It is generally accepted that the choice of severity distribution in loss distribution approach has a significant effect on the operational risk capital estimation. However, the usually used parametric approaches with predefined distribution assumption might be not able to fit the severity distribution accurately. The objective of this paper is to propose a nonparametric operational risk modeling approach based on CornishFisher expansion. In this approach, the samples of severity are generated by CornishFisher expansion and then used in the Monte Carlo simulation to sketch the annual operational loss distribution. In the experiment, the proposed approach is employed to calculate the operational risk capital charge for the overall Chinese banking. The experiment dataset is the most comprehensive operational risk dataset in China as far as we know. The results show that the proposed approach is able to use the information of high order moments and might be more effective and stable than the usually used parametric approach.
Operational loss is an important source of bank risk. Take the incident of Barings Bank as an example; Nick Leeson’s unauthorized trading activities at Singapore office result in a total loss of US $ 1.4 billon and eventually lead to the sudden bankruptcy of Barings bank. Nowadays, researchers, practitioners, and regulatory institutions are fully aware of the importance of operational risk. Basel committee on banking supervision (BCBS for short) formally defines it as the risk of loss resulting from inadequate or failed internal processes, people, and systems from external events [
BCBS also introduces three approaches to the quantification of operational risk in a continuum of increasing sophistication and risk sensitivity, that is, basic indicator approach (BIA), standardized approach (SA), and advanced measurement approach (AMA) [
Among the eligible variants of AMA, loss distribution approach (LDA) is the most popular methodology by far [
The choice of severity distributions is usually supposed to have a more pronounced effect on capital than the choice of frequency distributions in LDA models [
The parametric approaches have dominated the estimation of severity distribution so far. It assumes a particular distribution for severity and then estimates the parameters by using moment estimation, maximum likelihood estimation, and so on [
Unlike parametric approach, the nonparametric approach derives a loss amount at random from loss data to perform a simulation without assuming any particular severity distribution [
Under different severity distribution assumption, results from LDA are greatly different from each other [
The objective of this paper is to propose a nonparametric operational risk modeling approach. This approach uses CornishFisher expansion to estimate the severities under the framework of LDA. CornishFisher expansion is a wellknown mathematical expansion which is able to approximate the quantiles of a random variable based on its first few cumulants or moments [
The rest of this paper is organized as follows. Section
In this section, the proposed nonparametric approach using CornishFisher expansion in the framework of LDA is illustrated at length. Firstly, the CornishFisher expansion is introduced. Then the concept and specific steps of the proposed approach are presented.
The CornishFisher expansion, firstly proposed by Cornish and Fisher [
Then it can be proven that
Next, according to (
Finally, by combing (
LDA is a technique firstly estimating a frequency distribution for the occurrence of operational losses and a severity distribution for the economic impact of individual loss separately. Then in order to obtain the total distribution of operational losses, these two distributions are combined through
Because the multiple convolutions are usually analytically complex and do not lend themselves to implementation with closedform formulas, Monte Carlo simulation is commonly used to derive the final annual distribution of operational risk loss. The procedure of Monte Carlo simulation based LDA is as follows: (1) determine loss frequency and loss severity distribution; (2) generate a number
As described in (
The framework of the proposed approach is shown in Figure
The framework of the proposed operational risk modeling approach.
As shown in Figure
The procedure of the proposed operational risk modeling approach.
Assume that there are
In this section, the proposed approach is employed to calculate the operational risk capital charge for the overall Chinese banking based on the most comprehensive operational risk dataset as far as we know. Firstly, the dataset and its statistical characteristics are introduced. Then the experiment results on the dataset are presented.
Today, many financial institutions have started collecting data on their own operational loss experience, but it will take some time before the size and quality of most institution’s databases allow reliable estimation of the parameters in the models [
The summary statistics of operational risk loss severity are shown in Panel A of Table
Summary statistics of loss severity and logarithmic loss severity.
Min.  Max.  Median  Mean  SD  Skewness  Kurtosis 

Panel A: statistics of loss severity  
0.01  800000  155.03  11405.83  50083.25  8.51  91.63 


Panel B: statistics of logarithmic loss severity  
−4.61  13.59  5.04  5.32  3.31  0.15  2.18 
SD: standard deviation.
The closer the unknown distribution is to the standard normal distribution, the more accurate the CornishFisher expansion is. Therefore, we calculate the natural logarithm of loss severity to make the distribution closer to the standard normal distribution. After taking natural logarithm, the summary statistics of new data are shown in Panel B of Table
In this section, the results of the proposed approach on the operational risk dataset are presented. Firstly, we will find a proper discrete distribution for frequency distribution. Poisson, negative binomial, and geometric distributions are three commonly used distributions in operational risk modeling [
Estimated parameters and goodnessoffit test results of frequency distribution.
Distribution  Parameters  KS test  




Poisson 

0.45  0.00 
Negative binomial 

0.20  0.43 
Geometric 

0.28  0.10 
For negative binomial distribution,
As for KS test, the larger the
Then we normalize the logarithmic operational risk severity by its mean 5.32 and standard deviation 3.31. The moments and cumulants of operational risk severity after normalization are shown in Table
Estimated parameters for CornishFisher expansion.




 

Moments  0  1  0.15  2.18  0.69 






 


Cumulants  0  1  0.15  −0.82  −0.83 
The larger the number of simulations is, the more accurate the results are and the longer the computational time required is. In order to balance simulation accuracy and time cost, we follow other studies and set the number of simulations as 100000 [
VaR of operational risk by using CornishFisher expansion
Orders  CornishFisher expansion  VaR (99.9%) 

1 order 

3380 
2 order 

13290 
3 order 

84 
4 order 

82 
5 order 

67 
6 order 

82 
7 order 

82 
Table
Among the parametric distributions, lognormal distribution is undoubtedly the most frequently used one for modeling operational risk severity [
In our published book the proposed approach drew the conclusion that the capital charge for operational risk is 31 billion CNY in 2007 [
In this paper, a nonparametric operational risk modeling approach based on CornishFisher expansion and loss distribution approach is proposed. This approach does not need to assume a distribution for severity beforehand. Only the cumulants or moments of the severities are required in Monte Carlo simulation process. In the experiment, based on the most comprehensive operational risk dataset as far as we know, the proposed approach is employed to calculate the operational risk capital charge for the overall Chinese banking.
The experiment shows that the resulting VaR values range from 67 billion CNY to 13290 billion CNY. The expansions with low order moments lead to large VaR values of 3390 and 13290 billion CNY. When higher order moments, that is, fourth and fifth moments, are added in the expansion, VaR converges to around 82 billion CNY. The widely used lognormal distribution only uses the information of the first and second moments, while the proposed approach is able to include the information of high order moments. Therefore, the proposed approach is supposed to model the operational risk in a more effective way.
The authors declare that there is no conflict of interests regarding the publication of this paper.
This research has been supported by Grants from the National Natural Science Foundation of China (71071148 and 71301087), Key Research Program of Institute of Policy and Management and Youth Innovation Promotion Association of the Chinese Academy of Sciences.