CreditRisk+

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Features of CreditRisk+

• Applies an actuarial science framework to the derivation of the lossdistribution of a bond/loan portfolio.

• Only default risk is modelled, not downgrade risk.

• Default risk is not related to the capital structure of the firm. No assumption is made about the causes of default. An obligor A iseither in default with probability PA and that not in default withprobability 1−PA.

Assumptions on probability of default

The probability distribution for the number of defaults during a givenperiod, is approximated by Poisson distribution

where

µ = average number of defaults over the period

,2,1,0for !

)defaults ( !==−

nn

enP

n µµ

.∑=A

AP

• Stationary assumption on the probability of default.

• Number of defaults that occur in any given period is independent ofthe number of defaults that occur in any other period.

CreditRisk+ risk measurement framework

Input • default rates • exposure• default rates/volatilities • recovery rates

Stage One What is the frequency

of defaults?

What is the severity

of the losses?

Distribution ofdefault losses

Stage Two

Frequency of default events

Credit Rating Average (%) Standard deviation (%)Aaa 0.00 0.0

Aa 0.03 0.1

A 0.01 0.0

Baa 0.13 0.3

Ba 1.42 1.3

B 7.62 5.1

One year default rate

source: Carty and Lieberman (1996)

Rating B: standard deviation of default rate

CreditRisk+ assumes that the mean default rate is Gamma distributed.

1.5 76.262.7 versus=

Distribution of default events

Historical default rates of corporate bond issuers

S en io rit y a n d secu rit y A v era g e S ta n d a rd D eriv a t io nS e n io r s e c u re d b a n k lo a n s 71.18 21.09S e n io r s e c u re d p u b lic d e b t 63.45 26.21S e n io r u n s e c u re d p u b lic d e b t 47.54 26.29S e n io r s u b o rd in a te d p u b lic d e b t 38.28 24.74S u b o rd in a te d p u b lic d e b t 28.29 20.09Ju n io r s u b o rd in a te d p u b lic d e b t 14.66 8.67

1 9 2 0 -1 9 9 6

Severity of the lossesIn CreditRisk+, the exposure for each obligor is adjusted by the anticipated recovery rate in order to produce a loss given default (exogenous to the model).

O b lig o r Exp o s u re Cre d it ra t in g1 358,475 H2 1,082,473 A3 2,124,342 B4 135,423 G5 278,477 D

Example of data set

Credit rating Mean default rate Standard deviation

A 1.50% 0.75%B 1.65% 0.80%C 3.00% 1.50%D 5.00% 2.50%E 7.50% 3.75%F 10.00% 5.00%G 15.00% 7.50%

Division into exposure bandsLosses (exposures, net of recovery) are divided into bands, with thelevel of exposure in each band being approximated by a single number.

Notation

Obligor A

Exposure (net ofrecovery)

LA

Probability of default PA

Expected loss λA = LA × PA

Obligor A Exposure ($) Exposure Round-off Band j(loss given (in $100,000) exposure

default) (in $100,000)

1 150,000 1.5 2 22 460,000 4.6 5 53 435,000 4.35 5 54 370,000 3.7 4 45 190,000 1.9 2 26 480,000 4.8 5 5

jvjv

*L = $100,000

number of obligors1 30 1.5 1.52 40 8 43 50 6 24 70 25.2 6.35 100 35 76 60 14.4 2.4

j∈ jµjv

jv = common exposure in band j in units of L

j∈ = expected loss in band j in units of L

jµ = expected number of defaults in band j

• Expected loss for obligor A in units of

• The expected loss over one-year in band j

.L

L AA

λ==e

∑=∈==∈

jA vvAAj

:

.

• The expected number of default in band j

./ jjj v=∈= µ

• The portfolio is divided into m exposure bands to simplify calculations.

• The expected number of defaults in the portfolio

.1

∑=

==m

jjµµ

Probability generating functionsThe probability generating function of a discrete random variable K is a function of the auxiliary variable z such that the probability that K = nis given by the coefficient of zn in the polynomial expansion of the probability generating function.

• The pgf of the sum K1 + K2 is simply the product of the two pgf’s.

Example

For a single obligor, FA(z) = (1 − PA) + PAz.

For the whole portfolio, .)defaults ()(0

∑∞

=

=n

nznpzF

Probability generating function of the distribution of loss amounts forband j

( ).exp!

)defaults ()loss()(

0

00

jj

j

j

vjj

n

nvnj

nv

n

n

nj

zzn

e

znPznLPzG

µµµµ

+−==

====

∑

∑∑∞

=

−

∞

=

∞

=

Probability generating function for the entire portfolio

( ).exp)(1

∏=

+−==m

j

v

jjjzzG µµ

We then have

.,2,1 ,)(

!

1) of loss(

0

!===

ndz

zGd

nnLP

z

n

n

From ,exp)(1 1

+−= ∑ ∑

= =

m

j

vm

jjj

jzzG µµ

we write ,)(

1

1

∑

∑

=

=

∈

∈

=m

j j

j

m

j

v

j

j

v

zv

zp

j

then .)( ]1)([ −= zpezG µ

Here, µ gives the Poisson randomness of the incidence of default events and p(z) gives the variability of exposure amounts withinthe portfolio.

Correlation in defaults

Two obligors are sensitive to the same set of background factors (with

differing weights), their default probabilities will move together.

These co-movements in probabilities give rise to correlations in

defaults.

•

• Observed default probabilities are volatile over time, even for obligors having comparable credit quality. The variability of default probabilities can be related to underlying variability in a number of background factors, like the state of the economy.

• CreditRisk+ does not attempt to model correlations explicitly but captures the same concentration effects through the use of defaultrate volatilities and sector analysis.

Sector Analysis

• Write Sk, k = 1, …, n for the sectors, each of which should be thought of as a subset of the set of obligors.

• Each sector is driven by a single underlying factor, which explainsthe variability over time in the average total default rate measuredfor that sector.

• The underlying factor influences the sector through the total averagerate of defaults in that sector, which is modeled as a random variablexk with mean µk and standard deviation σk.

Let xk be Gamma distributed with mean µk and standard deviation σk.

.)(

1)()( 1dxxedxxfdxxxxp

x

k−−

Γ==+≤< α

αβ

αβNow,

kx

zp

pdx

xee

dxxfxnp

znpzF

k

kzx

nx

nk

α

α

α

αβ

β

−−=

Γ=

=

=

−−∞ −

∞

=

∞

=

∞

=

∫

∑∫

∑

1

1

)(

)()defaults (

)defaults ()(

1

0

)1(

00

0n

where . and , )1(2

2

2

2

k

kk

k

kk

k

kk p β

β

µσβ

σµα +===

The pgf for default events from the whole portfolio

.1

1)()(

11∏∏

==

−−==

n

k k

kn

kk

k

zp

pzFzF

α

General sector analysisThe default rate of an individual obligor depends on more than onefactors.Sector decompositionAssignment of θAK represents the judgement of the extent to whichthe state of sector k influence the fortunes of obligor A.

.∑∑

=

=

AAAkk

AAAkk

σθσ

µθµ

Pairwise correlation

( ) .

otherwise0

horizon me within tidefaults obligor if1 where),(

1

2

21 ∑

=

=

==

n

k k

kBkAkBAAB

ABAAB

AIII

µσθθµµρ

ρρ

If obligors A and B have no sector in common, then ρAB =0.

Risk contribution

The risk contribution of obligor A having exposure EA is the marginaleffect of the presence of EA on the standard deviation of the distribution of the portfolio credit loss

. 2

∈

+=

∂∂

=

∑k

Akkk

kA

p

AA

A

pAA

EE

EERC

θµσ

σµ

σ

Advantages and limitations of CreditRisk+

• Closed form expressions are derived for the probability of portfolioband/loan losses. Marginal risk contributions by obligor can be easily computed.

• Focuses only on default, requiring relatively few inputs to estimate.

• Assumes no market risk.

• Ignores migration risk so that the exposure for each obligor is fixedand does not depend on eventual changes in credit quality.

• Credit exposures are taken to be constant.

C red itM etr ics C red itR isk +

· M etho d lo lo gy and d ataset · M etho d o lo gy

· C o stly C red itM anager so ftw are · Imp lemented in sp read sheet

· S imulatio n- b ased p o rtfo lio ap p ro ach · A nalytic - b ased p o rtfo lio ap p ro ach

· Based o n p ro b ab ilities o f ra tings · Based o n d efault ra te asso cia ted w ith

transitio ns and co rre la tio ns o f these ra tings and the vo la tility o f these ra tes

p ro b ab ilities

· G enera tes sk ew ed lo ss d istrib utio n · same

fo r ca lcula tio n o f exp ected lo ss,

unexp ected lo ss and risk cap ita l

Extension ITo incorporate the effects of ratings changes in CreditRisk+ (withoutthe need of Monte Carlo simulation as in CreditMetrics).

• Profits and losses are used as net exposures.

• Default rate corresponds to the migration rate.

Reference “Good migrations,” by B. Rolfes and F. Broeker, Risk,(Nov. 1998) p.72-73.

Extension IICorrelated credit events such as defaults can be studied and analyzed ina closed form fashion without the need of simulations.

Data setN: The number of different types of exposures. The type of the

exposure could be based on the rating of the exposure (A or BB, for example), the sector of the exposure (banking or aerospace, forexample), or the geographical region or a combination of all three.

ni: The number of exposures of the ith asset type.eik: The dollar amount of the kth exposure of the ith asset type.pi: The probability of default for assets of ith type.cij: The default correlation between exposures of ith type and jth type.qij: The joint default probability of an exposure of type i and an

exposure of type j. Given the definitions above, the following identity holds:

.)1()1( jjiiijjiij ppppcppq −−+=

If the portfolio is made up of N asset types, then the loss distributionunder correlated defaults can be obtained by combining loss distribution of at most 2N scenarios.

References1. K. M. Nagpal and R. Bahar, “An analytical approach for credit risk

analysis under correlated defaults,” CreditMetrics Monitor (April 1999) p.51-74.

2. K. M. Nagpal and R. Bahar, “Modelling default correlation,”Risk (April 2001) p.85-89.

Review articles

• S. Paul-Choudhury, “Choosing the right box of credit tricks,”Risk (Nov. 1997).

• H. U. Koyluoglu and A. Hickman, “Reconcilable differences,”Risk (Oct. 1998) p.56-62.

• M. Crouchy, D. Galai and R. Mark, “A comparative analysis of current credit risk models,” Journal of Banking and Finance,vol. 24 (2000) p.59-117.