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.