Estimation of Truncated Data Samples in Operational Risk Modeling

• Author: Konstantin Pavlikov,Stan Uryasev,Bakhodir Ergashev,Evangelos Sekeris
• DOI: http://doi.org/10.1111/jori.12062
• Date: 01 September 2016
• Published date: 01 September 2016

©2015 The Journal of Risk and Insurance. Vol.83, No. 3, 613–640 (2016).

DOI: 10.1111/jori.12062

Estimation of Truncated Data Samples in

Operational Risk Modeling

Bakhodir Ergashev

Konstantin Pavlikov

Stan Uryasev

Evangelos Sekeris

Abstract

This article addresses challenges of estimating operational risk regulatory

capital when a loss sample is truncated from below at a data collection

threshold. Recent operational risk literature reports that the attempts to

estimate loss distributions by the maximum likelihood method are not

always successful under the truncation approach that accounts for the

existence of censored losses—the likelihood surface is sometimes ascending

with no global solution. The literature offers an alternative called the shifting

approach, which estimates the loss distribution without taking into account

censored losses. Wepresent a necessary and sufficient condition for the exis-

tence of the global solution to the likelihood maximization problem under the

truncation approach when the true loss distribution is lognormal, and derive

a practically explicit expression for the global solution. Weshow by a simula-

tion study that, as the sample size increases, the capital bias by the truncation

approach declines while the bias by the shifting approach does not.

Introduction

Available databases of operational losses usually do not store records below some

data collection thresholds at the event level. Having a data collection threshold helps

Bakhodir Ergashev is at the Office of the Comptroller of the Currency, 400 7th Street SW,

Washington,DC 20024. Ergashev can be contacted via e-mail: bakhodir.ergashev@occ.treas.gov.

Konstantin Pavlikov and Stan Uryasev are at the Risk Management and Financial Engineering

Lab, Department of Industrial and Systems Engineering, 303 Weil Hall, University of Florida,

Gainesville, FL 32611. Pavlikov and Uryasev can be contacted via e-mail: kpavlikov@ufl.edu

and uryasev@ufl.edu. Evangelos Sekeris is at AON, 9841 Broken Land Parkway, Suite 305,

Columbia, MD 21046. Sekeris can be contacted via e-mail: evangelos.sekeris@aon.com. The

authors would like to thank Valery Valyaev for his valuable assistance in the development of

the paper as well as Konstantin Kalinchenko and Azamat Abdymomunov for reviewing the

paper and making helpful comments. We are also grateful to the two anonymous referees for

their very constructive and valuable suggestions. The views expressed in this paper are those

of the authors and do not necessarily reflect the position the Office of the Comptroller of the

Currency.

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614 The Journal of Risk and Insurance

to avoid difficulties with recording and storing too many small loss events. However,

omitting data falling below the threshold will most likely make the difficultproblem of

modeling operational risk accurately even more challenging. An important challenge

is whether one needs to take into account the fact that the data are censoredfrom below

at the data collection threshold. Theoretically, it would be more appropriate to account

for the existence of censored losses by explicitly acknowledging the truncated nature

of the data samples while modeling both severity and frequency distributions under

the widely accepted loss distribution approach (LDA). However, the operational risk

literature reports that attempts to fit operational losses using truncated severity distri-

butions by the method of maximum likelihood estimation are not always successful.

It turns out that, in many cases, the likelihood surface is nearly flat or even ascending

with no global maximum, which forces standardoptimization algorithms to encounter

numerical problems.Also, unconditional frequency estimates are quite high, reflecting

a large number of censored small losses and making the convolution algorithms (such

as Panjer recursion, Fourier transform, or Monte Carlo simulation) computationally

intense. To avoid the above-mentioned difficulties, some researchers suggest the use

of the so-called shifting approach. Under the shifting approach,a loss sample is shifted

to the left by the threshold value, the shifted sample is fitted by a non-truncated distri-

bution, and the resulting distribution is shifted to the right to derive the estimated loss

distribution. Proponents of the shifting approach present the following arguments,

among others, to support it. The shifting approach eliminates the numerical diffi-

culties with fitting that the truncation approach often encounters. Under the shifting

approach, fitting resultsare more stable and convolution algorithms are more efficient.

Also, in cases of very heavy-tailed severity distributions, omitting censored data leads

to negligible changes in capital estimates. Supporters of the truncation approach argue

that the shifting approach leads to stability of estimates at the expense of adding signif-

icant bias to parameter as well as capital estimates. So far,the operational risk literature

could not present clear evidence favoring one approach over the other. For instance,

using a simulation study,Shevchenko (2010) shows that for light-tail lognormal sever-

ity distributions, the shifting approach might induce significant bias in comparison to

the truncation approach, but this bias becomes insignificant for heavy-tail lognormal

distributions. Meanwhile, simulation studies performed by Cavallo et al. (2012) under

the shifting approach reveal that the overstatement or understatement of the severity

of an individual loss in the extreme right tail depends on the characteristics of the

data sample. In addition, the literature is not clear about the reasons as to why one

sample generates stable and reasonable fitting results under the truncation approach

while another sample, with similar characteristics, might lead to unstable and unrea-

sonable results. Also, it is not clear whether the trade-off between stability and bias

under the shifting approach is tolerable. In this article, we focus on the challenges of

estimating the parameters of the lognormal severity and Poisson frequency distribu-

tions under the truncation approach, and derive a specific, necessary, and sufficient

condition for the existence of the global solution to the severity parameter estima-

tion problem. In the sequel, we also call this condition the regularity condition. An

important implication of this result is that if the regularity condition is not satis-

fied, the maximum likelihood estimate does not exist, meaning the loss data sample

under consideration does not support the lognormal model and a different model

needs to be used. Violations of the regularity condition are the main reason leading