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• 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 migration risk modelling - Johannes Rebel

2

Agenda - II

• Mathematical Interpretation

� Modified transition matrix

� Markov property

� Speed and direction

� Time-(in)homogeneity

� Generators� Generators

• Pricing

� Risk-neutral measure

� Calibration to the market

� Generator-based simulation

� Jumps

� Jump-diffusion model

� Pricing credit correlation products


3

Agenda - III

• Models

� Merton model

� Multi-factor Merton

� Correlation and credit contagion

� Brownian bridge

� Model outline - revisited� Model outline - revisited

• Back-tests


4

Disclaimer: The views expressed in this material are those of the author

and do not necessarily reflect the position of Nykredit

Scope

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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”


6

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”• ”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”


7

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 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!


8

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”“buckets”

• The buckets have sizes equal to integer multiples of, say, 1 month

• The model times is the discrete set


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{ } Ttttttttt NNoNNo =<<<<= −− 1111 0 where,,,, KK

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 • The constant level of risk assumption is considered a trading strategy, Σ, so that for each time

• and at intermediate times

• The trading strategy doesn’t change positions before the end of their respective liquidity horizons


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( )11

,++

ΠΣ=Πiii ttt R

[ )1,for , +∈Π=Π iitt ttti

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 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)


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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 • 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


12

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 timetime

• 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


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Data

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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

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

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


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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 …

• Conveys more information about the sample

• But many more rare events – poor estimates of ”true” probabilities*


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AA 0.54 … …

*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 • Rating volatility is higher for lower rated issuers and vice versa

• Ratings are effective indicators of relative default risk


17

Transition rates data – low default rates

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

• Could use linear regression on a logarithmic scale*


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*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

DrawbacksDrawbacks

• 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


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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 timeand 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


20

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 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.


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*[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 position3. 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.


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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)


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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 issuer – assume some simple rule, e.g. constant notch spread


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Mathematical Interpretation

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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 • 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


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* 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

and the map from rating categories to S

The modified transition matrix

• is square

=

Kpp

P MOM

K 111

{ } { }8,...,2,1,..., →DAAA

},...,2,1{ KS =

• is square

• has non-negative elements

• each row sums to 1

These properties characterise a (right) stochastic matrix, which describes a (stationary) Markov chain, ηt, over S.


27

=

KKK pp

P

K

MOM

1

{ }ijp ttij === + ηη |Pr 1

Speed and direction of migrations

[AV08] introduce the concepts of speed and direction of rating migrations.

The direction is defined as

and measures the general tendency of ratings to drift upwards or downward, typically during economic

[ ]1,11

1

1−∈

−

−∑ ∑∑

−

= ><

K

ppK

i ij

ij

ij

ij


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upwards or downward, typically during economic expansions and contractions respectively

The speed is defined as

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

∑

∑∑−

=

= =

−

1

1

1 1

||

K

k

K

i

K

j

ij

k

pji

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 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


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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 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.


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Time-homogeneity

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

• But there is no simple extension of this rule to

Ν∈== − nIPPPP nn anyfor, 01

• But there is no simple extension of this rule to intermediate periods – the square root of a matrix for example is not unique


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Generator – Derivative of the transition probability matrix

• We can use the derivative of P instead!

• Note that (Chapman-Kolmogorov equation)

• - 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

∑=

+ =K

k

tsts jkPkiPjiP1

),(),(),(

from state i at time 0 to state j at time t+s

• Differentiate with respect to s

• and set s=0


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∑=

+ =K

k

tsts jkPkiPjiP1

'' ),(),(),(

∑=

=K

k

tt jkPkiPjiP1

'

0

' ),(),(),(

Generator - Solution to the Kolmogorov backward equation

• Define the generator matrix as

• Then we can write

• which is a matrix (ordinary) differential equation with boundary condition

'

0PG =

equation)backwardv(Kolmogoro'

tt GPP =

boundary condition

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


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IP =0

The matrix exponential - Definition

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

for any positive integer n. This is just the Taylor series

∑∞

=

× ≡ℜ∈∀0 !

)exp(:k

knn

k

XXX

for any positive integer n. This is just the Taylor series expansion (around 0) of .

Note that for n>1 in general:

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


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)exp(x

≠

=

)exp()exp(

)exp()exp(

exp)exp(

1

111

1

111

nnn

n

nnn

n

xx

xx

xx

xx

X

L

MOM

L

L

MOM

L

Generator matrix properties

• For P=exp(tG) to be a stochastic matrix for all t, G has to satisfy

1.

2.

3.

• Any matrix satisfying having the above properties is

i allfor 0 ∞<−≤ iig

ji allfor 0 ≠≥ijg

∑ =j

ijg i allfor 0

• 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


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The matrix exponential -Calculation

• The naïve approach to calculating the matrix exponential is just to calculate the truncated sum

• Unfortunately it is not numerically stable (adding large quantities with opposite signs), but the

largelysufficientNfor!

)exp(0

∑=

≈N

k

k

k

XX

large quantities with opposite signs), but the following diagonal adjustment overcomes this problem*:

• Choose

• and note that (since xIK and X commute)

• then the last exponent has all elements positive


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},...,1|:max{| Kixx ii ==

)exp()exp()exp()exp()exp()exp( XxIxXXxIXxI KKK +−=⇒=+

*see [Lando04] Appendix C

The matrix exponential - solution to an ODE

The matrix exponential,

solves an ordinary differential equation (ODE)

with boundary condition (identity matrix).

)()( tXytydt

d=

Iy =)0(

)exp(tX

with boundary condition (identity matrix).

So

and thus

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


37

)1()exp( yX =

Iy =)0(

)()exp( tytX =

The matrix logarithm

• But how do we find the generator G itself ? Taking the logarithm ? Almost …

• [IRW01] suggest

• which corresponds to the series expansion of the logarithm function in the scalar case.

k

IPG

k

k

k )()1(

1

1 −−=∑

∞

=

+

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


38

The generator matrix in action

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


39

The generator matrix in action - II


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Survival probabilities generated over a 12-month horizon

Pricing

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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


42

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

ijijij ptttq )()1,( π=+

• or the generator matrix by

• Note that the process under Q need not be Markovian nor time-homogenous, but is usually assumed to be at least Markovian


43

ijijij

U(t)G (t)G~

=

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

• where each row in the generator matrix is scaled up

)1),(,),((diag)( 11 tttU K−= µµ K

• 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!


44

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 iarrival intensities to any state other than i

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


45

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

• This is an exponentially distributed random variable

)/( ijguLNt =

• 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).


46

Generator – Simulation II

• Given that a move has occurred, the probability that the move is to state j (<>i) is

• Partition the unit interval into subintervals of these lengths for all j<>i

∑=

≠=K

k

ikikij qq1

}{1

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.


47

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.


48

Alternative approach - Translated asset value process

• [ML00] translate the true distribution (normal) of asset returns by a risk premium

• where is the correlation with the market (CAPM).

• Denoting the risk-neutral probabilities by we get

ρθρ

ijq

• where

• define interval boundaries (thresholds)


49

{ }111 where},{}{ +++ <<=<+<=<<= jjijjjjjij bRbQqbRbQbRbPp ρθ

∞=<<=∞− +11 Kbb K

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 spreadscredit 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


50

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– 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


51

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 probabilitiesfor default probabilities


52

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 probabilitiesdefault probabilities


53

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


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*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

• 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

TtdWVdtVdV ttVtVt ≤≤+= 0,σµ

the debt at Τ

• Classic extensions of the model include

– Stochastic interest rates

– Jumps

– Default barrier

– Less simplistic capital structure incl. coupons


56

( )[ ]tT

tTrQ

t FBVeS |)( +−− −Ε=

Incorporating rating transitions

• Incorporating rating levels is easy, define thresholds

• so that making the transition from the current rating, i, to rating, j, is

∞=<<<<=∞− + 011 bbbb KK K

{ }1+≤<= jTjij bVbΡrp


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

{ }1+≤<= jTjij bVbΡrp

The multi-factor Merton Model

• Asset value log-returns of m obligors over a given horizon Τ is

• 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

mirV

Viiii

T ,,1for ,ln0

K=+Φ==

εβ

i i i i

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!


58

Composite factors - breakdown

• The composite factors are composed of industry-and country-specific factors, Ψk, with corresponding weights

• The industry- and country-specific factors in turn

miwK

k

kkii ,,1for ,1

, K=Ψ=Φ ∑=

• The industry- and country-specific factors in turn are represented by a weighted sum of independent global factors

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

∑=

=+Γ=ΨN

n

knnkk Kkb1

, ,,1for , Kδ

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!• Intrinsic risk that cannot be diversified away!

• Could maybe be ignored for large retail credit portfolios, but what about the trading book ?


60

* 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 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


61

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 handledhandled

• 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*


62

*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!


63

*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 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


64

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=ti1. Mark all positions to model using current time and 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 distributionIncremental Risk Charge - Credit migration risk modelling - Johannes Rebel

65

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 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”


66

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 • [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


67

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 • [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 Vecchiato, John Wiley & Sons Ltd. 2004


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Incremental Risk Charge - Credit Migration Risk

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