1. Eric on Credit Risk Economic
Capital Modeling
-Concept
-Model Introduction
-Applications
2009/05/04
Eric Kuo

2. Banks hold „Capital‟ to protect against “Unexpected Loss” and to maintain
„Solvency‟, not for the regulatory compliance.
AGENDA OF TODAY
1. Role of bank Capital is not just for the regulatory requirement.
capital Capital is a cushion to absorb unexpected loss.
Basel 2 ‘s capital equation is a simplified ‘EC’ model, it comprises 4 factors
:
2. Interpreting
 Vasicek model
Basel 2 capital  Correlation
equation  EL
 Tenor adjustment
3.Model This model follows Basel’s approach and incorporate simulation skill to
Introduction & measure the EC.
Applications
Economic capital is wildly utilized in all aspects of bank’s management.
Three key management applications :
4. Role of EC in
 Risk governance
Pillar 2
 External communication
 Internal management
2009 Eric Confidential
2

3. Bank capital serves as cushion to absorb losses and maintain solvency.
Profit generation Credit loss is tiny Credit loss is huge
When economy is healthy and When economy is stable or become In the extreme stressed situation, bank
credit quality is solid, risk taking vulnerable, bank may encounter credit suffers huge credit loss that not only
will generate profit and will loss and erode the profit. In this case, erode the profit but also consume all the
strengthen bank capital. bank’s available capital is served as available capital that causes bank
cushion and absorb the loss. Therefore, insolvent problem. Capital injection is
bank capital decreases and this may immediate needed.
affect bank rating or business.
40 Need to raise
60 40 30 80 capital
immediately
30 40 30
40
-10
-50
30 10
20
-10
Revenue Credit Profit Bank Bank Revenue Credit Profit Bank Bank Revenue
Credit Profit Bank Bank
after Op loss available remaining after Op loss available remaining after Op
loss
available remaining
cost capital capital cost capital capital cost capital capital
Before add Before Before
profit absorb absorb
loss loss
2009 Eric Confidential
3

4. Basel 2 promote the risk measurement that based on bank‟s historical data.
The EL is considered as cost of doing business, while as capital serves as reserve
to protected against unexpected loss. Conceptual
Generate risk parameters (PD,LGD,EAD)
from historical loss data. Bank capital (risk or economic capital) is
prepared as cushion to absorb the
The expected loss estimation is the cost of
doing loan business. unexpected credit losses.
Capital is used to Target rating
cover these Credit loss
Credit Loss extraordinary loss
Risk Appetite
A
Bank’s actual loss
Unexpec- Risk
experience
ted loss Capital
Average
credit loss
EL =PD * LGD *EAD
EL
Time Probability
Note : Expected Loss = PD*LGD *EAD
EL doesn’t necessary equal to the historical loss experience, due to the portfolio component may change. 2009 Eric Confidential
4

5. More capital a bank has, allows bank to absorb more unexpected loss…
…but how much capital is sufficient ?
Conceptual
In theory the unexpected loss estimation is kind of prediction
whether if obligors will default or even default together.
Current Year End
Obligors
The potential outcome in the year end
Assuming a 3 # of Default No LGD Loss Possi- 1. How can
obligors, A B C , default default bility we
0 A,B,C 0 44% estimate
each has
the joint
1 A B,C 50 10% default
• 100 of exposure
B A,C 50 10% event ?
• 50% of LGD C A,B 50% 50 10% 2. What is the
31.5 possibility
• 10% of PD 2 A,B C 100 5%
of potential of each
A,C B 100 5% event ?
 EL= 15 credit loss
B,C A 100 5%
3 A,B,C 150 1%
Note: EL = PD*LGD*EAD 2009 Eric Confidential
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6. The amount of capital held by a bank reflects the risk appetite of a bank.
Illustrative
Based on the above example, this bank
need to reserve
Probability of • 15 for expected loss
loss • 31.5 for unexpected loss
Bank also can reserve more provision and
Loss Distribution capital for the future uncertainty.
A
Credit
Losses
0
Tail Risk
Expected Capital need to hold to
loss protect Unexpected loss
15 31.5 253.5
2009 Eric Confidential
6

7. Basel committee generates a general form of unexpected loss formula for banks to
gauge the unexpected loss and use as capital minimum requirement.
Factors in Basel2 Inverse of the
Standard normal distribution standard normal
(N) applied to threshold and distribution (G)
1 Year PD is considered, conservative value of applied to PD to
PD derive default
instead of cumulative PD systematic factor
threshold
Subtract EL
based
K= Correlation
Based on historical data provision
LGD
   R 
0.5
 
LGD N 1  R  GPD     G0.999  PD  LGD
0.5
 
 1 R  
 
 
 Inverse of the 
Current status of EAD standard normal
EAD
 1  1.5  b  1  M  2.5  b
1 distribution (G)
applied to
confidence level to
derive conservative
Tenor adjustment value of systematic
factor
Tenor B= 0.11852  0.05478  ln( PD)2 RWA = K * 12.50 * EAD
R=
Asset 1  e 50PD     1  e 50PD  
0.12   
50   0.241   50

 Capital = RWA * BIS Ratio
Correlation  1 e    1 e 
It might over or under estimated the risk. 2009 Eric Confidential
7

8. Basel 2 capital equation is a simplified method of EC and is originally from
Vasicek‟s model.
AGENDA OF TODAY
1. Role of bank Capital is not just for the regulatory requirement.
capital Capital is a cushion to absorb unexpected loss.
Basel 2 „s capital equation is a simplified „EC‟ model, it
comprises 4 factors :
2. Interpreting
 Vasicek model
Basel 2 capital
 Correlation
equation
 EL
 Tenor adjustment
3.Model This model follows Basel’s approach and incorporate simulation skill to
Introduction & measure the EC.
Applications
Economic capital is wildly utilized in all aspects of bank’s management.
Three key management applications :
4. Role of EC in
 Risk governance
Pillar 2
 External communication
 Internal management
2009 Eric Confidential
8

9. Vasicek generates a equation to estimate the probability distribution of the %
loss on the bank loan portfolio in 1987.
  G( PD )  R * G(99.9%)  
 PD  LGD * 1  1.5  b   1  M  2.5  b
1
k   LGD  N  

  1 R  

1 2 3 4
Asset correlation Vasicek Model Expected loss Tenor adjustment
subtraction
where
N = cumulative standard normal distribution
G= inverse standard normal distribution
R= asset correlation, used to gauge the relationship between obligors’ return and
systematic risk (common factor)
the G(99.9%) represents for the confidence level that bank capital is able to
sustain the credit losses.
2009 Eric Confidential
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10. Recall the CAPM theory that there is a one almighty factor that can explain the
asset relationship across assets.
Definition of default correlation
If Fannie Mae files chapter 11.. ..will Lehman also has insolvency problem?
Asset correlation
Correlation
between
corporations
Vasicek Model
..will this event affect to ..if Lehman did bankrupted and cause
auto industry?... Lehman’s employer unemployed..
Expected loss
Correlation
between
corporation and
individuals
Correlation
Tenor adjustment between
GM announces 500
individuals and
temporary layoffs
corporation
2009 Eric Confidential
10

11. Asset correlation is the dependence of asset value of obligors on the general
state of economy. All obligors are linked to each other by this systematic factors.
Let’s use the ‘SUN’ as an example…
‘Sun’ has almighty power to all creatures on the earth. In different seasons or timing, ‘Sun’ has different
influences on earth.
The clothes Plant Sun becomes less burdensome in the winter or
..and of setting
course to the
human being
Snowman Insect
•The impact to the different creatures can be considered •In the different stage of economy, same
as asset correlation. obligors or asset classes also perform different
•Different borrowers or asset classes show different dependency.
degrees of dependency on the state of economy.
2009 Eric Confidential
11

12. Highly correlated with state of economy implies that the PD of a company is
highly dependent on the economy situation.
•100% of asset correlation means Implication
the company’s asset return is
perfect link to the economy. •Company defaults in the bad state of
•No firm’s specific risk existed 100 % of correlation
economy
•Company shows strong financial
High return
100% 100% performance in the booming economy
Systematic Company’s PD
risk Company’s Bad increasing
asset economy Good along with the
return economy event of bad
economy
Company State of
economy Low return
State of economy
Bad economy Good economy
2009 Eric Confidential
12

13. Rating agencies do try to utilize the actual the joint default probability out of their
data base to estimate the asset correlation….
S & P‟s historical default study Asset correlation estimation
Year # of # obligors Joint Statistics suggests the asset correlation is a
defaults default function of joint default events and obligor
rate
defaults, depict as below
1981 0 1070 0.0000%
Jo int_ Default _ Pr obability ij
1982 2 1099 0.0002%
 ( Default _ threshold i , Defaulr _ threshold j
1983 1 1122 0.0000%
, Asset _ Correlatio nij )
1984 2 1181 0.0001%
1985 0 1216 0.0000%
 = Bivariate standard normal distribution
….. …. … …
2003 3 2998 0.0001% In this case, the asset correlation from
2004 0 3117 0.0000% S & P ‘s study is 3.88%
2005 1 3264 0.0000%
…but past experience tends to be less useful for the current and future,
in terms of correlation estimation.
Source: Diane Vazza and Devi Aurora: Annual 2005 Global Corporate Default Study And
Rating Transitions, 2006, Standard & Poors 2009 Eric Confidential
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14. Basel committee adopts the single factor model to modeling the asset
correlation. The asset correlation is a matter of systematic risk.
One Factor Model Rationale of joint default event
•Joint default event is driven by the correlation
This is the ‘factor’
•Borrower i’s asset value Ai depends on the common
factor Z and an idiosyncratic factor εi
Ai  Wi Z  1  Wi  i 2
•The parameter Wi represents the obligor’s
dependence to the market index.
asset return of .. explained by •The Wi can be estimated from an OLS regression by
.. the remainder
a single state of the utilizing the Equity return of obligor i
borrower over a economy (√1-W 2 ) is idiosyncratic
Correlation
one year (systematic factor) risk (firm’s specific risk)
‘Z’ at ‘W’ % Systematic factor
horizon can be
(market factor)
..
cov( i ,  j )  0
where _ i  j; idiosyncratic
cov( Z ,  i )  0,  i risk
Obligor 1 Obligor 2
Note : A detailed explanation of the capital requirement formula can be found in Basel Committee on
Banking Supervision, 2005, An Explanatory Note on the Basel II IRB Risk Weight Functions, Basel. 2009 Eric Confidential
14

15. Market practice leverages equity correlation for the estimation of asset
correlation.
•Credit Metrics suggests to utilize the MSCI to estimate the equity correlation.
•Suggest equity correlation is a good proxy of asset correlation.
It is also possible to
utilize the Taiwan
stock index to proxy
the asset correlation
for CBG.
Source: J.P. Morgan,1997.Introduction to CreditMetrics technical document. 2009 Eric Confidential
15

16. Moodys KMV estimate the asset value by adding the market cap with firm‟s
liability. Moreover, KMV extends the one factor model into multiple factors model
to estimate the asset correlation.
KMV begins at collecting equity indices cross ..KMV finally identified 120 factors in gauging the
globe and estimate the market cap for each firm systematic risks.
before adding firm‟s liability to come up with The systematic factors are the sources of „correlation‟ .
firm‟s asset value..
Systematic Risk
Firm Specific
Country Risk Industry Risk Risk
14 45 61
US Electronic rk   kf rf   kc  c   ki  i   k
UK Manufacturing
f 1 c 1 i 1
Taiwan Service
Korea Real estate 14 Common 45 Countries 61 Industries
… macro
…
factors
2009 Eric Confidential
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17. Higher the asset correlation easier to be influenced by the state of the economy.
Asset correlation between obligor and the
state of the economy Asset correlation between 2 obligors
Joint default
probability of
2 obligors
Default correlation between
2 obligors
Systematic
Risk
Can be
Further
Diversified
Through
Firm Specific
Add more
risk
obligors
2009 Eric Confidential
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18. Asset correlation estimation can be translate into default correlation and joint
probability of default.
Market Value of Illustrative
Assets
Taiwan Business bank Default Point Insurance Auto
has a high asset
correlation (51.57%)
Taiwan Business Bank
High Asset Correlation is
with the state of
bad, but low PD creates a
economy
very low probability of both
defaulting --- resulting in
lower default correlation
PD of
Taiwan Business Bank
=0.87% Default Point
Taiwan Business bank
Market Value of
Insurance Auto
Assets
Joint probability of default Insurance Auto has a
=0.00445% low asset correlation
Measures the probability
of both companies defaulting (11.29%) with the state
PD of Insurance Auto
at the same time =0.24% of economy
2009 Eric Confidential
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19. With asset correlation, bank can then compare the retail segments with corporate
customer.
Prob of default
together is
0.0079%
Asset correlation
The asset between TSMC
return of & Mail Loan is
TSMC is 16%
highly related
to global
asset return Default
correlation
between TSMC
& Mail Loan is
0.4%
Country risk
Systematic
Industry risk Risk
Sector risk
Region risk
Global risk
Firm Specific
risk
2009 Eric Confidential
19

20. Even the mortgage segment has the same correlation with mail loan segment, the
higher PD of mail loan result in a high joint PD.
Prob of default
together is
0.0079%
Asset correlation
between
Mortgage & Mail
Loan is 5%
Default
correlation
between
Mortgage & Mail
Loan is 1.3%
2009 Eric Confidential
20

21. Low the PD accompany with low asset correlation, generate a lower the joint EDF
and default correlation.
Lower the PD
lower the joint
EDF and default
correlation
Mortgage Mortgage
2009 Eric Confidential
21

22. Correlation plays important role in identifying the „Diversification Effect‟ and
serves as a driver to estimate the economic capital.
- Illustrative -
Case1 :Different industry but in same country 26 % This portion of the
Saving risk has been
diversified away
Diversified within the portfolio.
Basel 2 5.2 MM
19.7 MM
Obligor‟s Diversifiable
11.5 MM
Standalone
AIRB Capital
+ Firm Specific Economic Capital = 14.5 MM
8.2 MM
Risk Contribution of
Systematic
China Airline =3 MM
Bank Airline
Case2 :Different industry and different country
29.5 % This portion of the
Saving risk has been
Diversified diversified away
4.1 MM within the portfolio.
13.9
Obligor‟s total + Diversifiable
Economic Capital 8.2 MM Economic Capital = 9.8 MM
5.7 MM Firm Specific
(stand-alone)
Systematic Risk Contribution of Insurance
Airline Insurance (US) Auto = 1.6 MM
Assume : Lend NT 100 MM to each borrowers at LGD=45% of collateral, given their different PD. 2009 Eric Confidential
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23. For retail portfolios, leverage the historical time series of default or loss
information (by product, segments etc.) is the major practice on developing asset
correlations for retailing .
Default Rate
Default Rate by Product – Client Example
0.35%
Installment Line of Credit Real Estate Credit Card
0.30%
0.25%
0.20%
0.15%
0.10%
0.05%
0.00%
Jul-01
Jul-02
Jul-00
Jan-03
Mar-03
Jan-02
Jan-01
Mar-01
Mar-02
Sep-01
May-01
May-03
Nov-02
Sep-02
Sep-00
Jan-00
Nov-00
Nov-01
Mar-00
Sep-00
May-02
Nov-00
May-00
2009 Eric Confidential
23

24. Empirical studies show that : Higher the correlation will have high relationship with
the global economics, result in a higher impact to obligor‟s business. Therefore,
requires more capital to protect against „unexpected loss‟.
Higher the correlation higher the UL, therefore, bank
needs to reserve higher capital requirement
Low correlation
High correlation
Basel Committee generates the asset correlation through „Reverse Engineering‟ –
Empirical experimental through several banks‟ EC.
2009 Eric Confidential
Source : An Explanatory Note on the Basel II IRB Risk Weight Functions, Basel 2
24

25. Different types of loan assets have different asset correlation.
2009 Eric Confidential
25

26. Basel has difficulty to estimate correlation for different countries and industries.
Therefore, Basel come up with a equation for corporate obligor based on rating
and suggests a constant for retail products. •Asset correlation decrease with increasing PDs. This is
based on empirical evidence.
Better grade has higher asset correlation •Intuitively, higher the PDs, higher the firm specific risk.
Default risk depends less on the overall state of the
under Basel committee‟s assumption.
economy.
25% 23.82% ORR_Grade PD Correlation
1.AA- or better 0.03% 23.82%
2.A+~A- 0.10% 23.41%
20%
Mortgage 3.BBB+ 0.16% 23.08%
4.BBB 0.26% 22.54%
15% 15%
5.BBB- 0.42% 21.73%
Asset
Correlation 6.BBB- negative
0.61% 20.85%
perspective
10%
7.BB+ 0.90% 19.65%
Revolving Product
8.BB 1.35% 18.11%
5% 9.BB- 2.04% 16.33%
4%
10.BB- negative
3.15% 14.48%
perspective
0%
1 2 3 4 5 6 7 8 9 10 11 12 13
11.B+ 4.93% 13.02%
12.B 7.82% 12.24%
CTCB Rating Grates
13.B- 0.1261 12.02%
2009 Eric Confidential
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27. How does the Basel Committee come up with the correlation equation???
Technically, Basel utilize the CAPM theory…
1 Recall the CAPM model that : 3 Defined A, Z, ε are normalized
random variables
ri=αi+βirm+εi ri   i   i E (rm )
where A 
ri = return of firm i.
i
rm= return of the market rm  E (rm ) i
εi= firm’s specific risk- a random error Z 
m  ( i )
Manipulating CPM by Re write the
CAPM tells us that .. Define the variables
normalizing equation
4
m  ( i )
2 A  ( i )Z  
Normalizing the above equation, we have: i i
ri   i   i E (rm )  m rm  E (rm )  ( i )  i Where
 ( i ) 
i i m  i  ( i )  m 2  ( i ) 2
1  ( i ) ( )
Where  i ,  m ,  ( i ) i i
are the standard variance of firm i asset, the
return of market and the random error,
respectively.
2009 Eric Confidential
27

28. … to derive the one factor correlation model.
5 7
A ordinary regression Correlation One factor model is
between I and derived from CAPM
Yi=α+βiXm+ε
m is
 im A  WZ  1  W 2 
 i 
 i  im  m i
m2
 W 2 = asset correlation
W  ( i m ) =correlation weight
i
Recall the regression in Re-arranging the ..finally, we have One
statistics parameters.. factor model
6
m  ( i )
A  ( i )Z  
i i
 m 2  ( i ) 2
Because 1  (  i ) ( )
i i
    We’ll have
(  i m ) 2  ( im m ) 2  ( im ) 2   i
2
i m i
2
 m i  ( i ) 2 
( )  1  (  i m ) 2  1  i2  1  W 2
m i i
Set W  ( i )
i  ( i )
the  1W 2
i
2009 Eric Confidential
28

29. The empirical asset correlation estimation done by Thomas and Zhiqiang support
Basel‟s correlation equation.
Hugh Thomas and Zhiqiang Wang examine
the average asset correlation of United In the first draft of Basel Accord, Basel suggested a constant
States by using the one factor model and asset correlation 20% , for all corporate obligors. Later the
banks in US and Europe provide their survey and empirical
found the current asset correlation is close to
study to Basel and urged to refine the asset correlation.
their finding..
2
  
v  i 2   i m
 


 i  20%
2
 20% 
 1 * 
 46% 
 18.9% Asset
where the market return Correlation
volatility
m = 20 % per annum;
the average stock volatility is
i = 46 % per annum
i =1
PD
Source: Hugh Thomas and Zhiqiang Wang "Interpreting the Internal Ratings-
Based Capital Requirements in Basel II" Journal of Banking Regulation, 2005. 2009 Eric Confidential
29

30. Nobel prize winner Merton suggests that default can be viewed as a function of
the underlying asset value of company…and Vasicek modified Merton‟s model
and became the cornerstone of Basel‟s capital equation.
Definition of default correlation
Under Merton’s model assumption, underlying firm value is random with
Asset correlation
normal distribution. If the value of assets decreases below the default
threshold (amount of liabilities outstanding), it will be impossible for the firm to
satisfy its obligations and it will thus default.
Vasicek Model The PD of an obligor can be translated into the default threshold.
For example: the threshold of PD= 2% is -2.053
Default threshold
Expected loss
2% of probability
will fall below 98%
of confidence
interval
Tenor adjustment
NORMSINV(2%)= -2.053
Note: Vasicek is one of the founders of KMV. 2009 Eric Confidential
30

31. Vasicek combines Merton‟s model with one factor correlation model to measure
the credit loss.
1 2 3
The company defaults on its But we’d like to know the PD of an Basel set the Z (state of economy) at the
loan if the value of its assets obligor fluctuate with the economy; poorest condition (99.9%), to test how
drops below the contractual therefore, we rewrite the equation by many capital reserve is required by bank
value of its obligations α incorporating the state of the to sustain in the worst economic situation.
economy: Therefore we have
(default threshold) payable at
PDi=P(Ai<αi)
one year horizon. We thus
Z =N-1(99.9%)=G(99.9%)
have:
Wi Z  1 Wi  i <αi)
2
= P(
Recall the Wi  
PDi=P(Ai<αi)   Wi Z i
= P( i  ) And Basel use the symbol ‘R’ to represent
1  Wi 2
for asset correlation
The default threshold can be Therefore, we have
derived :
Substituting the α and expressed Wi    R
αi=N-1(PDi)
(1  Wi )  1  R
the above with cumulative standard 2
normal distribution:
PDi= P(Ai<αi) And finally we have the Basel’s K factor =
Note:  G( PD )  R * G(99.9%) 
Vasicek, O, 1987. Probability of loss on loan
=N ( N 1 ( PD )  Wi Z ) N 
portfolio. KMV Corporation.
 1 R 
Vasicek, O, 1991. Limiting loan loss probability 1  Wi 2
distribution. KMV Corporation. 2009 Eric Confidential
31

32. Basel defined that the adequate capital must be sufficient at 99.9% of the time…
to make up the Vasicek‟s assumption.
Major assumption of Vasicek‟s model for calculate the
State of economy loan loss distribution
•Consider a portfolio consisting of n loans in equal dollar
amounts : Diversified and no concentration risk existed.
•Let the probability of default on any one loan be ‘P’ : each
Poor state Good state of obligor has the same PD
of economy economy •Assume that the values of the borrowing companies’
assets are correlated with a coefficient ρ for any two
0.1% of probability
companies : constant asset correlation
will fall below 99.9%
•Assuming that the loan generates no income: Net interest
of confidence
interval income and fee income is not considered.
• When a loan goes into default, there is no recovery:
LGD=100%
•Firm’s asset returns are normally distributed : Normal
0
distribution
-NORMSINV(99.9%)= •The company defaults on its loan if the value of its assets
NORMSINV(0.1%) = -3.09 Mean of state of economy drops below the contractual value : Merton’s model
Vasicek assumes if a bank has diversified portfolio : same
Asset value follows Normal distribution is a strong
PD, LGD, EAD and same correlation. The only thing
assumption. To prevent from this flaw, Basel set the
needs to worry is the ECONOMY, borrowers have high
capital cushion at a high level that can protect bank at probability in the bad state of economy : source of
99.9% of probability. unexpected loss. 2009 Eric Confidential
32

33. EL based provision comes with a proper reason. If banks are not comply with EL based
provision then banks may need to prepare more capital to protect credit losses.
Banks should price the expected loss based o the risk and as provision
reserve. Otherwise, banks need to hold more capital to prevent from
insolvency.
Asset correlation
Basel set at 99.9% of confidence
Vasicek Model
=100% - 99.9% = 0.1%
Expected loss
Subtract EL
based
Correlation
provision
   R 
0.5
 
LGD N 1  R  GPD    G0.999  PD  LGD
0.5
Tenor adjustment 
K=  
 1 R  
 
 
 
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34. Longer term credit facilities are riskier than short-term credits. As a
consequence, the capital requirement should increase with maturity.
The final adjustment in the IRB formula is the adjustment for the
average maturity.
Asset correlation
  G( PD )  R * G(99.9%)  
 PD  LGD * 1  1.5  b   1  M  2.5  b
1
k   LGD  N  

  1 R  

Vasicek Model
b= 0.11852  0.05478  ln( PD)2
As we all know that the longer term credit facilities are riskier
Expected loss than short-term credits. As a consequence, the capital
requirement should increase with maturity. The maturity
adjustments can be interpreted as anticipations of additional
capital requirements due to downgrades. Downgrades are
more likely in case of long-term credits and hence the
Tenor adjustment anticipated capital requirements will be higher than for short-
term credits.
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35. 4 objectives in this topic : Rational of model development, Demo the model,
understanding the importance of risk parameters and estimation of CTCB‟s EC.
AGENDA OF TODAY
1. Role of bank Capital is not just for the regulatory requirement.
capital Capital is a cushion to absorb unexpected loss.
Basel 2 ‘s capital equation is a simplified ‘EC’ model, it comprises 4 factors
:
2. Interpreting  Vasicek model
Basel 2 capital  Correlation
equation  EL
 Tenor adjustment
3.Model This model follows Basel‟s approach and incorporate
Introduction &
simulation skill to measure the EC.
Applications
Economic capital is wildly utilized in all aspects of bank’s management.
Three key management applications :
4. Role of EC in
 Risk governance
Pillar 2
 External communication
 Internal management
2009 Eric Confidential
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36. Model introduction & application
Ai  Wi Z  1  Wi 2  i
Theory introduction
Pi (Z )  P( Ai   i | Z )  P(Wi Z  1  Wi 2  i   i | Z )
Model demonstration  P(Wi Z  1  Wi 2  i  N 1 ( PDi ))
Conditional PD  Wi Z  1  Wi * N ( i )
2 1
Model applications
Wi Z  1  Wi 2 * N 1 ( i )  N 1 ( PDi )
Limitation
2009 Eric Confidential
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37. Integrate the bank „internal rating system‟ into one factor model with simulation
skill to depict the loan loss distribution.
Applying one-factor Estimating the Simulating and
Data Estimating Default
model to estimate the credit loss in the aggregating all
Requirement threshold
default event event of default scenarios
Obligor based or segment based For example :
• PD ‘AAA’ Corporation has
• LGD - 1% PD based on bank’s
• EAD This model requires to treat AAA as
Internal rating
• PD=1%
system
• EAD=200
- 2 credit lines
• LGD = 75%
1. EAD=100 with
LGD =50%
2. EAD =100 with
LGD =100%
2009 Eric Confidential
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38. Estimating default threshold means to translate the unconditional PD into
threshold value.
Applying one-factor Estimating the Simulating and
Data Estimating Default
model to estimate the credit loss in the aggregating all
Requirement threshold
default event event of default scenarios
Log of
market Probability
value distribution Default threshold
of asset
values
Expected 2% of probability
asset values will fall below 98%
over time
Today’s of confidence
firm interval
value
Default
point
Refers to this area
Default threshold = NORMSINV(2%)= -2.053
This area
translate into
PD,say 2%
The company defaults on its loan if the value of its assets drops
The structured portfolio model doesn’t mean that below the contractual value of its obligations α (default threshold)
we need to model the default probability by using payable at one year horizon. We thus have: PDi=P(Ai<αi)
the concept of Merton. Instead we apply the The default threshold can be derived :
concept into our portfolio model development. αi=N-1(PDi)
2009 Eric Confidential
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39. The obligor‟s PD is significantly influenced by the state of economy.
A PD that considered the state of economy is called „Conditional PD‟
Applying one-factor Estimating the Simulating and
Data Estimating Default
model to estimate the credit loss in the aggregating all
Requirement threshold
default event event of default scenarios
The conditional default threshold is the default probability
Recall the One Factor Model describe conditional on the systematic factor Z: a function of the
as below : idiosyncratic component of obligor i and the credit worthiness
index, then we have :
Ai  Wi Z  1  Wi 2  i The PD of obligor i considering
the state of economy
1 Pi (Z )  P( Ai   i | Z )  P(Wi Z  1  Wi 2  i   i | Z )
Asset State of the W 2 : asset Idiosyncratic
return of economy correlation risk (firm’s  P(Wi Z  1  Wi 2  i  N 1 ( PDi ))
obligor i (systematic between specific risk)
2 The conditional PD of obligor i can be described as :
factor) obligor i and
Conditional PD  Wi Z  1  Wi * N ( i )
2 1
state of
economy
Where : 3 We call obligor default if :
cov( i ,  j )  0 Wi Z  1  Wi 2 * N 1 ( i )  N 1 ( PDi )
where _ i  j; 4 To model and to simulate the conditional PD of
cov( Z ,  i )  0,  i obligor i, we simulate ‘Z’ & ‘ε’
2009 Eric Confidential
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40. Conditional PD reflects the state of economy.
Unconditional PD refers to the „Through The Cycle‟ PD estimation.
Conceptual
State of Description Occurrence Probability
economy of default
2%
Bad a recession at 2%
the risk
¼
horizon
Good expansion 0.4%
¼ Weighted average= 1.1%
1%
Neutral ordinary 1%
times
½
0.40%
Weighted 1.1%
average
Bad Good Neutral
Note: More detail can be found in Basel Committee : Credit risk
modeling: Current practices and applications. Page 28.April , 1999. 2009 Eric Confidential
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41. Applying the single factor correlation model to simulate the Conditional default.
Model of firm value and its default threshold Conceptual
Probability
of density
Simulation will try all valid
Default threshold
combinations of the current
given obligor’s PD
portfolio, given the PD, LGD,
EAD and asset correlation
Default area
-2.07 -1.74 -1.28 -0.01 0 …………………………
Asset return within one year:
Estimated by using the firm‟s PD Estimated by applying the one factor model through simulation :
 i  N 1 ( PDi )  N 1 (10%)  1.28 Conditional PD = W i * NORMSINV(RAND()) + √(1- W 2i) *
NORMSINV(RAND())
2009 Eric Confidential
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42. Apply LGD * EAD in the event of default in each scenario.
Illustrative
Applying one-factor Estimating the Simulating and
Data Estimating Default
model to estimate the credit loss in the aggregating all
Requirement threshold
default event event of default scenarios
Default Simulation Loss Aggregation
Assume a constant 10% of probability of default for 100 segments ..We can also see there are events of joint default, that explain
or obligors. After a 10 thousands of simulation, the following table when some obligors /segments insolvent , will influence other
illustrates the default event under the given correlation…. obligors / segments. These joint default event will occurs
‘UNEXPECTED LOSS’
Loan 1 2 3 … 100 Loan 1 2 3 … 100
PD 10% 10% 10% 10% LGD 50% 50% 50% 50%
Scenario Total Scenario Total
1 1 0 0 .. 0 1 1 50 0 0 .. 0 50
2 0 1 0 .. 0 1 2 0 50 0 .. 0 50
3 0 1 1 .. 0 2 3 0 50 50 .. 0 100
4 1 1 1 .. 0 3 4 50 50 50 .. 0 100
5 0 0 0 .. 0 1 5 0 0 0 .. 0 ..
... ...
10,000 0 0 1 .. 0 1 10,000 0 0 50 .. 0 50
# of Default 1,000 1,000 1,000 … 1,000 xxx $ of Loss 50,000 50,000 50,000 … 50,000 xxx
# of Survive 9,000 9,000 9,000 … 9,000 xxxx
Avg Default ~10% ~10% ~10% .. ~10% xx
Frequency Avg LGD ~50% ~50% ~50% .. ~50% xx
Note: Assume EAD=100 for all obligors
Based on portfolio simulation model 2009 Eric Confidential
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43. The outcome of simulation can allow bank to draw „Loss distribution‟ and
„Joint default event distribution‟ as well.
Illustrative
Applying one-factor Estimating the Simulating and
Data Estimating Default
model to estimate the credit loss in the aggregating all
Requirement threshold
default event event of default scenarios
Default Simulation Loss Aggregation
Diagnosing the joint default event allows bank to understand By summing up the individual loan losses, we then can obtain
the potential default contagion driven by the state of economy. the portfolio loss. Further considering the number of
occurrences and its associated loss amount, we can derive the
Furthermore, bank can draw joint default distribution.
portfolio loss distribution.
Loan 1 2 3 … 100 Loan 1 2 3 … 100
PD 10% 10% 10% 10% LGD 50% 50% 50% 50%
Scenario Total Scenario Total
1 1 0 0 .. 0 1 1 50 0 0 .. 0 50
2 0 1 0 .. 0 1 2 0 50 0 .. 0 50
3 0 1 1 .. 0 2 3 0 50 50 .. 0 100
4 1 1 1 .. 0 3 4 50 50 50 .. 0 100
5 0 0 0 .. 0 1 5 0 0 0 .. 0 ..
... ...
10,000 0 0 1 .. 0 1 10,000 0 0 50 .. 0 50
# of Default 1,000 1,000 1,000 … 1,000 xxx $ of Loss 50,000 50,000 50,000 … 50,000 xxx
# of Survive 9,000 9,000 9,000 … 9,000 xxxx
Avg Default ~10% ~10% ~10% .. ~10% xx
Frequency Avg LGD ~50% ~50% ~50% .. ~50% xx
Note: Assume EAD=100 for all obligors
Based on portfolio simulation model 2009 Eric Confidential
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44. Excel based and allows user to play around.
Theory introduction
Model demonstration
Model applications
Limitation
2009 Eric Confidential
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45. Confidence interval Credit VaR
# of
Simulation
Simulation
EC Systematic results
factor Z
This area allow user to
observe the joint
default event.
Default Limited to 255
Data input Weighting of columns though.
area correlation threshold
Conditional
PD threshold
Estimate Loss if
obligor default :
Call default if
LGD * EAD
Conditional PD < Default threshold
2009 Eric Confidential
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46. Output
Enter ‘Bucket’
Joint Default Distribution
Loss Distribution
2009 Eric Confidential
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47. Correlation plays an important role in determining the joint default event.
Effect of correlation
Theory introduction
Effect of concentration
Model demonstration
Effect of PD
Model applications
Effect of LGD
Limitation
2009 Eric Confidential
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48. This is a simple but still useful for bank to depict how the risk or EC will deviate
by changing the risk parameters.
Goal of this exercise
Assuming bank has a 100 obligors, each
has identical risk parameters describe as
below: • Observing how the EC change if change the
assumption of correlation. I’ll set the asset
Risk parameters Portfolio statistics
correlation at following and observe how the
PD=10% Portfolio EAD= 10,000
EC changes
LGD=50% EL =500
Asset Correlation Correlation Weight
EAD=100 Bank target rating = A
1. 6.25% 25%
2. 25% 50%
3. 1% 10%
4. 100% 100%
5. 0% 0%
Note : Defined ‘W 2 ‘ as asset correlation, ‘W‘ as correlation weight 2009 Eric Confidential
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49. Under a 6.25% of asset correlation , there are a max „33‟ segments/ obligors will
have a joint default event (0.1% of probability). The EC=900.
Illustrative
Joint Default Distribution Loss Distribution
Frequency of Joint Default Frequency of Loss
9% 10%
8.2% 經由模
9%
8% 擬發現, Target Rating =A
8% 有接近 Cumulative
7%
61% 的 probability =99.9%
There is a 0.1% of 7%
6% possibility that 33 情況,損 3.5% of
6% possibility result
segments or obligors 失不會 in a ‘800’
5% will default together 5% 超過EL potential loss
within 1 year. within 1 year
4% 4%
In other word, this
might happen once 3% 0.1% of possibility the
3% in 1,000 year loss will exceed 1,400 ,
2% Max loss=1,650
2%
0.8% 1%
1%
0.1% 0%
0
100
200
300
400
500
600
700
800
900
1000
1100
1200
1300
1400
0% 1,650
33
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
$ of Loss Amount
500
# of Joint Defaulted
Segment or Obligors EL Unexpected Loss = 1,400-500 =900 =
Economic Capital
6.25 % of correlation = 25% of correlation weight. 2009 Eric Confidential
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