Review and Implementation of Credit Risk Models · developed by Credit Suisse Financial Products...

of the losses on a portfolio of loans and other debt instru-ments. Being able to compute the loss distribution of a port-folio is critical, because it allows the estimation of the credit value at risk (VaR) and, therefore, the economic capital re-quired by credit operations.

In this chapter, we present the theoretical background that underpins one of the most frequently used models for loss dis-tribution determination: CreditRisk+. Th is model, originally

Over the last 15 years, we have witnessed major advances in the fi eld of modeling credit risk. Th ere are now three main approaches to quantitative credit risk modeling: the “Merton- style” approach, the purely empirical econometric approach, and the actuarial approach.1 Each of these approaches has, in turn, produced several models that are widely used by fi nan-cial institutions around the world. All these models share a common purpose: to determine the probability distribution

CHAPTER 10

Review and Implementation of Credit Risk Models

RENZO G. AVESANI • KEXUE LIU • ALIN MIRESTEAN • JEAN SALVATI

The chapter presents the basic CreditRisk+ model along with extensions suggested in the existing literature and proposes some modifi cations. The purpose of these models is to determine the probability distribution of the losses on a portfolio of loans and other debt instruments so that

they could be used for stress testing in the IMF ’s Financial Sector Assessment Program, as a benchmark for credit risk evaluations. First, we present the setting and basic defi nitions common to all the model specifi cations used in this chapter. Then, we proceed from the simplest model based on Bernoulli- distributed default events and known default probabilities to the full- fl edged CreditRisk+ implementation. The latter is based on the Poisson approximation and uncertain default probabilities determined by mutually in de pen dent risk factors. As an extension, we present a Credit-Risk+ specifi cation with correlated risk factors as in Giese (2003). Finally, we illustrate the characteristics and the results obtained from the diff erent models using a specifi c portfolio of obligors.

METHOD SUMMARY

Overview The CreditRisk+ model along with extensions suggested in the existing literature can be used to determine the probability distribution of the losses on a portfolio of loans and other debt instruments.

Application The ability to compute the loss distribution of a portfolio enables the determination of the credit value at risk and, there-fore, the economic capital required by credit operations.

Nature of approach Actuarial approach.

Data requirements Individual exposures, the mean individual default probabilities, the standard deviations of the individual default probabili-ties, and the matrix of weights representing the exposure of each obligor to the risk factors.

Strengths Under certain assumptions, the model provides an analytical solution for determining the loss distribution.

Weaknesses • The default probabilities are assumed to be given by a linear multifactor model. • Unlike a probit or logit model, this model cannot restrict default probabilities from falling outside the [0,1] interval. • The analytical approximation may be imprecise if the probabilities of default are too high, that is, if they exceed single

digits.

Tool The Excel add- in is available in the toolkit, which is on the companion CD at www.elibrary.imf .org/stress-test-toolkit. Contact author: A. Mirestean.

Th is chapter is an abridged version of IMF Working Paper 06/134 (Avesani and others, 2006).1 See Koyluoglu and Hickman (1998) and Crouhy, Galai, and Mark (2000) for more details.

536-57355_IMF_StressTestHBk_ch01_3P.indd 135 11/5/14 2:03 AM

Review and Implementation of Credit Risk Models 136

GD, n(z) = P(Dn = x ) z x .x=0

Because Dn can take only two values (0 and 1), GD,n(z) can be rewritten as follows:

GD,n (z) = (1 pn) z + pn z = (1 pn) + pn z.

B. Losses

Th e loss on obligor n can be represented by the random vari-able Ln = Dn En. Th e probability distribution of Ln is given by P(Ln = En) = pn and P(Ln = 0) = 1 pn. Th e total loss on the portfolio is represented by the random variable L:

L= Lnn=1

N

= Dnn=1

N

En .

Th e objective is to determine the probability distribution of L under various sets of assumptions. Knowing the probability distribution of L will allow us to compute the VaR and other risk mea sures for the portfolio.

C. Normalized exposures

When implementing the model, it has become common prac-tice to normalize and round the individual exposures and then group them in exposure bands. Th e pro cess of normal-ization and rounding limits the number of possible values for L and hence reduces the time required to compute the prob-ability distribution of L. When the normalization factor is small relative to the total portfolio exposure, the rounding error is negligible. Let F be the normalization factor.4 Th e rounded normalized exposure of obligor n is denoted n and is defi ned by the following:

n = ceil (En / F ). (10.1)

D. Normalized losses

Th e normalized loss on obligor n is denoted by n and is defi ned by the following:

n = Dn νn,

where Dn is the default and n is the normalized exposure for obligor n. Hence, n is a random variable that takes value n with probability pn, and value 0 with probability 1 pn. Th e total normalized loss on the portfolio is represented by the random variable :

= nn=1

N

.

Finding the probability distribution of is equivalent to fi nding the probability distribution of L.

4 In a portfolio with high variation of the exposure levels, the smallest exposure could be chosen as the normalization factor.

developed by Credit Suisse Financial Products (CSFP), is based on the actuarial approach and has quickly become one of the fi nancial industry benchmarks in the fi eld of credit risk modeling. Its popularity spilled over into the regulatory and supervisory community, prompting some supervisory author-ities to start using it in their monitoring activities.2 Th ere are several reasons why this model has become so pop u lar:

1. It requires a limited amount of input data and assumptions.

2. It uses as basic input the same data as required by the Basel II Internal Ratings Based (IRB) approach.

3. It provides an analytical solution for determining the loss distribution.

4. It brings out one of the most important credit risk drivers— concentration.

We illustrate our implementation of the model and suggest that it could be used as a toolbox in the diff erent surveillance activities of the IMF. We also analyze the problems that may arise by directly applying CreditRisk+, in its original formu-lation, to certain portfolio compositions. We subsequently propose some solutions.

Th e chapter is or ga nized as follows. Initially, we present the setting and basic defi nitions common to all the model specifi -cations used in this chapter. Th en, we gradually proceed from the simplest model based on Bernoulli- distributed default events and default probabilities known with certainty to the full- fl edged version of CreditRisk+. Th e latter is based on the Poisson approximation and uncertain default probabilities de-termined by mutually in de pen dent risk factors. We then go beyond CreditRisk+ by presenting a specifi cation that allows for correlation among risk factors, as in Giese (2003). We also apply the implemented models to a specifi c portfolio of expo-sures to illustrate their characteristics and discuss in detail the results. Finally, we present some conclusions.

1. THE BASIC MODEL SETTINGIn this section, we present the setting and basic defi nitions common to all the model specifi cations used in the chapter. We consider a portfolio of N obligors indexed by n = 1,…,N. Obligor n constitutes an exposure En. Th e probability of de-fault of obligor n over the period considered is pn.

A. Default events

Th e default of obligor n can be represented by a Bernoulli random variable Dn such that the probability of default over a given period of time is P(Dn = 1) = pn, while the probability of survival over the same span is P(Dn = 0) = 1 pn. Th en the Probability Generating Function (PGF ) of Dn is given by the following equation:3

2 See Austrian Financial Market Authority and Oesterreichische Nation-albank (2004).

3 Probability concepts are reviewed in the Appendix.


Renzo G. Avesani, Kexue Liu, Alin Mirestean, and Jean Salvati 137

G (z)= IFFT {FFT [G , n(z)]}n=1

N

G , B (z),

where IFFT is the inverse fast Fourier transform. Th is algo-rithm can be effi ciently implemented as long as the portfolio does not contain more than a few thousand obligors.5 Th is is about as far as the model can be analytically developed under the assumption that the probability distributions of default events are Bernoulli distributions. In order to fi nd a closed- form solution for the PGF of , when default events are rep-resented by Bernoulli random variables, it is also necessary to assume that default events are in de pen dent among obli-gors and that default probabilities are known with certainty (nonrandom). Finding an analytical solution when proba-bilities are random and default events are no longer in de-pen dent implies using an approximation for the distribution of the default events. Th is is precisely the path taken by the CreditRisk+ model when deriving closed- form expressions for the PGF of .

3. THE POISSON APPROXIMATIONIn this section, we describe in detail one of the essential as-sumptions of the CreditRisk+ model: the individual proba-bilities of default are assumed to be suffi ciently small for the compound Bernoulli distribution of the default events to be approximated by a Poisson distribution. Th is assumption makes it possible to obtain an analytical solution even when the default probabilities are not known with certainty.

Under the assumption that default events follow Bernoulli distribution, the PGF of obligor n’s default is

GD,n (z) = (1 pn) + pn z = 1 + pn (z 1).

Equivalently:

GD , n(z) = exp ln(GD , n(z)) = exp ln(1+ pn(z 1))[ ]. (10.2)

If we assume that pn is very small, then pn(z 1) is also very small under the assumption that |z| 1. Defi ning w = pn(z 1), we can perform a Taylor expansion of ln(1 + w) in the vicinity of w = 0:

ln(1+ w ) = w w 2

2+ w

3

3.

Neglecting the terms of order 2 and above and going back to the original notation yields the following:

ln[l + pn (z 1)] pn (z 1). (10.3)

Th e assumption that justifi es neglecting the terms of order 2 and above is that pn is “small”: the smaller pn, the smaller the (absolute) diff erence between ln[1 + pn(z 1)] and pn(z 1).

Going back to equation (10.2) and using the approxima-tion from equation (10.3), we have

GD,n (z) exp[pn (z 1)].

5 See Section 9 for more details.

E. Exposure bands

Th e use of exposure bands has become a common technique used in the literature to simplify the computational pro cess. Once the individual exposures have been normalized and rounded, as shown in equation (10.1), common exposure bands can be defi ned in the following manner. Th e total num-ber of exposure bands, J, is given by the highest value of the normalized individual exposures, J = max{ n }n=1

N . Let j rep-resent the index for the exposure bands. Th en, the common exposure in band j is j = j. Th e distribution of obligors among exposure bands is set such that each obligor n is assigned to band j if n = j = j. Th en, the expected number of defaults in band j, j , is given by

j = pn .n , n= j

Consequently, the total expected number of defaults in the portfolio, , is given by

= jj=1

J

.

2. A SIMPLE MODEL WITH NONRANDOM DEFAULT PROBABILITIESIn this section, we present a simple model with Bernoulli- distributed default events and nonrandom default proba-bilities. Th e advantage of this model is that it relies on the smallest set of assumptions. As a result, the loss distribu-tion in this model can be effi ciently computed as a simple convolution, without making any approximation for the distribution of default events. Th is approach is particularly appropriate when default probabilities are high and when there is little uncertainty concerning the values of these probabilities.

Th e key assumptions of the model are that individual de-fault probabilities over the period considered are known with certainty and that default events are in de pen dent among ob-ligors. Th e objective is to determine the probability distribu-tion of the total normalized loss, , or, equivalently, the PGF of . Given the assumption of in de pen dence among obligors, the PGF of the total normalized loss, , can easily be com-puted from the PGF of the individual normalized loss, n. As

n can take only the values 0 and n, the PGF of n is given by the following:

G , n(z) = P( = 0) z 0 +P( = n ) z n = (1 pn )+ pn z n .

Taking in account that is the sum of n over n and because default events are in de pen dent among obligors, the PGF of is simply given by

G (z)= G ,nn=1

N

(z)= [(1 pn )+ pn z n ].n=1

N

Th is product of the individual loss PGF is a simple convolu-tion that can be computed using the fast Fourier transform (FFT). From the convolution theorem,



Replacing G ,n(z) in equation (10.9) with the expression from equation (10.8), we have

G (z)= exp[pn(z n 1)]n=1

N

= exp pnn=1

N

(z n 1) .

Using the defi nition of the exposure bands, as presented in Section 1, G (z) can be written as follows:

G (z) = exp pn z j 1( )n , n= Jj=1

J

.

Th is expression can be fi nally simplifi ed using the defi nitions of the expected number of defaults in band j, j and the ex-pected total number of defaults in the portfolio, :

G (z) = exp jj=1

J

z j 1( )

= exp j z j 1j=1

J

G , FIXED (z).

(10.10)

Equation (10.10) defi nes the probability distribution of the total loss on the portfolio when default events are mutually in de pen dent, default probabilities are known with certainty, and Poisson distributions are used to approximate the distri-butions of individual default events.

5. RANDOM DEFAULT PROBABILITIESIn this section, we present the full- fl edged version of Credit-Risk+. We do so by removing the assumptions used so far—that is, that the default probabilities are known with certainty and that default events are unconditionally mutually in de-pen dent. Instead, we assume that individual default proba-bilities are random and are infl uenced by a common set of Gamma- distributed systematic risk factors. Consequently, the default events are assumed to be mutually in de pen dent only conditional on the realizations of the risk factors. Under these assumptions, the use of the Poisson approximation still allows us to obtain an analytical solution for the loss distribution.

Let us assume that there are K risk factors indexed by k. Each factor k is associated with a “sector” (an industry or a geographic zone, for example) and is represented by a random variable k. Th e probability distribution of k is assumed to be a Gamma distribution with shape pa ram e ter k =1/ k

2 and scale pa ram e ter k = k

2 . Th erefore, the mean and the vari-ance are given by the following:

E( k ) = k k = 1

Var( k ) = k k2 = k

2 .

Th e moment generating function (MGF ) of k is the function defi ned by

M , k (z) = E[exp( k z)] = (1 k z) k = (1 k2 z) 1/ k

2.

Performing again a Taylor expansion of exp[pn(z 1)] in the vicinity of pn = 0 fi nally yields the following:

GD, n(z) exp[pn(z 1)] = exp( pn )pnx

x !z x .

x=0

(10.4)

Th e last member of equation (10.4) is the PGF of a Poisson distribution with intensity pn. Th erefore, for small values of pn, the Bernoulli distribution of Dn can be approximated by a Poisson distribution with intensity pn.

Using the Poisson approximation, the probability distri-bution of Dn is then given by

P(Dn = x ) = exp( pn )pnx

x !.

A new expression for the PGF of the individual normalized loss n can also be derived using the Poisson approximation in equation (10.4). Th e PGF of n is defi ned by

G , n(z) = E(z n ) = E(zDn n ), (10.5)

where E is the expectation operator. Given that n is not ran-dom, equation (10.5) can be written as follows:

G , n(z)= P(Dn = x ) zDn n

x=0

= exp( pn )pnx

x !(z n )x .

x=0

(10.6)

Given that Dn takes only two values (0 and 1), we can express the PGF of n as

G , n(z) = (1+ pn z n )exp( pn ). (10.7)

Th e second term of the right side in equation (10.7) is the fi rst- order Taylor expansion of exp( pnz n), that is, exp( pnz n ) 1+ pnz n. Th erefore, for small values of probabilities pn, the PGF of the individual normalized loss, n, can be approxi-mated by

G , n(z)= exp[pn(z n 1)]. (10.8)

4. THE MODEL WITH KNOWN PROBABILITIES REVISITEDIn this section, we apply the Poisson approximation to the basic model with known default probabilities, as presented in Section 2. Hence, we derive a new expression for the PGF of the total normalized loss on the portfolio, G (z), based on the expression for G ,n(z) from equation (10.8). Individual default probabilities are still assumed to be known with cer-tainty (nonrandom), and default events are still assumed to be in de pen dent among obligors.

Given that individual losses are mutually in de pen dent, G (z) is simply the product of the individual loss PGF, as in Section 2:

G (z)= G , nn=1

N

(z). (10.9)



PGF of the total normalized loss, denoted by G (z), is the expectation of G (z | ) under probability distribution:

G (z)= E G z |( )

= E exp kk=1

K

Pk (z) .

Defi ne P(z) = [P1(z),… ,Pk(z),… ,PK(z)]. Using the defi nition of the joint MGF of , M , we can write G (z) as follows:

G (z) = E . {exp [P(z) ' ]} = M [P(z)]. (10.13)

It is important to note that this equation does not rely on the fact that k is mutually in de pen dent.

Let us consider now a vector = ( 1,… , k,… , K) of auxil-iary variables such that 0 k < 1 for all k = 1,… ,K. Th e joint MGF of the vector is given by

M ( ) = E [exp( )] = E exp k kk=1

K

= E exp( k k )k=1

K

. (10.14)

Recall that the MGF of k is defi ned by

M , k ( k ) = E , k [exp( k k )] = 1 k2

k( ) 1/ k2

.

Given that variables k are mutually in de pen dent, M ( ) can be rewritten as follows:

M ( ) = E , k

k=1

K

exp k k( )

= M , k k( )k=1

K

.

Th erefore, setting k = Pk(z), G (z) is given by

G (z) = M , k

k=1

K

[Pk (z)]

= 1 k2 Pk (z)[ ] 1/ k

2

k=1

K

. (10.15)

Equation (10.15) can be further transformed as follows:

G (z) = exp ln G (z)( )= exp

1

k2

ln 1 k2 Pk (z)[ ]

k=1

K

G ,CR+ (z).

Hence, we obtain the PGF of the total normalized loss on the portfolio for the CreditRisk+ model, G ,CR+(z). It is worth noting that G ,CR+(z) is obtained under the following as-sumptions: default probabilities are random; the factors that determine default probabilities are Gamma distributed and mutually in de pen dent; and default events are mutually in de-pen dent conditional on these factors.

6. LATENT FACTORSIn CreditRisk+, the factors k are treated as latent variables. Th at is to say, the factors that infl uence the default probabilities are assumed to be unobservable. It is further assumed that

Th e random variables k are assumed to be mutually in de pen-dent. In addition, the default probability of obligor n is assumed to be given by the following model:

pn = pn k k , nk=1

K

, (10.11)

where pn is the average default probability of obligor n, and k,n is the share of obligor n’s exposure in sector k (or the

share of obligor n’s debt that is exposed to factor k). Accord-ing to this model, the default probability of obligor n is a random variable with mean pn .

It is important to note that exposure to common risk fac-tors introduces unconditional correlation between individual default events. Consequently, default events are no longer unconditionally mutually in de pen dent.6 However, individual default events— and therefore individual losses— are assumed to be mutually in de pen dent conditional on the set of factors = ( 1,… , k,… , k).

As in Sections 3 and 4, Dn is assumed to follow a Poisson distribution with intensity pn, and the PGF of the individual normalized loss n is assumed to be

G , n(z) = exp pn z n 1( ) .

Using the multifactor model for pn defi ned in equation (10.11), G ,n(z) can be written as follows:

G , n(z) = exp pn k k , nk=1

K

z n 1( )

= expk=1

K

pn k k , n z n 1( ) .

Let G (z | ) be the PGF of the total normalized loss, condi-tional on . Given that individual losses are mutually in de-pen dent conditional on , G (z | ) is simply the product of the individual loss PGF conditional on :

G z |( )= G , n z |( )n=1

N

= exp pn k k , n z n 1( )k=1

K

n=1

N

= exp pn k k , n z n 1( )k=1

K

n=1

N

.

If we defi ne Pk(z) as

Pk (z) = pnn=1

N

k , n(z n 1), (10.12)

then G (z | ) can be written as

G z |( ) = exp kk=1

K

Pk (z) .

Let E denote the expectation operator under the probability distribution of , and let E ,k denote the expectation operator under the probability distribution of k. Th e unconditional

6 An approximate method for computing the correlation matrix of indi-vidual default events is presented in CSFP (1997).



Th is expression is actually not consistent with the linear mul-tifactor model for individual default probabilities used in Cre-ditRisk+, and it results in an underestimation of k. However, in practice, it gives reasonably accurate results, as will be shown in Section 10.

An alternative is to use least squares to estimate k from n. According to the multifactor model for the default prob-

abilities, k2 is the solution of the following linear system:

n2 = pn2 k

2k, n

k=1

K

, n = 1,…,N.

If N K, then a solution to this system can be found using constrained least squares. However, there is no way to guar-antee that k

2, k = 1… K will all be strictly positive. Th e least- squares method and the weighted average method are compared in Section 10.

7. EXTENSION OF CREDITRISK+ WITH CORRELATED FACTORSOne of the limitations of CreditRisk+ is the assumption that the factors determining the default probabilities are mutu-ally in de pen dent. In this section, following the approach de-veloped by Giese (2003), we integrate the factors correlation in the model. One of the main limitations of CreditRisk+ is the assumption that the factors that determine default prob-abilities are mutually in de pen dent. Giese (2003) has derived an extension of CreditRisk+ that allows for some form of correlation between the factors.

In CreditRisk+, the kth factor k follows a Gamma distri-bution with nonrandom shape and scale pa ram e ters ( k and

k, respectively). In other words, CreditRisk+ assumes that the means and variances of k are nonrandom. Giese (2003) intro-duces an additional factor, which will be denoted by , that aff ects the distributions of all k, thereby introducing some correlation between the factors. is assumed to follow a Gamma distribution with shape pa ram e ter = 1/ 2 and scale pa ram e ter = 2. Consequently, the mean and variance are given by

E( ) = = 1

Var( ) = 2 = 2 .

Now variables k are assumed to be mutually in de pen dent conditional on .

Th e probability distribution of k is still a Gamma distribu-tion with shape pa ram e ter k and scale pa ram e ter k . How-ever, while k is still constant, k is now a function of and hence a random variable:

k = k ,

where k is a constant.Let E denote the expectation under the distribution,

and let E ,k denote the expectation under the k distribution conditional on . Similarly, let Var denote the variance under the distribution, and let Var ,k denote the variance under the k distribution conditional on .

the means and standard deviations of the default probabilities, as well as their sensitivities to the latent factors, are known or can be estimated without using observations of the factors.

Th e latent factors approach is required by the linear mul-tifactor model for the default probabilities (equation (10.11)). Th is model has the advantage of enabling the derivation of an analytical solution for the loss distribution. However, un-like a probit or logit model, this model has little empirical relevance, because it allows default probabilities to exceed one (albeit with very low probability). If equation (10.11) were to be estimated using empirical observations of the factors

k, it is likely that the default probabilities implied by the eco-nometric model would often fall outside the interval [0,1]. To avoid having to deal with unreasonable values for the default probabilities, instead of estimating equation (10.11) using ob-servations of the factors k, these factors are treated as unob-served latent variables.

In CreditRisk+, the means of the latent factors can be normalized to one without loss of generality. As a result, the latent factors play a role only via their standard deviations after normalization ( k, k = 1,… ,K ), which can be estimated from the means, standard deviations, and sensitivities of the default probabilities. Th erefore, when CreditRisk+ is imple-mented in practice, the inputs of the model are the mean individual default probabilities, denoted by pn , n = 1,…, N ; the standard deviations of the individual default probabilities, denoted by n, n = 1,… ,N; and the matrix of weights { k,n} for k = 1… K, n = 1… N, representing the exposure of each obligor n to each factor k. Th en the k are estimated from the n.

Let us assume that some of the obligors in the portfolio are exposed to a single factor. For example, consider an obligor n who is exposed only to factor :

k, n = 1, k =

k, n = 0, k .

Th e default probability of this obligor is given by

pn = pn .

Th erefore,

n = pn .

Note that this equation implies that the ratio n/pn is the same for all obligors such that ,n = 1. Nevertheless, because this relationship holds for any obligor n such that n = 1, can be computed from n as follows:

= 1N

n

pn,

, =1

where N is the number of obligors such that ,n = 1. In the case where all obligors are exposed to more than one factor, the original CreditRisk+ implementation uses weighted averages of n to estimate k:

k =1

kk, n

n=1

N

n =1

kk, n

n=1

N

pnn

pn

with k = k, n pn .n=1

N

(10.16)



where M ,k( | ) is the MGF of k conditional on :

M , k (z | ) = E , k [exp( k z) | ] = (1 k z) k = (1 k z) k .

M (Z ) can now be rewritten as follows:

M (Z ) = E exp log M , kk=1

K

(zk | )

= E exp logk=1

K

[M , k (zk | ]

= E exp kk=1

K

ln(1 k zk )

= E exp1

k

ln(1 k zk )k=1

K

.

Defi ne a new auxiliary variable t:

t = 1

k

ln(1 k zk ).k=1

K

Using t, M (Z) can be rewritten as the MGF of , denoted by M :

M (Z ) = E [exp( t )] = M (t ) = 12

ln(1 2 t ).

Replacing t with its expression gives us the fi nal expression for M (Z):

M (Z ) = exp1

2ln 1+ 2 1

k

ln(1 k zk )k=1

K

.

Recall equation (10.13) established in Section 5:

G (z) = M [P(z)].

Th is equation is still valid in the context of this section, and, when M is replaced with its new expression, it gives us the PGF of the total normalized loss in the model with correlated sectors:

G (z) = exp1

2ln 1+ 2 1

k

ln(1 k Pk (z))k=1

K

G ,CORR (z),

with

Pk (z) = pnn=1

N

kn(z n 1).

8. MODEL SUMMARYIn this section, we provide a summary pre sen ta tion for the loss distributions of the various models described so far. Th e following equations show the four expressions for the PGF of the normalized loss on the portfolio. Each expression cor-responds to the PGF of a par tic u lar model:

G , B (z) = IFFT FFT [G ,n(z)]{ }n=1

N

G , FIXED (z) = exp j z j 1j=1

J

Th e expectation of k conditional on is given by

E , k k |( ) = k k = k k .

Th e unconditional expectation of k is given by

E( k ) = E E , k k |( ) = E ( k k ) = E k k( ).Given that k and k are not random, E( k) can be rewritten as follows:

E( k ) = k k E ( ) = k k .

Th e unconditional expectation of k is assumed, without loss of generality, to be equal to 1, so that k = 1/ k . Th e variance of k conditional on is given by

k2 =Var , k k |( )= k k

2 = k k2 = / k .

Th e unconditional variance of k is given by

Var( k ) = E Var , k k |( ) + Var E , k k |( )

= E 1

k

+ Var k k( )= k E ( ) + k k Var ( ) = k + 2 .

Th e unconditional covariance between any two factors k and l is given by

Cov( k , l ) = E ( k 1) E ( k ) E ( l )

= E E , k, l ( k l | ) E ( k ) E ( l )

Given that k and l are in de pen dent conditional on , this expression becomes

Cov( k , l ) = E E , k k |( ) E , k k |( ) E( k ) E( l )

= k k l l [E( 2 ) 1]

= k k l l [Var( )+ E( )2 1].

Given that k k = l l = 1 and E( )2 = 1, the uncondi-tional covariance between k and l is simply represented as Cov( k , l ) = 2 > 0.

Consider a vector Z = (z1,… ,zk,… ,zK) of auxiliary variables such that 0 zk < 1 for k = 1,… ,K. Th e joint MGF of is the function M defi ned by

M (Z ) = E exp k zkk=1

K

.

Using the properties of conditional expectations, M (Z) can be rewritten as follows:

M (Z )= E E exp k zkk=1

K

,

where E is the expectation under the joint distribution.Given that k is mutually in de pen dent conditional on ,

M (Z) can be further rewritten as follows:

M (Z ) = E E , kk=1

K

[exp( k zk ) | ]

= E M , kk=1

K

(zk | ) ,



• Second, the Panjer recursion is numerically unstable in the sense that numerical errors accumulate as more terms in the recursion are computed. Th is can result in signifi cant errors in the upper tail of the loss dis-tribution and hence in the computation of the port-folio’s VaR.

Th is section briefl y presents two alternative algorithms that can be used to implement CreditRisk+ and the other models based on the Poisson approximation.8

• Th e fi rst algorithm is an alternative recursive scheme introduced by Giese (2003). Haaf, Reiss, and Schoen-makers (2003) have shown that this algorithm is numerically stable, in the sense that precision errors are not propagated and amplifi ed by the recursive for-mulas. Th is algorithm can provide a unifi ed imple-mentation of all versions of the model. However, like the Panjer recursion, this algorithm can fail for very large numbers of obligors.

• Th e second algorithm is attributable to Melchiori (2004). It is based on the fast Fourier transform (FFT) and can deal with very large numbers of obligors. It can easily be applied to the model with nonrandom default probabilities and to the model with random probabilities and uncorrelated factors.

A. Alternative recursive scheme

Recall the defi nition of the PGF of the total normalized loss on the portfolio:

G (z)= P( = x ) z x .x=0

G (z) is a polynomial function of the auxiliary variable z. Th e coeffi cients of this polynomial are the probabilities associated with all the possible values for . One way to implement a par tic u lar version of the model is to derive a polynomial repre-sen ta tion for the corresponding version of G (z) and to com-pute the coeffi cients of that polynomial.

Note that all three versions of G (z) derived using the Pois-son approximation involve exponential and/or logarithmic transformations of polynomials in z.9 Th e fi rst step in the computation of G ,FIXED(z) is the computation of the coeffi -cients of Gl(z). Similarly, the fi rst step in the computation of G ,CR+(z) and G ,CORR(z) is the computation of the coeffi cients of Pk(z), for k = 1,… ,K. Once these coeffi cients have been determined, exponential and logarithmic transformations of Gl(z) and/or Pk(z) must be computed.

Exponential and logarithmic transformations of polyno-mials can be computed using recursive formulas derived from

8 Gordy (2002) presents a third algorithm based on saddlepoint approxi-mation of the loss distribution. Th at algorithm computes only the cumu-lative distribution function of the loss distribution— not the probability distribution function. Furthermore, it is much more accurate for large portfolios than for small ones.

9 See Section 8.

G ,CR+ (z) = exp1

k2

ln[1 k2 Pk (z)]

k=1

K

G ,CORR (z) = exp1

2ln 1+ 2 1

k

lnk=1

K

(1 k Pk (z))

• G ,B(z) is the PGF of the normalized loss on the port-folio in the “basic” model with Bernoulli default events and nonrandom default probabilities;

• G ,FIXED(z) is the PGF of the normalized loss on the portfolio in the model with fi xed default probabilities as approximated by a Poisson distribution;

• G ,CR+(z) is the PGF of the normalized loss on the portfolio in the model with random probabilities and mutually in de pen dent factors; this is the CreditRisk+ model;

• G ,CORR(z) is the PGF of the normalized loss on the portfolio in the model with random probabilities and correlated factors; this is an extension of the Credit-Risk+ model.

Th e basic model with Bernoulli default events and nonran-dom default probabilities is the only one that does not rely on the Poisson approximation. Both G ,B(z) and G ,FIXED(z) were derived under the assumption that default probabilities are nonrandom. Th e Poisson approximation is the only source of discrepancies between the resulting loss distributions in these two models. Th erefore, the accuracy of that approxi-mation can be evaluated by comparing G ,FIXED(z) and G ,B(z).

9. NUMERICAL IMPLEMENTATIONIn this section, we discuss numerical issues that arise in the implementation of the models described above, and then we present two algorithms that address these problems. As explained in Section 2, the PGF of the portfolio loss in the basic model with Bernoulli default events and fi xed proba-bilities is a simple convolution of the individual loss PGFs. Th e models based on the Poisson approximation (includ-ing CreditRisk+) are implemented using more sophisticated algorithms.

Th e original algorithm proposed by CSFP (1997) to im-plement the CreditRisk+ model is based on a recursive for-mula known as the Panjer recursion.7 In the context of credit risk models, the usefulness of the Panjer recursion is limited by two numerical issues. Both issues arise from the fact that a computer does not have infi nite precision.

• First, the Panjer recursion cannot deal with arbitrarily large numbers of obligors: as the expected number of defaults in the portfolio increases, the computation of the fi rst term of the recursion becomes increasingly imprecise; above a certain value for the expected num-ber of defaults in the portfolio, the value found for the fi rst term of the recursion becomes meaningless.

7 See CSFP (1997).



Th e characteristic function of a random variable X is de-fi ned as

X (z) = EX [exp (i X z)] = GX [exp (i z)], 0 z <1,

where EX is the expectation under the X probability distribu-tion, and GX is the PGF of X.

Using equation (10.18) and this defi nition, we obtain the following expression for the characteristic function of the normalized portfolio loss:

, FIXED (Z ) = G , FIXED[exp(i z)]

= exp Gl (exp(i z)) 1[ ]{ }= exp l (z) 1[ ]{ }, (10.19)

where l(z) is the characteristic function of l.Th e characteristic function of a random variable can also

be defi ned as the Fourier transform of its density. Th is fact and equation (10.19) suggest a simple and effi cient algorithm for computing the portfolio loss distribution.

Let denote the vector of probabilities representing the distribution of , and let = ( 1,… , j,… , J) denote the vector of probabilities representing the distribution of l. can be computed as follows:

= IFFT [exp{ [FFT( ) 1]}],

where FFT is the fast Fourier transform, and IFFT is the inverse fast Fourier transform. Th is algorithm can easily be extended to the model with random default probabilities and uncor-related factors. It has been successfully applied to a portfolio containing 679,000 obligors.

10. NUMERICAL EXAMPLES USING THE CREDIT RISK TOOLBOXAll the models discussed in this chapter may be implemented in MATLAB. In this section, we are going to present a very short description of the capabilities of the toolbox, and then we will off er some numerical examples. To illustrate the models presented in this chapter, we use the same portfolio as in the CreditRisk+ demonstration spreadsheet from CSFP. Th is port-folio is presented in Table 10.1:

• In order to run the basic model with fi xed probabili-ties, whether the distribution of the default events is assumed to be Bernoulli or Poisson, one needs to have a set of individual exposures and a set of individual default probabilities.10 Th is corresponds to the fi rst two columns in Table 10.1. If data are available only in aggregate form for diff erent classes of obligors, the pro-grams can still be used by making additional assump-tions. For example, obligors can be classifi ed according

10 Th ere are three main approaches to estimating the probabilities of de-fault. One approach is to use historical frequencies of default events for diff erent classes of obligors in order to infer the probability of default for each class. Alternatively, econometric models such as logit and probit could be used to estimate probabilities of default. Finally, when available, one could use probabilities implied by market rates and spreads.

the power series repre sen ta tions of the exponential and loga-rithm functions.

Consider two polynomials of degree xmax in z, Q(z) and R(z):

Q(z) = qx z xx=0

xmax

R(z) = rx z xx=0

xmax

.

If R(z) = exp[Q(z)], the coeffi cients of R(z) can be computed using the following recursive formula:

r0 = exp(q0 )

rm = sm

qs rm ss=1

m

.

If R(z) = ln[Q(z)], the coeffi cients of R(z) can be computed using the following recursive formula:

r0 = ln(q0 )

rm = 1q0

qmsm

qs rm ss=1

m 1

.

Th ese recursive formulas can be used to compute the coeffi -cients of G ,FIXED(z), G ,CR+(z), and G ,CORR(z).

Th is algorithm produces accurate results for portfolios such that

kn

n=1

N

pn < kmax 750 k = 1,…,K . (10.17)

For example, if K = 5, pn = 0.05 for all obligors, and n,k = 0.2 for all obligors and all sectors, then the maximum number of obligors in the portfolio is approximately 75,000.

B. Fast Fourier transform– based algorithm

To describe the FFT- based algorithm proposed by Melchi-ori (2004), we will use the PGF of the portfolio loss for the model with Poisson default events and nonrandom default probabilities:

G , FIXED (z)= exp j z j 1j=1

J

.

Defi ne the function Gl(z) as follows:

Gl (z)=j z j .

j=1

J

Note that Gl(z) can be interpreted as the PGF of a random variable l such that

P(l = v j ) =j

j .

Using the defi nition of Gl(z), G ,FIXED(z) can be rewritten as follows:

G ,FIXED (z) = exp { [Gl (z) 1]}. (10.18)



the matrix of weights, representing the exposure of each obligor n to each factor k. As mentioned above, if individual data are not available, the programs can use aggregate data for diff erent classes of obligors by making additional assumptions. CreditRisk+ has also been implemented using both algorithms presented in Section 9.

• When running the CreditRisk+ model with its ex-tension as described in Section 7, in addition to the data required to implement CreditRisk+, this model requires a value for the intersector covariance. Th us, one needs to have available the mean individual de-fault probabilities, the standard deviations of the in-dividual default probabilities, the matrix of weights, and a value for the intersector covariance. As men-tioned above, if individual data are not available, the programs can use aggregate data for diff erent classes of obligors by making additional assumptions. Th e model with correlated factors of Section 7 has been implemented using the numerically stable recursive algorithm. Consequently, it presents a limitation on the total number of obligors it can handle.11

Th e exposures in Table 10.1 correspond to En in this study; the mean default rates correspond to pn in the case with non-random default probabilities and to p n in the case with random default probabilities; the standard deviations corre-

11 See Section 9 for more details.

to exposure, rating, or type (corporate vs. individual, for instance). When the average exposure and the aver-age default probability are the only data available in each class, then the program assumes that all obligors in a given class have the same exposure and the same default probability.– Computation time for the model with Bernoulli

default events and fi xed probabilities is determined by three variables: the number of obligors, the total normalized exposure, and the granularity of the exposure bands. Th e last two factors can be ad-justed to reduce computation time, at the expense of accuracy. Generally speaking, the convolution can be effi ciently computed in MATLAB as long as the number of obligors does not exceed a few thousand.

– Th is model has the advantage of being the most accurate when the default probabilities can be treated as nonrandom. As shown later on in this section, this is an important consideration when the default probabilities are relatively high. Th e model of Section 4 with fi xed probabilities and Poisson default events has been implemented using both algorithms presented in Section 9.

• When running the CreditRisk+ model as described in Sections 5 and 6, one needs to have available the mean individual default probabilities, the standard deviations of the individual default probabilities, and

TABLE 10.1

Credit Suisse Financial Products Reference Portfolio

Obligor Exposure

Mean Default Rate

Standard Deviation of Default Rate

Factor Weights

Sector 1 Sector 2 Sector3 Sector4 Total

358,475 30.00% 15.00% 50% 30% 10% 10% 100%1,089,819 30.00% 15.00% 25% 25% 25% 25% 100%1,799,710 10.00% 5.00% 25% 25% 20% 30% 100%1,933,116 15.00% 7.50% 75% 5% 10% 10% 100%2,317,327 15.00% 7.50% 50% 10% 10% 30% 100%2,410,929 15.00% 7.50% 50% 20% 10% 20% 100%2,652,184 30.00% 15.00% 25% 10% 10% 55% 100%2,957,685 15.00% 7.50% 25% 25% 20% 30% 100%3,137,989 5.00% 2.50% 25% 25% 25% 25% 100%

3,204,044 5.00% 2.50% 75% 10% 5% 10% 100%4,727,724 1.50% 0.75% 50% 10% 10% 30% 100%4,830,517 5.00% 2.50% 50% 20% 10% 20% 100%4,912,097 5.00% 2.50% 25% 25% 25% 25% 100%4,928,989 30.00% 15.00% 25% 10% 10% 55% 100%5,042,312 10.00% 5.00% 25% 25% 30% 20% 100%5,320,364 7.50% 3.75% 75% 10% 5% 10% 100%5,435,457 5.00% 2.50% 50% 20% 10% 20% 100%5,517,586 3.00% 1.50% 50% 10% 10% 30% 100%5,764,596 7.50% 3.75% 25% 25% 20% 30% 100%5,847,845 3.00% 1.50% 25% 10% 10% 55% 100%6,466,533 30.00% 15.00% 25% 25% 20% 30% 100%6,480,322 30.00% 15.00% 75% 10% 5% 10% 100%7,727,651 1.60% 0.80% 25% 25% 20% 30% 100%

15,410,906 10.00% 5.00% 50% 20% 10% 20% 100%20,238,895 7.50% 3.75% 75% 10% 10% 5% 100%

Source: CSFP (1997).



Th e Poisson approximation generally results in an overesti-mation of the VaR. Th is result is observed for portfolios with very diff erent structures.

An additional numerical experiment was performed to illustrate the relationship between the magnitude of the de-fault probabilities and the error due to the Poisson approxi-mation. Th is experiment uses the same exposures as in the CSFP portfolio, but it assumes that all obligors have the same default probability. Th e ratio between the VaRb when the defaults are assumed to follow a Bernoulli distribution and the VaRp when the defaults are assumed to follow a Poisson distribution, (VaRb/VaRp), is computed for multiple values of

spond to the values of n in the models with random probabili-ties. Th e toolbox treats the factors as latent variables, and the values for k are estimated from the values of n. Two estima-tion methods are compared: least squares (LS) on the one hand and the weighted averages used by CSFP on the other hand. Th e standard deviations and the sector weights are required only to implement the models with random default probabili-ties; the models with nonrandom default probabilities use only the exposures and the mean default probabilities. Also, in gen-eral, individual exposures and default probabilities are not re-quired to run the model; aggregate data per class of obligor can be used instead by making additional assumptions.

Th e loss distribution for the CSFP sample portfolio is com-puted using fi ve models. Th e assumptions underlying these models are summarized in Table 10.2. Model 3 is CSFP’s CreditRisk+. By default, the CSFP implementation treats Sec-tor 1 as a special sector representing diversifi able idiosyncratic risk. Th is cannot be done in the model with correlated fac-tors. Th erefore, all the models discussed in this section treat Sector 1 as a regular sector (which is also an option in CSFP’s CreditRisk+). Th e portfolio loss distribution is also computed by Monte Carlo simulation under the following assumptions: Bernoulli default events; random default probabilities; and in de pen dent Gamma- distributed factors, with k

2 estimated by LS. Th is loss distribution computed by Monte Carlo sim-ulation constitutes a benchmark. It can be used to assess the impact of the Poisson approximation and of the k

2 estimation method on the models’ accuracy.

A. Eff ects of Poisson approximation: nonrandom default probabilities

To assess the impact of the Poisson approximation, we fi rst compare Models 1 and 2. Th e only diff erence between these two models is the distribution of default events: Bernoulli for Model 1, Poisson for Model 2. Figures 10.1 and 10.2 pres-ent the portfolio’s loss distributions for these two models, respectively. Th ey also show the VaR at the 99 percent level for each model. Th e VaR is 8.67 percent larger in the model with Poisson defaults than in the model with Bernoulli de-faults. Th is is the error introduced by the Poisson approxima-tion for this par tic u lar portfolio with 25 obligors and with default probabilities ranging from 3 percent to 30 percent.

TABLE 10.2

Summary of the Assumptions Used in the Diff erent Models

ModelDefault Events

Default Probabilities

Correlated Factors

Latent Factors

k2

Estimation Method

1 Bernoulli Fixed N/A N/A N/A2 Poisson Fixed N/A N/A N/A3 Poisson Random No Yes Averages4 Poisson Random No Yes LS5 Poisson Random Yes Yes Averages

Source: Authors.Note: LS = least squares.

Loss0 2 4 6 8 10 12

x 107

0.04

0.03

0.02

0.01

0Pr

obab

ility

VaR at 99% = 42829849(32% of total exposure)

Source: Authors.Note: VaR = value at risk.

Figure 10.1 Model 1: Fixed Probabilities, Bernoulli Defaults

Loss0 2 4 6 8 10 12

x 107

0.04

0.03

0.02

0.01

0

Prob

abili

ty



Figure 10.2 Model 2: Fixed Probabilities, Poisson Defaults



the common default probability, ranging from 1 percent to 30 percent. Th e results of this experiment are presented in Fig-ure 10.3. One can see that VaRb/VaRp decreases steadily from 1 to 0.86 as the common default probability increases from 1 percent to 30 percent. In other words, as could be expected from the derivation of the Poisson approximation, the error due to this approximation increases with the magnitude of the default probability.

B. Random default probabilities, uncorrelated factors

Th is section compares the loss distributions for Models 3 and 4 to the loss distribution computed by Monte Carlo simula-tion. Models 3 and 4 diff er only by the method used to esti-mate k: Model 3 uses the CSFP weighted average; Model 4 uses LS. With CSFP’s portfolio, k

2 estimated by least squares is strictly positive. Both models assume Poisson default events and random default probabilities driven by Gamma- distributed factors.

Monte Carlo simulation is used to estimate the loss distri-bution of a model with random default probabilities driven by Gamma- distributed factors but with Bernoulli default events. Monte Carlo simulation is required because there is no analytical solution for the loss distribution when Bernoulli defaults are combined with random default probabilities. To perform the Monte Carlo simulation, k is estimated from n using least squares. Five thousand random draws of k are then performed to determine pn. For each set of k, 5,000 random draws of Dn (the Bernoulli random variables representing de-fault events) are then generated. Th erefore, overall, 25 million random combinations are used for the Monte Carlo simula-tion. Th e loss distribution computed by Monte Carlo simu-lation is presented in Figure 10.4.

Th e loss distributions for Models 3 and 4 are presented in Figures 10.5 and 10.6, respectively:

• Th e VaR for Model 4 is 9 percent higher than the VaR computed by Monte Carlo simulation. Th is is

more evidence of the fact that the Poisson estimation results in an overestimation of the VaR: the only dif-ference between Model 3 and the model used for the Monte Carlo simulations is the distribution of default events.

• Th e VaR in Model 3 (which does not use a rigorous method to estimate n) is only 1 percent higher than the VaR computed by Monte Carlo simulation. Th is result is actually not surprising. Using weighted aver-ages of n to estimate k leads to an underestimation of

k and hence to an underestimation of the VaR. How-ever, this underestimation partially off sets the overes-timation resulting from the Poisson approximation.

Overall, using the simple method suggested by CSFP to estimate the k gives a value for the VaR that is very close to the value computed by Monte Carlo simulation. Note that the VaR is higher in all the models with random default proba-bilities than in the models with nonrandom probabilities. Th is refl ects the additional risk resulting from the uncertainty concerning the default probabilities.

1.05

1

0.95

0.9

0.85

0.8

VaR

ratio

300 600 900 1200 1500 1800 2100 2400 2700Individual probability of default (basis points)

VaRb/VaRp ratio for CSFP portfolio

VaR ratioVaR ratio

Source: Authors.Note: CSFP = Credit Suisse Financial Products; VaR = value at risk.

Figure 10.3 Ratio of Bernoulli VaR to Poisson VaR for CSFP Portfolio

Loss0 2 4 6 8 10 12

x 107

0.04

0.05

0.06

0.07

0.03

0.02

0.01

0

Prob

abili

ty



Figure 10.4 Bernoulli Defaults, Random Probabilities, Monte Carlo Simulation

Loss0 2 4 6 8 10 12

x 107

0.04

0.05

0.03

0.02

0.01

0

Prob

abili

ty



Figure 10.5 Model 3: Poisson Defaults, Uncorrelated Factors, Weighted Average



C. Random default probabilities, correlated factors

In this section, we present the portfolio loss distribution com-puted using Model 5, that is, using the model with random default probabilities and correlated factors described in Sec-tion 7. Th e only diff erence between Model 5 and Model 3 is the presence of the common factor . Th is common factor aff ects the distributions of k. In par tic u lar, it introduces some correlation between these factors. Th e portfolio loss distribu-tion was computed for two diff erent values of the variance of

(which is also the covariance between any two factors): 0.1 and 0.2. Th e results of these numerical experiments are pre-sented in Figures 10.7 and 10.8, respectively. Not surprisingly, the VaR is higher in Model 5 than in Model 3, and it increases with the variance of . Th is refl ects the additional risk of in-curring a large loss resulting from the positive intersector correlation, as well as the increased uncertainty concerning the default probabilities.

11. CONCLUSIONEach of the models presented here has specifi c features that make them useful in par tic u lar situations. Th e basic model with known probabilities and Bernoulli- distributed default events is useful when there is little uncertainty concerning default probabilities, when default probabilities are relatively high, and when the portfolio does not contain more than a few thousand obligors. At the cost of some approximations, CreditRisk+ and its extensions provide quasi- instantaneous solutions— even for very large portfolios— when default prob-abilities are infl uenced by a number of random latent factors. Th e alternative to using these models is to perform time- consuming Monte Carlo simulations. For our sample port-folio, the results of CreditRisk+ are very close to those of Monte Carlo simulations, even though this portfolio contains only 25 obligors with default probabilities as high as 30 percent.

Th is chapter provides a toolbox that has been used in the IMF ’s Financial Sector Assessment Program exercises to esti-mate credit risk pa ram e ters, including expected losses and credit VaR. Th e latter is the fundamental risk mea sure used to determine the economic capital required by a certain portfo-lio. Th is mea sure plays an important role in the IMF ’s bilat-eral fi nancial surveillance of member countries— the need to understand how the gap between regulatory and economic capital could be bridged continues to be an important aspect of IMF staff ’s analysis. Th e instruments presented in this chapter add a rigorous quantitative dimension to this pro cess.

REFERENCESAustrian Financial Market Authority and Oesterreichische Nation-

albank, 2004, “New Quantitative Models of Banking Supervi-sion” (Vienna, July).

Avesani, Renzo G., Kexue Liu, Alin Mirestean, and Jean Salvati, 2006, “Review and Implementation of Credit Risk Models in the Financial Sector Assessment Program,” IMF Working Paper

Loss

0 2 4 6 8 10 12

x 107

0.06

0.08

0.04

0.02

0

Prob

abili

ty



Figure 10.6 Model 4: Poisson Defaults, Uncorrelated Factors, Least Squares

Loss

0 2 4 6 8 10 12

x 107

0.06

0.08

0.04

0.02

0

Prob

abili

ty VaR at 99% = 52871760(40% of total exposure)


Figure 10.7 Model 5: Correlated Sectors, Intersector Covariance = 0.1

Loss

0 2 4 6 8 10 12

x 107

0.06

0.1

0.08

0.04

0.02

0

Prob

abili

ty



Figure 10.8 Model 5: Correlated Sectors, Intersector Covariance = 0.2



Gordy, Michael B., 2002, “Saddlepoint Approximation of Credit Risk,” Journal of Banking and Finance, Vol. 26, No. 6, pp. 1335– 53.

Haaf, Hermann, Oliver Reiss, and John Schoenmakers, 2003, “Nu-merically Stable Computation of CreditRisk+,” Weierstrass- Institut für Angewandte Analysis und Stochastik, Vol. 846, Berlin.

Koyluoglu, H. Ugur, and Andrew Hickman, 1998, “A Generalized Framework for Credit Risk Portfolio Models,” DefaultRisk.com, September 14. Available via the Internet: http:// www .defaultrisk .com /pp _model _17 .htm

Melchiori, Mario R., 2004, “CreditRisk+ by FFT,” Working Paper (Santa Fe: Universidad Nacional del Litoral).

06/134 (Washington: International Monetary Fund). Available via the Internet: http:// www .imf .org /external /pubs /cat /longres .aspx ?sk=19111

Credit Suisse Financial Products, 1997, “CreditRisk+: A Credit Risk Management Framework,” Technical Report (London: Credit Suisse First Boston). Available via the Internet: http:// www .csfb .com /institutional /research /assets /creditrisk .pdf

Crouhy, Michel, Dan Galai, and Robert Mark, 2000, “A Compara-tive Analysis of Current Credit Risk Models,” Journal of Banking and Finance, Vol. 24, Nos. 1– 2, pp. 59– 117.

Giese, Gotz, 2003, “Enhancing CreditRisk+,” Risk, Vol. 16, No. 4, pp. 73– 77.


AppendixProbability and Moment

Generating Functions

Consider a discrete random variable X that can take nonnegative integer values. Th e probability generating function (PGF) of X is the function GX defi ned by

GX (z) = E(z x ) = P(X = x ) z x ,x=0

with 0 z < 1 if z is real and |z| < 1 if z is complex. GX(z) is simply a polynomial whose coeffi cients are given by the probability distribution of X. A PGF uniquely identifi es a probability distribution: if two probability distributions have the same PGF, then they are identical.

Consider a second discrete random variable Y that can take nonnegative integer values. If X and Y are in de pen dent, then the PGF of X + Y is given by

GX + Y (z) = GX (z) GY (z).

Th e moment generating function (MGF) of X is the function defi ned by

Mx (z) = E(ezX )= P(X = x ) ezXx=0

.

As its name indicates, the MGF of X can be used to compute the moments of X. Th e mth moment of X about the origin is given by the mth derivative of MX valued at 0. Th is implies in par tic u lar that E (X ) = Mx

' (0) and Var(x) = MX" (0) MX

’ (0)[ ]2 .Th e joint MGF of two random variables X and Y is defi ned as

MX ,Y (z 1 , z2 ) = E(ez1 X + z2 Y ),

where z1 and z2 are two auxiliary variables with the same properties as z.If X and Y are two in de pen dent random variables, then the MGF of X + Y is given by:

MX + Y (z) = MX (z) MY (z),

and the joint MGF of X and Y becomes

MX,Y (z1, z2 ) = E(ez1 X ) E(ez2 Y ) = MX (z1) MY (z2 ).

If z1 = z2, then MX,Y(z1,z2) = MX + Y(z).



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