Transcript Corporate Default Modelling

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  EDit 
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 e
1
 S it
, where S it    1S1it  ...   n S nit ,
• 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