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Corporate Default Modelling Forecasting defaults and analysing the interaction between defaults and the real economy David Tysk Central Bank of Iceland June 16, 2010 Questions to be answered today • What is probability of default (PD)? • Are corporate defaults relevant for financial stability? • Does defaults interact with the real economy? • What data is available to model PD? • How to model PD? • How good is the PD model? • Is it possible to make long term forecasts? • What does the crystal ball say? What is probability of default? • Probability of default (PD) is a quantitative assessment of the likelihood that an obligor (e.g., a corporation) will default within a specified period of time • Default rate (DR) is the ratio of defaulted corporations over the total number of corporations in a specific period of time • The average PD is an estimate of the default rate PD – Here and there • PD period – There: The length of the period is often one year; e.g., in Basel II – Here: Set to one quarter to enable analysis of quarterly variations (results are however sometimes annualised) • Default definition: – There: The Basel II definition of default is – simplified – equal to >90 days past due – Here: A corporation is defined as defaulted if it has filed for bankruptcy Are corporate defaults relevant for financial stability? • Default rate: Annual corporate default rate • Loan loss ratio: Annual loan losses over loans and receivables to customers for the three main banks 0,050 0,045 0,040 0,035 Default rate 0,030 0,025 Loan loss ratio Default rate 0,020 0,015 Loan loss ratio (new banks) 0,010 0,005 0,000 1999 2001 2001 2003 2003 200520052007 2007 2009 2009 Arguments for the corporate default rate as a measure of the risk of financial instability • The Icelandic corporate sector represents the largest credit risk in the Icelandic banking system • Neither loan losses nor defaults are leading or lagging the other variable • Loan losses are more complex to forecast due to operational risks and changes in accounting rules • Macro prudential – it captures the systematic risk in the banking system Does the default rate interact with the real economy? • The Icelandic economy is modelled as a small, open economy with a vector autoregressive (VAR) model – Estimated on 1999/Q1 to 2009/Q4 data • Variables – Endogenous: lag 1-2 quarters, output gap, inflation, real exchange rate, CBI monetary policy interest rate – Exogenous: lag 0-2 quarters, foreign: output gap, inflation, short term interest rate • Test statistics – Stationary (largest |unit root| is 0.89) – Lag order selection criteria suggest more lags – Residuals are normal and without auto-correlation Default rate causes GAP, RS, (INF) – Include the default rate as endogenous in the VAR-model to analyse interaction • Granger causality test • Impulse response – Default rate shock Response to Cholesky One S.D. Innovations ± 2 S.E. Response of GAP_SA to DR_AV .008 .01 .004 .00 .000 -.01 -.004 -.02 -.008 -.03 -.012 1 – Granger causality test indicates that DR causes • output gap (GAP_SA) • policy interest rate (RS) • inflation (INF) p-value=0.1 Response of INF to DR_AV .02 2 3 4 5 6 7 8 9 1 10 2 Response of RS to DR_AV 3 4 5 6 7 8 9 10 9 10 Response of REX to DR_AV .004 .04 .02 .000 .00 -.004 -.02 -.04 -.008 -.06 1 2 3 4 5 6 7 8 9 10 9 10 Response of DR_AV to DR_AV .0005 .0004 .0003 .0002 .0001 .0000 -.0001 -.0002 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 Default rate is the preferred measure of the financial stance of the economy • Block-exogeneity test – Evaluate the predictive power of some commonly used measures of the financial stance of the economy – Default rate and loan losses shows highest predictive power – ...but loan losses does not “Granger cause” any of the other variables Description 4-quarter rolling Default Rate Seansonally adj Default Rate Term structure Growth in Equity prices Growth in Housing prices 4-quarter rolling Loan Loss ratio Variable DR_AV DR_SA RLRS EQPG PHG LL p-value Predictive power 0,0% Yes 1,2% Yes 5,2% (Yes) 74,6% No 33,4% No 0,0% Yes THE PD MODEL What data is available to model PD? • Default data – From 1985 • Annual accounts – 1997 to 2008 accounts • Macro data – QMM database • Exclusions – Corporations that have not reported their accounts during the previous year are excluded, e.g. a company is excluded in 2005 if it has not by then reported the 2003 accounts More than half a million quarterly observations of individual corporations • Dependent – Default indicator: Default or not? • Independent – 28 micro variables • Age variable • Ratios derived from balance sheets and income statements • Lagged 2 years – 12 macro (only domestic) • Lagged 2 quarters – 3 dummies to model quarterly variations and one trend variable were defined • 1999/Q1 to 2009/Q4 used to estimate the PD-model PD – The definition • Probability of default (PD) – Let Dit be the default indicator of corporation i in period t. – Dit = 1 if i has defaulted in t, and zero otherwise – Probability of default, PDit, in period t is given by PDit EDit i.e., Dit is a binary variable with parameter PDit – The default rate, rt, in period t is given by nt rt Dit nt i 1 where nt is the number of corporations in period t. Logistic regression is used to model PD • Generalised linear model for binomial regression • Dit ...dependent variable, default/non-default • Sjit ...independent variables, micro and macro • PD given the information S is modelled with the logistic function PDit E Dit S it 1 1 eS it , where S it 1S1it ... n Snit , • Fitting: Maximum likelihood using an iteratively reweighted least squares algorithm Some financial variables behave badly IE/EBIT: IE/EBIT: Value PDand given Default Scorerate 1. Calculate default rate PD f v v 1 0,015 0,015 PD 2. Estimate PD = f(v) 0,02 0,02 0,01 0,01 1 e v v v 3. Calculate the score S 0,005 0,005 00 -5 -3 -3 PD S PD ln 1 PD -1 -1 11 99 88 77 66 55 44 33 22 11 00 Score 1 1 e s s s 55 IE/EBIT: Value to Score 5. Estimate PD = f(s) PD f s s 33 Value Value 4. Derive value-to-score s g (v) PD=f(v) PD=f(v) Default rate Default rate PD=f(s) -5 -3 -1 -1 s(PD) s(PD) s=g(v) 11 Value Value 33 55 ...but macro variables are fairly nice 0,008 0,008 0,007 0,007 0,006 0,006 0,005 0,005 0,004 -0,04 0,004 0,003 Default rate 0,003 Default rate 0,002 PD=f(v) 0,002 PD=f(v) 0,001 0,001 0 0 -0,02 0 0,02 0,04 0,06 0,7 0,8 0,9 Value 1 Value Policy interest rate: PD given Value 0,007 0,006 0,005 PD -0,06 Real exchange rate: PD given Value PD PD Output GAP: PD given Value 0,004 0,003 Default rate 0,002 PD=f(v) 0,001 0 0 0,05 0,1 0,15 Value 0,2 0,25 1,1 1,2 Automated factor selection process to reduce the risk of over-fitting • Single factor analysis – exclusions – Factors with incorrect sign are excluded; e.g., GDP growth – Factors with “complex” behaviour are excluded; e.g., size • Regression – exclusions – Factors with coefficients with incorrect sign; e.g., EBITDA/revenues – Factors with insignificant coefficients; e.g., inflation • K-fold cross validation – exclusions – Factors with high variance in the coefficient; e.g., dividend • Marginal contribution – exclusion – Factors with negative marginal contribution Increased output gap, a stronger króna, and a lower interest rate reduce the PD • 43 variables are reduced to 15 – 9 micro: Age, unpaid taxes and liquidity most important – 3 macro: Real exchange rate most important – 3 dummies for quarterly variations K-fold Score/ Type Variable Variable Value Coefficient p-value std1 AR MC2 Age AGE Score -0,99 0,0000 4% 0,015 (Liquid assets-current liabilities)/Revenues (LA-CL)/REV Score -0,46 0,0000 4% 0,011 Accounts payable/Total assets AP/TA Score -0,41 0,0000 5% 0,008 Adjusted Equity/Total assets EQ*/TA Score -0,18 0,0010 13% 0,001 Micro Interest expenses/Earnings before interest and tax IC: IE/EBIT Score -0,25 0,0000 8% 0,001 Inventories/Revenue INV/REV Score -0,23 0,0021 17% 0,000 Liquid assets/Total liabilities LA/TL Score -0,42 0,0000 3% 0,007 Net income/Total assets NI/TA Score -0,21 0,0001 15% 0,002 Unpaid taxes/Total assets TAXU/TA Score -0,79 0,0000 4% 0,028 Output gap seasonally adjusted GAP_SA Value -4,07 0,0008 18% 0,001 Macro Real exchange rate REX Value -1,34 0,0000 11% 0,004 CBI monetary policy interest rate RS Value 3,15 0,0000 16% 0,003 Seasonal dummy for Q1 D1 Value -0,30 0,0000 11% 0,002 Dummy Seasonal dummy for Q3 D3 Value -0,40 0,0000 8% 0,002 Seasonal dummy for Q4 D4 Value 0,29 0,0000 8% 0,003 1) Relative standard deviation of the coefficients from the K-fold cross validation 2) Marginal contribution to Accuracy Ratio, calculated by re-estimating the model with the variable excluded. Validation of the calibration is more difficult than of the discriminatory power • Micro level – Discriminatory power: accuracy ratio (AR) • Takes on value 1 if the model is perfect and 0 if the model has no discriminatory power – Calibration: Binomial test • Defaults are assumed to be independent • Aggregate level – Calibration: Binomial test • Defaults are assumed to be independent – Time-varying changes : R-square • α-value = 5% and two-sided confidence intervals Discriminatory power is stable over time Accuracy Ratio - Annual average 1 Accuracy ratio Mean accuracy ratio 95% confidence interval 0.9 0.8 0.7 AR 0.6 0.5 0.4 0.3 0.2 0.1 0 1999 2001 2003 2005 t 2007 2009 The model is well calibrated Probability of Default 0 Distribution 10 0.2 All Defaults 0.18 -1 10 0.16 0.14 Default rate -2 10 0.12 0.1 -3 10 0.08 0.06 -4 10 0.04 PD=Default rate Model 95% confidence interval -5 10 -5 10 -4 10 -3 -2 10 10 PD -1 10 0.02 0 10 0 -5 10 -4 10 -3 -2 10 10 PD -1 10 0 10 Time variations are well modelled Probability of Default - Annual 0.02 0.018 0.016 0.014 PD 0.012 0.01 0.008 0.006 0.004 Default rate 0.002 PD 95% confidence interval 0 1999 2001 2003 2005 2007 2009 Quarterly variations are well modelled Probability of Default - Quarterly 0.01 0.018 0.009 0.016 0.008 0.014 0.007 0.012 0.006 PD PD Probability of Default - Annual 0.02 0.01 0.005 0.008 0.004 0.006 0.003 0.004 0.002 Default rate 0.002 0.001 PD 95% confidence interval 0 1999 2001 2003 2005 2007 2009 0 99Q1 00Q2 01Q3 02Q4 04Q1 05Q2 06Q3 07Q4 09Q1 10Q2 TTC intends to minimise pro-cyclicality • Two canonical approaches to PD-model design • “Point-in-time” (PIT) – PIT will tend to adjust the PD quickly to macro changes – Gives time varying capital requirements – PD is calibrated to the default rate at each point in time • “Through-the-cycle” (TTC) – More-or-less constant even as macro changes over time – Gives less time varying capital requirements – At any time, PD is calibrated to the long-term default rate • Validation of either design requires a long time series of data Macro gives PIT characteristics • Base model • Re-estimated model excluding macro • “Point-in-time” • “Through-the-cycle” Probability of Default - Annual: Re-estimated model excluding macro 0.02 0.02 0.018 0.018 0.016 0.016 0.014 0.014 0.012 0.012 PD PD Probability of Default - Annual 0.01 0.01 0.008 0.008 0.006 0.006 0.004 0.004 0.002 0.002 0 1999 2001 2003 2005 2007 2009 0 1999 2001 2003 2005 2007 2009 The micro-macro approach is superior to other approaches • The value-to-score transformation increases discriminatory power significantly • Macro increases the aggregate performance • PD model has as high or higher R-square than other models Model Dependent Independent Value-to-Score* Logit Default Macro Micro Yes Logit Default Macro Micro No Logit Default Micro Yes Logit Default Micro No Linear OLS Default rate Macro Linear OLS sign** Default rate Macro VAR*** SOE Default rate Macro**** VAR*** Default rate Macro * Only on micro. ** Only variables with significant coefficients. *** Seasonally adjusted PD has been used to avoid the usage of seasonal dummies. **** Same variables as in the small, open economy model R^2 0,74 0,66 0,36 0,42 0,74 0,65 0,74 0,68 AR 0,56 0,28 0,54 0,25 N/A N/A N/A N/A Is it possible to make long term forecasts of the default rate? • Independent variables need to be forecasted • Two options to forecast macro – The SOE VAR-model with/without DR as exogenous – The Central-Bank of Iceland's (CBI) baseline forecast • Forecasting micro is much more challenging – No obvious method – Is the portfolio mix stable? Corporations are born, grow older (and die?) • Age variable kept constant – Account variables are modelled using a VAR-model • Endogenous: lag 1-(2) quarters, micro variables • Exogenous: lag 0 quarters, macro variables Forecast validation – model selection • Forecasts – 3-year forecasts – Total 39 forecasts • Forecast validation 2000Q2 to 2003Q2 2000Q3 to 2003Q3 2000Q4 to 2003Q4 2001Q1 to 2004Q1 … – Focus on aggregate performance, i.e., the default rate – Average R2 Macro • Selection of forecast – Macro forecast – Account forecast PD model Micro forecast Default rate forecast The macro model generates accurate forecasts • Macro forecast: CBI’s baseline forecast is preferred – The small, open economy VAR-model gives as accurate default rate forecasts as actual macro data – Including DR in the VAR-model doesn’t improve forecasts • Micro forecast: VAR(1,0)-model is preferred – A VAR-model with few lags is preferred over static accounts and a VAR-model with more lags Start first Start last Years Nr Macro Accounts Acc lags** 2000Q2 2009Q4 3 39 Data Static 2000Q2 2009Q4 3 39 Data Model 1-2, 0 2000Q2 2009Q4 3 39 Data Model 1, 0 2000Q2 2009Q4 3 39 Model* Static 2000Q2 2009Q4 3 39 Model Static * SOE VAR-model with DR as exogenous ** Lags for Account model's enodgenous and exogenous variables R^2 0,68 0,63 0,71 0,58 0,70 What does the crystal ball say? • Given CBI’s baseline forecast the default rate is expected to be slightly higher in 2010 than 2009 • ...and reach average levels first in 2012 0,025 0,02 0,015 0,01 0,005 0 1999 2001 2003 2005 Annual default rate 2007 2009 2011 95% conf. int. Default rate forecast is based on CBI's baseline macroeconomic and inflation forecast, Monetary bulletin 2010/2. The 95% confidence intervals do not take the uncertainty of this forecast into account. Main conclusions • Corporate defaults are relevant for financial stability • The default rate shows highly significant predictive power for the real economy • Predictive power increases with the micro-macro approach and the value-to-score transformation • Macro dramatically increase the aggregate performance of the PD model • An increased output gap, a stronger króna, and a lower policy interest rate reduce the PD • The PD model performs well under extraordinary conditions This is not the end, just the beginning... • Applications – Model and stress-test regulatory capital requirements and credit losses – Simulation of the banking sector’s capital position and profitability, especially from a macroprudential perspective – Industry and large exposure analysis • Research – Does the predictive power vary across industries? – Does un-lagged forecasted variables improve the performance? – Further link the default rate and financial stability – Further link monetary policy and financial stability