Bivariate Copula Trees for Gross Loss Aggregation with Positively Dependent Risks
Abstract
:1. Introduction
 Groundup loss (total loss without any policy conditions applied)
 Retained loss (loss retained by insured party)
 Gross loss (loss to an insurer after application of policy financial terms)
2. Preliminaries
2.1. Basic Concepts
2.2. Financial Terms
2.3. Ordering the Gross Loss Sums
3. Partial Sums and Aggregation Trees
3.1. Copulas at Summation Nodes
Algorithm 1 Estimate copula parameter and add two risks in groundup pass 
INPUT: Pmfs ${p}_{X}$, ${p}_{Y}$ with support sizes ${N}_{x}$, ${N}_{y}$, copula $C(u,v;\theta )$, (if available) partial derivative ${C}^{\prime}(u,v;\theta )=\partial C(u,v;\theta )/\partial \theta $, initial ${\theta}_{0}$ and bounds ${\theta}_{min}$, ${\theta}_{max}$, correlation $\rho (X,Y)$, maximum iteration ${t}_{max}$, numeric tolerance $\u03f5$. OUTPUT: ${\theta}^{*}$, ${p}_{S}$.

3.2. Covariance Scaling
3.3. A Comment on Back Allocation
4. Computational Aspects
Algorithm 2 Add two risks in gross loss pass 
INPUT: Pmfs ${p}_{{\varphi}_{X}\left(X\right)}$, ${p}_{{\varphi}_{Y}\left(Y\right)}$ with support sizes ${N}_{x}$, ${N}_{y}$, parameterized copula model $C(u,v;{\theta}^{\star})$, decomposition flag. OUTPUT: Pmf ${p}_{\varphi \left(S\right)}$ of the gross loss sum $\varphi \left(S\right)={\varphi}_{X}\left(X\right)+{\varphi}_{Y}\left(Y\right)$.

Algorithm 3 Add two risks in gross loss pass using Fréchet copula with covariance scaling 
INPUT: Pmfs ${p}_{X}$ and ${p}_{Y}$, financial terms ${\varphi}_{X}$ and ${\varphi}_{Y}$, covariance $\mathrm{Cov}[X,Y]$. OUTPUT: Pmf of the sum ${p}_{\varphi \left(S\right)}$.

5. Results
6. Conclusions
7. Future Research
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A. Bivariate Copulas and Their Partial Derivatives
Appendix A.1. Joe Copula
Appendix A.2. Gumbel Copula
Appendix A.3. Morgenstern Copula
Appendix A.4. Student’s t Copula, ν=1 (Cauchy)
Appendix B. Copula pmf on Finite Precision Machine
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Name  $\mathit{C}(\mathit{u},\mathit{v};\mathit{\theta})$  Parameter $\mathit{\theta}$ 

Fréchet  $(1\theta )uv+\theta min(u,v)$  $\theta \in [0,1]$ 
Gaussian *  ${\Phi}_{\Sigma}\left[{\Phi}^{1}\left(u\right),{\Phi}^{1}\left(v\right);\theta \right]$  $\theta \in [0,1]$ 
Student’s t **  ${\int}_{\infty}^{x}{\int}_{\infty}^{y}{\left(2\pi \sqrt{1{\theta}^{2}}\right)}^{1}{\left[1+\frac{{x}^{2}2\theta xy+{y}^{2}}{\nu (1{\theta}^{2})}\right]}^{(\nu +2)/2}dxdy$, $x={P}_{\nu}^{1}\left(u\right)$, $y={P}_{\nu}^{1}\left(v\right)$  $\theta \in [1,1]$ 
Gumbel  $exp\left\{{\left({[ln\left(u\right)]}^{\theta}+{[ln\left(v\right)]}^{\theta}\right)}^{1/\theta}\right\}$  $\theta \in [1,+\infty )$ 
Joe  $1{\left[{(1u)}^{\theta}+{(1v)}^{\theta}{(1u)}^{\theta}{(1v)}^{\theta}\right]}^{1/\theta}$  $\theta \in [1,+\infty )$ 
Morgenstern  $uv[1+\theta (1u\left)\right(1v\left)\right]$  $\theta \in [0,1]$ 
$\mathit{a}+\mathit{b}$  $\mathbf{a}\mathbf{b}$  $exp\left(\mathit{a}\right)$  $ln\mathit{a}$  ${\mathit{a}}^{\mathit{b}}$  $arctan\left(\mathit{a}\right)$ 

5593 (1)  6130 (1.1)  315,567 (56.4)  209,969 (37.5)  670,001 (119.8)  130,785 (23.4) 
Statistic  Portfolio 1 (Large)  Portfolio 2 (Medium)  Portfolio 3 (Small) 

Event peril  Hurricane  Hurricane  Earthquake 
# of risks  31,896  9056  1209 
# of sublimits  3364  412  19 
# of layers  1778  398  14 
# of policies  1676  398  14 
Total replacement value ()MM USD)  671,191  25,811  14,350 
Total groundup loss  410  1198  427 
Total groundup damage ratio  0.06%  4.64%  0.41% 
Portfolio 1 (Large)  Portfolio 2 (Medium)  Portfolio 3 (Small)  

[MM $]  $\mu $  $\sigma $  TVaR${}_{\mathbf{1}\%}$  TVaR${}_{\mathbf{5}\%}$  TVaR${}_{\mathbf{10}\%}$  $\mu $  $\sigma $  TVaR${}_{\mathbf{1}\%}$  TVaR${}_{\mathbf{5}\%}$  TVaR${}_{\mathbf{10}\%}$  $\mu $  $\sigma $  TVaR${}_{\mathbf{1}\%}$  TVaR${}_{\mathbf{5}\%}$  TVaR${}_{\mathbf{10}\%}$ 
Multivariate Fréchet  293.2 (1.00)  276.5 (1.00)  1167.6 (1.00)  730.4 (1.00)  647.5 (1.00)  36.1 (1.00)  4.8 (1.00)  53.6 (1.00)  48.5 (1.00)  45.8 (1.00)  16.0 (1.00)  6.3 (1.00)  37.5 (1.00)  32.7 (1.00)  29.9 (1.00) 
Fréchet  291.0 (0.99)  271.3 (0.98)  2000.7 (1.71)  1035.3 (1.42)  795.4 (1.23)  36.1 (1.00)  5.4 (1.12)  53.7 (1.00)  49.2 (1.01)  46.7 (1.02)  15.9 (1.00)  6.3 (1.00)  37.5 (1.00)  32.6 (1.00)  29.8 (1.00) 
Gaussian  290.1 (0.99)  285.6 (1.03)  1626.9 (1.39)  1129.6 (1.55)  931.2 (1.44)  36.2 (1.00)  6.0 (1.25)  56.3 (1.05)  51.0 (1.05)  48.2 (1.05)  15.9 (1.00)  6.3 (1.00)  37.4 (1.00)  32.6 (1.00)  29.8 (1.00) 
Gaussian decomp  290.6 (0.99)  279.9 (1.01)  2091.4 (1.79)  1085.4 (1.49)  823.6 (1.27)  36.2 (1.00)  6.0 (1.25)  57.4 (1.07)  51.2 (1.06)  48.2 (1.05)  15.9 (1.00)  6.3 (1.00)  37.6 (1.00)  32.6 (1.00)  29.8 (1.00) 
Joe  291.2 (0.99)  183.6 (0.66)  1141.2 (0.98)  765.9 (1.05)  651.5 (1.01)  36.2 (1.00)  5.4 (1.13)  54.3 (1.01)  49.5 (1.02)  46.9 (1.02)  15.9 (1.00)  6.3 (1.00)  37.6 (1.00)  32.6 (1.00)  29.8 (1.00) 
Joe decomp  291.4 (0.99)  183.5 (0.66)  1155.2 (0.99)  769.5 (1.05)  652.7 (1.01)  36.3 (1.00)  5.4 (1.13)  54.1 (1.01)  49.4 (1.02)  47.0 (1.03)  15.9 (1.00)  6.3 (1.00)  37.6 (1.00)  32.6 (1.00)  29.8 (1.00) 
Gumbel  291.3 (0.99)  184.5 (0.67)  1138.9 (0.98)  767.9 (1.05)  654.1 (1.01)  36.3 (1.01)  5.5 (1.15)  55.1 (1.03)  50.0 (1.03)  47.4 (1.03)  15.9 (1.00)  6.3 (1.00)  37.6 (1.00)  32.6 (1.00)  29.8 (1.00) 
Gumbel decomp  291.4 (0.99)  185.0 (0.67)  1175.8 (1.01)  773.9 (1.06)  654.8 (1.01)  36.1 (1.00)  5.5 (1.15)  54.1 (1.01)  49.5 (1.02)  47.0 (1.03)  15.9 (1.00)  6.3 (1.00)  37.6 (1.00)  32.6 (1.00)  29.8 (1.00) 
Morgenstern  293.2 (1.00)  255.9 (0.93)  1101.3 (0.94)  908.7 (1.24)  812.4 (1.25)  36.2 (1.00)  6.0 (1.25)  55.4 (1.03)  50.7 (1.04)  48.1 (1.05)  15.9 (1.00)  6.3 (1.00)  37.4 (1.00)  32.6 (1.00)  29.8 (1.00) 
Morgenstern decomp  293.2 (1.00)  238.9 (0.86)  1718.7 (1.47)  971.8 (1.33)  765.7 (1.18)  36.2 (1.00)  6.0 (1.25)  57.1 (1.06)  51.1 (1.05)  48.3 (1.05)  15.9 (1.00)  6.3 (1.00)  37.6 (1.00)  32.7 (1.00)  29.8 (1.00) 
Student’s t, $\nu =1$  290.7 (0.99)  150.3 (0.54)  794.7 (0.68)  656.9 (0.90)  582.3 (0.90)  36.8 (1.02)  4.4 (0.93)  54.2 (1.01)  49.2 (1.01)  46.4 (1.01)  16.0 (1.00)  5.8 (0.92)  36.7 (0.98)  32.0 (0.98)  29.0 (0.97) 
Student’s t decomp, $\nu =1$  290.7 (0.99)  154.1 (0.56)  788.2 (0.68)  656.9 (0.90)  585.1 (0.90)  36.3 (1.01)  5.1 (1.06)  53.4 (1.00)  48.9 (1.01)  46.6 (1.02)  16.0 (1.00)  5.8 (0.92)  36.7 (0.98)  32.0 (0.98)  29.0 (0.97) 
Student’s t, $\nu =3$  290.9 (0.99)  151.1 (0.55)  916.9 (0.79)  677.7 (0.93)  592.9 (0.92)  36.5 (1.01)  4.3 (0.90)  55.6 (1.04)  49.4 (1.02)  46.1 (1.01)  16.0 (1.00)  6.0 (0.95)  37.1 (0.99)  32.3 (0.99)  29.4 (0.98) 
Student’s t decomp, $\nu =3$  291.0 (0.99)  154.4 (0.56)  791.3 (0.68)  658.1 (0.90)  586.2 (0.91)  36.3 (1.00)  5.2 (1.08)  53.4 (1.00)  49.0 (1.01)  46.7 (1.02)  16.0 (1.00)  6.0 (0.96)  37.2 (0.99)  32.3 (0.99)  29.4 (0.98) 
Student’s t, $\nu =10$  290.9 (0.99)  177.8 (0.64)  1194.1 (1.02)  794.4 (1.09)  663.0 (1.02)  36.7 (1.02)  5.2 (1.09)  60.0 (1.12)  51.8 (1.07)  48.2 (1.05)  15.9 (1.00)  6.2 (0.99)  37.5 (1.00)  32.6 (1.00)  29.7 (0.99) 
Student’s t decomp, $\nu =10$  291.1 (0.99)  181.6 (0.66)  1124.2 (0.96)  761.4 (1.04)  649.1 (1.00)  36.3 (1.00)  5.3 (1.11)  53.8 (1.00)  49.2 (1.01)  46.9 (1.02)  15.9 (1.00)  6.3 (1.00)  37.5 (1.00)  32.6 (1.00)  29.7 (0.99) 
Student’s t, $\nu =30$  290.9 (0.99)  256.2 (0.93)  1596.6 (1.37)  1056.0 (1.45)  854.9 (1.32)  36.1 (1.00)  5.8 (1.21)  57.7 (1.08)  51.3 (1.06)  48.2 (1.05)  15.8 (0.99)  6.4 (1.01)  37.9 (1.01)  32.7 (1.00)  29.7 (1.00) 
Student’s t decomp, $\nu =30$  291.2 (0.99)  257.1 (0.93)  1884.1 (1.61)  1007.5 (1.38)  782.8 (1.21)  36.2 (1.00)  5.9 (1.22)  56.6 (1.06)  50.8 (1.05)  48.0 (1.05)  15.8 (0.99)  6.4 (1.01)  37.6 (1.00)  32.7 (1.00)  29.7 (1.00) 
Name  Portfolio 1 (Large)  Portfolio 2 (Medium)  Portfolio 3 (Small) 

Multivariate Fréchet  0.43 (1.0)  0.11 (1.0)  0.01 (1.0) 
Fréchet  0.54 (1.3)  0.15 (1.3)  0.01 (1.1) 
Gaussian  22.57 (52.5)  16.81 (152.8)  0.28 (28.0) 
Gaussian decomp  22.72 (52.8)  18.44 (167.6)  0.29 (29.4) 
Joe  34.89 (81.1)  15.12 (137.5)  0.45 (44.5) 
Joe decomp  35.02 (81.4)  15.91 (144.6)  0.45 (44.8) 
Gumbel  52.66 (122.5)  23.40 (212.7)  0.63 (63.3) 
Gumbel decomp  52.91 (123.0)  24.04 (218.6)  0.65 (65.4) 
Morgenstern  1.03 (2.4)  0.49 (4.4)  0.02 (2.0) 
Morgenstern decomp  1.10 (2.6)  0.50 (4.5)  0.02 (2.2) 
Student’s t, $\nu =1$  15.52 (36.1)  5.28 (48.0)  0.32 (32.1) 
Student’s t decomp, $\nu =1$  15.80 (36.7)  5.85 (53.2)  0.35 (34.7) 
Student’s t, $\nu =3$  40.25 (93.6)  16.30 (148.2)  1.11 (110.6) 
Student’s t decomp, $\nu =3$  40.95 (95.2)  17.26 (156.9)  1.15 (114.6) 
Student’s t, $\nu =10$  81.55 (189.7)  34.18 (310.7)  2.20 (220.3) 
Student’s t decomp, $\nu =10$  83.40 (194.0)  34.55 (314.1)  2.37 (236.8) 
Student’s t, $\nu =30$  104.66 (243.4)  35.76 (325.1)  3.04 (303.7) 
Student’s t decomp, $\nu =30$  105.52 (245.4)  35.83 (325.7)  3.07 (307.2) 
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Wójcik, R.; Liu, C.W. Bivariate Copula Trees for Gross Loss Aggregation with Positively Dependent Risks. Risks 2022, 10, 144. https://doi.org/10.3390/risks10080144
Wójcik R, Liu CW. Bivariate Copula Trees for Gross Loss Aggregation with Positively Dependent Risks. Risks. 2022; 10(8):144. https://doi.org/10.3390/risks10080144
Chicago/Turabian StyleWójcik, Rafał, and Charlie Wusuo Liu. 2022. "Bivariate Copula Trees for Gross Loss Aggregation with Positively Dependent Risks" Risks 10, no. 8: 144. https://doi.org/10.3390/risks10080144