TW3421x - An Introduction to Credit Risk Management
Default Probabilities
C-VaR and F-IRB Capital Requirements
!
Dr. Pasquale Cirillo
Week 6
Lesson 2
Introduction
✤
Assume that a bank has a large number of counterparties.!
✤
Each obligor i has a 1-year probability of default equal to PDi.!
✤
Assume that the correlation between each pair of obligors is ⇢ .
WCDR
✤
Using a one-factor Copula model, banks usually compute a quantity called
“worst case default rate”, or WCDR.!
✤
WCDR indicates the “worst case probability of default” and it is defined as the
99.9% quantile of the default rate distribution.
WCDR
The formula for the computation of WCDR for obligor i is
✤
W CDRi =
✓
1
p
(P Di ) + ⇢
p
1 ⇢
1
(0.999)
◆
From WCDR to C-VaR
✤
In Week 3 we have studied VaR, and we have said that in credit risk
management, VaR is often called C-VaR.!
✤
If we have the exact distribution of credit losses, C-VaR is computed as usual,
according to the techniques we have seen together.
From WCDR to C-VaR
If we assume that all the n obligors of a bank have the same pairwise correlation
✤
⇢ (or very similar, so that we can average it) , it can be shown (Gordy, 2003)
that, for the entire portfolio,
C
V
1 year
aR0.999
=
n
X
i=1
EADi ⇥ LGDi ⇥ W CDRi
Expected Loss
✤
What is the expected credit loss for the bank’s portfolio?!
✤
The answer is clearly
n
X
i=1
EADi ⇥ LGDi ⇥ P Di
Flashback
✤
Do you remember what we have said about the IRB approaches in Week 2?!
✤
Do you remember risk factors and risk-weight functions?!
✤
We are now ready to understand a little more about all that!
Corporate, Sovereign and
Bank Exposures
In the Basel framework, on the basis of some empirical research, the correlation
✤
parameter for corporate, sovereign and bank exposures is computed as
✤
⇢i = 0.12(1 + e
Notice that as PDi increases, ⇢i decreases.
50⇥P Di
)
Corporate, Sovereign and
Bank Exposures
The formula used for defining the capital requirements for counterparty i is
✤
EADi ⇥ LGDi ⇥ (W CDRi
P D i ) ⇥ M Ai
Corporate, Sovereign and
Bank Exposures
The formula used for defining the capital requirements for counterparty i is
✤
EADi ⇥ LGDi ⇥ (W CDRi
P D i ) ⇥ M Ai
1 + (Mi 2.5) ⇥ bi
M Ai =
1 1.5 ⇥ bi
bi = (0.11852
0.05478 ⇥ log(P Di ))2
Corporate, Sovereign and
Bank Exposures
✤
total
the portfolio
The formula used for defining the capital requirements for counterparty i is
n
X
i=1
EADi ⇥ LGDi ⇥ (W CDRi
P D i ) ⇥ M Ai
Corporate, Sovereign and
Bank Exposures
✤
total
the portfolio
The formula used for defining the capital requirements for counterparty i is
CapReq
n
X
i=1
EADi ⇥ LGDi ⇥ (W CDRi
P D i ) ⇥ M Ai
This corresponds to 8% of RWA, so that
RW A = CapReq ⇥ 12.5
Exercise
✤
Suppose that the assets of a bank include £100 million of loans to A-rated
corporations.
The PD of all these corporations is estimated to be 0.1%.
Historical data give an average LGD equal to 60%.
The maturity for all loans is 2.5 years.!
✤
What are the RWA? And capital requirements?
Exercise
✤
We first compute !⇢ = 0.12(1 + e
✤
And! b = (0.11852
✤
Then
✤
50⇥0.001
) = 0.234
0.05478 ⇥ log(0.001))2 = 0.247
MA =
1
Using R, we compute WCDR, as
1
= 1.59
1.5 ⇥ 0.247
Exercise
✤
Therefore we have:!
!
✤
RW A = 12.5 ⇥ 100 ⇥ 0.6 ⇥ (0.034
0.001) ⇥ 1.59 = 39.35
And capital requirements are approximately:
CapReq = RW A ⇤ 0.08 = 3.15
Exercise
✤
Therefore we have:!
!
✤
In the STA approach, this would be 50
RW A = 12.5 ⇥ 100 ⇥ 0.6 ⇥ (0.034
0.001) ⇥ 1.59 = 39.35
And capital requirements are approximately:
CapReq = RW A ⇤ 0.08 = 3.15
Retail Exposures
For retail exposures, we have
✤
⇢i = 0.03 + 0.13e
35⇥P Di
Capital requirements for obligor i are
✤
EAD
⇥
LGD
⇥
(W
CDR
i
i
i
P Di )
Notice that there is no maturity adjustment.!
✤
Then we aggregate, as for corporate exposures.
Thank You