# Bayesian Modelling, Monte Carlo Sampling and Capital Allocation of Insurance Risks

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## Abstract

**:**

## 1. Introduction

## 2. Risk Allocation for the Swiss Solvency Test

**Remark**

**1.**

**Remark**

**2.**

## 3. SMC Samplers and Capital Allocation

#### 3.1. A Brief Introduction to SMC Methods

#### 3.1.1. SMC Algorithm

**Remark**

**3.**

#### 3.1.2. SMC Samplers

Algorithm 1: SMC sampler algorithm. |

#### 3.2. Allocations for the Marginalized Model

#### 3.2.1. Reaching a Rare Event Using Intermediate Steps

**Remark**

**4.**

#### 3.3. Allocations for The Conditional Model

#### 3.3.1. Single Auxiliary Variable Method

- ${\mathbf{\theta}}_{s}\sim {f}_{\mathbf{\theta}}\left({\mathbf{\theta}}_{s}\right)$;
- ${\overline{\mathit{z}}}_{s}\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}{\overline{\mathit{z}}}_{s+1}\sim {\overline{L}}_{s}({\overline{\mathit{z}}}_{s+1},\phantom{\rule{0.166667em}{0ex}}{\overline{\mathit{z}}}_{s}\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}{\mathbf{\theta}}_{s})$.

**Remark**

**5.**

#### 3.3.2. Multiple Auxiliary Variable

## 4. Swiss Solvency Test and Claims Development

**Remark**

**6.**

#### 4.1. Conditional Predictive Model

**Remark**

**7.**

#### 4.2. Marginalized Predictive Model

#### 4.3. Solvency Capital Requirement (SCR)

#### 4.3.1. SCR for the Conditional Model

#### 4.3.2. SCR for the Marginalized Model

**Remark**

**8.**

**Remark**

**9.**

## 5. Modelling of Individual LoBs PY Claims

**Model**

**Assumptions**

**1.**

- (a)
- Conditionally, given $\mathbf{\varphi}=({\varphi}_{0},\cdots ,{\varphi}_{J-1})$ and $\mathbf{\sigma}=({\sigma}_{0},\cdots ,{\sigma}_{J-1})$, cumulative claims ${\left({C}_{i,j}\right)}_{j=0,\cdots ,J}$ are independent (in accident year i) Markov processes (in development year j) with$${C}_{i,j+1}\phantom{\rule{4pt}{0ex}}|\phantom{\rule{4pt}{0ex}}\{\mathcal{F}(i+j),\phantom{\rule{4pt}{0ex}}\mathbf{\varphi},\phantom{\rule{4pt}{0ex}}\mathbf{\sigma}\}\sim \Gamma \left({C}_{i,j}{\sigma}_{j}^{-2},\phantom{\rule{4pt}{0ex}}{\varphi}_{j}{\sigma}_{j}^{-2}\right),$$
- (b)
- The parameter vectors $\mathbf{\varphi}$ and $\mathbf{\sigma}$ are independent.
- (c)
- For given hyper-parameters ${f}_{j}>0$ the components of $\mathbf{\varphi}$ are independent such that$${\varphi}_{j}\sim \underset{{\gamma}_{j}\to 1}{lim}\Gamma \left({\gamma}_{j},\phantom{\rule{4pt}{0ex}}{f}_{j}({\gamma}_{j}-1)\right),$$
- (d)
- The components ${\sigma}_{j}$ of $\mathbf{\sigma}$ are independent and ${F}_{{\sigma}_{j}}$-distributed, having support in $(0,{d}_{j})$ for given constants $0<{d}_{j}<\infty $ for all $0\le j\le J-1$.
- (e)
- $\mathbf{\varphi}$, $\mathbf{\sigma}$ and ${C}_{1,0},\cdots ,{C}_{t,0}$ are independent and $\mathbb{P}[{C}_{i,0}>0]=1$, for all $1\le i\le t$.

#### 5.1. MSEP Results Conditional on $\sigma $

**Remark**

**10.**

**Remark**

**11.**

#### 5.2. Marginalized MSEP Results

**Lemma**

**1.**

**Proof.**

**Proposition**

**1.**

**Proof.**

**Remark**

**12.**

#### 5.3. Statistical Model of PY Risk in the SST

#### 5.3.1. Conditional PY Model

**Model**

**Assumptions**

**2**

**.**We assume that

#### 5.3.2. Marginalized PY Model

**Model**

**Assumptions**

**3**

**.**We assume that

## 6. Modelling of Individual LoBs CY Claims

#### 6.1. Modelling of Small CY Claims

**Model**

**Assumptions**

**4**

**.**For known constants $v,\phantom{\rule{0.166667em}{0ex}}{r}_{s}>0$ and $\mathbb{E}\left[{Z}_{CY,s}\phantom{\rule{0.166667em}{0ex}}\right|\phantom{\rule{0.166667em}{0ex}}\mathcal{F}\left(t\right)]$ we set

#### 6.2. Modelling of Large CY Claims

#### 6.2.1. SST Model for Cumulated Claims

**Model**

**Assumptions**

**5**

**.**For ${\alpha}_{mkt}$, ${\beta}_{mkt}$ and $\gamma $ provided by the regulator in FINMA (2016), $\beta \in \{1,\phantom{\rule{0.166667em}{0ex}}5\}$, $m\in (0,1)$,

**Remark**

**13.**

#### 6.2.2. SST Model for Individual Claims

**Model**

**Assumptions**

**6**

**.**For ${\alpha}_{\beta}$, ${p}_{1}$ and $\gamma $ provided by the regulator in FINMA (2016), $\beta \in \{1,\phantom{\rule{0.166667em}{0ex}}5\}$ and ${\lambda}_{CY}>0$,

## 7. Joint Distribution of PY and CY Claims

**Remark**

**14.**

#### 7.1. Conditional Joint Model

**Model**

**Assumptions**

**7**

**.**Based on Model Assumptions 2 and 4 we link the marginals of the conditional model through a Gaussian copula with correlation matrix $\overline{\Omega}$, with elements ${\left(\overline{\Omega}\right)}_{i,j}={\overline{\omega}}_{i,j}$. More formally, given $\mathcal{F}\left(t\right)$ and $\mathbf{\sigma},$ the joint distribution of $\overline{\mathit{Z}}$ is given by

**Remark**

**15.**

#### 7.2. Marginalized Joint Model

**Model**

**Assumptions**

**8**

**.**Based on Model Assumptions 3 and 4 we link the marginal distributions of the marginalized model through a Gaussian copula with correlation matrix $\Omega $, with elements ${(\Omega )}_{i,j}={\omega}_{i,j}$. More formally, given $\mathcal{F}\left(t\right),$ the joint distribution of $\mathit{Z}$ is given by

## 8. Data Description and Parameter Estimation

#### 8.1. Hyperparameters for ${\varphi}_{j}$

#### 8.2. Current Year Small and Large Claims

#### 8.3. Parameter Estimation

#### 8.4. The Correlation Matrices

## 9. Details of the SMC Algorithm

#### 9.1. Selection of Intermediate Sets

#### 9.2. Marginalized Model

#### 9.2.1. The Forward Kernel

#### 9.2.2. The Backward Kernel

#### 9.2.3. The MCMC Move Kernel

#### 9.3. Conditional Model

#### 9.3.1. The Forward Kernel

#### 9.3.2. The Backward Kernel

#### 9.3.3. The MCMC Move Kernel

## 10. Results

## 11. Conclusions

## Author Contributions

## Conflicts of Interest

## Appendix A. Posterior Distributions

**Lemma**

**A1.**

## Appendix B. Correlation Bounds in the Log-Normal–Gaussian Copula Model

**Lemma**

**A2**

**.**Let $({X}_{1},{X}_{2})$ be a bivariate random variable with marginal distributions ${F}_{1}$ and ${F}_{2}$. Then the correlation between ${X}_{1}$ and ${X}_{2}$ is bounded by

**Figure A1.**Lower (

**left**) and upper (

**right**) bound for correlations in a Gaussian-copula model with Log-Normal marginal distributions, as a function of the scale parameters ${\sigma}_{1}$ and ${\sigma}_{2}$.

## Appendix C. Data Generating Process

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**Figure 1.**Quantile-Quantile plots for the different lines of business (LoBs) comparing (vertical axis) the empirical distribution of ${\overline{Z}}_{PY}\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}\mathbf{\sigma},\phantom{\rule{0.166667em}{0ex}}\mathcal{F}\left(t\right)$ based on Model Assumptions 1 and (horizontal axis) the log-normal approximation from Model Assumptions 2. Based on 1000 samples.

**Figure 2.**Quantile-Quantile plots for the different LoBs comparing (vertical axis) the empirical distribution of ${Z}_{PY}\phantom{\rule{0.166667em}{0ex}}|\phantom{\rule{0.166667em}{0ex}}\mathcal{F}\left(t\right)$ based on Model Assumptions 1 and (horizontal axis) the log-normal approximation from Model Assumptions 3 and using posterior samples as in Figure 5 and Figure 6. Based on 1000 samples.

**Figure 3.**Posterior distributions for ${\sigma}_{j}$ for the (

**a**) Motor Third Part Liability (MTPL) (

**b**) Property and (

**c**) Motor Hull lines of business. One sees solid lines representing the unnormalized posteriors, the histogram of the Markov Chain Monte Carlo (MCMC) outputs and a red dashed line indicating the CL standard deviation estimate. Note that for LoB MTPL we only plot selected development periods: $j\in \{0,7,14,21,28\}$.

**Figure 4.**Cumulative claims payment (in millions of CHF). Lighter colours represent more recent accident years.

**Figure 5.**Histogram of the parameter ${\overline{\sigma}}_{PY}$ for the conditional model. Red dashed line: ${\sigma}_{PY}$.

**Figure 6.**Histogram of the parameter ${\overline{\mu}}_{PY}$ for the conditional model. Red dashed line: ${\mu}_{PY}$.

**Figure 7.**Histograms levels used in the SMC sampler algorithm with ${p}_{0}=0.5$ in the marginalized model. The red dashed bar represents the true value of the $\alpha $ quantile.

**Figure 8.**Histograms levels used in the Sequential Monte Carlo (SMC) sampler algorithm with ${p}_{0}=0.5$ in the conditional model. The red dashed bar represents the true value of the $\alpha $ quantile.

**Figure 9.**Bias for the marginalized model. Note that although the bias for some of the CY large claims is around 10% their allocated capital is rather small, as seen in Figure 11 (a).

**Figure 10.**Bias for the conditional model. Note that although the bias for some of the current year (CY) large claims is around 10% their allocated capital is rather small, as seen in Figure 11 (b).

**Figure 11.**Comparison between the “true” allocations (calculated via a large Monte Carlo procedure) and the SMC sampler solution for the (

**a**) marginalized and (

**b**) conditional models.

**Figure 14.**Relative bias in the marginalized model as a function of (

**a**) the parameter ${p}_{0}$ and (

**b**) the sample size in the SMC sampler, ${N}_{SMC}$.

LoB | Reserves | Premium |
---|---|---|

1 MTPL | 2391.64 | 503.14 |

2 Motor Hull | 99.08 | 573.26 |

3 Property | 449.26 | 748.76 |

4 Liability | 870.27 | 299.73 |

5 Workers Compensation (UVG) | 1104.66 | 338.63 |

6 Commercial Health | 271.54 | 254.21 |

7 Private Health | 7.32 | 7.20 |

8 Credit and Surety | 49.50 | 34.64 |

9 Others | 67.64 | 46.28 |

Total | 5310.92 | 2805.87 |

LoB | Reserve/ | $\mathit{\sigma}$ | $\mathit{\mu}$ | CoVa | Expectation | Standalone | Marginalized | Conditional | |||||

Premium | ES${}_{99\%}$ | SCR | ES${}_{99\%}$ | SCR | Div. Benefit | ES${}_{99\%}$ | SCR | Div. Benefit | |||||

1 | 2365.44 | 0.0287 | 7.7659 | 2.87% | 2365.44 | 2546.31 | 180.87 | 2489.85 | 124.41 | 31.22% | 2492.05 | 126.61 | 30.00% |

2 | 99.37 | 0.2164 | 4.5755 | 21.90% | 99.37 | 173.23 | 73.86 | 131.73 | 32.36 | 56.19% | 132.59 | 33.21 | 55.03% |

3 | 405.99 | 0.1142 | 5.9998 | 11.46% | 405.99 | 547.25 | 141.26 | 479.11 | 73.12 | 48.24% | 485.27 | 79.28 | 43.88% |

4 | 870.19 | 0.0315 | 6.7682 | 3.15% | 870.19 | 946.06 | 75.87 | 905.48 | 35.29 | 53.49% | 905.29 | 35.10 | 53.73% |

5 | 1105.95 | 0.0193 | 7.0083 | 1.93% | 1105.95 | 1,164.04 | 58.09 | 1137.06 | 31.11 | 46.44% | 1136.88 | 30.93 | 46.76% |

6 | 274.91 | 0.0410 | 5.6156 | 4.10% | 274.91 | 306.43 | 31.52 | 287.33 | 12.42 | 60.59% | 286.97 | 12.06 | 61.74% |

7 | 7.150 | 0.0547 | 1.9657 | 5.48% | 7.15 | 8.26 | 1.11 | 7.45 | 0.30 | 73.27% | 7.43 | 0.28 | 74.50% |

8 | 48.18 | 0.0493 | 3.8738 | 4.93% | 48.18 | 54.89 | 6.71 | 50.51 | 2.32 | 65.36% | 50.43 | 2.25 | 66.44% |

9 | 72.20 | 0.1332 | 4.2706 | 13.38% | 72.2 | 102.16 | 29.96 | 85.32 | 13.12 | 56.21% | 85.15 | 12.95 | 56.77% |

Total PY | 5249.38 | 5249.38 | 5848.63 | 599.25 | 5573.84 | 324.45 | 45.86% | 5582.06 | 332.67 | 44.49% | |||

1 | 503.14 | 0.0685 | 6.0958 | 6.86% | 448.94 | 533.07 | 84.13 | 499.16 | 50.21 | 40.32% | 498.37 | 49.43 | 41.25% |

2 | 573.26 | 0.0702 | 6.0356 | 7.03% | 402.87 | 504.20 | 101.33 | 472.25 | 69.38 | 31.53% | 471.66 | 68.79 | 32.11% |

3 | 748.76 | 0.0683 | 6.3013 | 6.84% | 547.23 | 654.38 | 107.15 | 603.36 | 56.13 | 47.62% | 602.61 | 55.38 | 48.31% |

4 | 299.73 | 0.0923 | 5.3596 | 9.25% | 216.70 | 272.05 | 55.35 | 239.69 | 22.99 | 58.47% | 239.57 | 22.87 | 58.69% |

5 | 338.63 | 0.0648 | 5.6841 | 6.49% | 303.77 | 349.69 | 45.92 | 319.17 | 15.40 | 66.47% | 318.71 | 14.94 | 67.45% |

6 | 254.21 | 0.0804 | 5.4296 | 8.05% | 228.79 | 282.62 | 53.83 | 249.63 | 20.85 | 61.28% | 249.31 | 20.52 | 61.88% |

7 | 7.20 | 0.1047 | 1.8628 | 10.5% | 6.48 | 8.52 | 2.04 | 7.01 | 0.53 | 73.84% | 7.01 | 0.53 | 74.06% |

8 | 34.64 | 0.0981 | 3.3172 | 9.84% | 27.72 | 35.84 | 8.13 | 30.32 | 2.60 | 67.95% | 30.28 | 2.57 | 68.44% |

9 | 46.28 | 0.1004 | 3.6066 | 10.06% | 37.03 | 48.16 | 11.14 | 41.83 | 4.81 | 56.83% | 41.79 | 4.77 | 57.19% |

Total CY,s | 2805.85 | 2219.53 | 2688.53 | 469.02 | 2462.42 | 242.9 | 48.21% | 2459.31 | 239.8 | 48.87% | |||

Peril | ${\mathbf{\beta}}^{\left(\mathbf{5}\right)}$ | $\mathbf{\gamma}$ | $\mathbf{\alpha}$ | CoVa | Expectation | Standalone | Marginalized | Conditional | |||||

ES${}_{\mathbf{99}\%}$ | SCR | ES${}_{\mathbf{99}\%}$ | SCR | Div. benefit | ES${}_{\mathbf{99}\%}$ | SCR | Div. benefit | ||||||

1 | 2.50 | 2.80 | 3.89 | 20.14 | 16.25 | 4.03 | 0.15 | 99.1% | 4.01 | 0.12 | 99.27% | ||

2 | 13.35 | 300 | 1.85 | 27.08 | 191.21 | 164.13 | 39.96 | 12.88 | 92.15% | 39.61 | 12.53 | 92.36% | |

3 | 6.28 | 100 | 1.50 | 14.34 | 84.31 | 69.97 | 16.5 | 2.16 | 96.91% | 16.45 | 2.11 | 96.98% | |

4 | 3.88 | 100 | 1.80 | 8.10 | 61.34 | 53.24 | 8.94 | 0.84 | 98.42% | 8.91 | 0.81 | 98.48% | |

5 | 0.50 | 2.00 | 1.00 | 10.00 | 9.00 | 1.07 | 0.07 | 99.19% | 1.12 | 0.12 | 98.69% | ||

Total CY,l | 54.41 | 367 | 312.59 | 70.5 | 16.1 | 94.85% | 70.1 | 15.69 | 94.98% | ||||

Total | 8055.26 | 7523.32 | 8904.18 | 1380.86 | 8106.77 | 583.45 | 57.75% | 8111.5 | 588.18 | 57.40% |

**Table 3.**Swiss Solvency Test (SST)’s (2015) standard development patterns for claims provision (normalized to have at most 30 development years and rounded to 2 digits).

LoB | Year 0 | Year 1 | Year 2 | Year 3 | Year 4 | Year 5 | Year 6 | Year 7 | Year 8 | Year 9 | Year 10 | Year 11 | Year 12 | Year 13 | Year 14 | Year 15 |

1 | 30.18% | 15.63% | 5.78% | 4.94% | 4.43% | 4.34% | 4.09% | 3.92% | 3.66% | 3.50% | 3.08% | 2.64% | 2.16% | 1.86% | 1.50% | 1.30% |

2 | 81.08% | 18.67% | 0.24% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% |

3 | 58.24% | 35.06% | 4.36% | 1.37% | 0.64% | 0.33% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% |

4 | 26.55% | 23.53% | 8.33% | 6.18% | 4.79% | 4.15% | 3.63% | 3.14% | 2.55% | 2.11% | 1.80% | 1.59% | 1.35% | 1.20% | 1.12% | 1.02% |

5 | 40.62% | 24.92% | 7.14% | 4.86% | 4.43% | 3.13% | 2.57% | 1.67% | 1.31% | 1.22% | 1.05% | 0.69% | 0.60% | 0.56% | 0.51% | 0.47% |

6 | 36.83% | 47.68% | 14.20% | 0.88% | 0.28% | 0.14% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% |

7 | 46.26% | 38.05% | 10.78% | 2.94% | 1.27% | 0.69% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% |

8 | 45.85% | 35.28% | 11.35% | 3.72% | 1.62% | 0.91% | 0.52% | 0.32% | 0.20% | 0.13% | 0.10% | 0% | 0% | 0% | 0% | 0% |

9 | 58.24% | 35.06% | 4.36% | 1.37% | 0.64% | 0.33% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% |

LoB | Year 16 | Year 17 | Year 18 | Year 19 | Year 20 | Year 21 | Year 22 | Year 23 | Year 24 | Year 25 | Year 26 | Year 27 | Year 28 | Year 29 | Year 30 | |

1 | 1.06% | 0.88% | 0.73% | 0.64% | 0.60% | 0.53% | 0.47% | 0.44% | 0.41% | 0.37% | 0.29% | 0.21% | 0.15% | 0.12% | 0.10% | |

2 | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | |

3 | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | |

4 | 0.88% | 0.77% | 0.72% | 0.66% | 0.60% | 0.55% | 0.52% | 0.49% | 0.45% | 0.4% | 0.31% | 0.22% | 0.16% | 0.13% | 0.11% | |

5 | 0.43% | 0.40% | 0.37% | 0.35% | 0.33% | 0.31% | 0.29% | 0.27% | 0.26% | 0.24% | 0.23% | 0.22% | 0.20% | 0.19% | 0.18% | |

6 | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | |

7 | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | |

8 | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | |

9 | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% | 0% |

**Table 4.**Mack’s standard deviation parameter estimates, ${s}_{j}$, based on exogenous triangles and for the development lengths given in Table 3.

LoB | Year 0 | Year 1 | Year 2 | Year 3 | Year 4 | Year 5 | Year 6 | Year 7 | Year 8 | Year 9 | Year 10 | Year 11 | Year 12 | Year 13 | Year 14 |

1 | 0.5673 | 0.2280 | 0.1922 | 0.2681 | 0.2683 | 0.3949 | 0.2652 | 0.2641 | 0.2789 | 0.3055 | 0.1458 | 0.1577 | 0.2140 | 0.1001 | 0.1016 |

2 | 0.6640 | 0.0659 | |||||||||||||

3 | 1.3614 | 0.4921 | 0.3215 | 0.0875 | 0.0666 | ||||||||||

4 | 0.8248 | 0.4328 | 0.4021 | 0.3644 | 0.3772 | 0.2729 | 0.5268 | 0.244 | 0.2786 | 0.1559 | 0.2660 | 0.0776 | 0.0757 | 0.1220 | 0.0418 |

5 | 0.9914 | 0.3317 | 0.1807 | 0.1072 | 0.0740 | 0.0444 | 0.0359 | 0.0255 | 0.0190 | 0.0106 | 0.0166 | 0.0094 | 0.0040 | 0.0105 | 0.0040 |

6 | 0.6069 | 0.2405 | 0.0597 | 0.0371 | 0.0172 | ||||||||||

7 | 0.1053 | 0.0450 | 0.0157 | 0.0113 | 0.0091 | ||||||||||

8 | 0.3098 | 0.0737 | 0.0310 | 0.0203 | 0.0137 | 0.0051 | 0.0020 | 0.0026 | 0.0020 | 0.0014 | 0.0011 | ||||

9 | 0.9163 | 0.1910 | 0.1248 | 0.0340 | 0.0258 | ||||||||||

LoB | Year 15 | Year 16 | Year 17 | Year 18 | Year 19 | Year 20 | Year 21 | Year 22 | Year 23 | Year 24 | Year 25 | Year 26 | Year 27 | Year 28 | Year 29 |

1 | 0.0466 | 0.1097 | 0.1081 | 0.0583 | 0.1353 | 0.0916 | 0.0916 | 0.0916 | 0.0916 | 0.0916 | 0.0916 | 0.0916 | 0.0916 | 0.0916 | 0.0916 |

2 | |||||||||||||||

3 | |||||||||||||||

4 | 0.0272 | 0.0886 | 0.0422 | 0.0190 | 0.0238 | 0.0190 | 0.0152 | 0.0122 | 0.0097 | 0.0078 | 0.0062 | 0.0050 | 0.0040 | 0.0032 | 0.0025 |

5 | 0.0040 | 0.0040 | 0.0040 | 0.0040 | 0.0040 | 0.0040 | 0.0040 | 0.0040 | 0.0040 | 0.0040 | 0.0040 | 0.0040 | 0.0040 | 0.0040 | 0.0040 |

6 | |||||||||||||||

7 | |||||||||||||||

8 | |||||||||||||||

9 |

LoB | Claims Ratio | Average Claim Amount | Market Share |
---|---|---|---|

1 | 90% | 0.005 | |

2 | 75% | 0.003 | 20% |

3 | 75% | 0.004 | |

4 | 75% | 0.004 | |

5 | 90% | 0.004 | 10% |

6 | 90% | 0.003 | |

7 | 90% | 0.002 | |

8 | 80% | 0.003 | |

9 | 80% | 0.003 |

$\mathbf{\Omega}=\left[\begin{array}{ccc}{\mathbf{\Omega}}_{PY}& {\mathbf{\Omega}}_{PY,\phantom{\rule{0.166667em}{0ex}}CY,s}& {\mathbf{0}}_{L\times P}\\ & {\mathbf{\Omega}}_{CY,s}& {\mathbf{0}}_{L\times P}\\ & & {\mathbf{I}}_{P\times P}\end{array}\right]$ |
---|

LoB | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

1 | 1 | 0.1517 | 0.1505 | 0.2501 | 0.5001 | 0.2501 | 0.1501 | 0.2502 | 0.2511 |

2 | 1 | 0.1520 | 0.1517 | 0.1517 | 0.1517 | 0.1517 | 0.1517 | 0.2532 | |

3 | 1 | 0.1505 | 0.1505 | 0.1505 | 0.1505 | 0.1505 | 0.2515 | ||

4 | 1 | 0.2501 | 0.1501 | 0.1501 | 0.1501 | 0.2511 | |||

5 | 1 | 0.2501 | 0.1501 | 0.2501 | 0.2511 | ||||

6 | 1 | 0.1501 | 0.2502 | 0.2511 | |||||

7 | 1 | 0.1502 | 0.2511 | ||||||

8 | 1 | 0.2511 | |||||||

9 | 1 |

**Table 8.**Correlation block for the marginalized model: ${\mathbf{\Omega}}_{PY,\phantom{\rule{0.166667em}{0ex}}CY,s}$.

LoB | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

1 | 0.5004 | 0.5005 | 0.1502 | 0.2505 | 0.2503 | 0.2504 | 0.1504 | 0.2506 | 0.2506 |

2 | 0.5046 | 0.5046 | 0.2528 | 0.1519 | 0.2528 | 0.1518 | 0.1519 | 0.1519 | 0.2529 |

3 | 0.1506 | 0.2509 | 0.5013 | 0.2510 | 0.1506 | 0.1506 | 0.1508 | 0.1507 | 0.2511 |

4 | 0.2503 | 0.1502 | 0.2503 | 0.5008 | 0.1502 | 0.1503 | 0.1504 | 0.1504 | 0.2506 |

5 | 0.2503 | 0.2503 | 0.1502 | 0.1503 | 0.5004 | 0.2504 | 0.1504 | 0.2506 | 0.2506 |

6 | 0.2503 | 0.1502 | 0.1502 | 0.1503 | 0.2503 | 0.5006 | 0.2507 | 0.2506 | 0.2506 |

7 | 0.1502 | 0.1503 | 0.1502 | 0.1504 | 0.1502 | 0.2505 | 0.5010 | 0.1504 | 0.2506 |

8 | 0.2503 | 0.1502 | 0.1502 | 0.1504 | 0.2503 | 0.2504 | 0.1504 | 0.5009 | 0.2506 |

9 | 0.2511 | 0.2511 | 0.2511 | 0.2513 | 0.2511 | 0.2512 | 0.2514 | 0.2513 | 0.5018 |

LoB | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 |
---|---|---|---|---|---|---|---|---|---|

1 | 1 | 0.5006 | 0.1503 | 0.2506 | 0.2504 | 0.1504 | 0.1505 | 0.1505 | 0.2507 |

2 | 1 | 0.2505 | 0.1504 | 0.2504 | 0.1504 | 0.1505 | 0.1505 | 0.2507 | |

3 | 1 | 0.2506 | 0.1503 | 0.1504 | 0.1505 | 0.1505 | 0.2507 | ||

4 | 1 | 0.1504 | 0.1505 | 0.1506 | 0.1506 | 0.2509 | |||

5 | 1 | 0.2505 | 0.1505 | 0.2507 | 0.2507 | ||||

6 | 1 | 0.2508 | 0.2508 | 0.2508 | |||||

7 | 1 | 0.1507 | 0.2510 | ||||||

8 | 1 | 0.2509 | |||||||

9 | 1 |

${\mathit{p}}_{0}$ | 1 | 2 | 3 | 4 | 5 | 6 | 7 |

0.4 | 0.6 | 0.84 | 0.936 | 0.9744 | 0.9898 | 0.9959 | |

0.5 | 0.5 | 0.75 | 0.875 | 0.9375 | 0.9688 | 0.9844 | 0.9922 |

0.7 | 0.3 | 0.51 | 0.657 | 0.7599 | 0.8319 | 0.8824 | 0.9176 |

${\mathit{p}}_{\mathbf{0}}$ | 8 | 9 | 10 | 11 | 12 | 13 | |

0.4 | |||||||

0.5 | |||||||

0.7 | 0.9424 | 0.9596 | 0.9718 | 0.9802 | 0.9862 | 0.9903 |

© 2017 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Peters, G.W.; Targino, R.S.; Wüthrich, M.V. Bayesian Modelling, Monte Carlo Sampling and Capital Allocation of Insurance Risks. *Risks* **2017**, *5*, 53.
https://doi.org/10.3390/risks5040053

**AMA Style**

Peters GW, Targino RS, Wüthrich MV. Bayesian Modelling, Monte Carlo Sampling and Capital Allocation of Insurance Risks. *Risks*. 2017; 5(4):53.
https://doi.org/10.3390/risks5040053

**Chicago/Turabian Style**

Peters, Gareth W., Rodrigo S. Targino, and Mario V. Wüthrich. 2017. "Bayesian Modelling, Monte Carlo Sampling and Capital Allocation of Insurance Risks" *Risks* 5, no. 4: 53.
https://doi.org/10.3390/risks5040053