# The Bayesian Approach to Capital Allocation at Operational Risk: A Combination of Statistical Data and Expert Opinion

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

- (1)
- The IMA approach based on a proportionality assumption between the expected loss and unexpected loss, presented by Akkizidis and Bouchereau (2005) and Cruz et al. (2015);
- (2)
- The scorecard approach based on calculating a score for the risks measured by an entity and acting on its changing values, presented by Niven (2006), Akkizidis and Bouchereau (2005), Figini and Giudici (2013), and Facchinetti et al. (2019);
- (3)
- The LDA approach based on the distribution of frequency and the severity of losses.

## 3. Methodology

#### 3.1. The Risk Appetite Process

- (1)
- A Bottom-up analysis of the company’s current risk profile;
- (2)
- Interviews with the board of directors regarding the level of risk tolerance;
- (3)
- Alignment of risk appetite with the company’s goal and strategy;
- (4)
- Formalization of the risk appetite statement with approval from the board of directors;
- (5)
- Establishment of risk policies, risk limits, and risk-monitoring processes consistent with risk appetite;
- (6)
- Design and implementation of a risk-mitigation plan consistent with risk appetite;
- (7)
- Communication with local senior management for their buy in.

- (1)
- The risk profile;
- (2)
- The risk tolerance process;
- (3)
- The process for defining operational risk limits.

#### 3.2. The Process of Capital Allocation

#### 3.3. The Risk Mapping and Capital Requirements

#### 3.3.1. The Risk Mapping

- (1)
- The generating factor of the risk (hazard), which constitutes the factors that favor the occurrence of the risk incident for inexperienced personnel and the malfunction of the control device;
- (2)
- The operational risk event (incident), which constitutes the single incident whose occurrence can generates losses for the bank as internal fraud and external fraud;
- (3)
- The impact (loss), which it constitutes the amount of financial damage resulting from an event.

#### 3.3.2. The Operational Risk Categories

#### 3.3.3. The Business Lines

#### 3.3.4. Capital Requirements

- -
- $P{I}_{\mathrm{i}}$ and $C{I}_{i}$ are, respectively, the Interest Income and the Interest Expense for the year $\left(\mathrm{i}\right);$$AP{I}_{i}$ is the Interest Earning Assets for the year $\left(\mathrm{i}\right);$${D}_{i}$ is the Dividend Income for the year ($\mathrm{i})$;
- -
- $AC{E}_{i}$ and $AP{E}_{i}$ are the other operating income and the other operating expense for the year $\left(\mathrm{i}\right)$;
- -
- $PH{C}_{i}$ and $CH{C}_{i}$ are, respectively, the Fee Income Fee Expenses for the year $\left(\mathrm{i}\right);$
- -
- $PL{T}_{i}$ is the Net P&L Trading Book for the year $\left(\mathrm{i}\right)$;
- -
- $PL{B}_{i}$ is the Net P&L Banking Book for year $\left(\mathrm{i}\right)$.

#### 3.4. The LDA Approach and the $VaR$ of Operational Risk

#### 3.4.1. The Loss Distribution Approach $L\mathrm{DA}$

#### The Classical $LDA$ Model

- -
- ${X}_{i}$ is the random variable that represents the individual impact of operational risk incidents;
- -
- $N$ is the random variable that represents the number of occurrences on a horizon $T$.

#### The Pure Bayesian $LDA$ Approach

- $\pi \left(\theta \right)$ is the probability density of the parameter $\theta $ called the “prior density function”;
- $\pi \left(\theta /Y\right)$ is the conditional probability density function of the parameter $\theta $ knowing $Y$, which is called “posterior density”;
- $f\left(Y,\theta \right)$ is a probability density function of the couple $\left(Y,\theta \right)$;
- $f\left(Y/\theta \right)$ is the conditional density function of $Y$ knowing $\theta $; this is the likelihood function $f\left(Y/\theta \right)={{\displaystyle \prod}}_{i=1}^{m}{f}_{i}\left({Y}_{i}/\theta \right)$ with ${f}_{i}\left({Y}_{i}/\theta \right)$ as the conditional probability density function of ${Y}_{i}$;
- $f\left(Y\right)$ is the marginal density of $Y$ that can be written as $\int}f\left(Y/\theta \right)\pi \left(\theta \right)d\theta $.

- ii.
- The Bayesian Estimator ${\widehat{\theta}}_{Bay}$The parameter ($\theta $) can be univariate or multivariate. The estimate of the Bayesian posterior mean ${\widehat{\theta}}_{Bay}$ of $\theta $ is defined as follows:
- iii.
- If parameter ($\theta $) is univariate, the estimate of the Bayesian posterior mean of $\theta $, denoted as ${\widehat{\theta}}_{Bay}$, is a conditional expectation of $\theta $ knowing $Y$, defined by$${\widehat{\theta}}_{Bay}=E\left(\theta /Y\right)={\displaystyle \int}\theta \times \pi \left(\theta /Y\right)d\theta =\frac{{\displaystyle \int}\theta \times f\left(Y/\theta \right)\pi \left(\theta \right)d\theta}{f\left(Y\right)}.$$
- iv.
- In a multidimensional context, where $\theta =\left({\theta}_{1},{\theta}_{2},\dots ,{\theta}_{p}\right)$, the estimate of the Bayesian posterior mean of θ, denoted as ${\widehat{\theta}}_{Bay}$, is a conditional expectation of vector $\theta $ knowing $Y$, defined by$$\begin{array}{ll}{\widehat{\theta}}_{Bay}& =E\left(\theta /Y\right)=\left(E\left({\theta}_{1}/Y\right),E\left({\theta}_{2}/Y\right),\dots ,E\left({\theta}_{p}/Y\right)\right)\\ & =\left({\displaystyle \int}{\theta}_{1}\times \pi \left({\theta}_{1}/X\right)d{\theta}_{1},{\displaystyle \int}{\theta}_{2}\times \pi \left({\theta}_{2}/X\right)d{\theta}_{2},\dots .,{\displaystyle \int}{\theta}_{p}\times \pi \left({\theta}_{p}/X\right)d{\theta}_{p}\right).\end{array}$$

#### 3.4.2. Value at Risk of Operational Risk

#### Presentation of Value at Risk ($VaR$)

#### Definition of the Capital at Operational Risk

#### 3.5. Bayesian Modelling of the Expert Opinion

#### 3.5.1. Collecting and Modelling the Expert Opinion

#### Organization of the Process for Collecting the Expert Opinion

#### Presentation of the Delphi Method

- (1)
- Selection of issues or questions and development of questionnaires;
- (2)
- Selection of experts who are most knowledgeable about issues or questions of concern;
- (3)
- Issue familiarization of experts by providing sufficient details on the issues via questionnaires;
- (4)
- Elicitation of experts about the pertinent issues. The experts might not know who the other respondents are;
- (5)
- Aggregation and presentation of the results in the form of median values and using an inter-quartile range (i.e., 25% and 75% values);
- (6)
- Review of results and revision of the initial answers by experts. This iterative re-examination of issues sometimes increases the accuracy of results. Respondents who provide answers outside the inter-quartile range need to provide written justifications or arguments during the second cycle of completing the questionnaires;
- (7)
- Revision of results and re-review for another cycle. The process should be repeated until a complete consensus is achieved. Typically, the Delphi method requires two to four cycles or iterations;
- (8)
- A summary of the results is prepared with an argument summary for out of inter-quartile range values.

#### 3.5.2. Summary Presentation of the Process for Collecting Expert Opinions

- (1)
- Definition of the information requested;
- (2)
- Definition of interveners in the data collecting process;
- (3)
- Identification of problems, information sources and insufficiencies;
- (4)
- Analysis and collecting of pertinent information;
- (5)
- Choice of interveners in the data collecting process;
- (6)
- Knowledge of the operation’s objectives by the experts and a formation of those objectives.
- (7)
- Soliciting and collecting opinions;
- (8)
- Simulation, revision of assumptions, and estimates. If the expert provides his consent, we pass to the next step; otherwise, we repeat steps 6, 7, and 8;
- (9)
- Aggregation of estimates and overall validation;
- (10)
- Preparation of reporting and determination of results.

#### Definition of the Information Requested

- (1)
- The first consists in modelling the law a priori of the frequency and the severity of data by risk category. Indeed, the expert must provide the forms of the priori laws of frequency and severity and an estimation of their parameters (${\lambda}_{e},{\mu}_{e},{\sigma}_{e})$;
- (2)
- The second objective is the estimation of the expert weighting with the control functions (internal audit and permanent control).

- (1)
- The estimation of parameter ${\mu}_{\mathrm{e}}$ of the lognormal law $LN\left(\mu ,\sigma \right)$, which models the severity ${X}_{i}$ by risk category by knowing that σ is a constant and $\mu \text{}~\text{}N\left({\mu}_{\mathrm{e}},{\sigma}_{0}\right)$;
- (2)
- The estimation of parameter ${\lambda}_{e}$ of the Poisson’s law $P\left(\lambda \right)$, which models the frequency $N$ by risk category over a horizon ($T$) knowing that $\lambda ~$gamma (${a}_{0},\text{}{b}_{0}$).

#### Definition of Interveners in the Data Collecting Process

#### Identification of Problems, Information Sources, and Insufficiencies

#### Analysis and Collecting of Pertinent Information

- (1)
- The evolution of the bank’s size in terms of net banking income, the number of transactions, the number of incidents, the size of the banking network, and the number of customer claims.
- (2)
- The organizational and business changes, such as the introduction of new products, the industrialization of sales, control and treatment processes, external audits, control activities, and outsourcing of activities.
- (3)
- The major losses suffered and the action plans implemented, as well as their impact on the control and risk management device.
- (4)
- The formation programmers of operational risk and their frequency.

#### Choice of Interveners in the Data Collecting Process

- (1)
- Relevant expertise, academic and professional formation as well as professional experience;
- (2)
- The number of risk incidents declared and treated;
- (3)
- Knowledge and mastery of the control device;
- (4)
- The level of formation and, the knowledge of operational risk;
- (5)
- The level of knowledge of descriptive and inferential statistics;
- (6)
- Excellent communication abilities, flexibility, impartiality, and a capacity to generalize and simplify.

- (1)
- Relevant expertise, academic and professional formation, as well as professional experience;
- (2)
- The number of control and audit missions conducted annually;
- (3)
- The level of formation and knowledge of operational risk;
- (4)
- The level of knowledge of descriptive and inferential statistics;
- (5)
- Excellent communication abilities, flexibility, impartiality, and a capacity to generalize and simplify.

- (1)
- The description of the operation’s objectives;
- (2)
- The list of experts from the operating entities and person in charge of incident reporting, as well as hierarchical managers and the evaluators for internal audit and permanent control;
- (3)
- A summary description of the risks, tools, and operating system, as well as the organization and controls;
- (4)
- Basic terminology, definitions that should include probability density, arithmetic and weighted mean, standard deviation, mode, median, etc.;
- (5)
- A detailed description of the process by which meetings and workshops to collect expert opinions are conducted and the average duration of their conduct;
- (6)
- Methods for aggregating expert opinions.

- (1)
- The expert estimates the average loss per risk category that will be used to determine the parameters of the frequency law $P\left({\widehat{\lambda}}_{expert}\right)$ and severity law $LN\left({\widehat{\mu}}_{expert},{\widehat{\sigma}}_{expert}\right)$ knowing that ${\widehat{\sigma}}_{expert}$ is equal to $\sigma $, as determined by the likelihood. These parameters will be used to simulate, via Monte Carlo, three samples of the realizations concerning, respectively, the individual loss ${X}_{i}$, the frequency $N$, and the annual loss $P\left({{\displaystyle \sum}}_{i=1}^{n}{X}_{i}\right)$ Then, we analyze the characteristics of these samples with the expert, particularly the average, median, maximum, minimum, and maximum values, etc.;
- (2)
- If the expert accepts the simulations and their characteristics, the estimation of the parameters ${\widehat{\lambda}}_{expert},$ ${\widehat{\mu}}_{expert}$, and ${\widehat{\sigma}}_{expert}$ will be validated;
- (3)
- If the expert rejects the simulations, we will eliminate the outliers rejected by the expert and revise the expert’s initial estimates and proposed simulations in an iterative manner until the expert’s consent is obtained.

#### 3.5.3. Determination of the Bayesian Estimator

- (1)
- For frequency, Formula (19) defines the Bayesian estimator of $\lambda $ by$${\widehat{\lambda}}_{Bay}={\epsilon}_{1}\times {\lambda}_{\mathrm{expert}}+\left(1-{\epsilon}_{1}\right)\times {\lambda}_{\mathrm{observe}}.$$
- (2)
- For severity, Formula (20) defines the Bayesian estimator of $\mu $ by$${\widehat{\mu}}_{Bay}={\epsilon}_{2}\times {\mu}_{\mathrm{expert}}+\left(1-{\epsilon}_{2}\right)\times {\mu}_{\mathrm{observe}}.$$

## 4. Results

#### 4.1. Data Description

#### 4.2. The LDA Approach

#### 4.2.1. Statistical Estimation of Parameters

#### The Parameters of the Severity

#### The Parameters of the Frequency

#### 4.2.2. Experts’ Estimates

#### Expert Estimation ${\lambda}_{e}$ of Parameter $\lambda $

#### Estimation of the Parameter ${\mathsf{\mu}}_{\mathrm{e}}$

#### 4.2.3. The Bayesian Estimators of Parameters

#### 4.2.4. Determination of the $VaR$ by Risk Category

#### 4.3. Capital Allocation

- For $R{T}_{1}$, the database only includes proven losses, while risk events are generally adjusted without an accounting impact. However, they can have consequences if the losses recorded are not recovered;
- For $R{T}_{6}$, the experts believe that fraud attempts to target large amounts of money, especially those that have not been successful. However, if they are successful, the impact will be great.

- The bank studied is a medium-sized bank whose main activity is the granting of bank loans. Therefore, the use of simple and easy to implement approaches is its principal concern. However, other allocation approaches can be used to refine the allocation process;
- The approach we propose is based on the average loss per risk category, which favors the category $R{T}_{7}$. However, the collection approach used by the bank may bias the results because the bank accounts for the losses per fraud file even if a fraud is composed of different amounts distributed over several years;
- We have defined a list of criteria to score the experts and define their weighting, which makes the process very sensitive to the choice of scoring tool.

## 5. Discussion and Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Simulation of Aggregate Operational Losses

#### Appendix A.1. Presentation of the Simulation by the Inverse Cumulative Distribution Function

- (1)
- Simulate a realization ${n}_{j}$ of frequency $N$ from the chosen law of frequency $\left(P\left(\lambda \right)\text{}or\text{}BN\left(a,b\right)\right);$
- (2)
- Simulate ${n}_{j}$ realizations ${x}_{i}$, $1\le i\le {n}_{j},$ of severity $X$, from the chosen law of severity $\left(LN\left(\alpha ,\beta \right)\text{}or\text{}Wei\left(\alpha ,\beta \right)\right)$;
- (3)
- Calculate ${p}_{j}={{\displaystyle \sum}}_{i=1}^{{n}_{j}}{x}_{i}$, which will constitute a realization of the loss ${P}_{N}={{\displaystyle \sum}}_{i=1}^{N}{X}_{i}$.

**Theorem**

**A1.**

- Simulate a realization ${u}_{i}$ of the Uniform distribution $U\left[0,1\right]$;
- Calculate the inverse cumulative distribution function ${y}_{i}={F}^{-1}({u}_{i}$). Then, ${y}_{i}$ is considered to be a realization of $Y$.

#### Appendix A.1.1. Simulation of the Realizations ${n}_{j}$ for $1\le \mathrm{j}\le \mathrm{100,000}$

**(1) Simulation Poisson’s distribution.**

**Propriety:**Let ${\left({V}_{i}\right)}_{i\ge 1}$ be a sequence of exponential random variables of parameter $\lambda $. Then, the random variable is defined by

**Step 1: Simulation of**${\mathit{n}}_{\mathbf{1}}$

- We simulate a realization ${v}_{1}$ of the law $Exp\left(\lambda \right)$ by the inverse cumulative distribution function. For that, we must
- Simulate a realization ${u}_{1}$ of the Uniform law $U\left[0,1\right]$;
- Define the cumulative distribution function of the exponential law $Exp\left(\lambda \right)$ by ${F}^{-1}\left(u\right)=-\frac{ln\left(1-u\right)}{\lambda}$. We then deduct ${v}_{1}={F}^{-1}\left({u}_{1}\right)=-\frac{ln\left(1-{u}_{1}\right)}{\lambda}$

- If ${v}_{1}>1$ then ${n}_{1}=0$.

**Step j: A simulation of**$\mathit{n}}_{\mathit{j}},2\le \mathit{j}\le \mathrm{100,000$

**(2) Simulation of the laws**$\mathit{L}\mathit{N}\left(\mathit{\alpha},\mathit{\beta}\right)$.

- Simulate a realization ${u}_{i}$ of the Uniform law $U\left[0,\text{}1\right]$;
- Calculate ${x}_{i}={F}_{\left(\alpha ,\beta \right)}^{-1}\left({u}_{i}\right)$, where ${F}_{\left(\alpha ,\beta \right)}$ is a cumulative distribution function of the law $LN\left(\alpha ,\beta \right)$. As ${F}_{\left(\alpha ,\beta \right)}^{-1}\left({u}_{i}\right)$ has no analytical expression, we numerically simulate ${x}_{i}$.

#### Appendix A.1.2. Determination of Operating Losses

#### Appendix A.2. Calculation of the Capital at Operational Risk ($VaR$)

#### Appendix A.2.1. The Annual $VaR$ with Segmentation of the Database by Risk Category

- ✓
- ${N}_{c}$: The random variable that represents the frequency of losses of the risk category $R{T}_{c}$;
- ✓
- ${\mathrm{X}}_{ci}$: The random variable, for $1\le i\le {N}_{c}$, that represents the severity of the losses of the risk category $R{T}_{c}.$

#### Appendix A.2.2. The Modelling of the Loss Frequency for the Annual Horizon

#### Appendix A.2.3. The Modelling of the Loss Frequency for the Sub-Horizon ${T}_{k}=\frac{T}{k}$, $2\le k\le 12$

## References

- Abdymomunov, Azamat, and Ibrahim Ergen. 2017. Tail Dependence and Systemic Risk in Operational Losses of the US Banking. Industry. International Review of Finance 17. [Google Scholar] [CrossRef]
- Akkizidis, Ioannis S., and Vivianne Bouchereau. 2005. Guide to Optimal Operational & Basel II. Abingdon: Auerbach Publications. Taylor & Francis Group, p. 139. [Google Scholar]
- Alexander, Carol. 2003. Operational Risk: Regulation, Analysis and Management. London: FT Prentice Hall. [Google Scholar]
- Aumann, Robert J., and Lloyd S. Shapley. 1974. Value of Non-Atomic Games. Princeton: Princeton University Press. [Google Scholar]
- Ayyub, Bilal. 2001. A Practical Guide on Conducting Expert-Opinion Elicitation of Probabilities and Consequences for Corps Facilities. U.S. Army Corps of Engineers Institute. Available online: https://www.iwr.usace.army.mil/Portals/70/docs/iwrreports/01-R-01.pdf (accessed on 10 April 2019).
- BCBS. 2006. International Convergence of Capital Measurement and Capital Standards. Basel: Bank for International Settlements. [Google Scholar]
- BCBS. 2016. Standardised Measurement Approach for Operational Risk, Consultative Document. Basel: Bank for International Settlements. [Google Scholar]
- BCBS. 2017. Basel III: Finalising Post-Crisis Reforms. Basel: Bank for International Settlements. [Google Scholar]
- Bee, Marco. 2006. Estimating the Parameters in the Loss Distribution Approach: How Can We Deal with Truncated Data. The Advanced Measurement Approach to Operational Risk. London: E. Davis, Risk Books. [Google Scholar]
- Benbachir, Saâd, and Mohamed Habachi. 2018. Assessing the Impact of the Modelling on the Operational Risk Profile of Banks. International Journal of Applied Engineering Research 13: 9060–82. [Google Scholar]
- Boonen, Tim Jaij. 2019. Static and dynamic risk capital allocations with the Euler rule. Journal of Risk. [Google Scholar] [CrossRef]
- Brechmann, Eike, Claudia Czado, and Sandra Paterlin. 2013. Flexible dependence modeling of operational risk losses and its impact on total capital requirements. Journal of Banking & Finance 40: 271–85. [Google Scholar] [CrossRef] [Green Version]
- Chernobai, Anna, Christian Menn, Stefan Truck, and Svetiozar Rachev. 2005. A Note on the Estimation of the Frequency and Severity Distribution of Operational Losses. Mathematical Scientist 30: 87–97. [Google Scholar]
- Chernobai, Anna, Svetiozar Rachev, and Frank J. Fabozzi. 2007. Operational Risk. Hoboken: John Wiley & Sons, Inc. [Google Scholar]
- Cohen, Ruben. 2016. An assessment of operational loss data and its implications for risk capital modeling. Journal of Operational Risk 11: 71–95. [Google Scholar] [CrossRef]
- Cohen, Ruben. 2018. An operational risk capital model based on the loss distribution approach. Journal of Operational Risk 13: 59–81. [Google Scholar] [CrossRef]
- Cope, Eric, and Gianluca Antonini. 2008. Observed correlations and dependencies among operational losses in the ORX consortium database. The Journal of Operational Risk 3: 47–74. [Google Scholar] [CrossRef]
- Cruz, Marcelo G. 2002. Modeling, Measuring and Hedging Opérational Risk. Hoboken: John Wiley & Sons, Inc. [Google Scholar]
- Cruz, Marcelo G., Peters Gareth W., and Pavel V. Shevchenko. 2015. Fondamental Aspect of Operational Risk and Insurance Analytics. Hoboken: Wiley & Sons, Inc., p. 11. [Google Scholar]
- Dalla Valle, Luciana. 2009. Bayesian Copulae Distributions, with Application to Operational Risk Management. Methodology and Computing in Applied Probability 11: 95–115. [Google Scholar] [CrossRef]
- Dalla Valle, Luciana, and Paolo Giudici. 2008. A Bayesian approach to estimate the marginal loss distributions in operational risk management. Computational Statistics & Data Analysis 52: 3107–27. [Google Scholar] [CrossRef]
- Denault, Michel. 2001. Coherent allocation of risk capital. Journal of Risk 4: 1–34. [Google Scholar] [CrossRef] [Green Version]
- Dhaene, Jan, Andreas Tsanakas, Emiliano A. Valdez, and Steven Vanduffel. 2012. Optimal Capital Allocation Principles. Journal of Risk and Insurance 79: 1–28. [Google Scholar] [CrossRef] [Green Version]
- Driessen, Theodorus Stephanus Hubertus, and Stephanus Hendrikus Tijs. 1985. The Cost Gap Method and Other Cost Allocation Methods for Multipurpose Water Projects. Water Resources Research 21: 1469–75. [Google Scholar] [CrossRef]
- Facchinetti, Silvia, Paolo Giudici, and Silvia Angela Osmetti. 2019. Cyber risk measurement with ordinal data. Statistical Methods and Applications. [Google Scholar] [CrossRef]
- Figini, Silvia, and Paolo Giudici. 2013. Measuring risk with ordinal variables. The Journal of Operational Risk 8: 35–43. [Google Scholar] [CrossRef]
- Figini, Silvia, Lijun Gao, and Paolo Giudici. 2014. Bayesian operational risk models. Journal of Operational Risk 10: 45–60. [Google Scholar] [CrossRef]
- Frachot, Antoine, Pierre Georges, and Thierry Roncalli. 2001. Loss Distribution Approach for Operational Risk. Lyonnais: Groupe de Recherche Opérationnelle, Crédit Lyonnais. [Google Scholar] [CrossRef] [Green Version]
- Frachot, Antoine, Olivier Moudoulaud, and Thierry Roncalli. 2003. Loss Distribution Approach in Practice. In The Basel Handbook: A Guide for Financial Practioners. Chicago: Micheal Ong, Risk Books. [Google Scholar]
- Giudici, Paolo, and Annalisa Bilotta. 2004. Modelling operational losses: A bayesian approach. Quality and Reliability Engineering International 20: 407–17. [Google Scholar] [CrossRef]
- Groenewald, Andries. 2014. Practical methods of modelling operational risk. Paper presented at the Lecture, Actuarial, Society of South Africa’s 2014 Convention, Cape Town, South Africa, October 22–23; Available online: https://actuarialsociety.org.za/convention/convention2014/assets/pdf/papers/2014%20ASSA%20Groenewald.pdf (accessed on 23 May 2019).
- Hamlen, Susan S., William A. Hamlen Jr., and John T. Tschirhart. 1977. The Use of Core Theory in Evaluating Joint Cost Allocation Schemes. The Accounting Review 52: 616–27. [Google Scholar]
- Helmer, Olaf. 1968. Analysis of the Future: The Delphi Method. Available online: https://www.rand.org/pubs/papers/P3558.html (accessed on 15 January 2019).
- Jorion, Philippe. 2001. Value at Risk: The New Benchmark for Managing Financial Risk. New York: McGraw-Hill. [Google Scholar]
- King, Jack L. 2001. Operational Risk, Measurement and Modelling. New York: Wiley Finance. [Google Scholar]
- McConnell, Patrick. 2017. Standardized measurement approach: is comparability attainable? The Journal of Operational Risk 18: 71–110. [Google Scholar] [CrossRef]
- Mignola, Giuilio, Roberto Ugoccioni, and Eric Cope. 2016. Comments on the BCBS proposal for a new standardized approach for operational risk. The Journal of Operational Risk 11: 51–69. [Google Scholar] [CrossRef]
- Niven, Paul R. 2006. Balanced Scorecard Step-By-Step. Hoboken: John Wiley & Sons, Inc. [Google Scholar]
- Panjer, Harry H. 2002. Measurement of Risk, Solvency Requirements and Allocation of Capital within Financial Conglomerates. Available online: http://library.soa.org/files/pdf/measurement_risk.pdf (accessed on 20 February 2019).
- Peters, Gareth W., and Scott A. Sisson. 2006. Bayesian inference, Monte Carlo sampling and operational risk. The Journal of Operational Risk 1: 27–50. [Google Scholar] [CrossRef]
- Peters, Gareth W., Pavel V. Shevchenko, Bertrand Hassani, and Ariane Chapelle. 2016. Should the advanced measurement approach be replaced with the standardized measurement approach for operational risk? The Journal of Operational Risk 11: 1–49. [Google Scholar] [CrossRef] [Green Version]
- Shang, Kailan, and Zhen Chen. 2012. Risk Appetite Linkage with Strategy Planning. Available online: https://web.actuaries.ie/sites/default/files/erm-esources/research_risk_app_link_report.Pdf (accessed on 10 April 2019).
- Shapley, Lloyd Stowell. 1953. A value for n-person games. In Contributions to the Theory of Games II. Edited by H. W. Kuhn and A. W. Tucker. Princeton: Princeton University Press, pp. 307–17. [Google Scholar]
- Shevchenko, Pavel V. 2010. Implementing Loss Distribution Approach for Operational Risk. Applied Stochastic Models in Business and Industry 26: 277–307. [Google Scholar] [CrossRef] [Green Version]
- Shevchenko, Pavel V. 2011. Modelling Operational Risk Using Bayesian Inference. Springer. [Google Scholar]
- Shevchenko, Pavel V., and Grigory Temnov. 2009. Modeling operational risk data reported above a time-varying threshold. The Journal of Operational Risk 4: 19–42. [Google Scholar] [CrossRef] [Green Version]
- Tasche, Dirk. 2007. Capital Allocation to Business Units and Sub-Portfolios: The Euler Principle. Available online: http://arxiv.org/PScache/arxiv/pdf/0708/0708.2542v3.pdf (accessed on 13 March 2019).
- Urbina, Jilber, and Guillén Montserrat. 2014. An application of capital allocation principles to operational risk and the cost of fraud. Expert Systems with Applications 41: 7023–31. [Google Scholar] [CrossRef]

1 | The rubrics for calculating the BI are detailed in the Appendix A: the definition of the components of the BI of the Basel III reform. |

Qualification | Low | Medium | High |
---|---|---|---|

Rating | 1 | 2 | 3 |

Score | $\left(6\mathit{t}\mathit{o}7\right)$ | $\left(8\mathit{t}\mathit{o}9\right)$ | (10 to 11) | $\left(12\mathit{t}\mathit{o}14\right)$ | $\left(15\mathit{t}\mathit{o}18\right)$ |
---|---|---|---|---|---|

Weighting | 10% | 25% | 40% | 50% | 75% |

$\mathbf{Mean}\text{}\mathbf{of}\text{}\mathbf{Individual}\text{}\mathbf{Losses}\text{}{\mathit{X}}_{\mathit{i}}$ | Standard Deviation | Skewness | Kurtosis |
---|---|---|---|

468,730 | 8,719,755.32 | 36.28 | 1430.99 |

$\mathit{R}{\mathit{T}}_{\mathit{c}}$ | $\mathbf{Distribution}\text{}\mathbf{of}\text{}\mathbf{Individual}\text{}\mathbf{Losses}\text{}{\mathit{X}}_{\mathit{i}}\text{}\mathbf{in}\text{}\mathbf{Number}\text{}\left(\mathbf{in}\text{}\mathbf{MAD}\right)$ | $\mathbf{Mean}\text{}\mathbf{of}\text{}\mathbf{Individual}\text{}\mathbf{Losses}\text{}{\mathit{X}}_{\mathit{i}}$ | $\mathbf{Distribution}\text{}\mathbf{of}\text{}\mathbf{Individual}\text{}\mathbf{Losses}\text{}{\mathit{X}}_{\mathit{i}}\text{}\mathbf{by}\text{}\mathbf{Amount}$ | Standard Deviation | Skewness | Kurtosis |
---|---|---|---|---|---|---|

$R{T}_{1}$ | 12% | 781,175.04 | 16.8% | 9,355,985.39 | 15.33 | 234.22 |

$R{T}_{2}$ | 9% | 13,213.10 | 0.3% | 63,647.81 | 7.123 | 52.090 |

$R{T}_{3}$ | 45% | 34,573.71 | 0.7% | 363,852.27 | 20.47 | 457.38 |

$R{T}_{4}$ | 4% | 183,689.13 | 3.9% | 781,324.16 | 5.81 | 32.690 |

$R{T}_{5}$ | 2% | 199,146.42 | 4.3% | 84,075.40 | 1.853 | 4.475 |

$R{T}_{6}$ | 19% | 190,746.84 | 4.1% | 1,379,733.78 | 10.248 | 113.639 |

$R{T}_{7}$ | 10% | 3,249,849.84 | 69.9% | 25,490,889.73 | 13.408 | 184.914 |

$\mathit{R}{\mathit{T}}_{\mathit{c}}$ | Mean | Standard Deviation |
---|---|---|

$R{T}_{1}$ | 10.57 | 11 |

$R{T}_{2}$ | 11.87 | 14.54 |

$R{T}_{3}$ | 52.96 | 54.92 |

$R{T}_{4}$ | 3.17 | 2.71 |

$R{T}_{5}$ | 2.92 | 3.15 |

$R{T}_{6}$ | 38.72 | 52.82 |

$R{T}_{7}$ | 7.12 | 10.08 |

**Table 6.**Estimation and adjustment test of the $LN\left({\mu}_{h},{\sigma}_{h}\right)$ by risk category.

$\mathit{R}{\mathit{T}}_{\mathit{c}}$ | $\mathit{L}\mathit{N}\left({\mathit{\mu}}_{\mathit{h}},{\mathit{\sigma}}_{\mathit{h}}\right)$ | Kolmogorov-Smirnov Test | |
---|---|---|---|

${\mathit{\mu}}_{\mathit{h}}$ | ${\mathit{\sigma}}_{\mathit{h}}$ | p-Value | |

$R{T}_{1}$ | 10.60 | 1.67 | 0.084 |

$R{T}_{2}$ | 7.51 | 1.58 | 0.258 |

$R{T}_{3}$ | 8.59 | 1.49 | 0.419 |

$R{T}_{4}$ | 9.84 | 2.09 | 0.831 |

$R{T}_{5}$ | 12.14 | 0.35 | 0.649 |

$R{T}_{6}$ | 8.08 | 2.49 | <0.0001 |

$R{T}_{7}$ | 11.52 | 2.49 | 0.723 |

**Table 7.**Estimation and adjustment test of the $P\left({\lambda}_{h}\right)$ and $BN\left({a}_{h},{b}_{h}\right)$ by risk category.

$\mathit{R}{\mathit{T}}_{\mathit{c}}$ | $\mathbf{Poisson}\text{}\mathit{P}\left({\mathit{\lambda}}_{\mathit{h}}\right)$ | p-Value Chi-Square Test | $\mathbf{Negative}-\mathbf{Binomial}\text{}\mathit{B}\mathit{N}\left({\mathit{a}}_{\mathit{h}},{\mathit{b}}_{\mathit{h}}\right)$ | p-Value Chi-Square Test | |
---|---|---|---|---|---|

${\mathit{\lambda}}_{\mathit{h}}$ | ${\mathit{a}}_{\mathit{h}}$ | ${\mathit{b}}_{\mathit{h}}$ | |||

$R{T}_{1}$ | 10.57 | <0.0001 | 0.83 | 12.87 | 0.01 |

$R{T}_{2}$ | 11.87 | <0.0001 | 0.74 | 16.12 | 0.040 |

$R{T}_{3}$ | 52.96 | <0.0001 | 0.87 | 60.84 | 0.385 |

$R{T}_{4}$ | 3.17 | <0.0001 | 2.93 | 1.08 | 0.017 |

$R{T}_{5}$ | 2.92 | <0.0001 | 2.38 | 1.23 | 0.030 |

$R{T}_{6}$ | 38.72 | <0.0001 | 0.42 | 92.66 | 0.054 |

$R{T}_{7}$ | 7.12 | <0.0001 | 1.25 | 5.68 | <0.0001 |

$\mathit{R}{\mathit{T}}_{\mathit{c}}$ | $\mathbf{Mean}\text{}\mathbf{Annual}\text{}\mathbf{Empirical}\text{}\mathbf{Losses}\text{}({\mathit{P}}_{\mathit{M}}$$)\text{}\mathbf{by}\text{}\mathbf{Category}$ | Mean Losses Structure by Category | $\mathbf{Expert}\text{}\mathbf{Estimate}\text{}\mathbf{of}\text{}\mathbf{Mean}\text{}\mathbf{Losses}\text{}\left({\mathit{P}}_{\mathit{M}}\right)\text{}\mathbf{by}\text{}\mathbf{Category}$ |
---|---|---|---|

$R{T}_{1}$ | 18,966 | 31.6% | 21,313 |

$R{T}_{2}$ | 314 | 0.5% | 352 |

$R{T}_{3}$ | 2349 | 3.9% | 2640 |

$R{T}_{4}$ | 2266 | 3.8% | 2546 |

$R{T}_{5}$ | 1704 | 2.8% | 1915 |

$R{T}_{6}$ | 4706 | 7.8% | 5289 |

$R{T}_{7}$ | 29,762 | 49.5% | 33,445 |

$\mathit{R}{\mathit{T}}_{\mathit{c}}$ | ${\mathit{\lambda}}_{\mathit{e}}$ |
---|---|

$R{T}_{1}$ | 11.5 |

$R{T}_{2}$ | 14.3 |

$R{T}_{3}$ | 54.6 |

$R{T}_{4}$ | 4.07 |

$R{T}_{5}$ | 3.40 |

$R{T}_{6}$ | 5.8 |

$R{T}_{7}$ | 3.5 |

$\mathit{R}{\mathit{T}}_{\mathit{c}}$ | ${\mathit{P}}_{\mathit{M}}$ | ${\mathit{\lambda}}_{\mathit{e}}$ | ${\mathit{\mu}}_{\mathit{e}}$ |
---|---|---|---|

$R{T}_{1}$ | 21,313 | 11.5 | 6.13 |

$R{T}_{2}$ | 352 | 14.3 | 1.95 |

$R{T}_{3}$ | 2640 | 54.6 | 2.77 |

$R{T}_{4}$ | 2546 | 4.07 | 4.25 |

$R{T}_{5}$ | 1915 | 3.40 | 6.27 |

$R{T}_{6}$ | 5289 | 5.8 | 3.72 |

$R{T}_{7}$ | 33,445 | 3.5 | 6.06 |

$\mathit{R}{\mathit{T}}_{\mathit{c}}$ | $\mathit{P}\left(\mathit{\lambda}\right)$ | $\mathit{L}\mathit{N}(\mathit{\mu},\mathit{\sigma})$ | |
---|---|---|---|

${\widehat{\mathit{\lambda}}}_{\mathit{B}\mathit{a}\mathit{y}}$ | ${\widehat{\mathit{\mu}}}_{\mathit{B}\mathit{a}\mathit{y}}$ | ${\mathit{\sigma}}_{\mathit{h}}$ | |

$R{T}_{1}$ | 10.80 | 9.48 | 1.67 |

$R{T}_{2}$ | 12.48 | 6.12 | 1.58 |

$R{T}_{3}$ | 53.37 | 7.14 | 1.49 |

$R{T}_{4}$ | 3.40 | 8.44 | 2.09 |

$R{T}_{5}$ | 3.04 | 10.67 | 0.35 |

$R{T}_{6}$ | 30.49 | 6.99 | 2.49 |

$R{T}_{7}$ | 6.22 | 10.16 | 2.49 |

$R{T}_{c}$ | $Va{R}_{C,h}$$({\lambda}_{h},{\mu}_{h},{\sigma}_{h})$ | $Va{R}_{C,bay}$$({\widehat{\lambda}}_{Bay},{\widehat{\mu}}_{Bay},{\sigma}_{h})$ |

$R{T}_{1}$ | 45,526 | 14,449 |

$R{T}_{2}$ | 1560 | 367 |

$R{T}_{3}$ | 6926 | 1767 |

$R{T}_{4}$ | 45,526 | 11,764 |

$R{T}_{5}$ | 4026 | 955 |

$R{T}_{6}$ | 164,222 | 45554 |

$R{T}_{7}$ | 1,641,160 | 384680 |

$\mathit{R}{\mathit{T}}_{\mathit{c}}$ | $\mathbf{Percentage}\text{}\mathbf{of}\text{}\mathbf{Capital}\text{}\mathbf{Allocated}\text{}\mathbf{under}\text{}\mathbf{the}\text{}\mathbf{Classical}\text{}\mathit{L}\mathit{D}\mathit{A}\text{}(\%)$ | Percentage of Capital Allocated under the Bayesian LDA (%) | Deviation (%) |
---|---|---|---|

$R{T}_{1}$ | 2.38 | 3.14 | 31.93% |

$R{T}_{2}$ | 0.08 | 0.08 | 0.00% |

$R{T}_{3}$ | 0.36 | 0.38 | 5.56% |

$R{T}_{4}$ | 2.38 | 2.56 | 7.56% |

$R{T}_{5}$ | 0.21 | 0.21 | 0.00% |

$R{T}_{6}$ | 8.60 | 9.91 | 15.23% |

$R{T}_{7}$ | 85.97 | 83.71 | −2.63% |

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## Share and Cite

**MDPI and ACS Style**

Habachi, M.; Benbachir, S.
The Bayesian Approach to Capital Allocation at Operational Risk: A Combination of Statistical Data and Expert Opinion. *Int. J. Financial Stud.* **2020**, *8*, 9.
https://doi.org/10.3390/ijfs8010009

**AMA Style**

Habachi M, Benbachir S.
The Bayesian Approach to Capital Allocation at Operational Risk: A Combination of Statistical Data and Expert Opinion. *International Journal of Financial Studies*. 2020; 8(1):9.
https://doi.org/10.3390/ijfs8010009

**Chicago/Turabian Style**

Habachi, Mohamed, and Saâd Benbachir.
2020. "The Bayesian Approach to Capital Allocation at Operational Risk: A Combination of Statistical Data and Expert Opinion" *International Journal of Financial Studies* 8, no. 1: 9.
https://doi.org/10.3390/ijfs8010009