Presentation

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Presentation

Predictions based on certain uncertainties –

a Bayesian credit portfolio approach

Christoff Gössl

München, 03.03.2005

HVB C&M Fixed Income Portfolio Group

Dr. Christoff Gössl · Tel.: +44 20 7634 3923

. 03/06 – 1


Outline

Portfolio model and historic data

A Bayesian portfolio model

Application to Standard & Poor’s default data

Predictions

Summary and Outlook

HVB C&M Fixed Income Portfolio Group

Dr. Christoff Gössl · Tel.: +44 20 7634 3923

. 03/06 – 2


Portfolio model and historic data

Aim: Prediction of future loss figures or loss distribution

Means: Portfolio model, historical data

Problems:

Which is the right model to use?

Are the historical data good enough to allow for a reliant prediction?

Space of applicable models has been analysed and scrutinised very

thoroughly; properties have been compared and judged

Various data sources have been used for calibration and predictions,

however qualitative differences between sources make a comparison

almost infeasible

In general, model risk and data problems still play a prominent role in

the credit risk area

HVB C&M Fixed Income Portfolio Group

Dr. Christoff Gössl · Tel.: +44 20 7634 3923

. 03/06 – 3


Portfolio Model

Standard one factor asset value model

Portfolio of N credit risky instruments or assets ai, i = 1, . . . , N

Each asset falls into one of a finite number of risk classes

ri ∈ {R1, . . . , RK}

Each risk class is characterised by its individual one year probability

of default (PD) pRj , j = 1, . . . , K, with the asset PDs pi = pRj for

ri = Rj

The instrument’s asset value process Xi can be written as:

Xi = p 1ρ Y + p 1 − ρ Zi , i = 1, . . . , N ,

ρ ∈ [0, 1], Y ∼ N(0, 1), Zi ∼ N(0, 1) i.i.d.

Asset i defaults if its asset value falls below its liabilities: Xi < ki

HVB C&M Fixed Income Portfolio Group

Dr. Christoff Gössl · Tel.: +44 20 7634 3923

. 03/06 – 4


Portfolio Model

Therefore, for the default probabilities, it holds:

P ( asset i defaults ) = P (Xi < ki) = pi =⇒ ki = Φ −1 (pi) ,

and conditional on the portfolio factor Y this equation becomes

P ( asset i defaults | Y = y ) = P (Xi < ki | Y = y) = pi|y −1

Φ (pi) −

=⇒ pi|y = Φ

√ !

ρ × y


1 − ρ

The joint conditional likelihood of the portfolio’s risk classes is determined

by

P (L1 = l1, . . . , LK = lK| Y = y ) =

KY

j=1

B(nR j , p j|y, lR j ) ,

with L j|Y =y = P

r i =R j 1 Di |Y=y, j = 1, . . . , K, and B(n, p, k) the standard

Binomial distribution

HVB C&M Fixed Income Portfolio Group

Dr. Christoff Gössl · Tel.: +44 20 7634 3923

. 03/06 – 5


Standard & Poor’s default data

Default history for S&P rated corporates

Covering years 1981 to 2004

More than 5000 companies rated in 2004

Data for rating classes AAA to CCC

Problem I: Ratings as published by the agencies constitute an ordinal

ranking of the company’s likelihood to default; i.e. no fixed PDs are

associated with the classes

Problem II: Only one default in AAA and AA classes together

HVB C&M Fixed Income Portfolio Group

Dr. Christoff Gössl · Tel.: +44 20 7634 3923

. 03/06 – 6


Standard & Poor’s default data

%

0 600

0.02 0.06 0.10 0.14 Defaulted Companies Rating B (%)

1980 1985 1990 1995 2000

HVB C&M Fixed Income Portfolio Group

Dr. Christoff Gössl · Tel.: +44 20 7634 3923

Year

# rated Companies B

%

0 600

1980 1985 1990 1995 2000

0.0000 0.0010 0.0020 Defaulted Companies Rating A (%)

Year

# rated B companies # rated A companies

# rated companies A

. 03/06 – 7


Prediction

Predictions based, e.g., on maximum likelihood estimates derived from the assumed

model and historic data

Most applications make default predictions for several years necessary

PD

0.00 0.01 0.02 0.03 0.04


A

AA

AAA

HVB C&M Fixed Income Portfolio Group

Dr. Christoff Gössl · Tel.: +44 20 7634 3923

Historic PD Term Structure

2 4 6 8 10 12 14

Years

PD

0.0 0.1 0.2 0.3 0.4

Historic PD Term Structure

2 4 6 8 10 12 14

Years


B

BB

BBB

. 03/06 – 8


Prediction – Problems

Predictions based, e.g., on maximum likelihood estimates derived from the assumed

model and historic data

Most applications make default predictions for several years necessary

PD

0.00 0.01 0.02 0.03 0.04

Historic and ’Assumed’ PD Term Structure


A

AA

AAA

2 4 6 8 10 12 14

Years

0.0 0.1 0.2 0.3 0.4

Historic and ’Assumed’ PD Term Structure

2 4 6 8 10 12 14

Because of scarce data situation, frequently regularity or smoothness assumptions have

to be applied

In general, parameter estimates exhibit quite large standard errors and sensitivities to

changes in the data basis

HVB C&M Fixed Income Portfolio Group

Dr. Christoff Gössl · Tel.: +44 20 7634 3923

PD

Years


B

BB

BBB

. 03/06 – 9


Outline

Portfolio model and historic data

A Bayesian portfolio model

Application to Standard & Poor’s default data

Predictions

Summary and Outlook

HVB C&M Fixed Income Portfolio Group

Dr. Christoff Gössl · Tel.: +44 20 7634 3923

. 03/06 – 10


The Bayesian Approach

Bayes’ Theorem

Why Bayes?

p(β|data) =

p(data, β)

p(data)

= p(data|β) p(β)

R p(data, β) dβ ,

Established models for latent variable approaches and measurement

error settings

Bayesian methods allow for the formulation and evaluation of complex

hierarchical models and dependence structures

Use of non-informative prior distributions and use of the posterior mode

as estimator yields ML estimates

Estimation via Markov chain Monte Carlo (MCMC) methods

=⇒ posterior distribution sample available

HVB C&M Fixed Income Portfolio Group

Dr. Christoff Gössl · Tel.: +44 20 7634 3923

. 03/06 – 11


The Bayesian Approach

Requirements for a Bayesian credit portfolio model:

Likelihood function

Prior distributions of parameters involved

Likelihood function of the asset value model and the observed default

data L = (Lt,1, . . . , Lt,K)t=1,...,T :

P (L = l | n, p, ρ, y ) =

TY

t=1

KY

j=1

with pyt = (p 1|yt , . . . , p K|yt )′ and p j|y = Φ

B(nt,R j , p j|yt , lt,R j ).

−1

Φ (pRj )− √

ρ×y


1−ρ

Use of non-informative prior distributions for the remaining parameters:

HVB C&M Fixed Income Portfolio Group

Dr. Christoff Gössl · Tel.: +44 20 7634 3923



pRj ∼ U(0, 1), j = 1, . . . , K i.i.d.,

ρ ∼ U(0, 1)

yt ∼ N(0, 1), t = 1, . . . , T i.i.d.,

. 03/06 – 12


The Bayesian Approach –

The posterior distribution

The joint posterior distribution can be written down as follows:

p(p, ρ, y | n, l) =

with the above definitions

Q T

t=1 P (Lt = lt | nt, py t , ρ, yt ) p(p)p(ρ)p(y)

P (L = l | n)

Other, more complex parameterisations easily applicable

Another, more general Bayesian approach: McNeil and Wendin (2005),

”Bayesian Inference for GLMMs of Portfolio Credit Risk”

HVB C&M Fixed Income Portfolio Group

Dr. Christoff Gössl · Tel.: +44 20 7634 3923

. 03/06 – 13


Outline

Portfolio model and historic data

A Bayesian portfolio model

Application to Standard & Poor’s default data

Predictions

Summary and Outlook

HVB C&M Fixed Income Portfolio Group

Dr. Christoff Gössl · Tel.: +44 20 7634 3923

. 03/06 – 14


Application to Standard & Poor’s default data

Estimation of parameters for rating classes A to CCC

MCMC algorithm implemented in the BUGS software

(http://www.mrc-bsu.cam.ac/bugs)

Result: Sample of the joint posterior distribution

Posterior mean or posterior modes as standard point estimators

Highest posterior density intervals as analogue to classical confidence

intervals

Lower dimensional subsets of posterior distribution as samples of marginal

densities

HVB C&M Fixed Income Portfolio Group

Dr. Christoff Gössl · Tel.: +44 20 7634 3923

. 03/06 – 15


Results estimation – PD statistics

A BBB BB B CCC ρ

Hist. Mean 0,0004 0,0023 0,0107 0,0561 0,2813

Mean 0.0006 0.0029 0.0123 0.0591 0.2789 0.0941

Std. Dev. 0.0003 0.0010 0.0031 0.0095 0.0257 0.0357

2.5% Qu. 0.0002 0.0016 0.0079 0.0445 0.2328 0.0449

25% Qu. 0.0004 0.0022 0.0102 0.0526 0.2613 0.0690

Median 0.0005 0.0027 0.0118 0.0578 0.2772 0.0873

75% Qu. 0.0007 0.0034 0.0137 0.0642 0.2949 0.1115

97.5% Qu. 0.0014 0.0055 0.0198 0.0817 0.3341 0.1826

Descriptive statistics for the parameter posterior distributions of the

Bayesian Credit Portfolio Model.

HVB C&M Fixed Income Portfolio Group

Dr. Christoff Gössl · Tel.: +44 20 7634 3923

. 03/06 – 16


Results estimation – Marginal posterior densities

BB / ρ

BB / ρ

ρ

0.00 0.05 0.10 0.15 0.20 0.25

0.00 0.01 0.02 0.03 0.04 0.05

4000

KDE

2000

0

0.00

0.02

BB

BB

0.04

0.0

0.1

rho

0.2

BB / B

BB / B

B

0.00 0.02 0.04 0.06 0.08 0.10 0.12

0.000 0.005 0.010 0.015 0.020 0.025

Joint posterior distribution for selected parameters of the Bayesian credit portfolio model.

HVB C&M Fixed Income Portfolio Group

Dr. Christoff Gössl · Tel.: +44 20 7634 3923

10000

KDE

5000

0

0.00

0.01

BB

. 03/06 – 17

BB

0.02

0.00

0.05

0.10

B


Results estimation – Portfolio factor

Y

−1.0 −0.5 0.0 0.5 1.0 1.5

Portfolio Factor

1985 1990 1995 2000

Estimated time series of portfolio factors from the portfolio model

Year




97.5%

Mean

2.5%

Simultaneously estimated with the other model parameters

HVB C&M Fixed Income Portfolio Group

Dr. Christoff Gössl · Tel.: +44 20 7634 3923

. 03/06 – 18


Results estimation – Sensitivities

Data A BBB BB B CCC ρ

Mean 0.0006 0.0029 0.0123 0.0591 0.2789 0.0941

Original Std. Dev. 0.0003 0.0010 0.0031 0.0095 0.0257 0.0357

Median 0.0005 0.0027 0.0118 0.0578 0.2772 0.0873

w/o years Mean 0.0005 0.0024 0.0096 0.0546 0.2542 0.0792

’01 & ’02 Std. Dev. 0.0003 0.0009 0.0026 0.0089 0.0253 0.0343

Median 0.0005 0.0022 0.0091 0.0534 0.2524 0.0718

Descriptive statistics for the parameter posterior distributions of the

Bayesian Credit Portfolio Model.

HVB C&M Fixed Income Portfolio Group

Dr. Christoff Gössl · Tel.: +44 20 7634 3923

. 03/06 – 19


Outline

Portfolio model and historic data

A Bayesian portfolio model

Application to Standard & Poor’s default data

Predictions

Summary and Outlook

HVB C&M Fixed Income Portfolio Group

Dr. Christoff Gössl · Tel.: +44 20 7634 3923

. 03/06 – 20


Predictions

General problem: How could the uncertainty caused by the scarce data

sample be incorporated into predictions?

Because of the availability of a posterior distribution sample, the incorporation

within the Bayesian framework is straightforward

Let us consider homogeneous portfolios for each risk class

ri ∈ {R1, . . . , RK} consisting of an infinite number of assets

For each class for the fraction of defaulting assets lR k it holds:

HVB C&M Fixed Income Portfolio Group

Dr. Christoff Gössl · Tel.: +44 20 7634 3923

lR k (p ∗ , ρ ∗ , y) = Φ

Φ −1 (p ∗ ) − √ ρ ∗ × y

√ 1 − ρ ∗

y ∼ N(0, 1)

I classical approach : p ∗ = ˆpR k , ρ ∗ = ˆρ

II Bayesian approach : (p ∗ , ρ ∗ ) ∼ P (pR k , ρ | n, l, y)

. 03/06 – 21

!

,


Results prediction – Loss distributions

Frequency

Frequency

0 e+00 1 e+05

0 30000

Rating A

0.000 0.002 0.004 0.006

Loss

Rating BBB


− Bayes

classic

0.000 0.005 0.010 0.015 0.020

HVB C&M Fixed Income Portfolio Group

equency

Dr. Christoff Gössl · Tel.: +44 20 7634 3923

20000

Loss

Rating BB


− Bayes

classic


− Bayes

classic

Frequency

Frequency

equency

0 400 1000

0 300 600

200 500

Tails for Rating A

0.000 0.005 0.010 0.015

Loss

Tails for Rating BBB


− Bayes

classic

0.00 0.01 0.02 0.03 0.04 0.05

Loss

Tails for Rating BB


− Bayes

classic

. 03/06 – 22


− Bayes

classic


Results prediction – Shift of quantiles

A BBB BB B

Mean 0.0007 0.0041 0.0143 0.0613

Std. Dev. 0.0010 0.0045 0.0123 0.0377

1% Qu. 0.0000 0.0002 0.0013 0.0100

25% Qu. 0.0002 0.0015 0.0065 0.0354

Median 0.0004 0.0028 0.0110 0.0531

75% Qu. 0.0008 0.0051 0.0182 0.0778

90% Qu. 0.0015 0.0083 0.0281 0.1079

95% Qu. 0.0022 0.0116 0.0365 0.1320

99% Qu. 0.0046 0.0213 0.0609 0.1907

A BBB BB B

Mean 0.0007 0.0041 0.0143 0.0613

Std. Dev. 0.0008 0.0039 0.0112 0.0354

1% Qu. 0.0000 0.0003 0.0016 0.0115

25% Qu. 0.0002 0.0016 0.0066 0.0355

Median 0.0004 0.0029 0.0113 0.0536

75% Qu. 0.0009 0.0052 0.0187 0.0791

90% Qu. 0.0015 0.0086 0.0284 0.1086

95% Qu. 0.0021 0.0114 0.0363 0.1291

99% Qu. 0.0039 0.0184 0.0547 0.1749

Predictions of risk class dependent loss distributions for infinitely granular homogeneous

portfolios using the Bayesian (left) and the classical (right) approach.

Apparent shift of probability mass towards the tail in the Bayesian

approach

Shift becomes more significant for higher rating classes

Results for the CCC class differ only very slightly

HVB C&M Fixed Income Portfolio Group

Dr. Christoff Gössl · Tel.: +44 20 7634 3923

. 03/06 – 23


Results predictions – Shift of quantiles

rel. differrence

rel. differrence

0.6 0.8 1.0 1.2 1.4

0.6 0.8 1.0 1.2 1.4

HVB C&M Fixed Income Portfolio Group

1.2 1.4

Dr. Christoff Gössl · Tel.: +44 20 7634 3923

e

Rating A

0.0 0.2 0.4 0.6 0.8 1.0

Quantiles

Rating BB



Bayes

classic

0.0 0.2 0.4 0.6 0.8 1.0

Quantiles

Rating CCC



Bayes

classic

rel. differrence

rel. differrence

0.6 0.8 1.0 1.2 1.4

0.6 0.8 1.0 1.2 1.4

Rating BBB

0.0 0.2 0.4 0.6 0.8 1.0

Quantiles

Rating B



Bayes

classic

0.0 0.2 0.4 0.6 0.8 1.0

Quantiles



. 03/06 – 24

Bayes

classic


0

0.000 0.005 0.010 0.015

Loss

Results prediction – Pricing

Frequency

Frequency

y

0 300 600

0 200 500

400

Tails for Rating BBB

0.00 0.01 0.02 0.03 0.04 0.05

Loss

Tails for Rating BB

Tails for Rating B


− Bayes

classic

What are the effects for pricing a tranche in a CDO?

BBB portfolio, attachment point 1.90%, tranche thickness 1%

Results for the tranche on a one year horizon:

HVB C&M Fixed Income Portfolio Group

Dr. Christoff Gössl · Tel.: +44 20 7634 3923


− Bayes

classic

Bayes Classic

Prob. 0.00 of Default 0.04

Exp. Loss

0.08 88bps

50bps

0.12 50bps

23 bps

Loss

Loss given Default 57% 46%

. 03/06 – 25


Outline

Portfolio model and historic data

A Bayesian portfolio model

Application to Standard & Poor’s default data

Predictions

Summary and Outlook

HVB C&M Fixed Income Portfolio Group

Dr. Christoff Gössl · Tel.: +44 20 7634 3923

. 03/06 – 26


Summary

Scarce occurrence of default events and limited availability of default

data cause a non-negligible amount of uncertainty in estimated PDs and

portfolio correlation

Variation in the estimators proportional to the lack of information in the

data basis

Predictions are strongly affected by these sensitivities

Bayesian approach is a feasible way to account for the uncertainties

Effects of the data basis should be even stronger for multi-factor models,

or multi-year predictions

HVB C&M Fixed Income Portfolio Group

Dr. Christoff Gössl · Tel.: +44 20 7634 3923

. 03/06 – 27


Outlook

Application of a variety of prior distributions conceivable, e.g., random

walks for the portfolio factor (see also McNeil and Wendin, 2005)

Extension to robust multi-factor / multi-year models

Use of prior information to reduce sensitivities

Use of exogenous variables for predictions

...

HVB C&M Fixed Income Portfolio Group

Dr. Christoff Gössl · Tel.: +44 20 7634 3923

. 03/06 – 28

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