Técnicas de control de riego

The transition probabilities in this study use maximum likelihood estimates (MLE), i.e. taking an average of the history data. Researchers (Kadam and Lenk, 2008; Stefanescu et al., 2007) suggest that using the maximum likelihood method to estimate the sparse default probability entries results in large estimation errors, since MLE is suitable for a frequentist estimation framework, i.e. for a dataset consisting of numerous observations. By contrast, the Bayesian approach gives a greater mathematical basis for estimating the default probability. The idea is presented as follows. Assume the probability of default given a credit card holder in a state s (which can depend on behavioural score, credit limit, or other characteristics of the credit card holder) is P r(D|s). Then, according to Bayes’ rule, this probability can be written as:

P r(D|s) = P r(s|D)P r(D)

P r(s) (7.1) where P r(D) is the unconditional prior probability, P r(s|D) is the posterior probabil- ity that the credit card holder is in a state s given s/he defaults, and P r(S) is called the marginal probability of S. If we have a dataset with the credit card holders’ his- tories, we can estimate the posterior probability. Say there are ND default cases and

ND(s) of them are in default with a state s, then the posterior probability P r(s|D) is

equal to ND(s)

ND . The marginal probability is calculated as the sum of the product of

all probabilities of any state si and corresponding conditional probabilities P r(si), i.e.

P r(s) = P

iP r(s|si)P r(si). One can then set the prior probability P r(D) to a certain

value and calculate P r(D|s) in the first iteration. In the second iteration, one uses the computed posterior equation P r(s|D) as the prior posterior and repeats the calculation process. There is mathematical evidence to show one can find a posterior probability close to the real conditional probability after several iterations. However, researchers warn that the value of this posterior probability is sensitive to the prior probability.

Therefore, possible areas to explore in using Bayesian inference to estimate the default probability are: (1) The sensitivity of the default probabilities to the prior probabilities. (2) The difference in using the maximum likelihood method and the Bayesian method to estimate the default probabilities. (3) How to incorporate macroeconomic variables in estimating the default probabilities with Bayesian Inference.

Appendix A

UK Data

This Appendix provides some information about the UK data samples, including, the field specification, frequency distribution, coarse-classifying and chi-square goodnes-of-fit test results. All of these are excluded from the main thesis contents and are provided here as additional information for readers.

The file specifiication provided by the UK data provider. I. Fields definition

Name Description Contents Format

prodid MasterKey for linkage n 8 advlim Advised Credit Limit Pounds n 7

act\_bal Current balance

Pounds.Pence (eg

\pounds1 = 1.00) n 11.2 coff\_code Charge-off Reason Code c 2

days\_del Days delinquent

Numeric length 3 (eg 1

= 001) n 3

ext\_stat External status ext\_stat c 1

accopen Open date CCYYMMDD c 8

sortcode

The bank sortcode of the cardholder's checking account- holding branch unless an alternative sort code is used for

direct debit payments _{c 6}

cusbehsc Visa Behavioural score cusbehsc c 3

prodtype Product type c 3

attrsc Attrition score c 3

II. Further details

External Status (ext\_stat) Z Charge Off B Blocked / Bankrupt L Lost U Stolen U 5 + Cycles Delq U Never Active U Inactive U Transferred Account A Auth Prohibited C Closed E Revoked F Frozen I Interest Prohibited * **

*** Optional Scores (the behaviour score is added to the exception score to calculate an optional score)

Coff\_Code (following a 'Z' charge off) 00 Awaiting insolvency details 01-05 In house debt collectors (pre 1995) 07 Weekly fixed payers (or cash payers) 74 Fixed payers using Baines \& Ernst 75 Fixed payers using Gregory Pennington 76 All other fixed payers using external bureaux 88 Stolen charge off

89 Bankrupt charge off 90 Deceased

93 Being referred for or waiting abandon 94 Account outplaced to debt collector 95 Account outplaced to debt collector

Prodtype

ART ART Cards GCC Gold Credit Card GMC Gold Mastercard HN Harvey Nichols NT National Trust MC Master Card Pl Platinum Pr Premier V Visa

Designated Scores (the system does not calculate a behaviour score but uses the designated score during processing)

Behavioural Score (cusbehsc) Score Odds (/1) 780 3840 765 1920 750 960 735 480 720 240 705 120 690 60 675 30 660 15 645 7.5 630 3.75 615 1.87 600 0.93 585 0.46 570 0.23

Exclusions to Behaviour Scores

0 New account which has not yet cycled, so score not yet generated 1 Deceased 2 Not used 3 Not used 4 Bankrupt 5 Written Off 6 Not used

7 Involuntarily closed more than 6 months ago and balance = zero 8 Voluntarily closed more than 6 months ago and balance = zero

9 Attrition Score exclusion - Involuntarily closed more than 6 months ago and balance = zero 10 Attrition Score exclusion - Voluntarily closed more than 6 months ago and balance = zero 11 Never Active

12 Inactive 12+ months

13 Recently Reactivated in the last 2 months 14 Recently Acquired (less than 3 cycles on the books)

Score Odds Payment Projection 365 16 350 8 335 4 320 2 305 1 290 0.5 275 0.25 260 0.125 245 0.06 230 0.03 215 0.015 200 0.0075 (These scores are

applied if the account is under some stress, i.e. having these scores means the account is of worse condition than the rest). These scores predict the likelihood of payment.

NB Behaviour score forecasts the probability of accounts to become, within the next six months, 3 or more cycles delinquent (excluding fraudulent losses), or bankruptcy, where the customer's outstanding debt is \pounds10 or greater.

NB Behaviour score forecasts the probability of accounts to become, within the next six months, 3 or more cycles delinquent (excluding fraudulent losses), or bankruptcy, where the customer's outstanding debt is \pounds10 or greater.

Appendix B

UK Data - incorporating

macroeconomic variables

B.1

Unconditional transition matrices

Scorei Scorei0att+ 1 Row

att Closed Inactive Bad Risk Score1 Score2 Score3 Score4 Count Inactive 2.22 96.03 - - 1 0.68 0.06 0.01 458817 Risk 5.09 - 19.7953.68 21.44 - - - 3951 Score 1 0.76 0.37 0.08 0.49 84.25 11.73 2.2 0.13 379242 Score 2 0.69 1.68 0 - 12.15 67.81 15.69 1.98 323448 Score 3 0.78 1.46 - - 2.77 15.27 64.13 15.59 322344 Score 4 0.71 0.2 0 - 0.84 1.85 17.27 79.14 307728

”‘-”’ represents there is no sample observation.

”‘0”’ represents the transition probability is less than 0.0005. A bold value indicates the transition frequency is greater than 50% . The transition probabilities of all absorbing states (Closed and Bad) are not shown in the table.

Table B.1: Unconditional transition matrix (in percentage) for the UK data

The unconditional behavioural score transition matrix is diagonally dominated. Note that accounts in the Risk behavioural score state preserve the highest default probabilities (= 19.79%). 96.03% Inactive accounts remain Inactive after one time period which is

much higher than those (= 81.96%) of the HK dataset. The mobility of Score1 and Score4 accounts are low since around 80% of them remain in the same behavioural score state in the subsequency month.

Note that we use the account balance presented in Table 3.4 and r= 0.02 to estimate the unconditional account profit with result present in Table B.2.

Credit limit Score att(i)

att(l) Close Inactive Bad Risk Score 1 Score 2 Score 3 Score 4

Limit 1 0 0 -563 8.66 6.32 1.78 1.3 1.04 Limit 2 0 0 -761 14.06 9.8 2.38 1.46 1.08 Limit 3 0 0 -983 17.52 11.4 2.9 1.78 1.46 Limit 4 0 0 -1658 29.02 16.44 4.92 2.78 2.48 Limit 5 0 0 -2234 42.68 24.58 10.44 4.7 3.88 Limit 6 0 0 -3047 57.82 29.94 13.84 7.02 5.64 Limit 7 0 0 -3605 60.96 34.9 16.6 9.34 7.22 Limit 8 0 0 -5722 109.6 63.62 43.74 22.12 14.62 The profit value of Closed and Inactive are assumed to be 0.

Table B.2: Profit value used in the UK macroeconomic model