Issues in Credit Portfolio Modeling

MathFinance Colloquium, Frankfurt, February 2002

Ludger Overbeck

Research & Development

CIB/Credit Risk Management

Deutsche Bank AG

Frankfurt, Germany

Overview

� Basic Model

{ Static one period model

{ Losses only caused by defaults

{ Homogenes Approximation

{ Capital allocation

� Model with Migration

{ Losses also due to change in creditwor-

thiness of counterparty

{ E�ect of Maturity

{ Migration Matrix

� Default Times

Basic Model

Random variable describing defaults:

L =mXi=1

li1Di

m = # of counterparties

Di = Default of counterparty

li = Loss in the event of default

= Exposure * Loss Given Default

The exposure is viewed as a loan equivalent

exposure, if the transaction with a counter-

party is a traded product (Average Expected

Exposure)

Joint Default Probabilities

The determination of the probability of joint

defaults of each pair of obligors is important.

Single Defaults are derived from relative fre-

quency of defaults in uniform segments.

P [Di] =# Defaulted members in Segment A

All members in Segment A

Joint Defaults?

P [CP in Seg A defaults;CP in Seg B defaults]

=#?

#???

For joint defaults a model is needed!

Multivariate Ability to Pay

(A(i)1 )i=1;::;m ="Ability to pay" at year one of

the vector of obligors.

Default in one year is then be de�ned by the

event

Di = fA(i)1 < C(i)g;where C(i) is a calibrated "Default Point".

Remark: If the counterparty is an exchange traded �rm,

then A(i)t=Value of �rm's assets= F(E;L; t) where

E = (Et)t�0 value of equities

L = (Lt)t�0 Liability Process :

The function F may be derived by modeling the equity

as a contingent claim on the �rm ( Call option as in

Merton '74) or vice versa. "Default Point"= Function

of Liabilities.

Joint Defaults if A is normal

JDPij

:= P

�A(i)1 < C(i); A

(j)1 < C(j)

�

=

N�1(P [Di])Z�1

N�1(P [Dj])Z�1

1

2�q(1� r2ij)

exp

1

2(1� rij)(x21+ x22 � 2rijx1x2)

!dx1dx2

Default correlation

�ij = corr(1fA

(i)1 <C(i)g

;1fA

(j)1 <C(j)g

)

=JDPij � P [Di]P [Dj]q

P [Di](1� P [Di])P [Dj](1� P [Dj])

Economic Capital

Economic Capital is usually de�ned to be a

quantile of the distribution of Lminus the mean

of L.

EC(�) = q�(L)� E[L]:

Assuming that the returns on a single asset in

the portfolio are joint normal distributed, the

portfolio return L is also normal. Then the

economic capital is also given by multiples of

the standard deviation. These multipliers CM

are also called \Capital Multipliers".

EC(�) = �(L) � CM(�):

Alternative Capital De�nition

In light of the non-normality another Capital

De�nition should be considered. Economic Cap-

ital viewed as a Risk Measures should also sat-

isfy the Coherency Axioms formulated by Artzner

et al. A prominent example of a coherent risk

measure is similar to a kind of lower partial

moment

C(L) := E[LjL > K];

where K is a threshold, used to de�ne "Large

Losses", e.g.

K = q�0(L), then C(L) is coherent.

K =fraction of equity capital

K = experienced large losses

Contributory Economic Capital

This capital de�nition yields also a new de�ni-

tion of contributory economic capital

Ci(L) := E[1DijL > K]:

Average contribution of counterparty i to the

portfolio loss, when large losses occur.

Theorem:

� C(L) =Pmi=1 liCi

� If K =2 fPnk=1 likjfi1; ::; ing � f1; ::;mgg then

Ci =@C(L)

@+li

� Ci is a coherent risk measure on the prob-

ability space generated by the portfolio.

Remarks

� Ci < 1

� These �gures are a by-product if the loss

distribution is generated by a Monte-Carlo-

Simulation.

� First statistics of the distribution of Li given

L > K. Other statistics like variance could

be useful. ( Conditional variance is proba-

bly not coherent.)

� De�nition re ects a causality relation. If

counterparty i adds more to the overall loss

than counterparty j in bad situations for the

bank, also business with i should be more

costly ( asuming stand alone risk charac-

teristics are the same).

� A function C is a coherent risk measure i�

C is a generalized scenario, i.e. there is a

set of probability measures Q such that

C(X) = supfEQ[X]jQ 2 Qg:Since L;Li � 0 the capital allocation rule

carries over to all coherent risk measures.

Migration Risk

� Rating migration of a counterparty

{ Markov process with state space f1; : : : ; kg;k = Default.

{ Transition matrix � = (�jl)j;l2f1;::: ;kg

� Also migration behavior is modeled by the

multivariate normal random variable A.

� Rating migration in a portfolio

{ (Pt)t=0;1;::: = (Pt(1); :::; Pt(m))t=0;1;:::

{ State space f1; ::; kgm

{ For p = (p1; : : : ; pm); q = (q1; : : : ; qm) 2f1; ::; kgm,

P [Pt = qjPt�1 = p]

= N�(]�p1(q1 � 1); �p1(q1)]� : : :

�]�pm(qm � 1); �pm(qm)]);

where

1. N� denotes the m-dimensional stan-

dard normal distribution of the m Ability-

to-Pay processes.

2. N is the univariate standard normal

distribution with inverse N�1 and

3. �i(l) is de�ned by

�i(l) := N�1(lX

j=1

�i;j) (1)

for i; l 2 f0; : : : ; kg.

� Loss Distribution

{ L(i; l; n): Change in value of all trans-

actions the bank has with counterparty

n if he or she migrates from rating i to

rating j in one year time

{

L = �mXn=1

L(P0(n); P1(n); n) (2)

EC and other risk measures are de�ned in the

same way as for default only models.

Maturity e�ects

Since in general losses caused by migration to a non-default rating

class are much smaller than a "migration" to default, many banks

still ignore migration e�ects for their bank wide portfolio loss dis-

tribution. There is however one feature which is heavily discussed

in the new regulatory discussion, the maturity e�ect. Since de-

fault events are not in uenced by maturity, only the introduction

of migrations allows to measure e�ects of maturity.

� CEC(M;dp) Contributory Capital of a tran-

sation with maturity M years and default

probability dp

� Basel MtM proposal:

CEC(7;0:0003)

CEC(1;0:0003)= 5:75:

� Simulation study

{ Di�erent migration matrices (see below)

{ Di�erent spreads ! Di�erent L(i; j; n).

{ Di�erent assumptions on corr(A).

� Di�erent CEC de�nitions

{ Var/Covar: Not sensitive to quantile cho-

sen for EC

{ Shortfall Contribution: Sensitive to quan-

tile

{ As more severe losses should be covered

by EC, less in uence of maturity.

� Our recommendation:

CEC(7;0:0003)

CEC(1;0:0003)< 2:6:

� If capital, as reg cap should, is put aside for

extreme systemic risk, like 99.98% quan-

tile, migration and therefore maturity does

not in uence the risk. Extreme losses are

only driven by defaults. (btw already no-

ticed in the Credit Metrics Manual) High

maturity adjustments therefore misallocate

regulatory capital.

Stationary Portfolios

Let the relative frequency function

� : f1; ::; kgm !�k

be de�ned by

�(p)(j) =1

m

mXi=1

1fjg(p(i)); j = 1; ::; k (3)

A portfolio p is stationary i�

�(p)

(1� �(p)(k))=

�(p)

(1� �(p)(k))�:

(Equivalent to "Expected Frequencies are sta-

tionary".)

Sensitivity analysis:

� � = �KMV ;�S&P ;�GC

� Same maturity M .

� Spreads induced by these matrices for reval-

uation.

� EC(�;M; q) on the q-quantile

Results:

EC(�;7; q)

EC(�;1; q)< 1:5; for q � 0:999065:

Interpretation in terms of "Macroeconomic Pru-

dential".

Default Times

Given a default curve pi(t); t > 0 for each obligor,

where pi is a strictly increasing function (0;1)![0;1].

Associated random variable T (i) is the default

time of obligor i:

P [T (i) � t] = pi(t):

Goals: Construct T (i) as a simple �rst hitting

time of a process Y , the ability to pay process.

Condition (*)

9Y (i) = (Y(i)s )s�0; C 2 R;

s.t. with TC;Y (i) := inffsjY (i)s � Cg

P [TC;Y (i) � t] = pi(t)

Correlation

Condition (**): Given a set of one year joint

default probabilities (pij)i;j=1;::;m we have

P [TC;Y (i) � 1; T

C;Y (j) � 1] = pij;

8i; j = 1; ::;m; i 6= j:

Ansatz: Try to �nd a transformation Y =

F (W) of a Brownian motion W with corre-

lation matrix (�ij)i;j=1;::;m such that (*) and

(**) are satis�ed.

Theorem. There exist a correlation matrix R

and a vector of time changes T it such that Y

de�ned by Y it = W i

Tt, where W is a Brown-

ian motion with covariance matrix R, satis�es

conditions (*) and (**).

Proof: For a one-dimensional Brownian mo-

tion B the function

P [infs�t

Bs � C] = ~f(t; C)

is explicitly known, cf. Karatzas/Shreve:

~f(t; C) = 2N(C=pt):

Therefore for each Y it =W

T itwith a determin-

istic time change Tt we have

P [infs�t

Y is � C] = ~f(T it ; C):

Hence a time change de�ned by N�1(pi(t)=2)

b

!�2=: T it

yields condition (*) for each i.

For a given correlation �ij the joint default

probability of two time changed Brownian mo-

tions equals

P [TY i;C � 1; TY i;C � 1]

=

Z 1C

Z 1C

�f�x1; x2; �ij;min(T i1; T

j1)�

1� 2N

C � x2p

�

!!dx1dx2;

where �f(x1; x2; �ij; t) is the corresponding den-

sity without time change at time t ( as in

the paper of Zhou on default correlation) and

� = max(T i1; Tj1)�min(T i1; T

j1). �

Remarks

� A transformation of Brownian motion based

on deterministic time dependent volatility

Y =R�dW can also be written as a time

change of a Brownian motion B. Then

condition (*) can be met, however the cor-

relation structure of B and W are di�erent.

Therefore the solution to condition (**)

seems open.

� Open Problem:

Find a non-random drift g(i) such that with

the de�nition

Y it =W i

t �Z t

0gisds

conditions (*) and (**) can be met.