Applications of Probability Theory in
Credit Portfolio Management
C HRISTIAN B LUHM , C REDIT S UISSE , Z URICH
www.christian-bluhm.net
On Occasion of Professor Dietrich Kölzows 75th Birthday
Table of Contents
1. What is Credit Portfolio Management (CPM)?
2. The Role of Probability Theory in CPM
3. Further (Graphical) Examples
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1. What is Credit Portfolio Management (CPM)
Credit Portfolio Management aims at
• optimizing the bank’s profit on credit products by
• taking the risk inherent in these products and portfolios of such
products into account. Example:
[RAROC] =
[Margin] − [Expected Loss]
[Credit Risk Capital]
RAROC: Risk-Adjusted Return On Capital
• Hereby, banks buy (go ‘long’ in) credit risks as well as sell (go
‘short’ in) credit risks
• trying to achieve their target (‘tailor-made’) risk/return profile on
their credit portfolio.
• Credit risk: default and migration risk of borrowers
• Return: gross margin/earnings on credit products
• Credit risk capital: cushion against losses
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In order to enable risk & return steering, CPM has the responsibility
• to develop and implement mathematical models measuring risk
and return of credit-risky instruments,
• to backtest and validate these models, and
• to apply these models to the bank’s credit portfolio in order to
make value-adding recommendations to the bank’s senior management.
• These recommendations have to be compliant with regulatory
and accounting frameworks valid for the bank.
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2. The Role of Probability Theory in CPM
Client behaviour is uncertain:
• Does the client honour her/his financial obligations at any time?
• How does the credit quality of the client evolve over time?
• For stock exchange listed clients: what is the interplay of stock
market behaviour and the credit risk of clients (and vice versa)?
Market behaviour also is uncertain:
• Is the economy in an up- or down-swing?
• How do risk drivers like interest rates or currency exchange rates
impact the bank’s credit portfolio?
In order to find answers, probability theory has to be exploited.
Some Examples:
• Client behaviour: Bernoulli mixture models (default indicators)
Li ∼ B(1; Pi),
P = (P1, ..., Pm) ∼ F
F support in [0, 1]m
Li|Pi=pi ∼ B(1; pi),
(Li|P =p)i=1,...,m
independent
P[L1 = l1, ..., Lm = lm] =
Z
m
Y
=
pili (1 − pi)1−li dF (p1, ..., pm)
[0,1]m i=1
E[Li] = E[Pi],
V[Li] = E[Pi] (1 − E[Pi])
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(i = 1, ..., m)
• Borrower migration bevaviour: Markov chains
X (t)
borrower state (credit quality) at time t
X (t) ∈ {1, 2, ..., D − 1, D},
default state D absorbing
(t)
mij = P[X (t) = j | X (0) = i]
(t)
M (t) = (mij )i,j=1,...,D
migration matrix for [0, t]
Markov approximation:
∃ Q ∈ RD×D , Q generator, etQ ≈ M (t)
Remarks:
– In general, client migrations can not expected to be Markov
chains
– However, in some cases Markov approximation yields accaptable approximations
(approximative embedding: (M (k))k=1,2,... ,→ (etQ)t≥0 )
– New approaches, e.g., by F RYDMAN AND S CHUERMANN’s
Markov mixture models will lead to models for non-Markov
migration behaviour
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• Asset value models for listed clients
(i)
At
(i)
(i)
= A0 exp[(µi − 21 σi2)t + σi Bt ]
i = 1, ..., m; t ∈ [0, T ]
(i)
where Brownian motions (Bt ) are correlated via a correlation
matrix Γ = (%ij )1≤i,j≤m ; define default indicators
(t)
Li = 1{A(i)≤ c̃(i)}
t
t
(i)
where c̃t denotes the default-critical threshold for borrower i
(‘default point’);
(t)
(i)
P[Li = 1] = P[B (i) ≤ ct ]
(i)
(i)
ct
with
(i)
ln(c̃t /A0 ) − (µi − 21 σi2)t
√
=
σi t
(i)
where B (i) ∼ B1 ∼ N (0, 1) Remarks:
– Stochastic dynamic of asset value processes drives default
risk of borrowers
– Link between equity market and credit market via optiontheoretic (M ERTON-type) interpretations of debt and equity
– Real first-passage time models are not wide-spread in credit
risk; due to the in general non-explicit density of Brownian
FPTs with non-affine barriers, such models have to rely on
heavy numerics (in the same way as barrier options in derivative markets)
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• Economic Cycle models
– Autoregressive models
– Macroeconomic variables: GDP, unemployment rates, etc.
– Decomposition of credit-risk driving variables in systematic
and idiosyncratic component by means of regressions
– Exploitation of industry indices, e.g., MSCI indices (downloads in Bloomberg)
• Interest and currency risks
– Interest rates influence borrower behaviour; example: prepayments in mortgage-backed lending
– Currency risks most often will be hedged, e.g., in structured
credit products
– Relationship between macroeconomy and interest rates is
complicated to integrate in credit risk models
– In general, the Credit Portfolio Management unit of the bank
takes credit risks in credit products, the Asset Liability Management unit takes interest and market risks
The role of probability theory in CPM therefore is clear:
Stochastic models constitute an integral part of modern banking.
Without probability theory, CPM would never have reached today’s
level of sophistication in terms of structured credit products, credit
derivatives, transfer pricing mechanisms, rating systems, portfolio
models, etc.
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2. Further (Graphical) Examples
Loss distribution of credit portfolios:
Markov term structure of default probabilities:
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Comonotonic approximations (noise reduction, simulation speed):
See B and Overbeck: Comonotonic Default Quote Paths for Basket Evaluation, to appear in RISK, August (2005)
Many more examples can be found in the literature:
see www.defaultrisk.com
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