Monte Carlo Simulation of the Moments of a Copula-Dependent Risk Process with Weibull Interwaiting Time
Abstract
:1. Introduction
2. Materials and Methods
2.1. Aggregate Risk Model
2.1.1. Weibull Counting Process
2.1.2. Copula
2.2. Recursive Moment Expressions
2.3. Monte Carlo Simulation
- Generate pairs of dependent random variates from multivariate distributions constructed from the chosen copula. The multivariate distributions are based on the best-fit distribution obtained from the insurance dataset.
- Compute the random time , from the accumulated as in Equation (2).
- Compute the aggregate discounted claims , by assuming deterministic .
- Stop running the iterations once the is above a pre-determined term of a policy contract.
- Repeat the process from step 1 to 4 for n simulations.
- Determine the moments, premium and VaR from the simulated risk process .
3. Results and Discussion
3.1. Results Verification
3.2. Fitting Distribution and Parameter Estimation of Insurance Datasets
3.2.1. The Claim Sizes Distribution
3.2.2. The Interwaiting Time Distribution
3.2.3. The Dependency between the Claim Sizes and the IWT
3.3. Risk Characteristic of the Risk Process with Overdispersed Claim Arrival
3.4. Scenario Analysis on Risk Characteristic of the Risk Process under Various Dispersion Effect on Claims Arrival
Premium Computation and VaR of the Risk Portfolio
4. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
Abbreviations
AD | Anderson-Darling |
AIC | Akaike information criterion |
BIC | Bayesian information criterion |
CDF | Cumulative distribution function |
FGM | Farlie-Gumbel-Mogenstern |
GOF | Goodness-of-fit |
IWT | Interwaiting time |
KS | Kolmogorov–Smirnov |
MDPI | Multidisciplinary Digital Publishing Institute |
SCR | Solvency capital requirement |
Std Dev | Standard deviation |
VaR | Value-at-Risk |
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Moments | Monte Carlo | Laplace Transform | Relative Deviation | |
---|---|---|---|---|
−0.999 | 479.23 | 477.66 | 0.330% | |
−0.9 | 474.83 | 475.23 | 0.084% | |
−0.5 | 469.16 | 465.43 | 0.803% | |
Mean | 0 | 452.79 | 453.17 | 0.084% |
0.5 | 440.59 | 440.92 | 0.074% | |
0.9 | 433.43 | 431.12 | 0.537% | |
0.999 | 430.23 | 428.69 | 0.359% | |
−0.999 | 106,554.00 | 106,351.84 | 0.190% | |
−0.9 | 105,554.20 | 103,929.50 | 1.563% | |
−0.5 | 94,099.89 | 94,253.78 | 0.163% | |
Variance | 0 | 80,630.85 | 82,420.23 | 2.171% |
0.5 | 69,601.11 | 70,874.44 | 1.797% | |
0.9 | 61,217.14 | 61,845.86 | 1.017% | |
0.999 | 59.182.13 | 59,638.74 | 0.766% |
Min | Max | Median | Mean | Estimated Std Dev | Estimated Skewness | Estimated Kurtosis |
---|---|---|---|---|---|---|
0.01 | 112 | 3.6 | 10.67252 | 17.14509 | 2.997792 | 14.29496 |
GOF Criterion | GOF Test | ||||
---|---|---|---|---|---|
Distribution | −2 Log Likelihood | AIC | BIC | KS Test (p-Value) | AD Test (p-Value) |
Exponential | 828.447 | 830.447 | 833.260 | 0.0000 | 0.0000 |
Lognormal | 787.928 | 791.928 | 797.552 | 0.9980 | 0.9775 |
Gamma | 804.195 | 808.195 | 813.820 | 0.0912 | 0.0389 |
Weibull | 796.037 | 800.037 | 805.662 | 0.4291 | 0.1797 |
Pareto | 788.541 | 792.541 | 798.165 | 0.9096 | 0.8201 |
Burr | 788.420 | 794.421 | 802.857 | 0.9339 | 0.8521 |
Min | Max | Median | Mean | Estimated Std Dev | Estimated Skewness | Estimated Kurtosis |
---|---|---|---|---|---|---|
0.0027 | 7.3922 | 0.1478 | 0.3786 | 0.8125 | 6.0819 | 49.9694 |
GOF Criterion | GOF Test | ||||
---|---|---|---|---|---|
Distribution | Estimated Parameters | AIC | BIC | KS Test (p-Value) | AD Test (p-Value) |
Weibull | = 0.700152 = 0.282022 | −27.4824 | −21.8580 | 0.5383 | 0.2877 |
Exponential | = 2.641138 | 9.08241 | 11.8946 | 0.0001 | 0.0000 |
Copula | Estimated Parameter, | Log Likelihood | AIC | BIC |
---|---|---|---|---|
Independence | 0 | 0 | 0 | 0 |
Gaussian | 0.19 | 2.02 | −2.04 | 0.77 |
T | 0.19 | 2.12 | −0.25 | 5.38 |
Clayton | 0.32 | 3.57 | −5.13 | −2.32 |
Gumbel | 1.09 | 0.92 | 0.17 | 2.98 |
Frank | 1.1 | 1.91 | −1.81 | 1 |
Joe | 1.06 | 0.17 | 1.65 | 4.47 |
IWT Distribution | Mean | Variance | Skewness | Kurtosis |
---|---|---|---|---|
Weibull | 169.343 | 21313.540 | 6.296 | 145.444 |
Exponential | 154.034 | 18239.790 | 7.242 | 150.931 |
IWT | VaR ($m) | Premium Amount ($m) | ||
---|---|---|---|---|
Distribution | 95% | 99.50% | Mean Principle | Std Dev Principle |
Weibull | 406.757 | 846.901 | 186.278 | 183.942 |
Exponential | 363.706 | 809.141 | 169.438 | 167.540 |
Overdispersed | Equidispersed | Underdispersed | |
---|---|---|---|
() | () | () | |
Mean | 4.243 | 4.243 | 4.243 |
Variance | 9.903 | 4.243 | 1.375 |
Overdispersed | ||||
---|---|---|---|---|
Mean | Variance | Skewness | Kurtosis | |
−1 | 66.359 | 10202.750 | 9.423 | 276.883 |
−0.5 | 58.933 | 7595.139 | 7.010 | 128.587 |
0 | 53.764 | 7168.824 | 10.130 | 294.856 |
0.32 | 51.266 | 7538.556 | 22.392 | 1599.203 |
10 | 26.722 | 1205.303 | 16.188 | 714.862 |
50 | 19.757 | 109.521 | 2.620 | 105.380 |
100 | 19.454 | 90.367 | −0.336 | −0.839 |
Equidispersed | ||||
Mean | Variance | Skewness | Kurtosis | |
−1 | 60.663 | 8240.178 | 7.779 | 162.108 |
−0.5 | 54.657 | 6749.611 | 8.895 | 190.364 |
0 | 50.751 | 5835.950 | 10.186 | 315.298 |
0.32 | 48.389 | 5557.512 | 13.696 | 576.241 |
10 | 34.768 | 2378.918 | 13. | 372.840 |
50 | 28.978 | 855.983 | 21.968 | 1236.261 |
100 | 27.540 | 533.322 | 25.754 | 1589.694 |
Underdispersed | ||||
Mean | Variance | Skewness | Kurtosis | |
−1 | 47.414 | 6147.893 | 8.180 | 148.807 |
−0.5 | 42.014 | 4487.100 | 11.205 | 347.476 |
0 | 39.587 | 3991.358 | 9.331 | 190.339 |
0.32 | 38.672 | 4477.052 | 13.927 | 431.441 |
10 | 32.096 | 2714.674 | 14.606 | 497.202 |
50 | 30.036 | 2288.304 | 22.567 | 1324.124 |
100 | 29.623 | 2289.057 | 25.280 | 1591.957 |
Overdispersed | ||||
---|---|---|---|---|
VaR ($m) | Premium Amount ($m) | |||
95% | 99.50% | Mean Principle | Std Dev Principle | |
−1 | 225.744 | 573.577 | 72.994 | 76.459 |
−0.5 | 197.729 | 498.658 | 64.827 | 67.649 |
0 | 180.155 | 473.747 | 59.140 | 62.231 |
0.32 | 167.568 | 454.264 | 56.392 | 59.948 |
10 | 63.731 | 199.876 | 29.394 | 30.194 |
50 | 34.147 | 41.335 | 21.733 | 20.804 |
100 | 32.855 | 37.793 | 21.399 | 20.405 |
Equidispersed | ||||
VaR ($m) | Premium Amount ($m) | |||
95% | 99.50% | Mean Principle | Std Dev Principle | |
−1 | 202.152 | 514.118 | 66.729 | 69.741 |
−0.5 | 176.053 | 466.067 | 60.122 | 62.872 |
0 | 161.369 | 443.094 | 55.827 | 58.391 |
0.32 | 151.449 | 424.326 | 53.228 | 55.844 |
10 | 88.749 | 288.379 | 38.245 | 39.646 |
50 | 58.319 | 170.823 | 31.876 | 31.904 |
100 | 52.508 | 120.991 | 30.293 | 29.849 |
Equidispersed | ||||
VaR ($m) | Premium Amount ($m) | |||
95% | 99.50% | Mean Principle | Std Dev Principle | |
−1 | 163.186 | 463.176 | 52.155 | 55.254 |
−0.5 | 138.329 | 380.708 | 46.215 | 48.712 |
0 | 128.152 | 374.841 | 43.546 | 45.905 |
0.32 | 121.294 | 364.802 | 42.540 | 45.363 |
10 | 91.612 | 298.824 | 35.306 | 37.307 |
50 | 80.044 | 269.078 | 33.039 | 34.819 |
100 | 77.195 | 256.812 | 32.585 | 34.407 |
Overdispersed | ||||
---|---|---|---|---|
VaR ($m) | Premium Amount ($m) | |||
95% | 99.50% | Mean Principle | Std Dev Principle | |
−100 | 225.816 | 566.751 | 73.363 | 76.605 |
−50 | 227.023 | 554.466 | 73.356 | 76.737 |
−10 | 222.041 | 544.253 | 72.451 | 75.522 |
0 | 179.770 | 475.843 | 59.577 | 62.559 |
10 | 74.381 | 223.736 | 31.722 | 32.858 |
50 | 34.979 | 45.038 | 21.940 | 21.018 |
100 | 33.045 | 38.072 | 21.427 | 20.435 |
Equidispersed | ||||
VaR ($m) | Premium Amount ($m) | |||
95% | 99.50% | Mean Principle | Std Dev Principle | |
−100 | 203.215 | 536.925 | 67.420 | 71.469 |
−50 | 204.435 | 532.854 | 67.100 | 70.079 |
−10 | 200.478 | 504.214 | 66.215 | 69.807 |
0 | 161.197 | 429.608 | 55.876 | 58.620 |
10 | 94.393 | 300.319 | 39.512 | 41.085 |
50 | 58.900 | 177.422 | 31.990 | 32.082 |
100 | 52.942 | 124.696 | 30.383 | 29.798 |
Underdispersed | ||||
VaR ($m) | Premium Amount ($m) | |||
95% | 99.50% | Mean Principle | Std Dev Principle | |
−100 | 162.916 | 459.141 | 52.387 | 55.664 |
−50 | 161.985 | 445.691 | 52.355 | 55.655 |
−10 | 153.385 | 432.808 | 50.560 | 53.662 |
0 | 128.944 | 368.501 | 44.232 | 47.049 |
10 | 92.540 | 297.590 | 35.405 | 37.388 |
50 | 81.145 | 275.849 | 33.254 | 35.059 |
100 | 77.572 | 252.967 | 32.445 | 33.838 |
Overdispersed | ||||
---|---|---|---|---|
VaR ($m) | Premium Amount ($m) | |||
95% | 99.50% | Mean Principle | Std Dev Principle | |
−0.999 | 204.390 | 515.814 | 66.564 | 69.693 |
−0.5 | 190.822 | 509.914 | 62.891 | 66.048 |
0 | 178.699 | 480.092 | 59.106 | 62.356 |
0.5 | 165.488 | 448.837 | 55.451 | 58.282 |
0.999 | 154.182 | 426.935 | 51.789 | 54.361 |
Equidispersed | ||||
VaR ($m) | Premium Amount ($m) | |||
95% | 99.50% | Mean Principle | Std Dev Principle | |
−0.999 | 178.922 | 492.664 | 60.988 | 64.017 |
−0.5 | 172.109 | 458.163 | 58.608 | 61.381 |
0 | 161.823 | 436.838 | 56.028 | 58.640 |
0.5 | 151.357 | 423.809 | 53.247 | 56.144 |
0.999 | 140.298 | 411.168 | 50.556 | 53.115 |
Underdispersed | ||||
VaR ($m) | Premium Amount ($m) | |||
95% | 99.50% | Mean Principle | Std Dev Principle | |
−0.999 | 140.496 | 403.169 | 46.940 | 49.749 |
−0.5 | 133.912 | 392.826 | 45.231 | 47.897 |
0 | 126.499 | 372.115 | 43.538 | 46.174 |
0.5 | 120.662 | 367.651 | 42.261 | 44.904 |
0.999 | 115.166 | 349.602 | 40.982 | 43.725 |
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Syed Yusoff Alhabshi, S.F.; Zamzuri, Z.H.; Mohd Ramli, S.N. Monte Carlo Simulation of the Moments of a Copula-Dependent Risk Process with Weibull Interwaiting Time. Risks 2021, 9, 109. https://doi.org/10.3390/risks9060109
Syed Yusoff Alhabshi SF, Zamzuri ZH, Mohd Ramli SN. Monte Carlo Simulation of the Moments of a Copula-Dependent Risk Process with Weibull Interwaiting Time. Risks. 2021; 9(6):109. https://doi.org/10.3390/risks9060109
Chicago/Turabian StyleSyed Yusoff Alhabshi, Sharifah Farah, Zamira Hasanah Zamzuri, and Siti Norafidah Mohd Ramli. 2021. "Monte Carlo Simulation of the Moments of a Copula-Dependent Risk Process with Weibull Interwaiting Time" Risks 9, no. 6: 109. https://doi.org/10.3390/risks9060109
APA StyleSyed Yusoff Alhabshi, S. F., Zamzuri, Z. H., & Mohd Ramli, S. N. (2021). Monte Carlo Simulation of the Moments of a Copula-Dependent Risk Process with Weibull Interwaiting Time. Risks, 9(6), 109. https://doi.org/10.3390/risks9060109