Download stress-testing models

JJ Mois Année
SMILOVICE
Jan Neckař
Dana Chromíková
B2 - CONCEPT
EXPECTED LOSS
Basel II concept differs
two types of loss:
UNEXPECTED LOSS
LOSSES IN TIME
8%
7%
6%
loss
5%
Unexpected loss
4%
3%
2%
Expected loss
1%
0%
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Frequency
Years
2
Scoring Unit (SCOR)
April 2008
B2 - CONCEPT
BASEL II CONCEPT – distribution of the losses
CAPITAL RESERVES
MAINTENANCE
STRESS TESTING
FREQUENCY
Creation of
Provisions
Covered by
SRC
VALUE AT RISK
Probability
99,9%
Probability
0,1%
LOSS
3
Scoring Unit (SCOR)
April 2008
B2 - CONCEPT
EXPECTED LOSS
EL = PD * LGD * EAD
PD – probability of default
- estimation of probability then client is longer than 90 days
delayed with payments, insolvency, …
LGD – loss given default
- estimation of the resulting economic loss after the recovery process
- conditional estimation in case of client is in default
EAD – exposure at default
- conditional estimation of exposures in case of client is in default
- average drawing at default is higher than outside default
EL = PD*E(loss|default) + (1-PD)*E(loss|nedefault) = PD * LGD * EAD
4
Scoring Unit (SCOR)
April 2008
B2 - CONCEPT
UNEXPECTED LOSS – CAPITAL REQUIREMENT
Tier1 + Tier2 ≥ 8% * Σ RW * EAD
&
Tier1 ≥ Tier2



R
  1
 1  M  2,5 bPD
RW  LGD  N 
 GPD  
 G0,999  PD 
12,50  Scaling Factor


1

R
1

1,5

b
PD
1

R



 

N(x) – distribution function of normalized normal distribution of random quantity
G(x) – inversion function to distribution function of normalized normal distribution
Scaling factor – according to direction of ČNB is equal to 1,06
Maturity (M) – Average maturity of the expected cash-flows (repayments)
Factor of maturity
2
b(PD)  0,11852 - 0,05478  ln PD 
Correlation factor R for retail exposures (excl. Mortgages = 0,15, qualifying revolving = 0,04):
 1  e 35PD 
1  e 35PD

R  0,03 
 0,16  1 
35 
1  e 35
1

e


Correlation factor for non-retail exposures:
 1  e 50PD 
1  e 50PD
 max(min( S ,50),5)  5 


R  0,12 

0,24

1

0,04

1 

50 

1  e 50
1

e
45




S – Annual sales for the consolidated group (million EUR)
5
Scoring Unit (SCOR)
April 2008
B2 - CONCEPT
UNEXPECTED LOSS – CAPITAL REQUIREMENT
6
Scoring Unit (SCOR)
April 2008
STRESS-TESTING
Behavior under stress is not easy to predict
7
Scoring Unit (SCOR)
April 2008
STRESS-TESTING
LOSSES IN TIME
16%
14%
BASELINE
12%
DEPRESSION
loss
10%
8%
6%
4%
2%
0%
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Frequency
Years
8
Scoring Unit (SCOR)
April 2008
STRESS-TESTING
STRESSED
CHARACTERISTICS
STRESS-TESTING
MODELS
STRESS
SCENARIOS
ECONOMETRIC
MODEL
9
Scoring Unit (SCOR)
April 2008
STRESS-TESTING
ECONOMETRIC
MODEL
&
STRESS-TESTING
SCENARIOS
Econometric model predicts the macroeconomic characteristic as:
 GDP
 unemployment
 interest rates
 inflation / deflation
 price of oil
…
These models have usually 50 – 100 formulas and above 200 parameters
There are several various of predictions, called as scenarios:
 baseline
 depression
 deep depression
 high inflation
…
The most probable scenario is selected for development of the model.
10
Scoring Unit (SCOR)
April 2008
STRESS-TESTING
STRESS-TESTING
MODELS
Stress testing
= a way how to measure risk of
extreme but realistic events
Modeled via scenarios for macroeconomics characteristics
We assume that portfolio depends on macroeconomic
situation and we need to find relation between stressed
variable (PD, LGD, CCF) and macroeconomic characteristics:
Example for stressing PD:
PDt = f (Mt1)
t ≥ t1, f (Mt1) function of macroeconomic characteristics
Two type of models:
Logistic regression
Factor model based on Merton’s model
11
Scoring Unit (SCOR)
April 2008
STRESS-TESTING
STRESS-TESTING
MODELS
Logistic regression
EY 
1
k
1  exp(  i X i )
i 0
Y is explained variable (indicator of default), EY is probability of default
X  ( X 1 ,..., X k )
is vector of explanatory variables (macro-economic indicators).
Main advantages of this model:
Basic statistical model used for modelling 0-1 variable with good
mathematical properties
12
Scoring Unit (SCOR)
April 2008
STRESS-TESTING
Factor model based on Merton’s model
STRESS-TESTING
MODELS
Rit   Ft  1  Uit
Where
Rit
is logarithmic change of client’s asset
Ft
is systematic factor
U it is specific factor
Ait
Rit  log
Ai (t 1)
Ft  N (0,1)
U it  N (0,1)
  cor ( Rit , R jt )
  cor ( Rit , Ft )
13
Scoring Unit (SCOR)
April 2008
STRESS-TESTING
STRESS-TESTING
MODELS
Factor model based on Merton’s model
P(Yit  1)  P( Rit  C )
Yit
J
1 in case of default
0 in case of non-default
C   0    j x jt
j 1
Probability of default
PDit  P(Yit  1)  P( Rit  C ) 
J
J
j 1
j 1
P(  Ft  1  U it   0    j x jt )  (  0    j x jt )
14
Scoring Unit (SCOR)
April 2008
STRESS-TESTING
STRESS-TESTING
MODELS
Factor model based on Merton’s model
Conditional probability of default:
J

 0    j x jt   f t

j 1
PDit ( f t )  P(Yit  1 | Ft  f t )  PU it 
1 








Likelihood function derivated from binomial distribution of default rate:
J



  0    j x jt   f t
T
n
  
j 1
l (  0 ,...,  J ,  )   log    t 
1 
t 1
 d t  




15
Scoring Unit (SCOR)






dt
J


  0    j x jt   f t

j 1
1  


1 










nt  d t



 ( f t )df t 



April 2008