1. Universidad Complutense de Madrid Máster en Ingeniería Matemática Curso 2007-2008 Modelling Week Second 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. 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) Credit Scoring Modelling for Retail Banking Sector

4. 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. 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. Credit Scoring Modelling for Retail Banking Sector

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

8. 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 Credit Scoring Modelling for Retail Banking Sector

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

10. 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. Credit Scoring Modelling for Retail Banking Sector

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

12. 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. Credit Scoring Modelling for Retail Banking Sector

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

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

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

17. 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. Credit Scoring Modelling for Retail Banking Sector

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

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

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

28. 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. Credit Scoring Modelling for Retail Banking Sector

29. 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. 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. Credit Scoring Modelling for Retail Banking Sector

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

34. 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. Credit Scoring Modelling for Retail Banking Sector

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

36. 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. Credit Scoring Modelling for Retail Banking Sector

37. 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 didnt 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. 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 ρ i, an a priori probabilty of default PD i, the loss given default LGD i and the number of customers N i. 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

42. 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 αs that minimise the expected loss. We have two approaches to solve this problem.

48. The Problem of Capital Allocation First we fix αs and find the VaR99 and Expected Loss for each set of αs (Black dots).

49. The Problem of Capital Allocation Then we find the αs 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.

51. The Problem of Capital Allocation Conclusions Analytical method outperformed the simulation of w i 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.

52. The Problem of Capital Allocation ¿Questions?