# The Cascade Bayesian Approach: Prior Transformation for a Controlled Integration of Internal Data, External Data and Scenarios

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## Abstract

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## 1. Introduction

**Box 1.**Mixing of outcomes from AMA sub-models.

## 2. A Bayesian Inference in Two Steps for Severity Estimation

- Prior ${\pi}_{0}$ by the scenarios and the likelihood component by external data as follows:$${\pi}_{1}(\varphi ;{x}_{1},\dots ,{x}_{k})\propto {\pi}_{0}\left(\varphi \right)\prod _{i=1}^{k}{f}_{i}\left({y}_{i}\right|\varphi ).$$
- The aforementioned posterior function ${\pi}_{1}$ is then sampled and used as prior, and the likelihood component is informed by internal data as follows:$${\pi}_{2}(\varphi ;{x}_{1},\dots ,{x}_{k})\propto {\pi}_{1}\left(\varphi \right)\prod _{i=1}^{k}{f}_{i}\left({x}_{i}\right|\varphi ).$$

## 3. Carrying Out the Cascade Approach in Practice

#### 3.1. The Data Sets

#### 3.2. The Priors

- Lognormal—a Gaussian and a gamma distribution,
- Weibull—gamma distributions for both the scale and the shape,
- Generalized Pareto distribution—a beta distribution for the shape and a gamma distribution on the scale.

#### 3.3. Estimation

- One can choose conjugate priors for the parameters in the first step of the Bayesian inference estimation. In this case, the (joint) distribution of the posterior parameters is directly known, and it is possible to sample directly from this distribution to recreate the marginal posterior distributions for each severity parameter. These may then be used to generate the posterior empirical densities required to compute a Bayesian point estimator of the parameters. The admissible estimators are the median, mean and mode of the posterior distribution. The mode (also called maximum a posteriori (MAP)) can be seen as the ’most probable estimator’ and ultimately coincides with the maximum likelihood estimator (Lehmann and Casella 1998). Despite having good asymptotic properties, finding the mode of an empirical distribution is not a trivial matter and often requires some additional techniques and hypotheses (e.g., smoothing). This paper, therefore, uses posterior means as point estimators. To the best of our knowledge, the only conjugate priors for continuous distributions were studied by Shevchenko (2011), for the Lognormal severity case. Conjugate approach requires some assumptions that may not be sustainable in practice, particularly for priors that are modelled with ’uncommon’ distributions (e.g., inverse-Chi-squared). This might lead to difficulties in the step known as ’elicitation’, i.e., calibrating the prior hyper-parameters from the chosen scenario values.
- Another solution is to release the conjugate prior assumption and use a Markov chain Monte Carlo approach in the first step to sample from the first posterior. One can then use a parametric or non-parametric method to compute the corresponding densities. This enables the posterior function to be evaluated (see Equation (6)). Maximising this function directly gives the MAP estimators of the severity parameters (see above). Even if this method is sufficient for computing values for the global parameters, it misses the purpose of the Bayesian inference, which is to provide distribution as a final result instead of a single value. In addition, it may also suffer from all the drawbacks that an optimisation algorithm may suffer (e.g., non-convex functions, sensitivity to starting values, etc.)
- The final alternative is to use a Markov chain Monte Carlo (MCMC) approach at each step of the aforementioned cascade Bayesian inference. This method is more challenging to implement but is the most powerful one, as it generates the entire distribution of the final severity parameters from which any credibility intervals and/or other statistics may be evaluated.

**Remark**

**1.**

## 4. Results

- Scenarios’ severity is derived from the calibration of the prior distributions with the scenario values. The theoretical means obtained from the calibrated priors provide scenario severity estimates.
- Intermediate severity refers to the estimation of the severity of the first obtained posterior, i.e., the mean of the posterior distribution obtained from scenario values updated with external loss data.
- Similarly, final severity represents the severity estimation of the second and last posterior distribution, which includes scenarios, external and internal loss data. A 95% Bayesian confidence interval (also known as a credible interval) derived from this final posterior distribution is also provided. It is worth noticing that this interval, which is formally defined as containing 95% of the distribution mass, is not unique and is chosen here as the narrowest possible interval. It is, therefore, not necessarily symmetric around the posterior mean estimator.

**Remark**

**2.**

**Remark**

**3.**

**Remark**

**4.**

**Remark**

**5.**

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## Appendix A. Epanechnikov Kernel

## Appendix B. Least Square Cross Validation

## Appendix C. The Metropolis–Hastings Algorithm

- Initialise ${\varphi}^{l=0}$ with any value within a support of $\pi \left(\varphi \right|x)$.
- For $l=1,\dots ,L.$
- (a)
- Set ${\varphi}^{l}={\varphi}^{l-1}.$
- (b)
- Generate a proposal ${\varphi}^{\ast}$ from $q\left({\varphi}^{\ast}\right|{\varphi}^{\left(l\right)}).$
- (c)
- Accept proposal with the acceptance probability:$$p({\varphi}^{\left(l\right)},{\varphi}^{(\ast )})=min\left\{1,\frac{\pi \left({\varphi}^{\left(l\right)}\right|x)q\left({\varphi}^{(\ast )}\right|{\varphi}^{\left(l\right)})}{\pi \left({\varphi}^{(\ast )}\right|x)q\left({\varphi}^{\left(l\right)}\right|{\varphi}^{(\ast )})}\right\}.$$

- Next, l (i.e., do an increment, $l=l+1$), and return to step 2.

## Appendix D. Risk Measure Evaluation

## References

- Álvarez, Gene. 2001. Operational Risk Quantification Mathematical Solutions for Analyzing Loss Data. Available online: https://www.bis.org/bcbs/ca/galva.pdf (accessed on 26 April 2018).
- Bayes, Thomas. 1763. An Essay towards Solving a Problem in the Doctrine of Chances. By the Late Rev. Mr. Bayes, F. R. S. Communicated by Mr. Price, in a Letter to John Canton, A. M. F. R. S. Philosophical Transactions (1683–1775) 53: 370–418. [Google Scholar] [CrossRef]
- BCBS. 2001. Working Paper on the Regulatory Treatment of Operational Risk. Basel: Bank for International Settlements. [Google Scholar]
- BCBS. 2010. Basel III: A Global Regulatory Framework for More Resilient Banks and Banking Systems. Basel: Bank for International Settlements. [Google Scholar]
- Berger, James O. 1985. Statistical Decision Theory and Bayesian Analysis. New York: Springer. [Google Scholar]
- Böcker, Klaus, and Claudia Klüppelberg. 2010. Operational VaR: A Closed-Form Approximation. Risk 18: 90–93. [Google Scholar]
- Bowman, Adrian W. 1984. An alternative method of cross-validation for the smoothing of density estimates. Biometrika 71: 353–60. [Google Scholar] [CrossRef]
- Box, George E. P., and George C. Tiao. 1992. Bayesian Inference in Statistical Analysis. New York, Chichester and Brisbane: Wiley Classics Library, JohnWiley & Sons. [Google Scholar]
- Chernobai, Anna S., Svetlozar T. Rachev, and Frank J. Fabozzi. 2007. Operational Risk: A Guide to Basel II Capital Requirements, Models, and Analysis. New York: John Wiley & Sons. [Google Scholar]
- Cowell, Robert G., Richard J. Verrall, and Y. K. Yoon. 2007. Modeling operational risk with bayesian networks. Journal of Risk and Insurance 74: 795–827. [Google Scholar] [CrossRef]
- Cruz, Marcelo G. 2004. Operational Risk Modelling and Analysis. London: Risk Books. [Google Scholar]
- Cruz, Marcelo G., Gareth W. Peters, and Pavel V. Shevchenko. 2014. Fundamental Aspects of Operational Risk and Insurance Analytics: A Handbook of Operational Risk. Hoboken: John Wiley & Sons. [Google Scholar]
- Frachot, Antoine, Pierre Georges, and Thierry Roncalli. 2001. Loss Distribution Approach for Operational Risk. Working paper. Paris: GRO, Crédit Lyonnais. [Google Scholar]
- Gilks, Walter R., Sylvia Richardson, and David Spiegelhalter. 1996. Markov Chain Monte Carlo in Practice. London: Chapman & Hall/CRC. [Google Scholar]
- Guegan, Dominique, and Bertrand K. Hassani. 2013. Multivariate vars for operational risk capital computation: A vine structure approach. International Journal of Risk Assessment and Management 17: 148–70. [Google Scholar] [CrossRef]
- Guégan, Dominique, and Bertrand Hassani. 2013. Using a time series approach to correct serial correlation in operational risk capital calculation. The Journal of Operational Risk 8: 31. [Google Scholar] [CrossRef]
- Guégan, Dominique, and Bertrand Hassani. 2014. A mathematical resurgence of risk management: An extreme modeling of expert opinions. Frontiers in Finance & Economics 11: 25–45. [Google Scholar]
- Guégan, Dominique, Bertrand K. Hassani, and Cédric Naud. 2011. An efficient threshold choice for the computation of operational risk capital. The Journal of Operational Risk 6: 3–19. [Google Scholar] [CrossRef]
- Hassani, Bertrand, and Bertrand K. Hassani. 2016. Scenario Analysis in Risk Management. Cham: Springer. [Google Scholar]
- Lehmann, Erich L., and George Casella. 1998. Theory of Point Estimation, 2nd ed. Berlin: Springer. [Google Scholar]
- Leone, Paola, and Pasqualina Porretta. 2018. Operational risk management: Regulatory framework and operational impact. In Measuring and Managing Operational Risk. Cham: Palgrave Macmillan, pp. 25–93. [Google Scholar]
- Mizgier, Kamil J., and Maximilian Wimmer. 2018. Incorporating single and multiple losses in operational risk: A multi-period perspective. Journal of the Operational Research Society 69: 358–71. [Google Scholar] [CrossRef]
- Parent, Eric, and Jacques Bernier. 2007. Le Raisonnement Bayésien, Modelisation Et Inférence. Paris: Springer. [Google Scholar]
- Peters, Gareth William, and Scott Sisson. 2006. Bayesian inference, Monte Carlo sampling and operational risk. Journal of Operational Risk 1. [Google Scholar] [CrossRef]
- Peters, Gareth W., and Pavel V. Shevchenko. 2015. Advances in Heavy Tailed Risk Modeling: A Handbook of Operational Risk. Hoboken: John Wiley & Sons. [Google Scholar]
- Rudemo, Mats. 1982. Empirical choice of histograms and kernel density estimators. Scandinavian Journal of Statistics 9: 65–78. [Google Scholar]
- Shevchenko, Pavel V. 2011. Modelling Operational Risk Using Bayesian Inference. Berlin: Springer. [Google Scholar]
- Valdés, Rosa María Arnaldo, V. Fernando Gómez Comendador, Luis Perez Sanz, and Alvaro Rodriguez Sanz. 2018. Prediction of aircraft safety incidents using Bayesian inference and hierarchical structures. Safety Science 104: 216–30. [Google Scholar] [CrossRef]

1. | External loss data usually overlap with both internal data and scenario analysis and are therefore represented as a link between the two previous components. |

2. | It is generally accepted that capital calculations are not particularly sensitive to the choice of the frequency distribution (Álvarez 2001). |

3. | These parameters are omitted here for simplicity of notation, these are the parameters of the densities used as a prior distribution (e.g., a Gamma or Beta distribution). |

4. | These take into account the three different data sources. |

**Figure 1.**Combination of internal loss data, external loss data and scenario analysis. The representation may be slightly different, and the components may overlap with each other depending on the risk profile or the quantity of data available.

**Figure 2.**Posterior distributions and convergence of the estimations obtained on internal fraud (Lognormal case). (

**left**) Obtained posterior distribution for each severity parameter ($\mu $ and $\sigma $); (

**right**) convergence of the severity estimation as the posterior mean in the Markov chain Monte Carlo (MCMC) sampling.

**Figure 3.**Comparison of capital charge value obtained from the three components on a standalone basis and in combination.

**Table 1.**The table presents the statistical moments of the internal loss data used in this paper, as well as other statistics such as the minimum value, the maximum value and the number of data point available. NB stands for number of data points used. Level 2 ’Other’ gathers ’Execution, Delivery and Process Management’ losses other than ’Financial Instruments’ and ’Payments’.

Level 1 | Level 2 | NB Used | Min | Median | Mean | Max | sd | Skewness | Kurtosis |
---|---|---|---|---|---|---|---|---|---|

Internal Fraud | Global | 665 | 4137 | 28,165 | 261,475 | 46,779,130 | 1.9 × 10^{6} | 21.4 | 407.1 |

External Fraud | Payments | 1567 | 4091 | 12,358 | 36,133 | 1,925,000 | 9.2 × 10^{4} | 11.7 | 185.2 |

Execution, Delivery & Process Management | Other | 3602 | 4084 | 10,789 | 96,620 | 30,435,400 | 9.5× 10^{5} | 24.8 | 653.8 |

**Table 2.**The table presents the statistical moments of the external loss data used in this paper, as well as other statistics such as the minimum value, the maximum value and the number of data available. NB stands for number of data points used. Level 2 ’Other’ gathers ’Execution, Delivery and Process Management’ losses other than ’Financial Instruments’ and ’Payments’.

Level 1 | Level 2 | NB Used | Min | Median | Mean | Max | sd | Skewness | Kurtosis |
---|---|---|---|---|---|---|---|---|---|

Internal Fraud | Global | 2956 | 20,001 | 88,691 | 697,005 | 130,715,800 | 4.5 × 10^{6} | 17.6 | 387.6 |

External Fraud | Payments | 1085 | 20,006 | 36,464 | 326,127 | 106,772,200 | 4.3 × 10^{6} | 20.8 | 461.2 |

Execution, Delivery & Process Management | Other | 31,126 | 20,004 | 47,428 | 271,974 | 585,000,000 | 4.1 × 10^{6} | 107.2 | 14,068.6 |

**Table 3.**The table presents the scenario values used in this paper: 1 in 10 and 1 in 40, respectively, denote the biggest loss that may occur in the next 10 and 40 years. Level 2 ’Other’ gathers ’Execution, Delivery and Process Management’ losses other than ’Financial Instruments’ and ’Payments’.

Level 1 | Level 2 | 1 in 10 | 1 in 40 |
---|---|---|---|

Internal Fraud | Global | 6.0 × 10^{6} | 5.2 × 10^{7} |

External Fraud | Payments | 1.5 × 10^{6} | 2.5 × 10^{7} |

Execution, Delivery & Process Management | Other | 2.5 × 10^{7} | 5.0 × 10^{7} |

**Table 4.**This table presents the following priors that were used to parametrise: 1—the Lognormal distribution, i.e., the Gaussian distribution for $\mu $ and the gamma distribution for $\sigma $; 2—the Weibull distribution, i.e., two gamma distributions for the shape and the scale; 3—the generalized Pareto distribution (GPD), i.e., a beta distribution for the shape and a gamma distribution for the scale. These are informed by the scenarios.

Label | Density | Parameters |
---|---|---|

Beta | $\frac{\Gamma (\alpha +\beta )}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{x}^{\alpha -1}{(1-x)}^{\beta -1}$ | $\alpha =shape$, $\beta =scale$ |

Gamma | $\frac{1}{\Gamma \left(\alpha \right){\beta}^{\alpha}}{x}^{\alpha -1}{e}^{-\frac{x}{\beta}}$ | $\alpha =shape$, $\beta =scale$ |

Gaussian | $\frac{1}{b\sqrt{2\pi}}{e}^{-\frac{{(x-a)}^{2}}{2{b}^{2}}}$ | $a=location$, $b=standard\phantom{\rule{0.166667em}{0ex}}Deviation$ |

**Table 5.**This table presents the standalone parameters estimated for each components. Internal fraud severities are modelled using a lognormal distribution, external fraud/payments using a Weibull and execution, delivery, and product management considering a mixture model combining a Lognormal distribution in the body and a generalized Pareto distribution in the tail. The first column presents the initial parameters estimated for the scenarios. ${\varphi}_{1}$ and ${\varphi}_{2}$ represent the severity parameters of the chosen distribution, i.e., (resp.) $\mu $ and $\sigma $ for the lognormal and shape and scale for the Weibull or GPD distribution. The second column shows the values obtained for the external data. The third column shows the parameters obtained for the internal data.

Label | Scenarios | External Data | Internal Data | |||||
---|---|---|---|---|---|---|---|---|

${\mathit{\varphi}}_{1}$ | ${\mathit{\varphi}}_{2}$ | ${\mathit{\varphi}}_{1}$ | ${\mathit{\varphi}}_{2}$ | Size | ${\mathit{\varphi}}_{1}$ | ${\mathit{\varphi}}_{2}$ | Size | |

Internal Fraud | 10.616031 | 2.024592 | 8.707405 | 2.693001 | 2956 | 9.347693 | 2.485755 | 665 |

(Global)—Lognormal | ||||||||

External Fraud | 0.37642 | 4.6951 × 10^{4} | 1.644924 | 5.4293 × 10^{4} | 1085 | 0.292442 | 976.922586 | 1567 |

(Payments)—Weibull | ||||||||

Execution, Delivery, | 0.4705893 | 1.5309 × 10^{6} | 0.732526 | 1.1532 × 10^{6} | 31126 | 0.82231 | 7.0388 × 10^{5} | 3602 |

and Product Management | ||||||||

(Financial Instruments)—GPD |

**Table 6.**The table presents the evolution of the parameters obtained by carrying out the cascade Bayesian approach. Internal fraud severities are modelled using a Lognormal distribution, external fraud/payments using a Weibull and execution, delivery, and product management considering a mixture model combining a Lognormal distribution in the body and a generalized Pareto distribution in the tail. The first column presents the initial parameters estimated from the scenarios. ${\varphi}_{1}$ and ${\varphi}_{2}$ represent the severity parameters of the chosen distribution, i.e., (resp.) $\mu $ and $\sigma $ for the Lognormal and shape and scale for the Weibull or GPD distribution. The second column shows the values obtained after the first refinement, i.e., after the incorporation of the external data. The third column shows the final parameters following the second refinement, i.e., after the integration of the internal data. The figures in brackets represent a 95% Bayesian credible interval obtained from the final posterior distribution.

Label | Scenarios Severity from Priors | Intermediate Severity (Scenarios + External Data) | Final Severity (Scenarios + External Data + Internal Data) | |||
---|---|---|---|---|---|---|

${\mathit{\varphi}}_{1}$ | ${\mathit{\varphi}}_{2}$ | ${\mathit{\varphi}}_{1}$ | ${\mathit{\varphi}}_{2}$ | ${\mathit{\varphi}}_{1}$ | ${\mathit{\varphi}}_{2}$ | |

Internal Fraud (Global)—Lognormal | 10.616031 | 2.024592 | 9.123964 | 2.539844 | 8.727855 | 2.684253 |

(95% Bayesian Credibility Interval) | - | - | - | - | [7.98176; 9.95824] | [1.23225; 4.42314] |

External Fraud (Payments)—Weibull | 0.3761 | 46942 | 0.38873 | 49,942 | 0.39910 | 46,622 |

(95% Bayesian Credibility Interval) | - | - | - | - | [0.23127; 0.50673] | [37,632; 58,981] |

Execution, Delivery, and Product Management | 0.4705893 | 1.5309 × 10^{6} | 0.4452626 | 1.5109 × 10^{6} | 0.6021 | 6.05 × 10^{5} |

(Financial Instruments)—GPD | ||||||

(95% Bayesian Credibility Interval) | - | - | - | - | [0.3313; 0.9089] | [4.52 × 10^{5}; 8.30 × 10^{5}] |

**Table 7.**Body distribution (Lognormal) parameters for execution, delivery, and product management/financial instruments.

Event Type | $\mathit{\mu}$ | $\mathit{\sigma}$ |
---|---|---|

Execution, Delivery, | 8.092849 | 1.882122 |

and Product Management | ||

(Financial Instruments)—Lognormal body |

**Table 8.**Frequency distribution parameters used in the capital charge calculations. GDP: generalized Pareto distribution.

Event Type | Initial$\mathit{\lambda}$ | Corrected$\mathit{\lambda}$ | ||

Internal Fraud | 133 | 261.5885 | ||

(Global)—Lognormal | ||||

External Fraud | 313.4 | 485.0353 | ||

(Payments)—Weibull | ||||

Event Type | Initial$\mathit{\lambda}$body | Initial$\mathit{\lambda}$tail | Initial Global$\mathit{\lambda}$ | Corrected Global$\mathit{\lambda}$ |

Execution, Delivery, | 144.08 | 9.2 | 153.28 | 1882.327 |

and Product Management | ||||

(Financial Instruments)—GPD |

**Table 9.**This table presents the stand alone value-at-risk (VaR) for each of the three components, as well as the VaR and the expected shortfall (ES) computed by combining the three elements with cascade Bayesian integration for each of the three different event types. Internal fraud severities are modelled using a Lognormal distribution, external fraud/payments using a Weibull and execution, delivery, and product management considering a mixture model combining a lognormal distribution in the body and a generalized Pareto distribution in the tail. The parameters of the distributions used to compute these values are shown in Table 5 and Table 6.

Label | Scenarios | External Data | Internal Data | Combination | |
---|---|---|---|---|---|

VaR | VaR | VaR | VaR | ES | |

Internal Fraud | 843,445,037 | 855,464,158 | 1,143,579,396 | 1,106,692,211 | 2,470,332,812 |

(Global) | |||||

External Fraud Payments | 128,243,367 | 13,664,057 | 3,945,116 | 119,637,592 | 126,935,364 |

(Payments) | |||||

Execution, Delivery, and Product Management | 122,544,708 | 583,519,888 | 137,456,938 | 213,982,300 | 281,157,384 |

(Financial Instruments) |

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## Share and Cite

**MDPI and ACS Style**

Hassani, B.K.; Renaudin, A.
The Cascade Bayesian Approach: Prior Transformation for a Controlled Integration of Internal Data, External Data and Scenarios. *Risks* **2018**, *6*, 47.
https://doi.org/10.3390/risks6020047

**AMA Style**

Hassani BK, Renaudin A.
The Cascade Bayesian Approach: Prior Transformation for a Controlled Integration of Internal Data, External Data and Scenarios. *Risks*. 2018; 6(2):47.
https://doi.org/10.3390/risks6020047

**Chicago/Turabian Style**

Hassani, Bertrand K., and Alexis Renaudin.
2018. "The Cascade Bayesian Approach: Prior Transformation for a Controlled Integration of Internal Data, External Data and Scenarios" *Risks* 6, no. 2: 47.
https://doi.org/10.3390/risks6020047