Universidad Complutense de Madrid

1. Universidad Complutense de Madrid Máster en Ingeniería MatemáticaCurso 2007-2008 Modelling WeekSecond Edition June 16 – June 24, 2008

2. Credit Scoring Modelling for Retail Banking Sector • Problem raised by Accenture. • Coordinators: • Ignacio Villanueva (UCM). • Estela Luna (Accenture).

3. Credit Scoring Modelling for Retail Banking Sector • 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)

4. Credit Scoring Modelling for Retail Banking Sector • Index • Introduction • Methodology and Data • Univariate and Multivariate Analysis • Model Creation • Validation • Calibration

5. Credit Scoring Modelling for Retail Banking Sector • Index • Introduction • Methodology and Data • Univariate and Multivariate Analysis • Model Creation • Validation • Calibration

6. Credit Scoring Modelling for Retail Banking Sector • Our problem isconcerned 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.

7. Credit Scoring Modelling for Retail Banking Sector • Index • Introduction • Methodology and Data • Univariate and Multivariate Analysis • Model Creation • Validation • Calibration

8. Credit Scoring Modelling for Retail Banking Sector • 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

9. Credit Scoring Modelling for Retail Banking Sector • Sample Selection • We split the sample into two parts • The modelling sample • The validation sample

10. Credit Scoring Modelling for Retail Banking Sector • 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.

11. Credit Scoring Modelling for Retail Banking Sector • Index • Introduction • Methodology and Data • Univariate and Multivariate Analysis • Model Creation • Validation • Calibration

12. Credit Scoring Modelling for Retail Banking Sector • 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.

13. Credit Scoring Modelling for Retail Banking Sector • Univariate Analysis • We’ve used SAS software to generate these statistics: output.htm

14. Credit Scoring Modelling for Retail Banking Sector

15. Credit Scoring Modelling for Retail Banking Sector • This kind of analysis is very useful to detect outliers or transcription mistakes.

16. Credit Scoring Modelling for Retail Banking Sector Multivariate Analysis • Correlations

17. Credit Scoring Modelling for Retail Banking Sector • 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.

18. Credit Scoring Modelling for Retail Banking Sector

19. Credit Scoring Modelling for Retail Banking Sector

20. Credit Scoring Modelling for Retail Banking Sector

21. Credit Scoring Modelling for Retail Banking Sector

22. Credit Scoring Modelling for Retail Banking Sector

23. Credit Scoring Modelling for Retail Banking Sector

24. Credit Scoring Modelling for Retail Banking Sector • Index • Introduction • Methodology and Data • Univariate and Multivariate Analysis • Model Creation • Validation • Calibration

25. Credit Scoring Modelling for Retail Banking Sector • According to the results of the Univariate and Multivariate Analysis, the variables we include in our model are: • Age • Income • Wealth • Marital Status • Maturity

26. Credit Scoring Modelling for Retail Banking Sector • We apply a logit model using proc logistic in SAS and glmfit in MATLAB as well, obtaining the same results.

27. Credit Scoring Modelling for Retail Banking Sector

28. Credit Scoring Modelling for Retail Banking Sector • 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.

29. Credit Scoring Modelling for Retail Banking Sector • So, our model is as follows:

30. Credit Scoring Modelling for Retail Banking Sector • Index • Introduction • Methodology and Data • Univariate and Multivariate Analysis • Model Creation • Validation • Calibration

31. Credit Scoring Modelling for Retail Banking Sector • Model statistics:

32. Credit Scoring Modelling for Retail Banking Sector • 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.

33. Credit Scoring Modelling for Retail Banking Sector • Powerstat (Gini Index):

34. Credit Scoring Modelling for Retail Banking Sector • 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.

35. Credit Scoring Modelling for Retail Banking Sector • Index • Introduction • Methodology and Data • Univariate and Multivariate Analysis • Model Creation • Validation • Calibration

36. Credit Scoring Modelling for Retail Banking Sector • 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.

37. Credit Scoring Modelling for Retail Banking Sector • 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

38. Credit Scoring Modelling for Retail Banking Sector • 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.

39. The Problem of Capital Allocation • Index • The Problem of Capital Allocation • Implementation • Conclusions

40. The Problem of Capital Allocation • Index • The Problem of Capital Allocation • Implementation • Conclusions

41. The Problem of Capital Allocation • 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 LGDiand 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.

42. 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:

43. 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:

44. The Problem of Capital Allocation • We define the loss distribution as: • As L is a sum of independent normal distributions,

45. 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.

46. The Problem of Capital Allocation • Index • The Problem of Capital Allocation • Implementation • Conclusions

47. The Problem of Capital Allocation • 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.

48. 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).

49. 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).

50. The Problem of Capital Allocation • Here we have the results for 5 blocks, which took considerably longer than with 3 blocks.