Week 6 Lesson 2

TW3421x - An Introduction to Credit Risk Management

Default Probabilities C-VaR and F-IRB Capital Requirements !

Dr. Pasquale Cirillo

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

WCDRi = �

✓��1(PDi) +

p⇢��1(0.999)

p1� ⇢

◆

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 aR1�year0.999 =

nX

i=1

EADi ⇥ LGDi ⇥WCDRi

Expected Loss

✤ What is the expected credit loss for the bank’s portfolio?!

✤ The answer is clearly

nX

i=1

EADi ⇥ LGDi ⇥ PDi

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

✤ Notice that as PDi increases, decreases.⇢i

⇢i = 0.12(1 + e�50⇥PDi)

Corporate, Sovereign and Bank Exposures

✤ The formula used for defining the capital requirements for counterparty i is

EADi ⇥ LGDi ⇥ (WCDRi � PDi)⇥MAi

Corporate, Sovereign and Bank Exposures

✤ The formula used for defining the capital requirements for counterparty i is

EADi ⇥ LGDi ⇥ (WCDRi � PDi)⇥MAi

bi = (0.11852� 0.05478⇥ log(PDi))2

MAi =1 + (Mi � 2.5)⇥ bi

1� 1.5⇥ bi

Corporate, Sovereign and Bank Exposures

✤ The formula used for defining the capital requirements for counterparty i is

EADi ⇥ LGDi ⇥ (WCDRi � PDi)⇥MAi

nX

i=1

total the portfolio

Corporate, Sovereign and Bank Exposures

✤ The formula used for defining the capital requirements for counterparty i is

EADi ⇥ LGDi ⇥ (WCDRi � PDi)⇥MAi

nX

i=1

total the portfolio

This corresponds to 8% of RWA, so that

RWA = CapReq ⇥ 12.5

CapReq

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 !

✤ And!

✤ Then

✤ Using R, we compute WCDR, as

MA =1

1� 1.5⇥ 0.247= 1.59

b = (0.11852� 0.05478⇥ log(0.001))2 = 0.247

⇢ = 0.12(1 + e�50⇥0.001) = 0.234

Exercise

✤ Therefore we have:!

!

✤ And capital requirements are approximately:

RWA = 12.5⇥ 100⇥ 0.6⇥ (0.034� 0.001)⇥ 1.59 = 39.35

CapReq = RWA ⇤ 0.08 = 3.15

Exercise

✤ Therefore we have:!

!

✤ And capital requirements are approximately:

RWA = 12.5⇥ 100⇥ 0.6⇥ (0.034� 0.001)⇥ 1.59 = 39.35

CapReq = RWA ⇤ 0.08 = 3.15

In the STA approach, this would be 50

✤ For retail exposures, we have

✤ Capital requirements for obligor i are Notice that there is no maturity adjustment.!

✤ Then we aggregate, as for corporate exposures.

Retail Exposures

⇢i = 0.03 + 0.13e�35⇥PDi

EADi ⇥ LGDi ⇥ (WCDRi � PDi)

Thank You