The Extended Exogenous Maturity Vintage Model Across the Consumer Credit Lifecycle

Transcription

1 The Extended Exogenous Maturity Vintage Model Across the Consumer Credit Lifecycle Malwandla, M. C. 1,2 Rajaratnam, K. 3 1 Clark, A. E Department of Statistical Sciences, University of Cape Town, Cape Town, South Africa 2. Standard Bank, Johannesburg, South Africa 3. Department of Finance & Tax and the African Collaboration for Quantitative Finance and Risk Research, University of Cape Town, Cape Town, South Africa 4. African Collaborative for Quantitative Finance and Risk Research, Cape Town, South Africa Presented By: Musa Malwandla

2 2 Research Problem and Aim Identifiability Problem in the EMV Model, and lack of behavioural data Aim: bypass the identifiability problem Aim: introduce behavioural data into model Credit risk models lack unification Aim: show how model can be used across all areas of credit risk Lack micro-foundation for aggregation model, understanding of systemic risk Aim: show how model aggregates to portfolio loss Aim: provide general formula for asset correlation coefficient The problem of aggregating risk across portfolios Aim: provide an approach for aggregating risk across portfolios

3 3 Agenda Model Specification Standard EMV Model Extended EMV Model Illustration: South African Portfolio Application Areas Application and Behavioural Scorecard Impairment Modelling Stress Testing Capital Management Extension: Survival Analysis Decomposition Systemic Risk & Aggregation Understanding the LHP approximation Understanding the Asset Correlation Coefficient General Asset Correlation Coefficient Formula Illustration: South African Portfolio Understanding Diversifiable Cross-Portfolio Aggregation A formula for cross-portfolio aggregation Other uses of the formula Illustration: South African Portfolio

4 4 Model Specification Standard EMV Model Extended EMV Model Illustration: South African Portfolio

5 Age (Maturity) 5 Standard EMV Model APC Model pp tt, ss = AA tt ss + PP tt + CC ss pp tt, ss = MM tt ss + EE tt + VV ss Vintage (Cohort) Widely applied in epidemiology: Age Period Cohort Model Dimensional interpretation in mortality studies: Age: effect of age on mortality (e.g, Gompertz Law, Accident Hump) Period: effect of period on mortality (e.g., war, plague, cultural influence) Cohort: effect of cohort on mortality (e.g., epigenetics, unnatural selection) EMV Model Exogenous: mainly assumed to capture macroeconomic/policy environment Maturity: mainly captures risk by maturity and selective effects Vintage: mainly captures effect of acquisition policy (e.g. scorecard and cut-offs, target market)

6 6 Extended EMV Model: Specification pp tt, ss, kk = ΦΦ αα + MM tt ss + EE tt + VV ss + BB kk + AA kk ΦΦ μμ + σσ MM MM tt ss + σσ EE EE tt + σσ VV VV ss + σσ BB BB kk + σσ BB AA kk Extend model beyond time dimensions, to behavioural dimensions Include behavioural score dimension / application score dimension Standardisation For convenience, standardise all component to mean = 0, std. dev = 1 Standard deviation of each component becomes parameter / significance measure Identifiability Application scorecard attempts to capture same effect as vintage Replace vintage with application scorecard Link functions Logit, Probit, CLogLog Prefer Probit: leads to Vasicek distribution for portfolio loss Alternative is CLogLog: leads to Log-Log-Normal distribution for portfolio loss

7 7 Fitting Illustration: Portfolio Description South African consumer loan portfolio Fixed-Rate Loans: interest rate fixed at the outset of the loan Variable-Rate Loans: interest rate varies with central bank rate (Prime Overdraft Rate) Observations 2.5m observations September 2005 to June 2014 Default Definition: 90 Days Past Due, Distressed Restructure, Litigation, Write Off

8 8 Fitting Illustration: Dimensionality Model Fixed Rate Loans (AIC) Variable Rate Loans (AIC) Model Fixed Rate Loans (AIC) Variable Rate Loans (AIC) emvba mba emva ma emvb mb emba vba ema va emb vb evba ba eva a evb b mvba emv mva ev mvb em eba mv ea e eb v m

9 9 Fitting Illustration: Components Behavioural Component Maturity Component 2 2 1,5 1, ,5 0-0, ,5 0-0,5-1 -1, , ,5 Fixed Rate Loans Variable Rate Loans Fixed Rate Loans Variable Rate Loans 3 2 Exogenous Component pp tt, ss, kk = ΦΦ μμ + σσ MM MM tt ss + σσ EE EE tt + σσ BB BB kk Parameters Fixed Rate Loans Variable Rate Loans Fixed Rate Loans Variable Rate Loans αα μμ BB μμ EE μμ MM μμ σσ BB σσ EE σσ MM

10 10 Fitting Illustration: Exogenous Model 2,5 2 1,5 1 0,5 0-0,5-1 -1,5-2 Predicted vs Expected PD: EMB Fixed (R 2 =64,9) Predicted vs Expected PD: EMB Variable (R 2 =72,6) Target Predicted Target Predicted Fixed-Rate Model Variable-Rate Model Variable Estimate P-Value VIF Variable Estimate Variable-Rate Model VIF Intercept Intercept Consumer Price Index Consumer Price Index Ratio: Consumption to GDP Ratio: Savings to GDP

11 11 Fitting Illustration: Validation Accuracy Across Range Accuracy Across Time ACTUAL 3 25% 2 15% 5% 5% 15% 2 25% 3 PREDICTED 16% 1 1 6% Predicted PD Actual PD Gini Statistic EMB EMV Fixed Rate Loans 37% 16% Variable Rate Loans 47% 1

12 12 Application Areas Application and Behavioural Scorecard Impairment Modelling Stress Testing Capital Management

13 13 Application Areas: Scorecards Linking risk scores to macroeconomic variables: Dynamic score cut-offs Reducing recalibration frequency PD Across Time 0,16 0,14 0,12 Observation Performance 0,1 Im plem entation 0,08 0,06 0,04 0, Predicted PD Actual PD IFRS 9 SICR swap-set problem can be resolved through EMBA model: pp tt, ss, kk = ΦΦ EE tt + RR tt ss,ll,kk where RR tt,ss,ll,kk = VV ll αα tt ss + BB kk 1 αα tt ss for monotonic αα tt ss

14 14 Application Areas: Impairments & Stress-Testing IFRS 9 Lifetime PD Use Cumulative Incidence Function approach: kk pp kk tt, RR tt,ss,ll,kk = pp 12 tt, RR tt+jj,ss,ll,kk 1 pp 12 tt, RR tt+jj 1,ss,ll,kk qq 12 tt, RR tt+jj 1,ss,ll,kk jj=1 Stress-Testing Acquisition strategy using application score component. Account management strategy, through behavioural score card.

15 15 Application Areas: Capital Management (Model Error) 1 Random Effect or Model Error Full Error = σσ EE 2 Residual Error = 1 rr 2 σσ EE 2 6% Predicted PD Actual PD

16 16 Extended EMV Model: Summary Application Area Modelled Parameter Point-in-Time / Forward-in- Time Through-the-Cycle Application / Behavioural Scoring / Stress Testing Probability of Default; of Rolling (Collections); Write Off (LGD) ΦΦ RR tt,ss,ll,kk rr 2 σσ EE 2 ΦΦ KK tt,ss,ll,kk 1 + σσ EE 2 IFRS 9 Impairment Modelling Lifetime Probability of Default Cumulative Incidence Function N/A Economic Capital Value-at-Risk / Quantile Function ΦΦ ρρ 1 ρρ ΦΦ 1 αα ρρ ΦΦ 1 pp tt Economic Capital Asset Correlation Coefficient σσ 2 EE 1 rr 2 σσ EE 1 + σσ 2 EE 1 rr σσ EE Economic Capital (Cross-Portfolio) Cross-Portfolio Loss Distribution NN μμ, σσ NN μμ, σσ 2 Short hand: RR tt,ss,ll,kk = μμ + σσ MM MM tt ss + σσ VV VV ll + σσ BB BB tt + σσ EE EE tt KK tt,ss,ll,kk = μμ + σσ MM MM tt ss + σσ VV VV ll + σσ BB BB kk

17 17 Systemic Risk & Aggregation Understanding the LHP approximation Understanding the Asset Correlation Coefficient General Asset Correlation Coefficient Formula Illustration: South African Portfolio Understanding Diversifiable

18 18 Understanding the LHP Approximation?? Condition on e Unknown distribution Unknown distribution Assume homogeneity Binomial distribution 1% 1% 3% 5% 5% 6% 7% 7% 9% 11% 11% 1 13% 13% % FYI: CLogLog link leads to log-log-normal distribution Assume large portfolio ΦΦ Vasicek Quantile Function ρρ 1 ρρ ΦΦ 1 αα ρρ ΦΦ 1 pp tt = Un-condition on e, Assume Gaussian e and Probit link function Vasicek Distribution Degenerate distribution 1% 1% 3% 5% 5% 6% 7% 7% 9% 11% 11% 1 13% 13% % 1% 1% 3% 5% 5% 6% 7% 7% 9% 11% 11% 1 13% 13% %

19 19 Understanding the Asset Correlation Coefficient The asset correlation coefficient is the shape measure of the Vasicek distribution: Influenced by level of correlation between portfolio, or Influenced by level systemic risk relative to idiosyncratic risk σσ_uuuuuuvv= σσ_ssvvss=9 ρρ=99% Asset Correlation - High Scenario σσ_uuuuuuvv=75% σσ_ssvvss=25% ρρ= Asset Correlation - Low Scenario Asset Value Default Threshold Asset Value Default Threshold Time Time

20 20 General Asset Correlation Coefficient Formula ρρ = uuuuuuuuuuuuuuuuuuuuuu vvvvvvvvvvvvvvvvvvvv tttttttttt vvvvvvvvvvvvvvvvvvvv = σσ EE 2 1 rr σσ EE 2 1 rr 2 = ββσσ SS 2 1 rr ββσσ SS 2 1 rr 2 The asset correlation coefficient is influenced by: 2 Level of risk within the system/economy σσ SS Pro-cyclicality of portfolio ββ Level of portfolio s systemic risk that is explained by the model rr 2 To get TTC confidence intervals, simply set rr ss = 0 The formula supersedes the generic formulae offered under Basel.

21 21 Illustration of Value-at-Risk Point-in-Time PD Confidence Interval: Fixed Rate Loans Point-in-Time PD Confidence Interval: Variable Rate Loans 25% 2 15% 5% 25% 2 15% 5% Actual PD Predicted PD CI Lower Bound CI Upper Bound Though-the-Cycle PD Confidence Interval: Fixed Rate Loans Actual PD Predicted PD CI Lower Bound CI Upper Bound 1 1 6% 1 1 6% Actual PD Predicted PD CI Lower Bound CI Upper Bound Through-the-Cycle PD Confidence Interval: Variable Rate Loans Actual PD Predicted PD CI Lower Bound CI Upper Bound

22 22 Understanding (Non-)Diversifiable Risk For homogenous population, diversifiable risk follows a scaled binomial distribution. Due to the asymptotic aspect of LHP assumption, the loss distribution ignores sampling error (diversifiable risk). However, observed PD subject to sampling error, which filters through to exogenous component. Therefore, the estimate asset correlation coefficient captures total risk. 1 6% Simulation: Diversifiable vs. Total Risk (Sample Size = 1 000) 9% 7% 6% 5% 3% 1% Simulation: Diversifiable vs. Total Risk (Sample Size = ) DR Lower.Total Upper.Total Lower.Diversifiable Upper.Diversifiable DR Lower.Total Upper.Total Lower.Diversifiable Upper.Diversifiable

23 23 Cross-Portfolio Aggregation A formula for cross-portfolio aggregation Other uses of the formula Illustration: South African Portfolio

24 24 Formula for Cross-Portfolio Aggregation Aim: to allow for diversification benefits across portfolios Assumptions: Assume exogenous error size is small So that Taylor approximation applies: pp kk ee kk,tt ΦΦ RR kk,tt + φφ RR kk,tt σσ kk,ee ee kk,tt Assume that portfolio errors normally distributed ee kk,tt and linearly correlate Joint loss distribution becomes multivariate normal Practical uses: Working out required capital for an entire bank Capital budgeting: Determining how much a portfolio contributes to the total risk System analysis: Determining how much a bank contributes to systemic risk

25 25 Illustration 1 16% 1 1 6% Combined PD Confidence Interval: Point-in-Time Lower Actual Upper 16% 1 1 6% Combined PD Confidence Interval: Through-the-Cycle Lower Actual Upper

26 26 Final Comments on Error Distribution Diversifiable vs. Total Risk (Sample Size = ) 1 Most talk about error distribution looks at the fatness of the tails (Kurtosis) i.e., attempt to capture extreme events. 6% This presentation suggests that: For poor-fitting macroeconomic models, autocorrelation matters DR Lower.Total Upper.Total Lower.Diversifiable Upper.Diversifiable For small portfolios with low default rates, skewness matters. For aggregating across portfolio, the joint error distribution matters % Point-in-Time PD Confidence Interval: Variable Rate Loans Actual PD Predicted PD CI Lower Bound CI Upper Bound

27 Conclusion 27 Extended EMV Model Solving (bypassing) the identifiability problem Survival analysis decomposition Unification Application in scoring, impairment, stress testing, capital management Systemic Risk Understanding loss aggregation and asset correlation General formula for asset correlation Cross-Portfolio Aggregation Aggregation across portfolio Further Research Error distribution, and joint error distribution Combining model with other risk types (e.g., liquidity, operational)

28 28 Appendix Application of EMV-type decomposition to Survival Analysis

29 29 Extension: Survival Analysis Decomposition h jj,ss tt = ΦΦ αα 1 bb tt + αα 2 φφ GGjj,ss + αα 3 ee ss oooo h jj,ss tt = ΦΦ αα 1 bb tt + αα 3 ee ss ee φφ GG jj,ss bb tt ee φφ GG jj,ss Objective: Survival analysis with time varying covariates Non-parametric estimation without need for partial-likelihood function Proportional Hazard Model with TIme-Varying Baseline Step 1: Baseline Step 2: Baseline + Macroeconomic Index Step 3: Baseline + Macroeconomic Index + Behavioural Index

30 30 Extension: Survival Analysis Decomposition 0,3 0,2 0,1 0-0,1-0,2-0,3-0,4-0,5-0,6 Baseline Hazard Cycle Behav ioural Score Cycle 0 0,8 0,6 0,4 0,2 0-0,2-0,4 Baseline Hazard Cycle Credit Risk Index Cycle 0 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0, ,4 0,3 0,2 0,1 0-0,1-0,2-0,

31 31 Extension: Survival Analysis Decomposition 16% 1 1 6% PD by Horizon: Cycle Observed PD Unadjusted PD Adjusted PD 1 6% PD (Horizon = 12) by Calendar Month: Cycle Observed PD Unadjusted PD Adjusted PD Observ ed Accuracy Accross Range: Cycle 0, Horizon 12 25% 2 15% 5% 5% 15% 2 Predicted Observ ed Accuracy Accross Range: Cycle 0, Horizon % 3 25% 2 15% 5% 5% 15% 2 25% 3 35% Predicted

32 32 EMV Survival Analysis Data Setup

33 33 Questions Please contact me at: