Atlantis Press Journal style


 

 

Collective Risk Generalization to Creditrisk+ 

 

Reza Habibi  

Iran Banking Institute, Central Bank of Iran, Tehran, Iran 

 

Received  19-June 19, 2018 

Accepted  July 25, 2018 

Abstract 

Using the collective risk models of actuarial science, the Creditrisk+ is extended to the case of random number 

obligors. First, mathematical methods to compute the distribution of total loss are studied. Then, the mathematical 

results are applied and verified numerically. The insufficiency data in risk management is a big problem. Thus, the 

case of data scarce is studied using a Bayesian approach. Finally, a concluding remarks section is also given.  

Keywords: Bayesian inference; Collective risk; Creditrisk+; Modelrisk; Data scarce problem; Moment generating 

function; Monte Carlo simulation 

1. Introduction 

In the last two decades, the credit risk modeling has 

been received considerable attentions in financial 

literatures (Avesani et al., 2006). Indeed, evaluation of 

the default probability of any borrower is the main 

concern of bankers when they lend to their clients (Liao 

et al.  2009). Almost for all debtors, the quantitative 

modeling of the credit risk is too important. To this end, 

some risk measures such as the credit value at risk are 

computed. There are two approaches for modeling 

credit value at risk: CreditMetrics and CreditRisk+ 

(Avesani et al., 2006). These approaches are applicable 

by regulators and risk managers which make decisions 

about the capital adequacy ratio. The CreditMetrics is 

used for rating (Lee, 2011). By this approach, the credit 

risk is defined as the risk that the security keepers don't 

materialize the security expected value because the 

borrower’s credit quality is deteriorated (Jarrow, 2011). 

Using CreditRisk+ the default models are constructed. 

That is, in this approach, credit risk is considered as risk  

 

that borrower of security failures on his/her promised 

obligations. Therefore, default of borrowers may make 

losses in the portfolio (Huang  and  Yu, 2010). Another 

difference is that the Creditrisk+ applies actuarial 

methodologies to derive the loss distribution of a 

financial portfolio. In this approach, just the default risk 

is modeled, and downgrade risk is not considered 

(Xiaohong et al., 2010) 

As it is stated, the Creditrisk+ provides an actuarial 

based framework for quantitative credit risk 

management. This software computes the portfolio and 

other debt instruments loss distributions which lead to 

determination of the required economic capital. Avesani 

et al. (2006) reviewed basic Creditrisk+ models exist in 

the literatures and proposed some generalizations 

including latent factors and random probabilities.  

Creditrisk+ basic model is very similar to the individual 

risk model of actuarial risk theory (see, Kaas et al., 

2008) at which, following notation Avesani et al. (2006), 

the total normalized loss λ of 𝑛  obligor's normalized 

losses 𝜆𝑖 = 𝐷𝑖 𝑣𝑖 , 𝑖 = 1, … , 𝑛  is given by Eq. (1), as 

follows 

𝜆 =   𝜆𝑖
𝑛
𝑖=1 =   𝐷𝑖 𝑣𝑖

𝑛
𝑖=1 ,                   (1) 

where the i -th default Di  of Eq.  

(1) occurs (is one) with probability 𝑝𝑖  and is zero with 

probability 1 − 𝑝𝑖  and 𝑣𝑖  is the 𝑖 -th normalized 

185

 
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Copyright © 2019, the Authors.  Published by Atlantis Press. 
This is an open access article under the CC BY-NC license (http://creativecommons.org/licenses/by-nc/4.0/).

Journal of Risk Analysis and Crisis Response, Vol. 8, No. 4 (December 2018) 185 -191



 

exposure. Although, Avesani et al. (2006) assumed that 

𝑣𝑖 's are fixed, however, it is possible to assume that they 

are random variables with gamma or log-normal 

(generally positive skewed) distributions.  

A natural extension (and different of latent factors 

considered by Avesani et al.) to the basic model of 

Creditrisk+ is the collective risk model where the 

number of obligors 𝑛 who contribute exposure, itself, is 

also a random variable denoted by 𝑁. It is assumed that 

𝑁 is independent of, 𝐷𝑖 's and 𝑣𝑖 's. However, usually 𝐷𝑖 's 

and 𝑣𝑖 's are correlated. In literatures, usually, selected 

distributions for 𝑁  are negative binomial and Poisson 

laws. Thus, like Eq. (1), the collective (credit) risk of a 

portfolio is presented as a random sum as 

 λ =   𝐷𝑖 𝑣𝑖 .
𝑁
𝑖=1  

When 𝑁 = 0, then λ=0. This fact implies that λ has a 

mixture distribution. Thus,  has a compound 

distribution.  

Although, Avesani et al. (2006) proposed an Excel add-

in MCM CR+, however, for computational purposes of 

some generalizations of Creditrisk+, even generaliza-

tions proposed by Avesani et al. (2006), the Modelrisk 

Excel add-in of Vose (2015) seems to be very useful. 

This paper considers the credit risk management. An 

interesting fact is that similar pattern of the above 

mentioned random sum occurs in banking operational 

risk management using the AMA method (see, 

Shevchenko, 2011).  

The rest of paper is organized as follows. In the next 

section, mathematical results are presented and 

Creditrisk+ is generalized to the case of collective 

Creditrisk+. The numerical examples are developed in 

the section 3. The big problem of risk management is 

the scarcity of data available in hand. Hence, the data 

scarce case is studied using Bayesian inferential tools. 

Finally, a concluding remarks section is also proposed. 

2. Mathematical Expression of Creditrisk 

To compute risk measure like value at risk (VaR), the 

distribution of λ is needed. As follows, some stylized 

facts about actuarial risk theory are updated and 

modified for Creditrisk+ designed for credit risk 

management. For each fact, necessary conditions are 

stated. 

(a) The first fact is about the moment generating 

function of collective impact of credit risk. To this end, 

assuming all 𝜆𝑖 's have the same distributions, one can 

see that the moment generating function of λ is given by 

Eq. (2), as follows  

𝑀λ  𝑥 = 𝑀𝑁 (log  𝑀𝜆𝑖  𝑥  ,                  (2) 

(Kaas et al., 2008, see page 43, Eq. (3.5)). For example, 

when 𝑁  has geometric distribution with parameter 

0 < 𝑝 < 1  and λ𝑖 's are exponentially distributed with 

parameter 1 and independent, then Eq. (2) reduces to 

𝑀𝜆  𝑥  as follows 

𝑀𝜆  𝑥 = 𝑝 + (1 − 𝑝)
𝑝

𝑝 − 𝑥
 

which is the moment generating function of a mixture 

distribution. Although, there is a closed form for 𝑀𝜆  𝑥 , 

however, this is not true, generally and Monte Carlo 

simulation should be used. 

(b) A natural method in actuarial science is the 

approximating the distribution of total claim. One can 

see that in collective risk models, the central limit 

theorem (CLT) approximation doesn't work well and 

two more accurate approximations for distribution of λ 

are translated gamma (TG) and normal power (NP) 

approximations. Traditionally, there exist two 

justifications for NP approximations. First, based on 

approximating the distribution of 𝑍 +
𝛾𝜆

6
(𝑍2 − 1) where 

𝑍  has standard normal distribution. The second 

justification is derived using Edgeworth expansion. To 

apply both TG and NP methods, the first three moments 

are needed. Let the cumulant generating function of λ be 

186

 
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Journal of Risk Analysis and Crisis Response, Vol. 8, No. 4 (December 2018) 185 -191



 

𝐾𝜆  𝑥 = log 𝑀λ  𝑥  . 

Indeed, in the case of identically distributed 𝜆𝑖 's, then 

the cumulant function is given by Eq. (3), as follows  

𝐾𝜆  𝑥 = 𝐾𝑁  𝐾𝜆𝑖  𝑥  .                           (3) 

Notice that the cumulant function of Eq. (3) is 

approximated by Eq. (4), as given in below 

𝐾𝜆  𝑥 = 𝐸 λ 𝑥 + 𝑣𝑎𝑟 λ 
𝑥 2

2
+ 𝐸   λ − E λ  

3
 

𝑥 3

6
+

𝑂 𝑥4 ,                                                         (4) 

(Kaas et al., 2008, see page 29). 

(c) Here, the mean and variance of 𝜆 are derived. They 

are given in Eq. (5) 

𝐸 λ = 𝐸( 𝐸(𝜆𝑖 )),

𝑁

𝑖=1

 

𝑣𝑎𝑟 λ = E( 𝑣𝑎𝑟(𝜆𝑖 )) + 𝑣𝑎𝑟( 𝐸(𝜆𝑖 )).
𝑁
𝑖=1

𝑁
𝑖=1         (5) 

Again, when 𝜆𝑖 's are identically distributed, then these 

moments are reduced to the Eq. (6) 

 
𝐸 𝜆 = 𝐸 𝑁 𝐸 𝜆𝑖  ,                                        

𝑣𝑎𝑟 λ = 𝐸 𝑁 𝑣𝑎𝑟 𝜆𝑖  + 𝐸
2  𝜆𝑖  𝑣𝑎𝑟 𝑁 ,

      (6) 

 (Kaas et al., 2008, see page 29, Eqs. (3.3, 3.4)). 

(d) As follows, the skew and kurtosis indices of  are 

derived. These values are necessary to find the 

distribution of . To this end, notice that the following 

Eq. (7) is correct that 

𝐸   λ − E λ  
3
 = 𝐸( 𝐸(𝜆𝑖 − 𝐸(𝜆𝑖 ))

3𝑁
𝑖=1 .            (7)  

When 𝜆𝑖 's are identically distributed, then equation (7) 

is changed to the Eq. (8), as follows 

𝐸   λ − E λ  
3
 = 𝐸 𝑁 𝐸(𝜆𝑖 − 𝐸(𝜆𝑖 ))

3,           (8) 

(Lehmann and Casella 1998, see page 29, Eq. (5.26)). 

Thus, to find the moments of , the marginal 

distributions of 𝜆𝑖 's are needed. To this end, notice that, 

each 𝜆𝑖  has a mixture distribution. To see this, notice 

that following Eq. (9) as follows, we have  

𝑀𝜆𝑖  𝑥 = 𝐸 𝐸 𝑒
𝐷𝑖𝑣𝑖𝑥  𝐷𝑖                        

               = 𝐸  𝑀𝑣𝑖  𝐷𝑖 𝑥   

                              = 𝑝𝑖 𝑀𝑣𝑖  𝑥 +  1 − 𝑝𝑖  .                    (9) 

This shows that the marginal distribution of 𝜆𝑖  is 

mixture of two components laws of 𝑣𝑖  and degenerate 

distribution on zero. To find moments of 𝜆𝑖 , it is seen 

that the following Eq. (10) is correct that 

 
𝐸 𝜆𝑖  =

𝜕 𝑀𝜆 𝑖
(𝑥 )

𝜕𝑥
|𝑥 =0 = 𝑝𝑖 𝐸 𝑣𝑖  ,

𝐸 𝜆𝑖
𝑘  = 𝑝𝑖 𝐸 𝑣𝑖

𝑘  , 𝑘 ≥ 2.               

                  (10) 

Thus, the variance of skewness of 𝜆𝑖  are given by Eq. 

(11), as follows 

 
𝑣𝑎𝑟 𝜆𝑖  = 𝑝𝑖 𝐸 𝑣𝑖

2  − 𝑝𝑖
2𝐸2 𝑣𝑖  ,                      

𝐸((λ𝑖 − 𝐸(λ𝑖 ))
3 = 𝑝𝑖 𝐸 𝑣𝑖

3 + 2𝑝𝑖
3𝐸3 𝑣𝑖  −

  

                    3𝑝𝑖 𝐸 𝑣𝑖  𝐸 𝑣𝑖
2 .

     (11) 

 (e) Here, the density function of 𝜆 is derived. To this 

end, let 𝑓𝜆1→𝑖
∗𝑖

 be the convolution of densities of 𝜆𝑗 , 𝑗 =

1, … , 𝑖  for 𝑖 ≥ 1 . Then, the density function 𝑓𝜆  𝑦 , is 

given by Eq. (12) as below 

𝑓𝜆  𝑦 = 𝑃 𝑁 = 0 +  𝑓𝜆1→𝑖
∗𝑖  𝑦 𝑃 𝑁 = 𝑖 .∞𝑖=1          (12) 

Assuming 𝜆𝑗  has normal distribution with parameter 𝜇 

and variance 𝜍2, then 𝑓𝜆1→𝑖
∗𝑖  𝑦  is the density of normal 

distribution with mean 𝑖𝜇 and 𝑖𝜍2 . The same result is 

correct for 𝜆𝑗  has Cauchy distributions, i.e. if 𝜆𝑗  is 

𝐶(𝜋, 𝛿), then 𝑓𝜆1→𝑖
∗𝑖  𝑦  is the density of 𝐶(𝑖𝜋, 𝑖𝛿), (Kaas 

et al., 2008, see page 44, Eq. (3.10)). To see why the 

formula 𝑓𝜆  𝑦  is correct, it is enough notice that the 

moment generating function of 𝜆  by Eq.(3) is given by 

Eq. (13), as follows 

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Journal of Risk Analysis and Crisis Response, Vol. 8, No. 4 (December 2018) 185 -191



 

𝑀λ  𝑥 = 𝐸 𝑒
𝑥𝜆    

= 𝐸  𝐸 𝑒 𝑥𝜆  𝑁    

=  𝑃 𝑁 = 0 +  𝐸 𝑒𝑥𝜆  𝑁 = 𝑖 𝑃 𝑁 = 𝑖 .        ∞𝑖=1 (13) 

 (f) Again, the density function of 𝜆 is derived in 

alternative method. Let 𝑓𝜆1→𝑖
∗𝑖  𝑦  be the result of the fast 

Fourier transform (FFT) procedure, when 𝑁 = 𝑖 is kept 

fixed and denote it by 𝐹𝐹𝑇𝑖 (𝑦), hence, it is given by Eq. 

(14)  

𝑓𝜆  𝑦 = 𝑃 𝑁 = 0 +  𝐹𝐹𝑇𝑖  𝑦 𝑃 𝑁 = 𝑖 .
∞
𝑖=1          (14) 

Indeed, for each 𝑖, Excel add-in MCM CR+ gives the 

𝐹𝐹𝑇𝑖 (𝑦)  and therefore, 𝑓𝜆  𝑦  is obtained. Since, 

𝑃 𝑁 = 𝑖  decreases as 𝑖 gets bigger, thus, it is enough to 

compute 𝐹𝐹𝑇𝑖 (𝑦) for suitable 𝑖 's. The maximum 𝑖 's is 

chosen such that 𝑃(𝑁 ≥ 𝑖) is negligible. The Aggregate 

Panjer option of  Modelrisk software computes the 

density 𝑓𝜆  𝑦  and thus the VaR is obtained. 

The following proposition summarizes the above 

discussion. 

Proposition. In a collective Creditrisk+ format, 

assuming λ𝑖 's are identically distributed, then 

(i) The moment generating function, mean and variance 

of  are given by Eq. (15), as follows 

 

𝑀𝜆  𝑥 = 𝑀𝑁 (log  𝑀λ𝑖  𝑥  ,                        

𝐸 λ = E N E 𝜆𝑖  ,                                        

𝑣𝑎𝑟 λ = E N var 𝜆𝑖  + 𝐸
2 𝜆𝑖  𝑣𝑎𝑟 𝑁 .

        (15) 

 (ii) Generally,  𝐸 𝜆𝑖
𝑘  = 𝑝𝑖 𝐸 𝑣𝑖

𝑘  , 𝑘 ≥ 1. 

(iii) NP approximation. For 𝜆∗ ≥ 1,  then NP 

approximation is given by Eq. (16) 

𝑃  
λ−𝐸 λ 

 𝑣𝑎𝑟  λ 
≤ 𝜆∗ +

𝛾λ

6
 𝜆∗2 − 1  ≈ 𝑃 𝑍 ≤ 𝜆∗ .       (16) 

Here, 𝛾λ  is the skewness of 𝜆 and 𝑍 has standard normal 

distribution with zero mean and variance one.  

(iv) TG approximation. Let 𝐹λ  be the distribution 

function of  and 𝐺(. , 𝛼, 𝛽) be the distribution function 

of gamma distribution with parameters 𝛼, 𝛽 . Then, 

𝐹λ  𝜆
∗  is approximated as Eq. (17) 

𝐹λ  𝜆
∗ ≈ 𝐺 𝜆∗ − 𝜆0 , 𝛼, 𝛽 .                       (17) 

Here, 𝜆0 , 𝛼, 𝛽  are chosen such that 𝐸 𝜆 = 𝜆0 + 𝛼𝛽, 

𝑣𝑎𝑟 𝜆 = 𝛼𝛽2 and 𝛾λ =
2

 𝛼
. 

(v) The density function 𝑓𝜆  𝑦 , is given by Eq. (18) 

𝑓𝜆  𝑦 = 𝑃 𝑁 = 0 +  𝐹𝐹𝑇𝑖  𝑦 𝑃 𝑁 = 𝑖 ,
∞
𝑖=1        (18)  

for each 𝑖, Excel add-in MCM CR+ gives the 𝐹𝐹𝑇𝑖  𝑦 . 

Remark 1. Avesani et al. (2006) showed that 𝐷𝑖  has 

Poisson distribution as 𝑝𝑖 = 𝑝 → 0. Let 𝑣𝑖 = 𝑣. Then,  

𝑀𝜆  𝑥 = 𝐸  𝑒
𝑥𝑣  𝐷𝑖

𝑁
𝑖=1  = 𝐸  𝐸  𝑒 𝑥𝑣  𝐷𝑖

𝑁
𝑖=1  𝑁  = 

𝐸 𝑒𝑁𝑝  𝑒
𝑥𝑣 −1  = 𝑒𝜃 (𝑒

𝑢 −1), 

where 𝑢 = 𝑝 𝑒 𝑥𝑣 − 1 . Also, let 𝜆 =  𝜆𝑖
𝑁
𝑖=1  and 𝑁 has 

Poisson distribution with parameter 𝜃 and 𝜃 is a gamma 

variable with parameters 𝛼, 𝛽, then, it is easy to see that 

the moment generating function is given by Eq. (19) 

𝑀𝜆  𝑥 =
1

(1−𝛽  𝑒 𝑔 𝑥  −1 )𝛼
,              (19) 

where 𝑔 𝑥 = 𝑙𝑜𝑔𝑀𝜆𝑖 (𝑥). Suppose that 𝜆𝑖 's have power 

series distributions with density given by 𝑓𝜑  𝜆𝑖  =

𝑒 𝜑 𝜆 𝑖 𝑕 (𝜆𝑖 )

𝑐 (𝜑 )
. The moment generating function is 𝑀𝜆𝑖  𝑥 =

𝑐(𝜑 +𝑥 )

𝑐 (𝜑 )
. It is easy to see that the moment generating 

function given by Eq. (20) 

𝑀𝜆  𝑥 = 𝑒
𝜃 (

𝑐 𝜑 +𝑥  

𝑐  𝜑  
−1)

.                 (20) 

Remark 2. An alternative method to approximate 

distribution of  is to use simultaneously both Monte 

Carlo or bootstrap (to simulate moments of 𝜆 ) and 

Edgeworth or Cornish-Fisher expansions (to simulate 𝐹λ , 

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Journal of Risk Analysis and Crisis Response, Vol. 8, No. 4 (December 2018) 185 -191



 

i.e., the distribution function of 𝜆). To this end, the Urn-

to-Urn method should be applied in running the Monte 

Carlo method to simulate mixture distributions.  

3. Examples  

Here, using some examples, the above results are 

applied and verified numerically.  

Example 1. Let 𝑁 = 1  is observed and 𝑣  takes two 

values 𝑎 and 𝑏. Here, 𝜆 takes three values 0, 𝑎, 𝑏. Let 

𝑃 𝑣 = 𝑎, 𝐷 = 1 = 𝑝𝑎  and 𝑃 𝑣 = 𝑏, 𝐷 = 1 = 𝑝𝑏  

where 𝑝𝑎 + 𝑝𝑏 = 𝑃 𝐷 = 1 = 𝑝. Then, 𝑃 𝜆 = 𝑎 = 𝑝𝑎  

and 𝑃 𝜆 = 𝑏 = 𝑝𝑏 . Thus, 𝑃 𝑣 = 𝑎|𝐷 = 1 =
𝑝𝑎

𝑝𝑎 +𝑝𝑏
=

1 − 𝑃 𝑣 = 𝑏|𝐷 = 1 . Next, suppose that 𝑁 has Poisson 

distribution with intensity parameter 10 and 𝑃 𝑣 =

𝑎|𝐷=1=0.1 and 𝑝=0.1. It is seen that 𝑝𝑎=0.01 and 

𝑝𝑏 = 0.09 . Indeed, 𝑃 𝜆 = 0 = 0.9 , 𝑃 𝜆 = 𝑎 = 0.01 

and 𝑃 𝜆 = 𝑏 = 0.09. For 𝑎 = 10 and 𝑏 = 20 thousand 

dollars, 𝐸 𝜆𝑖  = 1.8,  𝑣𝑎𝑟 𝜆𝑖  = 33.1  and 𝐸(𝜆𝑖 −

𝐸(𝜆𝑖))3=553.605. Thus, 𝐸λ=18, 𝑣𝑎𝑟λ=363.4 and 

𝛾λ = 29.07. To apply the TG approximation, it is seen 

that 𝛼 = 0.00473,  𝛽 = 277.18  and 𝜆0 = 16.69 . 

However, it doesn't work well, here. Using the NP 

approximation, the 0.99 VaR, in this case, is 317.8 

thousand dollars. The exact VaR is 240 thousand dollars. 

Example 2. Suppose that obligors are categorized to 

three categories. The probability of each category is 𝑝𝑖 . 

The amount of loss severity is 𝑣𝑖 = 0, 𝑎, 𝑏  for each 

category. Let 𝑁1 , 𝑁2 , 𝑁3 be the number of obligors in the 

first, second and third category with intensities 

parameters 𝜃1 , 𝜃2 , 𝜃3 , respectively. Thus, 

𝐸 𝜆 = 𝑎𝜃2𝑝2 + 𝑏𝜃3𝑝3  

𝑣𝑎𝑟 𝜆 = 𝑎2𝑣𝑎𝑟 𝑁2𝐷2 + 𝑏
2𝑣𝑎𝑟 𝑁3𝐷3 , 

where 𝑣𝑎𝑟 𝑁2𝐷2 = 𝜃2𝑝2 + 𝜃2
2𝑝2 (1 − 𝑝2 )  and 

𝑣𝑎𝑟 𝑁3𝐷3 = 𝜃3𝑝3 + 𝜃3
2𝑝3 (1 − 𝑝3 ) . Assuming 

𝑝2 = 0.02, 𝑝3 = 0.03, 𝑎 = 1, 𝑏 = 2, 𝜃2 = 10  and 

𝜃3 = 20, then the 99 percent VaR is . 

Example 3. Assume that a portfolio contains two types 

of loans. The probability of default in each type is 𝑝𝑖 . If 

there is a default, then the severity of default has 

distribution 𝑞𝑖 (𝑣𝑖 ). Assuming each type contain 

𝑁𝑖 , 𝑖 = 1,2  numbers of obligors having Poisson 

distributions with parameters 𝜃𝑖 , 𝑖 = 1,2.. Then,  

𝐸 𝜆 = 𝐸(𝑣1 )𝜃1𝑝1 + 𝐸(𝑣2 )𝜃2𝑝2  

𝑣𝑎𝑟 𝜆 = 𝑣𝑎𝑟 𝑁1𝐷1𝑣1 + 𝑣𝑎𝑟 𝑁2𝐷2𝑣2 . 

Let 𝜃1 = 10, 𝜃2 = 20, 𝑝1 = 0.01, 𝑝2 = 0.02 and losses 

𝑣1  and 𝑣2  are 5 or 10 dollars with probability of 0.5 and 

0.5. Then, the 99 percent VaR is . 

Example 4. Let 𝑣𝑖 = 0,2 with probability of 0.25 and 

0.75, respectively. Then the 99 percent VaR is . 

Suppose that 𝑃 𝑣𝑖 = 𝑣
∗ 𝐷𝑖 = 1 = 𝑞𝑖  and 𝑃 𝑣𝑖 ∈

𝑣,𝑣+𝑑𝑣=1−𝑞𝑖𝑣∗) for 0<𝑣<𝑣∗. Again, let 𝜆𝑖=𝑣𝑖𝐷𝑖 and 

𝑁 has Poisson distribution with intensity parameter 𝜃. 

Here, when 𝑞𝑖 = 0.3 , 𝑣
∗ = 5, 𝜃 = 20 , then the 99 

percent VaR is . 

Example 5. Let 𝜆𝑖 = 𝑣𝑖 𝐷𝑖  where 𝑣𝑖  has exponential 

distribution with scale parameter 𝛽 . Therefore, 𝜆𝑖 = 0 

with probability of 𝑝𝑖  and 𝜆𝑖  has density of 
1

𝛽
𝑒

−
𝜆 𝑖
𝛽  with 

probability of 1 − 𝑝𝑖 . For 𝑝𝑖 = 0.05,  𝛽 = 1,  and 𝑁  is 

Poisson with parameter , the 99 percent VaR is . 

4 Data scare case. Data scarce is a small sample 

problem. Both Shevchenko (2011) and Svensson (2015) 

studied the operational risk management when the data 

is scarce. Here, their approach in operational risk 

managements is extended to the credit risk management. 

The main idea of Bayesian method to compute 𝑓𝜆 , the 

density of 𝜆 =  𝜆𝑖
𝑁
𝑖=1 ,  assuming the priors for 𝑁  and 

𝜆𝑖 's are given, is to derive the posterior of them and then 

to simulate 𝜆, using  Monte Carlo or Panjer recursion 

methods. The Aggregate Panjer option of Modelrisk 

software is good tool, in this way. Also, the Excel add-

in MCM CR+ is a useful instrument to compute the VaR, 

directly. Here, throughout two examples, these methods 

are studied.  

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Remark 3. When we take Bayesian approach to deal 

with it, a priori knowledge is required. How to solve this 

problem is worth considering. Generally, there are two 

approaches. First, because of computational simplicity, 

the conjugate priors are selected with hyper-parameters 

that are estimated using historical data based on 

empirical Bayes method. To this end, some 

computational methods like EM algorithms may be used 

(Lehmann and Casella (1998)). These approaches are 

chosen in the current paper. The second approach 

searches for suitable prior distribution for parameters to 

derive the posterior distributions.  Some computational 

methods such as Monte Carlo Markov Chain (MCMC) 

are applied, in this case.  However, in this paper, since 

the aim is development of Creditrisk+ and mostly data 

sets are simulated one, therefore, the problem of prior 

selection and its subjective is not the main body of study. 

Example 6. Here, first, the Bayesian inference for 

frequency component of collective Creditrisk+ is given. 

Following Shevchenko and Wuthrich (2006), let 

𝑁1 , … , 𝑁𝑘  be a sequence of independent and identically 

distributed Poisson random variables with intensity 

parameter 𝜃, and suppose that 𝜃 has gamma distribution 

with parameters 𝛼, 𝛽. Shevchenko and Wuthrich (2006) 

showed that 𝜃 has the posterior density of gamma with 

parameters 𝛼 𝑘  and 𝛽 𝑘  given by 𝛼 𝑘 = 𝛼 +  𝑁𝑖
𝑘
𝑖=1 =

𝛼 𝑘−1 + 𝑁𝑘  and 𝛽 𝑘 =
𝛽

𝑘𝛽 +1
=

𝛽 𝑘−1

1+𝛽 𝑘−1
 and  

𝐸 𝑁𝑘 +1 𝑁1 , … , 𝑁𝑘  = 𝑧𝑁 +  1 − 𝑧 𝛼𝛽, 

where 𝑧  (based on credibility theory) is the credible 

factor given by 𝑧 =
𝑘𝛽

𝑘𝛽 +1
. Again, the Bayesian modeling 

of severities are proposed. Here, in spite of Shevchenko 

and Wuthrich (2006) which assumes a log-normal law 

for severity distribution, it is assumed that  𝜆𝑗 =

𝐷𝑗 𝑣𝑗 , 𝑗 = 1,2, … , 𝑀 where 𝐷𝑗  has Bernoulli distribution 

with parameter 𝑝  and 𝑝  has beta distribution with 

parameters 𝜔, 𝜑 . Thus, 𝑝  has beta distribution with 

parameters 𝜔 +  𝐷𝑗
𝑀
𝑗 =1  and 𝜑 + 𝑘 −  𝐷𝑗

𝑀
𝑗 =1 .  Also, 𝑣𝑖  

has log-normal distribution with parameters 𝜇 and 𝜍2 . 

Here, for simplicity arguments, it is assumed that 𝜍2 is 

known and 𝜇 has prior normal with mean к and standard 

deviation 𝜁 . Again, Shevchenko and Wuthrich (2006) 

showed that 𝜇  has normal distribution with mean 

M 𝜍 2 +𝜁 2  𝑣𝑗
𝑀
𝑗 =1

M 𝜍 2 +𝜁 2
 and variance 

𝜍 2𝜁 2

M 𝜍 2 +𝜁 2
. Indeed, using hyper-

parameters and real data, posteriors estimates of 

parameters are computed. Then, 𝐷𝑗 , 𝑣𝑗 , then 𝜆𝑗  as well 

as N are simulated. Finally, using the Panjer recursive 

"Eq. the density of λ are computed and the Bayesian 

risk measure VaR is computed 

Example 7. Following Shevchenko (2006), let 𝜆 =

𝜆𝑥 + 𝜆𝑦 . Here, 𝜆𝑥 =  𝜆𝑗
𝑥𝑁1

𝑗 =1  and 𝜆
𝑦 =  𝜆𝑗

𝑦𝑁2
𝑗 =1  with 

𝜆𝑗
𝑥 = 𝐷𝑗

𝑥 𝑣𝑗
𝑥  and 𝜆𝑗

𝑦
= 𝐷𝑗

𝑦
𝑣𝑗

𝑦
 where 𝑣𝑗

𝑥  and 𝑣𝑗
𝑦

 are 

correlated with correlation coefficient 𝜌 . Indeed, 

Gaussian, t, Clayton, Frank and Gumbel copulas are 

used to model dependency between  𝑣𝑗
𝑥  and 𝑣𝑗

𝑦
.  

4. Conclusions 

This paper considers the collective Creditrisk+ 

extensions to the regular Creditrisk+ version. Indeed, 

here, it is assumed that the number of default obligors 

are random obeys a specified distribution say Poisson or 

negative binomial. Also, the severity of exposures are 

assumed to be random. In the new format, some 

extended approaches are given to compute the density 

of total loss variable which is needed to calculate the 

VaR risk measure. The Bayesian solution is developed 

to the problem of scarcity of data as well as correlations 

between exposures. Extending the collective Creditrisk+ 

models to latent factors seems to be straightforward 

which is omitted.  

Acknowledgements 

Author thanks the referee for several improving 

suggestions. 

References  

R. G. Avesani, K. Liu, A. Mirestean and J. Salvati, Review 

and implementation of credit risk models in the financial 

sector assessment program, IMF Working Paper 06/134 

(Washington: International Monetary Fund). (2006). 

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Available via http:// www .imf .org /external /pubs /cat 

/longres.aspx ?sk=19111. 

S. J. Huang,  and  J. Yu, Bayesian  analysis  of  structural  

credit  risk models  with  microstructure noises.  Journal of 

Economic  Dynamics & Control 34, (2010) 2259-2272. 

R. A.  Jarrow,  Credit  market  equilibrium  theory  and  

evidence: Revisiting  the  structural  versus  reduced  form  

credit  risk  model debate. Finance Research Letters 8 

(2011), 2–7. 

R. Kaas, M. Goovaerts, J.  Dhaene, and M. Denuit, Modern 

actuarial risk theory using R (Springer, 2006, USA).  

W. C. Lee, Redefinition of the KMV model’s optimal default 

point based on genetic-algorithms–Evidence from Taiwan. 

Expert Systems with Applications 38 (2011), 1010-1022. 

H. H. Liao, T. K. Chen and C. W. Lu, Bank credit risk and 

structural credit models: Agency and information 

asymmetry perspectives. Journal of Banking & Finance 33 

(2009), 1520-1530. 

P. V. Shevchenko, Modeling operational risk. Technical report. 

CSIRO division of mathematical and information sciences, 

Sydney (2006).  Australia. 

P. V. Shevchenko, Modeling operational risk using Bayesian 

inference (Springer, 2011, USA).  

P. V. Shevchenko and M. V. Wüthrich, The structural 

modeling of operational risk via Bayesian inference: 

combining loss data with expert opinions. The Journal of 

Operational Risk 1 (2006), 3-26. 

K. P. Svensson. A Bayesian approach to modeling operational 

risk when data is scarce. Master's theses in mathematical 

sciences. Lund University. Sweden. (2015) 

D. Vose, Risk analysis. vailable:http://www.vosesoftware.com 

(2015). 

C. Xiaohong,  W.   Xiaoding,  and W.D  Desheng,  Credit  risk 

measurement  and  early  warning  of  SMEs:  An  

empirical  study  of listed SMEs in China. Decision 

Support Systems 49 (2010), 301–310. 

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