((

)exp()(1

)(

)exp(1

)exp(

)(),...,(

110

0

0

0

0

Intercept -1.85136 Age -0.02678 Income 0.10025 Wealth -0.01761 Marital Status 0.79651 Maturity 0.00892

There must be some diferences because we randomize the sample.



So, our model is as follows:

-

1P(Default/x)=

1+e

-1.85 -0.026*Age+0.1*Inc-0.017*Wlth+0.79*Marit+0.0089*MaturWhere


Model statistics:

Powerstat is a method to measure the likelihood of the model

The data is sorted from worse to better according to the probability of default calculated with our model.

The perfect model will have the total amount of defaults at the beginning.

We plot accumulated defaults against accumulated observations.

Powerstat compares the area between the perfect model, our model and a random model.


Powerstat (Gini Index):


Validation

Once the probability of default for each client is found, the question is how to choose the level that classifies if a client will default or not.

We use the validation data to predict with our model how many observations will default and compare with which of them are really did default.

Repeating the process with several random samples, the probability has very low deviation and rounds 0.77.


The expected Loss is defined as:

EL = PD * EAD * LGD

PD is the percentage of default. Is defined as default probability calibrated for a year. EAD is the exposition to default. LGD are losses on the exhibition.



Scoring allows us to sort people against default. However, these probabilities do not take into account when the

default happens. This is the reason for calibration. We want to obtain the yearly average probability of default We need a sample of people observed in periods of years. The model is applied and the sample is sorted by score. We obtain a default observed rate:

Minimizing the Least Squares Error with the MATLAB function fminsearch, we obtain the values:

A=0.0004 B=3.7410 C=2.7870

BscoreCARate )(


The Credit Scoring Model was solved quickly and didn’t cause too much difficult.

We asked Accenture to bring another, related problem.

We now introduce the Problem of Capital Allocation.

The Problem of Capital Allocation

Index The Problem of Capital Allocation Implementation Conclusions

In this problem a lender has a fixed amount of money to lend, EAD, between n blocks of similiar customers

¿How to distribute the money between the blocks to maximize the profit?

Each block has associated with it an interest rate ρi, an a priori probabilty of default PDi, the loss given default LGDi and the number of customers Ni.

If each customer in each block is independent of the rest then we can easily compute the probability of k defaults.


The Problem of Capital Allocation But the customers are correlated via the economy. We can use

Gaussian Copula to introduce a default random variable for each customer:

Then for a particular state of the economy we have that the independent probability of default for each customer is:

1

mi

i j j i ij

Z a Y rw

( )i iZ PD Default

1

1

( )m

ii j j

ji

i

PD a Y

pr

The Problem of Capital Allocation We use the binomial distribution:

When N is big enough (in the order of 10^3) we can aproximate this binomial with normal random variable Di:

( ) (1 ) ii N kki i

NP k defaults p p

k

( , (1 ))i i i i i iD N N p N p p


1

( ( ) )n

ii i i i i

i i

EADL LGDD N D

N

We define the loss distribution as:

As L is a sum of independent normal distributions,

2L( , )LL N

The Problem of Capital Allocation To measure risk we use Value at Risk (VaR) with a 99%

confidence level. So the problem becomes:

Where VaR99 is the fixed level of risk the lender is willing to take.

1

( )

: 1

-2.3262 + = 99

L

n

ii

L L

Minimise f

Subject to

VaR


Index The Problem of Capital Allocation Implementation Conclusions


We start with 3 blocks to make the problem easier.

We have to find the α’s that minimise the expected loss.

We have two approaches to solve this problem.

The Problem of Capital Allocation First we fix α’s and find the VaR99 and Expected Loss for

each set of α’s (Black dots).

The Problem of Capital Allocation Then we find the α’s that minimise the Expected Loss for any fixed

VaR99 (Red Dots) using the MATLAB function fmincon.

As we can see we got very good agreement between the two approaches, on the order of 10^(−4).

The Problem of Capital Allocation Here we have the results for 5 blocks, which took considerably

longer than with 3 blocks.


Conclusions

Analytical method outperformed the simulation of wi as expected.

Optimise for more than 3 blocks the choice of optimiser needs to be investigated furhter.

Another interesting question is to look at the relationship between the efficient border and the interest rates charged for each block.


¿Questions?

Universidad Complutense de Madrid Máster en Ingeniería Matemática Curso 2007-2008 Modelling Week...

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