# Application of a Vine Copula for Multi-Line Insurance Reserving

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## Abstract

**:**

## 1. Introduction

## 2. Proposed Methodology

- Marginal distribution for each of ${D}^{\left(n\right)}$;
- Optimal d-dimensional vine structure;
- Copula family for each of pair copula in the selected vine structure.

- Estimate the parameters with Bayesian inference and choose marginal distribution based on Bayesian model selection criteria, such as deviance information criterion (DIC) and the logarithm of the pseudomarginal likelihood (LPML). The DIC for each marginal distribution of ${n}^{th}$ line, proposed by Spiegelhalter et al. (2002), is defined as$$\begin{array}{c}\hfill {\mathrm{DIC}}^{\left(n\right)}=-4\int \ell \left({\theta}^{\left(n\right)}\right|{\mathbf{d}}^{\left(n\right)})\pi \left({\theta}^{\left(n\right)}\right|{\mathbf{d}}^{\left(n\right)})d\theta +2\ell \left({\tilde{\theta}}^{\left(n\right)}\right|{\mathbf{d}}^{\left(n\right)}),\end{array}$$$${\widehat{\mathrm{DIC}}}^{\left(n\right)}=-4\sum _{s=1}^{S}\ell \left({\theta}_{\left[s\right]}^{\left(n\right)}\right|{\mathbf{d}}^{\left(n\right)})+2\ell \left(\frac{1}{S}\sum _{s=1}^{S}{\theta}_{\left[s\right]}^{\left(n\right)}|\phantom{\rule{4pt}{0ex}}{\mathbf{d}}^{\left(n\right)}\right),$$LPML is calculated based on the conditional predictive ordinate (CPO), which was proposed by Gelfand et al. (1992) and Geisser (2017). The CPO for ${d}_{i,j}^{\left(n\right)}$ is defined as follows:$${\mathrm{CPO}}_{i,j}^{\left(n\right)}=\int f\left({d}_{i,j}^{\left(n\right)}\right|\theta )\pi \left({\theta}^{\left(n\right)}\right|{\mathbf{d}}_{(-i,j)}^{\left(n\right)})d{\theta}^{\left(n\right)},$$$${\widehat{\mathrm{CPO}}}_{i,j}^{\left(n\right)}=S{\left(\sum _{s=1}^{S}\frac{1}{f\left({d}_{(i,j)}^{\left(n\right)}\right|{\theta}_{\left[s\right]})}\right)}^{-1}.$$Finally, according to Ibrahim et al. (2014), ${\mathrm{CPO}}_{i,j}^{\left(n\right)}$ can be summarized as LPML as follows:$${\mathrm{LPML}}^{\left(n\right)}=\sum _{i=1}^{I}\sum _{j=2}^{J+1-I}log\left({\widehat{\mathrm{CPO}}}_{i,j}^{\left(n\right)}\right).$$We prefer models with larger LPML values.
- Based on the fitted marginal model and corresponding posterior means of marginal parameters ${\widehat{\theta}}^{\left(n\right)}$ for $n=1,\dots ,N$, generate probability integral transfrom (PIT) ${\widehat{U}}_{ij}^{\left(n\right)}={F}^{\left(n\right)}\left({d}_{ij}^{\left(n\right)}|\phantom{\rule{4pt}{0ex}}{\widehat{\theta}}^{\left(n\right)}\right)$ and optimize the following:$$\ell \left(\varphi \phantom{\rule{4pt}{0ex}}|\phantom{\rule{4pt}{0ex}}\widehat{U}\right)=\sum _{i=1}^{I}\sum _{j=2}^{J+1-I}\left[log{c}_{\varphi}\left({\widehat{U}}_{ij}^{\left(1\right)},\phantom{\rule{0.166667em}{0ex}}\dots ,\phantom{\rule{0.166667em}{0ex}}{\widehat{U}}_{ij}^{\left(N\right)}\right)\right],$$$$\underset{{E}_{j},{N}_{j}}{argmin}\sum _{e\in {E}_{j}}{w}_{e},$$In our search for the best family for each of pair copula, we use the aforementioned bivariate Gaussian, Frank, Clayton, Gumbel, and their rotated copulas. Note that this approach also utilizes the inference by margin (IFM) method proposed by Joe and Xu (1996), since it decomposes estimation of marginal distribution and copula structure separately for model selection a with lesser computational burden. It should be noted that such computational convenience is obtained at the expense of potential estimation bias, as shown in Louzada and Ferreira (2016).

## 3. Data Description

`ACE Limited 2013 Global Loss Triangles`, which consists of aggregated claim developments for insurance operations in North America (4 lines of business), insurance operations overseas (3 lines of business), and global reinsurance operations (2 lines of business). A description and the corresponding index for each line of business are given in Table 1. We incorporate a 9-dimensional vine copula structure to capture possible dependence among all lines of business. Indeed, the current COVID-19 situation could be a clear example of showing the inappropriateness of ignoring potential dependencies among different countries. Note that the dataset and code for data analysis are attached as Supplementary Materials.

- Homeowners/farmowners;
- Private passanger auto liability/medical;
- Commercial auto/truck liability/medical;
- Worker’s compensation;
- Special liability;
- Other liability;
- Fidelity/surety.

## 4. Model Selection and Parameter Estimation

`RStan`because of its flexibility. For example, one can incorporate the constraint in (8) by forcing a lower limit of ${\zeta}_{t}^{\left(n\right)}$ as 0 in

`RStan`and using a diffuse uniform prior on a positive real number for ${\zeta}_{t}^{\left(n\right)}$, $n=1,\dots ,9$, and $t=1,\dots ,J$. For the unconstrained models, a diffuse uniform prior on the positive real number is directly used for ${\eta}_{j+1}^{\left(n\right)}$, $n=1,\dots ,9$, and $j=2,\dots ,J$. For each marginal model, four chains with 1000 iterates are used; the first 500 iterates are discarded for burn-in, which usually requires computation time for MCMC sampling of less than a second.

`R`package

`VineCopula`, as a function

`RVineStructureSelect`, which explores both the optimal vine structure and the choice of family for each of the pair copulas sequentially. For a detailed explanation of such an implementation, see Chapter 8 of Czado (2019).

## 5. Validation and Actuarial Implication

- Generate a 9-dimensional uniform random vector $({u}_{ij}^{r:\left(1\right)},\dots ,{u}_{ij}^{r:\left(9\right)})$ based on the specified copula structure in a sequential way. For example, one can first generate $({u}_{ij}^{r:\left(1\right)},{u}_{ij}^{r:\left(2\right)})$ from ${C}_{12}$. After that, ${u}_{ij}^{r:\left(3\right)}$ is generated subsequently due to the following identity:$$\mathcal{U}[0,1]\stackrel{d}{=}{w}_{3}={C}_{3|12}\left({u}_{3}\right|{u}_{1},{u}_{2})={h}_{3|1;2}\left({C}_{3|2}\left({u}_{3}\right|{u}_{2}),{C}_{1|2}\left({u}_{1}\right|{u}_{2})\right),$$
`RVineSim`function in`R`package`VineCopula`. (For details, see Chapter 6 of Czado (2019).) - Based on the lognormal assumption of marginal components, $({D}_{i,j}^{r:\left(1\right)},\dots ,{D}_{i,j}^{r:\left(9\right)})$ are generated as follows; ${D}_{i,j}^{r:\left(n\right)}=exp\left({\widehat{\eta}}_{j}^{r:\left(n\right)}+{\mathrm{\Phi}}^{-1}\left({u}_{ij}^{r:\left(n\right)}\right){\widehat{\sigma}}^{r:\left(n\right)}\right)$ where ${\widehat{\eta}}_{j}^{r:\left(n\right)}$ and ${\widehat{\sigma}}^{r:\left(n\right)}$ are ${r}^{th}$ MC samples of ${\eta}_{j}^{\left(n\right)}$ and ${\sigma}^{\left(n\right)}$ from the marginal model for the ${n}^{th}$ line of business, respectively.
- Repeat steps 1 to 2 to get ${L}^{r:\left(n\right)}$, the MC samples of ${L}^{\left(n\right)}$ for $r=1,\dots ,R$ where$${L}^{r:\left(n\right)}=\sum _{i=2}^{9}{\widehat{Y}}_{i,11-i}^{r:\left(n\right)}-{Y}_{i,10-i}^{\left(n\right)}=\sum _{i=2}^{9}\left({D}_{i,11-i}^{r:\left(n\right)}-1\right){Y}_{i,10-i}^{\left(n\right)}\phantom{\rule{4pt}{0ex}}\phantom{\rule{4.pt}{0ex}}\mathrm{and}\phantom{\rule{4.pt}{0ex}}\phantom{\rule{4pt}{0ex}}{L}^{r:\u2022}=\sum _{n=1}^{9}{L}^{r:\left(n\right)}.$$

`R`packages such as

`statip`. Here, ${\widehat{H}}^{2}\left(L\right|\mathrm{Copula},L\left|\mathrm{Independent}\right)$, the square of the Hellinger distance between the kernel density of L under copula model with constraints (illustrated with blue solid line in Figure 4) and the density under the independent model with constraints (illustrated with black dotted line in Figure 4) is about $4.33\%$ while ${\widehat{H}}^{2}\left(L\right|\mathrm{Copula},L\left|\mathrm{Silo}\right)=13.54\%$. Therefore, one can see that the proposed vine copula structure can capture weak dependencies among the lines of business.

## 6. Practical Issues for Implementation

## 7. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

MCMC | Markov chain Monte Carlo |

DIC | Deviance information criterion |

LPML | Logarithm of the pseudo marginal likelihood |

CPO | Conditional predictive ordinate |

IFM | Inference by margin |

ERM | Enterprise risk management |

AY | Accident year |

RMSE | Root mean squared error |

MAE | Mean absolute error |

VaR | Value at risk |

CTE | Conditional tail expectation |

## Appendix A

Unconstrained Lognormal Model | |||||||||
---|---|---|---|---|---|---|---|---|---|

T1 | T2 | T3 | T4 | T5 | T6 | T7 | T8 | T9 | |

${\eta}_{2}$ | 1.000 | 1.001 | 1.000 | 1.000 | 1.001 | 1.002 | 1.002 | 1.000 | 1.003 |

${\eta}_{3}$ | 1.000 | 0.999 | 1.000 | 1.002 | 1.000 | 0.999 | 1.000 | 1.003 | 1.002 |

${\eta}_{4}$ | 1.000 | 1.001 | 0.999 | 1.000 | 1.002 | 0.999 | 1.001 | 1.000 | 1.002 |

${\eta}_{5}$ | 1.001 | 1.001 | 1.002 | 1.000 | 1.003 | 0.999 | 1.001 | 1.001 | 1.000 |

${\eta}_{6}$ | 1.006 | 1.001 | 0.999 | 1.003 | 0.999 | 1.000 | 1.002 | 1.000 | 1.000 |

${\eta}_{7}$ | 1.000 | 1.001 | 1.002 | 1.000 | 1.000 | 1.003 | 0.999 | 1.000 | 1.001 |

${\eta}_{8}$ | 0.999 | 1.002 | 1.000 | 1.006 | 0.999 | 1.001 | 1.000 | 1.000 | 1.000 |

${\eta}_{9}$ | 1.000 | 0.999 | 0.999 | 1.001 | 1.002 | 1.002 | 1.002 | 1.002 | 1.001 |

Unconstrained Gamma Model | |||||||||

T1 | T2 | T3 | T4 | T5 | T6 | T7 | T8 | T9 | |

${\eta}_{2}$ | 0.999 | 1.000 | 1.000 | 1.000 | 1.006 | 1.000 | 0.999 | 1.000 | 0.999 |

${\eta}_{3}$ | 1.000 | 1.000 | 0.999 | 1.002 | 1.001 | 0.999 | 1.004 | 1.000 | 1.003 |

${\eta}_{4}$ | 1.002 | 1.001 | 1.004 | 1.000 | 1.001 | 1.000 | 1.004 | 0.999 | 1.001 |

${\eta}_{5}$ | 1.004 | 1.000 | 1.002 | 1.002 | 0.999 | 1.001 | 1.000 | 1.003 | 1.002 |

${\eta}_{6}$ | 1.003 | 1.002 | 1.002 | 1.002 | 1.000 | 0.999 | 0.999 | 1.000 | 0.999 |

${\eta}_{7}$ | 1.000 | 1.004 | 1.001 | 1.001 | 1.002 | 0.999 | 1.001 | 1.000 | 1.000 |

${\eta}_{8}$ | 1.003 | 1.000 | 1.003 | 1.001 | 1.000 | 1.003 | 1.002 | 1.003 | 1.000 |

${\eta}_{9}$ | 1.000 | 1.002 | 1.000 | 1.002 | 0.999 | 1.004 | 1.011 | 1.000 | 1.001 |

Constrained Lognormal Model | |||||||||

T1 | T2 | T3 | T4 | T5 | T6 | T7 | T8 | T9 | |

${\eta}_{2}$ | 0.999 | 0.999 | 0.999 | 1.000 | 1.001 | 0.999 | 1.000 | 1.000 | 0.999 |

${\eta}_{3}$ | 1.002 | 1.002 | 1.001 | 1.001 | 1.001 | 1.002 | 1.003 | 1.000 | 1.002 |

${\eta}_{4}$ | 0.998 | 1.001 | 1.002 | 1.000 | 1.001 | 0.999 | 1.000 | 0.998 | 1.002 |

${\eta}_{5}$ | 1.003 | 1.000 | 0.999 | 0.999 | 0.999 | 1.001 | 1.001 | 1.000 | 1.003 |

${\eta}_{6}$ | 1.002 | 0.999 | 0.999 | 0.999 | 1.002 | 1.000 | 0.999 | 0.999 | 1.003 |

${\eta}_{7}$ | 0.999 | 1.001 | 1.000 | 1.000 | 0.999 | 0.999 | 0.999 | 0.999 | 1.000 |

${\eta}_{8}$ | 1.001 | 1.002 | 1.000 | 1.000 | 1.001 | 0.999 | 1.000 | 1.001 | 0.998 |

${\eta}_{9}$ | 1.001 | 1.000 | 1.000 | 1.001 | 1.000 | 0.999 | 1.001 | 1.000 | 0.999 |

Constrained Gamma Model | |||||||||

T1 | T2 | T3 | T4 | T5 | T6 | T7 | T8 | T9 | |

${\eta}_{2}$ | 0.999 | 1.003 | 0.999 | 1.002 | 1.000 | 1.002 | 0.999 | 0.999 | 1.000 |

${\eta}_{3}$ | 0.999 | 1.000 | 0.999 | 1.004 | 1.003 | 1.001 | 0.999 | 1.000 | 0.999 |

${\eta}_{4}$ | 1.000 | 0.999 | 1.003 | 1.001 | 1.004 | 1.002 | 0.999 | 0.999 | 1.004 |

${\eta}_{5}$ | 0.999 | 1.001 | 0.999 | 1.001 | 1.001 | 1.002 | 0.999 | 0.999 | 1.000 |

${\eta}_{6}$ | 0.999 | 1.001 | 1.000 | 1.000 | 1.000 | 0.998 | 0.999 | 1.000 | 1.000 |

${\eta}_{7}$ | 1.000 | 1.001 | 0.999 | 1.000 | 0.999 | 0.999 | 1.000 | 1.001 | 1.000 |

${\eta}_{8}$ | 1.000 | 0.999 | 1.000 | 1.000 | 0.998 | 1.000 | 1.000 | 1.001 | 0.998 |

${\eta}_{9}$ | 1.001 | 1.000 | 1.001 | 1.000 | 1.000 | 1.001 | 1.000 | 1.002 | 1.000 |

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Index | Description |
---|---|

1 | North American Workers’ Compensation |

2 | North American General Liability |

3 | North American Other Casualty |

4 | North American Non-Casualty |

5 | Overseas General Casualty |

6 | Overseas General Non-Casualty |

7 | Overseas General Personal Accident |

8 | Global Reinsurance Property |

9 | Global Reinsurance Non-Property |

Paid Losses | Incremental Development | |||||
---|---|---|---|---|---|---|

Year | Personal | Commerical | Aggregate | Personal | Commercial | Aggregate |

2 | 5,000,000 | 800,000 | 5,800,000 | - | - | - |

3 | 5,200,000 | 1,200,000 | 6,400,000 | 3.92% | 40.55% | 9.84% |

4 | 5,300,000 | 1,500,000 | 6,800,000 | 1.90% | 22.31% | 6.06% |

Unconstrained Lognormal Model | |||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

T1 | T2 | T3 | T4 | T5 | T6 | T7 | T8 | T9 | |||||||||||||||||||

Mean | 5% | 95% | Mean | 5% | 95% | Mean | 5% | 95% | Mean | 5% | 95% | Mean | 5% | 95% | Mean | 5% | 95% | Mean | 5% | 95% | Mean | 5% | 95% | Mean | 5% | 95% | |

${\eta}_{2}$ | 0.79 | 0.73 | 0.85 | 1.26 | 1.19 | 1.33 | 0.66 | 0.62 | 0.70 | 0.36 | 0.32 | 0.40 | 0.72 | 0.70 | 0.74 | 0.81 | 0.79 | 0.83 | 0.53 | 0.53 | 0.54 | 1.45 | 1.28 | 1.61 | 1.80 | 1.74 | 1.86 |

${\eta}_{3}$ | 0.32 | 0.26 | 0.38 | 0.61 | 0.54 | 0.69 | 0.22 | 0.18 | 0.26 | 0.05 | 0.02 | 0.09 | 0.27 | 0.24 | 0.29 | 0.19 | 0.18 | 0.21 | 0.09 | 0.08 | 0.09 | 0.28 | 0.10 | 0.47 | 0.68 | 0.62 | 0.74 |

${\eta}_{4}$ | 0.19 | 0.13 | 0.26 | 0.43 | 0.35 | 0.51 | 0.13 | 0.09 | 0.17 | 0.03 | 0.00 | 0.07 | 0.16 | 0.14 | 0.19 | 0.07 | 0.05 | 0.10 | 0.03 | 0.03 | 0.04 | 0.14 | 0.02 | 0.31 | 0.38 | 0.31 | 0.44 |

${\eta}_{5}$ | 0.14 | 0.07 | 0.21 | 0.24 | 0.16 | 0.33 | 0.08 | 0.03 | 0.13 | 0.03 | 0.00 | 0.07 | 0.11 | 0.08 | 0.14 | 0.03 | 0.01 | 0.05 | 0.01 | 0.01 | 0.02 | 0.12 | 0.01 | 0.28 | 0.26 | 0.19 | 0.33 |

${\eta}_{6}$ | 0.10 | 0.02 | 0.18 | 0.21 | 0.12 | 0.31 | 0.06 | 0.01 | 0.11 | 0.03 | 0.00 | 0.07 | 0.07 | 0.04 | 0.10 | 0.02 | 0.00 | 0.04 | 0.01 | 0.00 | 0.02 | 0.12 | 0.01 | 0.30 | 0.17 | 0.10 | 0.25 |

${\eta}_{7}$ | 0.08 | 0.01 | 0.16 | 0.10 | 0.02 | 0.21 | 0.04 | 0.00 | 0.09 | 0.03 | 0.00 | 0.08 | 0.05 | 0.01 | 0.08 | 0.02 | 0.00 | 0.04 | 0.01 | 0.00 | 0.02 | 0.14 | 0.01 | 0.34 | 0.12 | 0.03 | 0.20 |

${\eta}_{8}$ | 0.08 | 0.01 | 0.18 | 0.10 | 0.01 | 0.21 | 0.04 | 0.00 | 0.10 | 0.04 | 0.00 | 0.10 | 0.04 | 0.01 | 0.08 | 0.02 | 0.00 | 0.05 | 0.01 | 0.00 | 0.02 | 0.17 | 0.01 | 0.43 | 0.08 | 0.01 | 0.17 |

${\eta}_{9}$ | 0.10 | 0.01 | 0.24 | 0.11 | 0.01 | 0.26 | 0.06 | 0.01 | 0.13 | 0.06 | 0.00 | 0.14 | 0.04 | 0.00 | 0.09 | 0.03 | 0.00 | 0.06 | 0.01 | 0.00 | 0.03 | 0.23 | 0.02 | 0.57 | 0.10 | 0.01 | 0.22 |

Unconstrained Gamma Model | |||||||||||||||||||||||||||

T1 | T2 | T3 | T4 | T5 | T6 | T7 | T8 | T9 | |||||||||||||||||||

Mean | 5% | 95% | Mean | 5% | 95% | Mean | 5% | 95% | Mean | 5% | 95% | Mean | 5% | 95% | Mean | 5% | 95% | Mean | 5% | 95% | Mean | 5% | 95% | Mean | 5% | 95% | |

${\eta}_{2}$ | 0.81 | 0.76 | 0.86 | 1.27 | 1.21 | 1.34 | 0.67 | 0.63 | 0.70 | 0.37 | 0.33 | 0.41 | 0.72 | 0.70 | 0.74 | 0.81 | 0.79 | 0.83 | 0.53 | 0.53 | 0.54 | 1.60 | 1.43 | 1.76 | 1.81 | 1.76 | 1.87 |

${\eta}_{3}$ | 0.32 | 0.27 | 0.38 | 0.62 | 0.55 | 0.69 | 0.22 | 0.18 | 0.26 | 0.05 | 0.01 | 0.10 | 0.27 | 0.24 | 0.29 | 0.19 | 0.17 | 0.21 | 0.09 | 0.08 | 0.09 | 0.29 | 0.11 | 0.47 | 0.68 | 0.63 | 0.74 |

${\eta}_{4}$ | 0.19 | 0.13 | 0.25 | 0.44 | 0.36 | 0.51 | 0.13 | 0.09 | 0.17 | 0.03 | 0.00 | 0.07 | 0.16 | 0.14 | 0.19 | 0.08 | 0.05 | 0.10 | 0.03 | 0.03 | 0.04 | 0.14 | 0.02 | 0.31 | 0.38 | 0.32 | 0.44 |

${\eta}_{5}$ | 0.14 | 0.07 | 0.20 | 0.25 | 0.17 | 0.33 | 0.08 | 0.03 | 0.13 | 0.03 | 0.00 | 0.07 | 0.11 | 0.09 | 0.14 | 0.03 | 0.01 | 0.05 | 0.01 | 0.01 | 0.02 | 0.12 | 0.01 | 0.29 | 0.26 | 0.20 | 0.33 |

${\eta}_{6}$ | 0.09 | 0.02 | 0.17 | 0.21 | 0.13 | 0.30 | 0.06 | 0.02 | 0.11 | 0.03 | 0.00 | 0.07 | 0.07 | 0.04 | 0.10 | 0.02 | 0.00 | 0.04 | 0.01 | 0.00 | 0.02 | 0.13 | 0.01 | 0.31 | 0.17 | 0.10 | 0.25 |

${\eta}_{7}$ | 0.07 | 0.01 | 0.15 | 0.10 | 0.02 | 0.20 | 0.03 | 0.00 | 0.08 | 0.03 | 0.00 | 0.09 | 0.04 | 0.01 | 0.08 | 0.02 | 0.00 | 0.04 | 0.01 | 0.00 | 0.02 | 0.14 | 0.01 | 0.35 | 0.12 | 0.04 | 0.20 |

${\eta}_{8}$ | 0.08 | 0.01 | 0.17 | 0.10 | 0.01 | 0.20 | 0.04 | 0.00 | 0.10 | 0.04 | 0.00 | 0.11 | 0.04 | 0.01 | 0.08 | 0.02 | 0.00 | 0.05 | 0.01 | 0.00 | 0.02 | 0.18 | 0.01 | 0.45 | 0.08 | 0.01 | 0.17 |

${\eta}_{9}$ | 0.10 | 0.01 | 0.22 | 0.10 | 0.00 | 0.26 | 0.05 | 0.00 | 0.13 | 0.06 | 0.01 | 0.15 | 0.03 | 0.00 | 0.08 | 0.02 | 0.00 | 0.06 | 0.01 | 0.00 | 0.02 | 0.25 | 0.02 | 0.64 | 0.09 | 0.01 | 0.22 |

Constrained Lognormal Model | |||||||||||||||||||||||||||

T1 | T2 | T3 | T4 | T5 | T6 | T7 | T8 | T9 | |||||||||||||||||||

Mean | 5% | 95% | Mean | 5% | 95% | Mean | 5% | 95% | Mean | 5% | 95% | Mean | 5% | 95% | Mean | 5% | 95% | Mean | 5% | 95% | Mean | 5% | 95% | Mean | 5% | 95% | |

${\eta}_{2}$ | 0.79 | 0.74 | 0.85 | 1.26 | 1.19 | 1.32 | 0.66 | 0.63 | 0.70 | 0.36 | 0.32 | 0.40 | 0.72 | 0.70 | 0.74 | 0.81 | 0.79 | 0.83 | 0.53 | 0.53 | 0.54 | 1.45 | 1.29 | 1.61 | 1.80 | 1.75 | 1.85 |

${\eta}_{3}$ | 0.33 | 0.27 | 0.38 | 0.61 | 0.54 | 0.69 | 0.21 | 0.18 | 0.25 | 0.07 | 0.04 | 0.11 | 0.27 | 0.24 | 0.29 | 0.19 | 0.17 | 0.21 | 0.09 | 0.08 | 0.09 | 0.33 | 0.20 | 0.48 | 0.68 | 0.62 | 0.73 |

${\eta}_{4}$ | 0.20 | 0.15 | 0.26 | 0.43 | 0.36 | 0.50 | 0.14 | 0.10 | 0.17 | 0.05 | 0.03 | 0.08 | 0.16 | 0.14 | 0.19 | 0.08 | 0.05 | 0.10 | 0.03 | 0.03 | 0.04 | 0.21 | 0.12 | 0.33 | 0.38 | 0.32 | 0.44 |

${\eta}_{5}$ | 0.15 | 0.10 | 0.20 | 0.27 | 0.20 | 0.34 | 0.09 | 0.06 | 0.12 | 0.04 | 0.02 | 0.06 | 0.11 | 0.09 | 0.14 | 0.04 | 0.02 | 0.06 | 0.02 | 0.01 | 0.02 | 0.16 | 0.08 | 0.25 | 0.26 | 0.20 | 0.32 |

${\eta}_{6}$ | 0.11 | 0.07 | 0.16 | 0.20 | 0.13 | 0.27 | 0.06 | 0.04 | 0.09 | 0.03 | 0.01 | 0.05 | 0.08 | 0.05 | 0.10 | 0.02 | 0.01 | 0.04 | 0.01 | 0.01 | 0.02 | 0.12 | 0.05 | 0.20 | 0.18 | 0.13 | 0.24 |

${\eta}_{7}$ | 0.08 | 0.03 | 0.12 | 0.13 | 0.06 | 0.20 | 0.04 | 0.02 | 0.07 | 0.02 | 0.01 | 0.04 | 0.05 | 0.03 | 0.07 | 0.02 | 0.01 | 0.03 | 0.01 | 0.00 | 0.01 | 0.09 | 0.03 | 0.16 | 0.12 | 0.07 | 0.18 |

${\eta}_{8}$ | 0.05 | 0.01 | 0.10 | 0.08 | 0.02 | 0.15 | 0.03 | 0.01 | 0.05 | 0.01 | 0.00 | 0.03 | 0.03 | 0.01 | 0.06 | 0.01 | 0.00 | 0.02 | 0.00 | 0.00 | 0.01 | 0.06 | 0.01 | 0.12 | 0.08 | 0.03 | 0.13 |

${\eta}_{9}$ | 0.03 | 0.00 | 0.07 | 0.04 | 0.00 | 0.10 | 0.01 | 0.00 | 0.03 | 0.01 | 0.00 | 0.02 | 0.02 | 0.00 | 0.04 | 0.01 | 0.00 | 0.01 | 0.00 | 0.00 | 0.01 | 0.03 | 0.00 | 0.08 | 0.04 | 0.00 | 0.09 |

T1 | T2 | T3 | T4 | T5 | T6 | T7 | T8 | T9 | |||||||||||||||||||

Mean | 5% | 95% | Mean | 5% | 95% | Mean | 5% | 95% | Mean | 5% | 95% | Mean | 5% | 95% | Mean | 5% | 95% | Mean | 5% | 95% | Mean | 5% | 95% | Mean | 5% | 95% | |

${\eta}_{2}$ | 0.81 | 0.76 | 0.86 | 1.27 | 1.21 | 1.33 | 0.67 | 0.63 | 0.70 | 0.37 | 0.33 | 0.40 | 0.72 | 0.70 | 0.74 | 0.81 | 0.79 | 0.83 | 0.53 | 0.53 | 0.54 | 1.60 | 1.44 | 1.76 | 1.81 | 1.76 | 1.86 |

${\eta}_{3}$ | 0.33 | 0.27 | 0.38 | 0.62 | 0.55 | 0.69 | 0.22 | 0.18 | 0.25 | 0.07 | 0.04 | 0.10 | 0.27 | 0.24 | 0.29 | 0.19 | 0.18 | 0.21 | 0.09 | 0.08 | 0.09 | 0.33 | 0.21 | 0.48 | 0.68 | 0.63 | 0.74 |

${\eta}_{4}$ | 0.20 | 0.15 | 0.25 | 0.44 | 0.37 | 0.51 | 0.13 | 0.10 | 0.17 | 0.05 | 0.03 | 0.07 | 0.16 | 0.14 | 0.19 | 0.08 | 0.06 | 0.10 | 0.03 | 0.03 | 0.04 | 0.22 | 0.12 | 0.32 | 0.38 | 0.32 | 0.44 |

${\eta}_{5}$ | 0.15 | 0.10 | 0.19 | 0.27 | 0.20 | 0.34 | 0.09 | 0.06 | 0.12 | 0.04 | 0.02 | 0.06 | 0.11 | 0.09 | 0.14 | 0.04 | 0.02 | 0.06 | 0.02 | 0.01 | 0.02 | 0.16 | 0.08 | 0.25 | 0.26 | 0.21 | 0.32 |

${\eta}_{6}$ | 0.11 | 0.06 | 0.15 | 0.20 | 0.14 | 0.27 | 0.06 | 0.03 | 0.09 | 0.03 | 0.01 | 0.05 | 0.08 | 0.05 | 0.10 | 0.02 | 0.01 | 0.04 | 0.01 | 0.01 | 0.02 | 0.12 | 0.05 | 0.20 | 0.18 | 0.13 | 0.24 |

${\eta}_{7}$ | 0.08 | 0.04 | 0.12 | 0.13 | 0.06 | 0.19 | 0.04 | 0.02 | 0.07 | 0.02 | 0.01 | 0.04 | 0.05 | 0.03 | 0.07 | 0.02 | 0.01 | 0.03 | 0.01 | 0.00 | 0.01 | 0.09 | 0.03 | 0.16 | 0.12 | 0.07 | 0.18 |

${\eta}_{8}$ | 0.05 | 0.02 | 0.09 | 0.08 | 0.02 | 0.15 | 0.03 | 0.01 | 0.05 | 0.01 | 0.00 | 0.03 | 0.03 | 0.01 | 0.06 | 0.01 | 0.00 | 0.02 | 0.00 | 0.00 | 0.01 | 0.06 | 0.01 | 0.12 | 0.08 | 0.03 | 0.13 |

${\eta}_{9}$ | 0.03 | 0.00 | 0.06 | 0.04 | 0.00 | 0.10 | 0.01 | 0.00 | 0.04 | 0.01 | 0.00 | 0.02 | 0.02 | 0.00 | 0.04 | 0.01 | 0.00 | 0.01 | 0.00 | 0.00 | 0.01 | 0.03 | 0.00 | 0.08 | 0.04 | 0.00 | 0.09 |

T1 | T2 | T3 | T4 | T5 | T6 | T7 | T8 | T9 | ||
---|---|---|---|---|---|---|---|---|---|---|

DIC | LNU | −57.1 | −46.5 | −90.3 | −84.2 | −125.1 | −140.5 | −224 | 16.8 | −61.8 |

GamU | −38.6 | −7.7 | −74.4 | −75.3 | −105.9 | −123.6 | −214.1 | 47.6 | −14.9 | |

LNC | −62.9 | −50.5 | −95.9 | −88.4 | −129.2 | −144.1 | −228.5 | 11.9 | −66.7 | |

GamC | −44.2 | −11.8 | −79.2 | −79.6 | −110.5 | −127.3 | −218 | 42.8 | −19.2 | |

LPML | LNU | −326.2 | −129.3 | −117.2 | −291.7 | −83.2 | −127.3 | −43.2 | −241.8 | −155.2 |

GamU | −379.2 | −168.6 | −144.3 | −364.8 | −103.1 | −162 | −62.1 | −342.7 | −215.2 | |

LNC | −368.1 | −132.7 | −123.4 | −300.6 | −82.6 | −118.1 | −43.4 | −242 | −144.8 | |

GamC | −419.6 | −176.4 | −145.5 | −392.5 | −100.1 | −161.3 | −57.5 | −350.5 | −219.5 |

Tree | Edge | Family | $\mathit{\varphi}$ | $\mathit{\tau}$ |
---|---|---|---|---|

1 | 9,8 | Gumbel | 1.87 | 0.46 |

1 | 9,6 | ${90}^{\circ}$ rotated Clayton | −1.20 | −0.37 |

1 | 8,1 | Frank | 9.40 | 0.65 |

1 | 1,7 | ${270}^{\circ}$ rotated Clayton | −0.20 | −0.09 |

1 | 7,2 | ${90}^{\circ}$ rotated Clayton | −0.95 | −0.32 |

1 | 6,4 | Frank | 5.87 | 0.50 |

1 | 9,5 | ${270}^{\circ}$ rotated Gumbel | −1.24 | −0.20 |

1 | 4,3 | Survival Gumbel | 1.61 | 0.38 |

2 | 9,4;6 | ${90}^{\circ}$ rotated Gumbel | −1.35 | −0.26 |

3 | 9,7;8,1 | Survival Clayton | 0.52 | 0.21 |

Unconstrained | Constrained | ||||
---|---|---|---|---|---|

Copula | Silo | Copula | Silo | Actual | |

AY = 2005 | 529,426 | 219,222 | 148,767 | 81,657 | 90,662 |

AY = 2006 | 264,689 | 167,787 | 185,653 | 131,065 | 109,420 |

AY = 2007 | 299,024 | 196,099 | 319,881 | 227,435 | 209,649 |

AY = 2008 | 454,503 | 404,983 | 461,885 | 414,982 | 338,359 |

AY = 2009 | 405,808 | 386,094 | 447,697 | 409,172 | 304,544 |

AY = 2010 | 651,228 | 610,943 | 703,166 | 615,770 | 450,724 |

AY = 2011 | 942,324 | 1,071,867 | 986,753 | 1,070,549 | 762,328 |

AY = 2012 | 2,182,202 | 2,649,858 | 2,181,708 | 2,648,604 | 1,837,070 |

Total | 5,729,205 | 5,706,854 | 5,435,511 | 5,599,236 | 4,102,756 |

Unconstrained | Constrained | |||
---|---|---|---|---|

Copula | Silo | Copula | Silo | |

RMSE | 234,540 | 318,849 | 190,381 | 315,934 |

MAE | 203,306 | 203,900 | 166,594 | 189,311 |

VaR | CTE | |||||
---|---|---|---|---|---|---|

90% | 95% | 99% | 90% | 95% | 99% | |

Marginal T1 | 384,383 | 420,016 | 494,059 | 432,782 | 464,901 | 521,869 |

Marginal T2 | 1,403,464 | 1,520,387 | 1,720,617 | 1,560,576 | 1,657,167 | 1,893,831 |

Marginal T3 | 655,532 | 694,812 | 766,766 | 707,338 | 740,161 | 805,153 |

Marginal T4 | 1,657,500 | 1,802,081 | 2,126,876 | 1,858,735 | 1,993,388 | 2,313,396 |

Marginal T5 | 777,885 | 807,011 | 869,457 | 818,302 | 844,170 | 895,520 |

Marginal T6 | 691,142 | 714,401 | 757,403 | 721,538 | 741,557 | 791,119 |

Marginal T7 | 405,697 | 413,424 | 428,243 | 416,023 | 423,117 | 439,641 |

Marginal T8 | 586,402 | 696,629 | 1,009,717 | 757,769 | 878,869 | 1,199,343 |

Marginal T9 | 556,892 | 591,841 | 658,353 | 603,369 | 633,144 | 703,385 |

Marginal Total | 7,118,897 | 7,660,602 | 8,831,491 | 7,876,431 | 8,376,473 | 9,563,257 |

Independent Aggregate | 6,163,307 | 6,363,592 | 6,755,125 | 6,428,484 | 6,608,753 | 6,978,484 |

Copula Aggregate | 6,145,848 | 6,376,847 | 6,774,463 | 6,435,712 | 6,610,684 | 6,995,674 |

Silo Aggregate | 6,434,983 | 6,692,370 | 7,309,860 | 6,815,535 | 7,076,775 | 7,586,896 |

VaR | CTE | |||||
---|---|---|---|---|---|---|

90% | 95% | 99% | 90% | 95% | 99% | |

Independent | 955,590 | 1,297,010 | 2,076,366 | 1,447,947 | 1,767,720 | 2,584,773 |

Copula | 973,049 | 1,283,755 | 2,057,028 | 1,440,719 | 1,765,789 | 2,567,583 |

Silo | 683,914 | 968,232 | 1,521,631 | 1,060,896 | 1,299,698 | 1,976,361 |

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## Share and Cite

**MDPI and ACS Style**

Jeong, H.; Dey, D.
Application of a Vine Copula for Multi-Line Insurance Reserving. *Risks* **2020**, *8*, 111.
https://doi.org/10.3390/risks8040111

**AMA Style**

Jeong H, Dey D.
Application of a Vine Copula for Multi-Line Insurance Reserving. *Risks*. 2020; 8(4):111.
https://doi.org/10.3390/risks8040111

**Chicago/Turabian Style**

Jeong, Himchan, and Dipak Dey.
2020. "Application of a Vine Copula for Multi-Line Insurance Reserving" *Risks* 8, no. 4: 111.
https://doi.org/10.3390/risks8040111