Incremental Risk Charge - Credit Migration Risk

Bridging market & credit risk: Modelling the
Incremental Risk Charge

Credit Migration Risk Modelling
Johannes Rebel, Nykredit Bank

Agenda

• Scope
IRC proposal
Model requirements
Assumptions
Model outline
• Data
1-year transition matrix
Rating modifiers
Characteristics
Low default rates
Through-the-cycle vs. point-in-time
Internal vs. external ratings
Special issues
Incremental Risk Charge - Credit 2
migration risk modelling - Johannes Rebel

Agenda - II

• Mathematical Interpretation
Modified transition matrix
Markov property
Speed and direction
Time-(in)homogeneity
Generators
• Pricing
Risk-neutral measure
Calibration to the market
Generator-based simulation
Jumps
Jump-diffusion model
Pricing credit correlation products

Agenda - III

• Models
Merton model
Multi-factor Merton
Correlation and credit contagion
Brownian bridge
Model outline - revisited
• Back-tests

Disclaimer: The views expressed in this material are those of the author
and do not necessarily reflect the position of Nykredit


IRC – Credit Migration Risk

• ”Credit Migration Risk. This means the potential for
direct loss due to internal/external ratings
downgrade or upgrade as well as the potential for
indirect losses that may arise from a credit migration
event”


IRC – Model requirements

• ”…an estimate of the default and migration risks of
unsecuritised credit products over a one-year capital
horizon at a 99.9% confidence level, taking into
account the liquidity horizons of individual positions
or sets of positions.”
• ”Soundness standard comparable to IRB”
• “..achieve broad consistency between capital charges
for similar positions (adjusted for illiquidity) held in
the banking and trading books”
• ”Constant level of risk” (optional)
• ”Clustering of default and migration events”
• ”Reflect issuer and market concentration”
• ”Significant basis risks … should be reflected”

IRC – Correlation assumptions

• “Correlation assumptions must be supported by
analysis of objective data in a conceptually sound
framework. If a bank uses a multi-period model to
compute incremental risk, it should evaluate the
implied annual correlations to ensure they are
reasonable and in line with observed annual
correlations. A bank must validate that its modelling
approach for correlations is appropriate for its
portfolio, including the choice and weights of its
systematic risk factors.“

• ... the IRA,B&C of risk modelling!


Assumption – Liquidity buckets

• Denote all positions or sets of positions (the IRC
model positions) at time τ by Πτ.
• Today is τ=0 and the capital horizon (one year) is
denoted Τ
• All positions in Π0 have been assigned to liquidity
“buckets”
• The buckets have sizes equal to integer multiples of,
say, 1 month
• The model times is the discrete set

{to , t1 ,K, t N −1 , t N } where 0 = to < t1 < K < t N −1 < t N = T


Assumption - Constant level of risk

• Denote the universe of all obligors to be modelled by
Ο
• Each obligor οi has (internal/external) rating Ri,τ,
where Ri,0 is known

• The constant level of risk assumption is considered a
trading strategy, Σ, so that for each time
Π t i +1 = Σ (Π t i , R t i +1 )
• and at intermediate times
Π t = Π t i , for t ∈ [t i , t i + 1 )
• The trading strategy doesn’t change positions before
the end of their respective liquidity horizons


Assumption - Pricing

• There are pricing models for all IRC model positions
that calculate prices given
– Current time
– Current ratings (or full path)
• The models need to be calibrated to rating transition
probabilities under the pricing measure as well as
relevant market data
• Note that there is only a limited number of discrete
times and ratings so for most positions (not path-
dependent) there is a limited number of prices
• Note also that we need a P&L with the isolated credit
migration effect (exclude the effect of the passage of
time)


IRC – Model outline - 1st attempt

1. Define IRC model positions - Π0
2. Assign to liquidity buckets
3. Starting at t=t0 for each time t=ti
• Simulate stochastic process (to be defined) for the
whole universe of obligors until t=ti+1
• Mark all positions to model using current time and
ratings
• Calculate P&L
• Rebalance according to trading strategy (constant
level of risk)
• Redo until t=Τ
4. Redo step 3 “10000” times
5. Calculate 99.9% quantile of P&L distribution


IRC - Model considerations

• Credit migration risk should be treated both under
the objective/empirical measure and the risk-
neutral/pricing measure
• When marking-to-model under the risk-neutral
measure we need rating transitions in continuous
time
• We must model dependencies between rating
transitions at issuer level and under both probability
measures!
• The model (and annual implied correlations in
particular) should be broadly consistent with the IRB


Transition rates data – External
ratings for corporate issuers

• Annually updated (external) issuer ratings transition rates
available from the credit rating agencies
• Methodologies and rating systems are broadly similar across
agencies

S&P Corporate Global Average Transition Rates, 1981-2007 (%), One year

From/To AAA AA A BBB BB B CCC/C D NR

AAA 88.53 7.70 0.46 0.09 0.09 0.00 0.00 0.00 3.15

AA 0.60 87.50 7.33 0.54 0.06 0.10 0.02 0.01 3.84

A 0.04 2.07 87.21 5.36 0.39 0.16 0.03 0.06 4.67

BBB 0.01 0.17 3.96 84.13 4.03 0.72 0.16 0.23 6.61

BB 0.02 0.05 0.21 5.32 75.62 7.15 0.78 1.00 9.84

B 0.00 0.05 0.16 0.28 5.92 73.00 3.96 4.57 12.05

CCC/C 0.00 0.00 0.24 0.36 1.02 11.74 47.38 25.59 13.67


Transition rates data – rating
modifiers

• Data for rating modifiers (A+, A, A- etc.) are
also available
From/To AAA AA+ AA
AAA 88.53 4.29 2.78
AA+ 2.45 77.90 …
AA 0.54 … …

• Conveys more information about the sample
• But many more rare events – poor estimates of
”true” probabilities*

*see [CL02]


Transition rates data -
characteristics

• “Sudden defaults” do happen - jumps
• Ratings transitions exhibit mean reversion - A-/B-
rated issuers are twice as likely to be down-/up-
graded one notch than to be up-/down-graded one
notch
• Rating volatility is higher for lower rated issuers and
vice versa
• Ratings are effective indicators of relative default
risk


Transition rates data – low default
rates

• Basel minimum probability of default (PD) of 0.03%

• Could use linear regression on a logarithmic scale*

*see [BOW03] sect. 2.7

Transition rates data – Moody’s
database

• The complete transition history has been studied
([CL02]) using a continuous-time procedure
• Significantly improved confidence sets for rare
events

Drawbacks
• Probably not available for commercial purposes!
• Various issues have to be dealt with on a case-by-
case basis
– Special covenants
– Several transitions over a very short time-span e.g. B1
to Caa to D (interpreted as B1 directly to D)
– New debt issued after default

Through-the-cycle vs. Point-in-
time

• Agencies’ average rating history data generally
accepted to be through-the-cycle (TTC) – i.e. rating
transitions have been recorded during all phases of
the macroeconomic cycle
• Point-in-time data pertain to a specific point in time
and reflect the state of the economy at that time
• Some argue (e.g. Calyon in response to the IRC)
that TTC data are more in line with Basel II banking
book parameters and that the PIT data are too
volatile
• Others argue (ISDA among others) that the TTC data
could be too conservative


An approach to assigning internal
ratings

• 8 steps to assigning internal ratings to obligors*

• 7 steps to assign an Obligor Default Rating (ODR)
that identifies the probability of default and a final
step to (independent of the ODR) assign a Loss
Given Default Rating (LGDR) that identifies the risk
of loss in the event of default.

• It’s important that the rating categories are not too
broad, so that the obligors do not get clustered in a
few categories.

*[CGM06] chap. 10

8 steps to an internal rating

1. Financial assessment – financial reports, capital
markets, competitive position etc. This will put a
limit on the up-side.
2. Qualitative factor – management, day-to-day
operations etc.
3. Industry, industry/regional position
4. Financial statement quality
5. Country risk – cross-border restrictions etc.
6. Comparison to external ratings
7. Loan structure – covenants, term of the debt etc.
8. LGDR – collateral, risk mitigants etc.


Internal vs. External ratings

• Internal ratings have a number of advantages:
– adapt faster to changing economic conditions (external
ratings tend to lag)
– should be easier to calibrate pricing models
– all obligors can be rated

• Drawbacks
– rating triggers are triggered by external ratings (could
assume a simple relationship e.g. time-lag)


Special issues

• Lower-rated sovereign debt – external ratings
available
• Global averages –need to match the issuers in the
trading book
• Some issues are rated lower (or higher) than the
issuer – assume some simple rule, e.g. constant
notch spread


Mathematical Interpretation

25

The modified transition matrix

The transition data have to be modified a bit to be
useful for modelling purposes:
• The NR probabilities get redistributed to the other
categories(*) assuming that they convey ”no
information”
• A default (D) row is added to make the matrix
square
• Each row gets rescaled to 1 (100%) to iron out minor
inaccuracies

The result is the Modified Transition Matrix

* Redistribution to other rating categories is likely to inflate the default risk
according to Standard and Poor’s ([SP07])

The modified transition matrix –
Markov formalism

Consider a finite state space S S = {1,2,..., K }

and the map from rating categories to S {AAA,..., D} → {1,2,...,8}

The modified transition matrix
 p11 K p1K 
• is square  
P= M O M 
• has non-negative elements p
 K1 K pKK 

• each row sums to 1

These properties characterise a (right) pij = Pr{ t +1 = j | ηt = i}
η
stochastic matrix, which describes a
(stationary) Markov chain, ηt, over S.


Speed and direction of migrations

[AV08] introduce the concepts of speed and direction
of rating migrations. K −1  
The direction is defined as ∑  j <i
 ∑ pij − ∑ pij 
i =1 

j >i  ∈ [− 1,1]
K −1
and measures the general tendency of ratings to drift
upwards or downward, typically during economic
expansions and contractions respectively
K K

The speed is defined as
∑∑ | i − j | p
i =1 j =1
ij

K −1

∑k
k =1

and measures the speed at which ratings jump –
weighted by jump size

Constructing PIT migration
matrices

• [AV08] model the point-in-time default probabilities
as expectation of the PD conditional upon the state
of the economy, which in turn is modelled by a single
macroeconomic factor – the CFNAI index (*)
• The resulting PIT matrix exhibits over time frequent
changes in both direction and speed compared to its
TTC-counterpart (solid and dashed lines resp.)

* Other indexes or multiple indexes could also be used


The Markov assumption

• The initial rating states are the ratings at the
beginning of each year
• There is no information on the previous years, so
data are ”born” Markovian
• Note that the Markov assumption is tied to the
definition of states rather than to the behaviour of
ratings per se. With a full data set at our disposal,
rating categories like ”BBB(by upgrade)” or ”AA(by
downgrade” could be defined and transition
probabilities be estimated while the transition
process would ”still” be Markovian.


Time-homogeneity

• The one year transition matrix implies (by
assumption of time-homogeneity) multiple years’
transition matrices

Pn = Pn −1 P, P0 = I for any n ∈ Ν
• But there is no simple extension of this rule to
intermediate periods – the square root of a matrix
for example is not unique


Generator – Derivative of the
transition probability matrix

• We can use the derivative of P instead!
• Note that (Chapman-Kolmogorov equation)
K
Ps + t (i, j ) = ∑ Ps (i, k ) Pt (k , j )
k =1

• - we have to pass one of the K states on our way
from state i at time 0 to state j at time t+s
• Differentiate with respect to s
K
P (i, j ) = ∑ Ps' (i, k ) Pt (k , j )
'
s +t
k =1

• and set s=0
K
Pt (i, j ) = ∑ P0' (i, k ) Pt (k , j )
'

k =1


Generator - Solution to the
Kolmogorov backward equation

• Define the generator matrix as G = P0'
• Then we can write

Pt ' = GPt (Kolmogorov backward equation)
• which is a matrix (ordinary) differential equation with
boundary condition
P0 = I

• In the scalar case this would have been solved by
exp(tG) …


The matrix exponential - Definition

The matrix exponential is a matrix function on square
matrices that is defined as

Xk ∞
∀X ∈ ℜ n×n : exp( X ) ≡ ∑
k = 0 k!

for any positive integer n. This is just the Taylor series
expansion (around 0) of exp(x).
Note that for n>1 in general:
 x11 L x1n   exp( x11 ) L exp( x1n ) 
   
exp( X ) = exp M O M  ≠  M O M 
 x L x   exp( x ) L exp( x ) 
 n1 nn   n1 nn 

Except in rare circumstances, e.g. X a diagonal matrix.


Generator matrix properties

• For P=exp(tG) to be a stochastic matrix for all t, G
has to satisfy
1. 0 ≤ − g ii < ∞ for all i
2. g ij ≥ 0 for all i ≠ j
3. ∑ g ij = 0 for all i
j

• Any matrix satisfying having the above properties is
a generator matrix
• The true generator need not exist! (embedding
problem)
• and it need not be unique!!
• The set of admissible P is larger than the set of
exp(tG) for admissible G


The matrix exponential -
Calculation

• The naïve approach to calculating the matrix
exponential is just to calculate the truncated sum
N
Xk
exp( X ) ≈ ∑ for N sufficiently large
k =0 k!
• Unfortunately it is not numerically stable (adding
large quantities with opposite signs), but the
following diagonal adjustment overcomes this
problem*:
• Choose x = max{| xii |: i = 1,..., K}
• and note that (since xIK and X commute)
exp( xI K + X ) = exp( xI K ) exp( X ) ⇒ exp( X ) = exp(− x) exp( xI K + X )
• then the last exponent has all elements positive
*see [Lando04] Appendix C


The matrix exponential - solution
to an ODE

The matrix exponential, exp(tX )
solves an ordinary differential equation (ODE)
d
y (t ) = Xy (t )
dt
with boundary condition y (0) = I (identity matrix).

So exp(tX ) = y (t )
and thus exp( X ) = y (1)

This property implies an alternative to calculating the
infinite sum: solve the ODE by numerical methods


The matrix logarithm

• But how do we find the generator G itself ? Taking
the logarithm ? Almost …
∞
(P − I )k
G = ∑ (−1) k +1

• [IRW01] suggest k =1 k
• which corresponds to the series expansion of the
logarithm function in the scalar case.
• Unfortunately property 2 could be violated so an
adjustment is necessary
• Best practise is to add back negative values to their
row neighbours in proportion to their absolute values


The generator matrix in action

Survivalces
probabilities (1-default probability) generated for each
rating class over a 30-year horizon. Data and methodology as
in [JLT97].


The generator matrix in action - II

Survival probabilities generated over a 12-month horizon


Risk-neutral probabilities

• Until now all probabilities have been empirical
(objective probability measure, P)

• For pricing we need risk-neutral probabilities
(equivalent martingale measure, Q)

• The difference is the market price of risk


Risk-neutral transition matrix or
generator

• General idea: calibrate risk premia to market data
using time and state dependent factors

• Transform the transition matrix by
q ij ( t , t + 1) = π ij ( t ) p ij

• or the generator matrix by
~
G (t) = U(t)G
• Note that the process under Q need not be
Markovian nor time-homogenous, but is usually
assumed to be at least Markovian


Calibration to market data

• Determine U from e.g. credit spreads or CDS
spreads
• In their seminal paper [JLT97] use time and state-
dependent factors
U(t) = diag µ1(t),K, µK−1(t),1)
(
• where each row in the generator matrix is scaled up
by a risk premium – increases the transition
intensities so that the drift towards default is
accelerated
• Due to mean reversion lower rated debt could get
lower credit spreads!


Generator – Probabilistic
interpretation

Interpretation of generator matrix elements
• off-diagonal (g(i,j), where i<>j) elements are
intensities of independent Poisson processes of
transition from state i to state j.
• Diagonal elements (g(i,i)) are the negatives of
arrival intensities to any state other than i

This interpretation suggests how to simulate the rating
process ([Jones03]):


Generator – Simulation I

• Start from state i at t = 0
• Draw a uniform [0, 1] random variable u.
• Time to (first) transition from state i is computed as
t = LN (u / g ij )
• This is an exponentially distributed random variable
with mean −1/g(i,j).
• and it is the time between arrivals in a Poisson
distribution with intensity g(i,j).


Generator – Simulation II

• Given that a move has occurred, the probability that
the move is to state j (<>i) is
K
qij = ∑1{k ≠i} q ik
k =1

• Partition the unit interval into subintervals of these
lengths for all j<>i
• To determine which state the transition is to, now
draw another uniform random variable v. The
subinterval in which it falls gives the next state j.


Generator – Simulation III

• If the new state is default (and that is absorbing), or
if transition date exceeds the horizon T, this path is
done
• Otherwise update t, return to first step, and draw
the next transition time.


Alternative approach - Translated
asset value process

• [ML00] translate the true distribution (normal) of
asset returns by a risk premium
ρθ
• where ρ is the correlation with the market (CAPM).
• Denoting the risk-neutral probabilities by qij we get
pij = P{b j < R < b j +1} = Q{b j < R + ρθ < b j +1}, where qij = Q{b j < R < b j +1}

• where − ∞ = b1 < K < bK +1 = ∞
• define interval boundaries (thresholds)


Jumps - Motivation

• Introduces fat tails and skews
• Can model short-term transition probabilities much
more realistically than pure diffusion processes –
allow sudden defaults
• Can give a much better fit to term structures of
credit spreads
• Interpreted as lack of information- incomplete
accounting information
• Downgrades and defaults tend to cluster, not
upgrades
• Jumps in rating could also reflect contagion effects


Modelling rating transitions – jump
diffusion model

• [ML00] introduce a mean-reverting jump-
diffusion process for the 1-year default probability
• The parameters
– diffusion volatility
– mean-reversion level and speed
– Jump intensity, size and standard deviation
– rating thresholds (7)
– market risk premium
• The parameters get calibrated to both historical
transition data, multi-year cumulative default
probabilities and credit spreads


Generator - Estimation

• [CL02] use Maximum-Likelihood estimators to
estimate the generator directly (from Moody’s
database)

• Then they use simulation to arrive at confidence sets
for default probabilities


Time-heterogeneity

• [BO07] use a time-dependent generator to fit to
multi-year default probabilities

• The time-homogeneity property is sacrificed to
obtain a better fit to the whole term structure of
default probabilities


Pricing correlation products

• Correlations under the risk-neutral measure

• Have to use fairly simple factor models - not much
information in the market*

• Or use copulas - an abundance of literature exists on
copula approaches**

*see [ILS09]
**see [CLV04] chap. 7


The Merton model and beyond ...

55

The Merton model

• The asset value process is assumed to follow a geometric
Brownian motion

dVt = µV Vt dt + σ V Vt dWt , 0 ≤ t ≤ T
• The value of the firm’s equity is equivalent to a call option
on the assets with the strike rate set to the face value of
the debt at Τ

[
S t = Ε Q e − r (T − t ) (VT − B ) | Ft
+
]
• Classic extensions of the model include
– Stochastic interest rates
– Jumps
– Default barrier
– Less simplistic capital structure incl. coupons


Incorporating rating transitions

• Incorporating rating levels is easy, define thresholds
− ∞ = bK +1 < bK < K < b1 < b0 = ∞
• so that making the transition from the current rating,
i, to rating, j, is

pij = Ρr {b j < VT ≤ b j +1 }

• A choice has to be made whether default is only
recognised at Τ or at any intermediate time (default
barrier/first passage)


The multi-factor Merton Model

• Asset value log-returns of m obligors over a given
horizon Τ is  V T 
ln 
 V  = ri = β i Φ i + ε i , for i = 1, K , m

 0 
• where Φi is called the composite factor of obligor i
(weighted sum of several factors)
• βi captures the linear correlation of ri and Φi and εi is
a residual – analogous to the CAPM
• The formula represents a division into systematic
and specific risk
• The Φi and εi are all assumed to be independent, so
that the returns are exclusively correlated by means
of their composite factors
• The returns are then independent conditional upon
the realisation of the composite factors!

Composite factors - breakdown

• The composite factors are composed of industry-
and country-specific factors, Ψk, with corresponding
weights
K
Φ i = ∑ wi ,k Ψk , for i = 1, K, m
k =1

• The industry- and country-specific factors in turn
are represented by a weighted sum of independent
global factors
N
Ψk = ∑ bk ,n Γn + δ k , for k = 1, K , K
n =1

• The independent global factors are obtained from a
principal components analysis (PCA) of the
industry- and country-specific factors

Credit contagion - I

• Conditional independence framework usually leads to
default correlations between obligors that are too
low to explain large portfolios losses*
• Should deal with asymmetrical dependencies –
counterparty relations
• Intrinsic risk that cannot be diversified away!
• Could maybe be ignored for large retail credit
portfolios, but what about the trading book ?

* see [Lüt09] chap. 12


Credit contagion - II

• [RW08] present a model that divides obligors into
infecting and infected firms (e.g. a large corporation
and its suppliers)

• Defaults in the infecting group feed into the
creditworthiness of infected firms by increasing the
default probability

• Contagion channels within business sectors

• Finding: most infecting firms are investment grade
and most infected firms speculative grade


Existing models

• Many IRB models based on the Merton model have
been implemented
• Several commercial products available (Moody’s
KMV, CreditMetrics™ etc.)
• Focus is on default risk but ratings can usually be
handled
• A lot of time and effort has gone into modelling joint
annual default probabilities
• Most are based on one-year horizons and not all are
easily adaptable to a multi-period setting*

*see [Straumann09]


Brownian bridge – I

• The Brownian bridge is a method to construct a path
of a Brownian motion between known end points*

• Bridging market and credit risk ? Use a Brownian
bridge!
*see [Jäckel02] sect. 10.8.3


Brownian bridge - II

• The idea is to run a simulation very similar to the
simulation inherent in most IRB models – over a
one-year capital horizon
• For every realisation of the asset values at T create a
Brownian bridge connecting the start and end values
of the asset process so that we get to “know” the
asset values at all intermediate times
• This approach will ensure “broad consistency” with
the IRB
• Note that if default is only recognized at the capital
horizon Τ and if the firm is not in default at time Τ,
we have to reject paths that indicate default at τ<Τ


Model outline - revisited

1. Define IRC model positions - Π0
2. Assign to liquidity buckets
3. Simulate composite factors
1. Simulate asset values for all assets (obligors) at t=T
and create Brownian bridge.
2. Starting at t=t1 for each time t=ti
1. Mark all positions to model using current time and
ratings
2. Calculate P&L due to credit migration
3. Rebalance according to trading strategy (constant level
of risk)
4. Redo until t=Τ
3. Redo “1000000” times
4. Redo from step 3 “1000” times
5. Calculate 99.9% quantile of P&L distribution

Back-test

• Need to find a trading strategy to match your way of
trading or test at shorter capital horizons
• Need to attribute part of the P&L to credit migrations
• ... and many other issues!

• “Owing to the high confidence standard and long capital
horizon of the IRC, robust direct validation of the IRC
model through standard backtesting methods at the
99.9%/one-year soundness standard will not be possible.
Accordingly, validation of an IRC model necessarily must
rely more heavily on indirect methods including but not
limited to stress tests, sensitivity analyses and scenario
analyses, to assess its qualitative and quantitative
reasonableness, particularly with regard to the model’s
treatment of concentrations”

References - I

• [JLT97] - ”A Markov Model for the Term Structure of Credit Risk Spreads”, Robert
A. Jarrow, David Lando, Stuart M. Turnbull, The Review of Financial Studies
summer 1997 Vol. 10, No. 2, pp. 481-523
• [CGM06] – ”The essentials of risk management”, Michel Crouhy, Dan Galai, Robert
Mark, McGraw-Hill Companies, Inc.
• [AV08] – ”Credit Migration Risk Modelling”, Andreas Andersson, Paolo Vanini,
2008
• [SP07] – ”2007 Annual Global Corporate Default Study And Rating Transitions”,
Standard and Poor’s, February 5, 2008
• [CL02] – ”Confidence sets for continuous-time rating transition probabilities”, Jens
Christensen, David Lando, 2002
• [BOW03] – ”An introduction to Credit Risk Modelling”, Christian Bluhm, Ludger
Overbeck, Christoph Wagner, Chapman & Hall/CRC 2003
• [Lando04] – ”Credit Risk Modelling”, David Lando, Princeton University Press,
2004
• [IRW01] – ”Finding Generators for Markov Chains via Empirical Transition
Matrices, with Application to Credit Ratings”, Robert B. Israel, Jeffrey Rosenthal,
Jason Z. Wei, Mathematical Finance, 11 (April 2001)
• [BO07] – ”Calibration of PD term structures: to be Markov or not to be”, Christian
Bluhm, Ludger Overbeck, RISK magazine, November 2007


References - II

• [Jones03] – ”Simulating Continuous Time Rating Transitions”, Robert A. Jones,
2003
• [Merton74] – ”On the Pricing of Corporate Debt: The Risk Structure of Interest
rates”, Robert C. Merton, Journal of Finance, 2, 449, 470
• [ML00] – ”Modeling Credit Migration“, Cynthia McNulty, Ron Levin, RISK
magazine, February 2000.
• [RW08] – ”Estimating credit contagion in a standard factor model“, Daniel Rösch,
Birker Winterfeldt, RISK magazine, August 2008.
• [ILS09] – ”Factor models for credit correlation”, Stewart Inglis, Alex Lipton, Artur
Sepp, RISK magazine, April 2009
• [Lüt09] – ”Concentration Risk in Credit Portfolios”, Eva Lütkebohmert, Springer-
Verlag, 2009
• [Jäckel02] – ”Monte Carlo methods in finance”, Peter Jäckel, John Wiley & Sons
Ltd. 2002
• [Straumann09] - “What happened to my correlation?”, On the white board,
Daniel Straumann, 2009
• [CLV04] - “Copula methods in finance” , Umberto Cherubini, Elisa Luciano, Walter
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