Cat/Non-Cat Case Study PIR Exhibits

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pricing
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Cat/Non-Cat Scenario PIR Case Study Exhibits
Author

Stephen J. Mildenhall

Published

2024-01-24

Cat/Non-Cat Book Case Study Results

Exhibits by Chapter

  • Chapter 2: Basic loss statistics (A-C)
  • Chapter 4: VaR, TVaR and EPD statistics (D, E)
  • Chapter 7: Portfolio pricing, used for calibration (F, G)
  • Chapter 9: Classical portfolio and stand-alone pricing (H-L)
  • Chapter 11: Modern portfolio and stand-alone pricing (M-Q)
  • Chapter 13: Classical allocations (R, S)
  • Chapter 15: Modern allocations (T-Y)

See Section 1.28 for more details.

Table A

PIR Chapter 2, Tables 2.3, 2.5, 2.6, 2.7, Estimated mean, CV, skewness and kurtosis by line and in total, gross and net.

(A) Cat/Non-Cat estimated mean, CV, skewness and kurtosis by line and in total, gross and net. Net of 100% share of 80 xs 41 in the aggregate. applied to Cat.
statistic Gross: Cat Gross: NonCat Gross: Total Net: Cat Net: NonCat Net: Total
Mean 20 80 100 17.786 80 97.786
CV 1 0.15 0.233 0.737 0.15 0.182
Skewness 3.972 0.3 2.539 3.139 0.3 1.351
Kurtosis 35.933 0.135 19.173 55.22 0.135 16.336

Figure B

PIR Chapter 2, Figures 2.2, 2.4, 2.6, Gross and net densities on a linear and log scale.

(B) Cat/Non-Cat, gross (top) and net (bottom) densities on a nominal (left) and log (right) scale.

(B) Cat/Non-Cat, gross (top) and net (bottom) densities on a nominal (left) and log (right) scale.

Figure C

PIR Chapter 2, Figures 2.3, 2.5, 2.7, Bivariate densities: gross and net with gross sample.

(C) Cat/Non-Cat, bivariate densities: gross (left), net (center), and a sample from gross (right). Impact of reinsurance is clear in net plot.

(C) Cat/Non-Cat, bivariate densities: gross (left), net (center), and a sample from gross (right). Impact of reinsurance is clear in net plot.

Figure D

PIR Chapter 4, Figures 4.9, 4.10, 4.11, 4.12, TVaR, and VaR for unlimited and limited variables, gross and net.

(D) Figure 4.10: Cat/Non-Cat, TVaR, and VaR for unlimited and limited variables, gross (left) and net (right). Lower view uses a log return period horizontal axis.

(D) Figure 4.10: Cat/Non-Cat, TVaR, and VaR for unlimited and limited variables, gross (left) and net (right). Lower view uses a log return period horizontal axis.

Table E

PIR Chapter 4, Tables 4.6, 4.7, 4.8, Estimated VaR, TVaR, and EPD by line and in total, gross, and net.

(E) Cat/Non-Cat estimated VaR, TVaR and EPD by line and in total, gross and net. EPD shows assets required for indicated EPD percentage. Sum column shows sum of parts by line with no diversification and benefit shows percentage reduction compared to total. Net of 100% share of 80 xs 41 in the aggregate. applied to Cat.
statistic Gross: Cat Gross: NonCat Gross: Benefit Gross: Sum Gross: Total Net: Cat Net: NonCat Net: Benefit Net: Sum Net: Total
VaR 90.0 41.1 95.7 0.0857 136.8 126 41.1 95.7 0.132 136.8 120.8
VaR 95.0 55.6 100.7 0.117 156.3 139.9 41.1 100.7 0.108 141.8 128
VaR 97.5 72.3 105.2 0.138 177.5 156 41.1 105.2 0.0901 146.3 134.2
VaR 99.0 98.1 110.5 0.152 208.6 181.1 41.1 110.5 0.0686 151.6 141.9
VaR 99.6 128.7 115.4 0.156 244.1 211.1 49 115.4 0.0919 164.5 150.6
VaR 99.9 185.3 122.2 0.151 307.5 267.2 105.6 122.2 0.215 227.9 187.6
TVaR 90.0 65.3 102.4 0.122 167.7 149.4 43.2 102.4 0.108 145.6 131.3
TVaR 95.0 83.3 106.7 0.139 190.1 166.8 45.3 106.7 0.0974 152 138.5
TVaR 97.5 103.8 110.8 0.149 214.6 186.8 49.5 110.8 0.0953 160.2 146.3
TVaR 99.0 135.2 115.7 0.153 250.9 217.7 62 115.7 0.114 177.7 159.5
TVaR 99.6 172 120.3 0.15 292.3 254.1 92.4 120.3 0.177 212.7 180.8
TVaR 99.9 239.5 126.7 0.141 366.3 321.1 159.9 126.7 0.187 286.6 241.5
EPD 10.0 45.9 74.4 0.253 120.3 96 29.7 74.4 0.128 104.1 92.3
EPD 5.0 65 81.7 0.338 146.7 109.7 35.3 81.7 0.139 117 102.7
EPD 2.5 87.4 87.7 0.407 175.1 124.5 38.9 87.7 0.135 126.6 111.5
EPD 1.0 122.6 94.5 0.464 217.1 148.2 48 94.5 0.172 142.4 121.6
EPD 0.4 164.9 100.4 0.492 265.3 177.9 91.3 100.4 0.46 191.7 131.2
EPD 0.1 244.5 108.2 0.496 352.8 235.7 172.6 108.2 0.776 280.8 158.1

Table F

PIR Chapter 7, Table 7.2, Pricing summary.

(F) Pricing summary for Cat/Non-Cat using a a 0.999 capital standard and 10.0% constant cost of capital for all layers.
stat Gross Net
Loss 99.946 97.732
Margin 15.205 8.168
Premium 115.151 105.899
Loss Ratio 0.868 0.923
Capital 152.052 81.679
Rate of Return 0.1 0.1
Assets 267.203 187.578
Leverage 0.757 1.297

Table H

PIR Chapter 9, Tables 9.2, 9.5, 9.8, Classical pricing by method.

(H) Classical pricing by method for Cat/Non-Cat. Pricing calibrated to total gross portfolio and applied to each line on a stand-alone basis. Sorted by gross premium for Cat.
method Parameters: Value Cat: Net Cat: Gross NonCat: Gross Total: Net Total: Gross Total: Ceded
Net 17.8 20 80 97.8 100 2.214
Expected Value 0.152 20.5 23 92.1 112.6 115.2 2.55
VaR 0.818 30.1 30.1 90.7 113.4 115.1 1.75
Variance 0.028 22.6 31.1 84 106.6 115.2 8.6
Dutch 1.859 27.2 32 88.9 110.6 115.2 4.513
Semi-Variance 0.040 22.5 32.8 83.1 105.2 115.2 10
Standard Deviation 0.650 26.3 33 87.8 109.3 115.2 5.8
Fischer 0.776 26.2 33.9 86.8 108.4 115.2 6.8
Esscher 0.011 23.6 35.9 81.6 105.1 115.2 10.1
Exponential 0.014 24.8 39.7 81.1 105.6 115.2 9.5

Table I

PIR Chapter 9, Tables 9.3, 9.6, 9.9, Sum of parts (SoP) stand-alone vs. diversified classical pricing by method.

(I) Sum of parts (SoP) stand-alone vs. diversified classical pricing by method for Cat/Non-Cat. Delta columns show the difference.
method Total: Gross Total: Net SoP: Gross SoP: Net Delta: Gross Delta: Net
Net 100 97.8 100 97.8 0 0
Expected Value 115.2 112.6 115.2 112.6 0 0
VaR 115.1 113.4 120.8 120.8 5.7 7.5
Variance 115.2 106.6 115.2 106.6 0.002 0
Dutch 115.2 110.6 120.9 116.1 5.7 5.4
Semi-Variance 115.2 105.2 115.9 105.6 0.726 0.41
Standard Deviation 115.2 109.3 120.8 114.1 5.6 4.767
Fischer 115.2 108.4 120.8 113.1 5.6 4.712
Esscher 115.2 105.1 117.5 105.1 2.354 0.083
Exponential 115.2 105.6 120.8 105.8 5.6 0.2

Table J

PIR Chapter 9, Tables 9.4, 9.7, 9.10, Implied loss ratios from classical pricing by method.

(J) Implied loss ratios from classical pricing by method for Cat/Non-Cat. Pricing calibrated to total gross portfolio and applied to each line on a stand-alone basis.
method Cat: Net Cat: Gross NonCat: Gross Total: Net Total: Gross Total: Ceded
Net 1 1 1 1 1 1
Expected Value 0.868 0.868 0.868 0.868 0.868 0.868
VaR 0.591 0.664 0.882 0.862 0.869 1.265
Variance 0.788 0.642 0.952 0.917 0.868 0.258
Dutch 0.655 0.625 0.9 0.884 0.868 0.491
Semi-Variance 0.79 0.61 0.963 0.93 0.868 0.222
Standard Deviation 0.676 0.606 0.911 0.894 0.868 0.381
Fischer 0.678 0.59 0.921 0.902 0.868 0.327
Esscher 0.754 0.556 0.981 0.931 0.868 0.219
Exponential 0.718 0.504 0.987 0.926 0.868 0.232

Table K

PIR Chapter 9, Table 9.11, Comparison of stand-alone and sum of parts premium.

(K) Comparison of stand-alone and sum of parts (SoP) premium for Cat/Non-Cat.
Gross SoP Gross Total Gross Redn Net SoP Net Total Net Redn
No Default: Loss 100 100 -0.0% 97.8 97.8 -0.0%
No Default: Premium 118.9 115.2 -3.1% 109.6 105.9 -3.3%
No Default: Capital 188.7 152 -19.4% 118.3 81.6 -31.0%
With Default: Loss 99.9 99.9 0.0% 97.7 97.7 0.0%
With Default: Premium 118.8 115.2 -3.1% 109.6 105.9 -3.3%
With Default: Capital 188.7 152.1 -19.4% 118.3 81.7 -31.0%

Table L

PIR Chapter 9, Tables 9.12, 9.13, 9.14, Constant CoC pricing by unit for Case Study.

(L) Constant CoC pricing by unit for Cat/Non-Cat, with 0.1 cost of capital and p=0.999. The column sop shows the sum by unit. Net of 100% share of 80 xs 41 in the aggregate. All units produce the same rate of return, by construction.
Gross: Cat Gross: NonCat Gross: SoP Gross: Total Net: Cat Net: SoP Net: Total
No Default: Loss 20 80 100 100 17.8 97.8 97.8
No Default: Margin 15 3.841 18.9 15.2 8 11.8 8.2
No Default: Premium 35 83.8 118.9 115.2 25.8 109.6 105.9
No Default: Loss Ratio 0.571 0.954 0.841 0.868 0.69 0.892 0.923
No Default: Capital 150.3 38.4 188.7 152 79.9 118.3 81.6
No Default: Rate of Return 0.1 0.1 0.1 0.1 0.1 0.1 0.1
No Default: Leverage 0.233 2.183 0.63 0.758 0.323 0.927 1.298
No Default: Assets 185.3 122.2 307.5 267.2 105.6 227.9 187.6
With Default: Loss 19.9 80 99.9 99.9 17.7 97.7 97.7
With Default: Margin 15 3.841 18.9 15.2 8 11.8 8.2
With Default: Premium 35 83.8 118.8 115.2 25.7 109.6 105.9
With Default: Loss Ratio 0.57 0.954 0.841 0.868 0.689 0.892 0.923
With Default: Capital 150.3 38.4 188.7 152.1 79.9 118.3 81.7
With Default: Rate of Return 0.1 0.1 0.1 0.1 0.1 0.1 0.1
With Default: Leverage 0.233 2.183 0.63 0.757 0.322 0.926 1.297
With Default: Assets 185.3 122.2 307.5 267.2 105.6 227.9 187.6

Figure M

PIR Chapter 11, Figures 11.2, 11.3, 11.4,11.5, Distortion envelope for Case Study, gross.

(M) Distortion envelope for Cat/Non-Cat, gross. Left plot shows the distortion envelope, middle overlays the CCoC and TVaR distortions, right overlays proportional hazard, Wang, and dual moment distortions.

(M) Distortion envelope for Cat/Non-Cat, gross. Left plot shows the distortion envelope, middle overlays the CCoC and TVaR distortions, right overlays proportional hazard, Wang, and dual moment distortions.

Table N

PIR Chapter 11, Table 11.5, Parameters for the six SRMs and associated distortions.

(N) Parameter estimates for the five base spectral risk measures.
method Param Error Premium K Ι S
Ccoc 0.1 0 115.151 152.052 0.1 0.001
PH 0.596 0 115.151 152.052 0.1 0.001
Wang 0.611 0 115.151 152.052 0.1 0.001
Dual 2.463 -0 115.151 152.052 0.1 0.001
Tvar 0.482 0 115.151 152.052 0.1 0.001

Figure O

PIR Chapter 11, Figures 11.6, 11.7, 11.8, Variation in insurance statistics for six distortions as \(s\) varies.

(O) Cat/Non-Cat, variation in premium, loss ratio, markup (premium to loss), margin, discount rate, and premium to capital leverage for six distortions, shown in two groups of three. Top six plots show proportional hazard, Wang, and dual moment; lower six: CCoC, TVaR, and Blend.

(O) Cat/Non-Cat, variation in premium, loss ratio, markup (premium to loss), margin, discount rate, and premium to capital leverage for six distortions, shown in two groups of three. Top six plots show proportional hazard, Wang, and dual moment; lower six: CCoC, TVaR, and Blend.

Figure P

PIR Chapter 11, Figures 11.9, 11.10, 11.11, Variation in insurance statistics as the asset limit is varied.

(P) Cat/Non-Cat, variation in SRM properties as the asset limit (x-axis) is varied. Column 1: total premium and loss; 2: total assets, premium, and capital; 3; total and layer loss ratio; and 4: total and layer discount factor. By row CCoC, PH, Wang, Dual, TVaR, and Blend.

(P) Cat/Non-Cat, variation in SRM properties as the asset limit (x-axis) is varied. Column 1: total premium and loss; 2: total assets, premium, and capital; 3; total and layer loss ratio; and 4: total and layer discount factor. By row CCoC, PH, Wang, Dual, TVaR, and Blend.

Table Q

PIR Chapter 11, Tables 11.7, 11.8, 11.9, Pricing by unit and distortion for Case Study.

(Q) Pricing by unit and distortion for Cat/Non-Cat, calibrated to CCoC pricing with 0.1 cost of capital and p=0.999. Losses and assets are the same for all distortions. The column sop shows sum of parts by unit, the difference with the total shows the impact of diversification. Net of 100% share of 80 xs 41 in the aggregate.
Gross: Cat Gross: NonCat Gross: SoP Gross: Total Net: Cat Net: SoP Net: Total
Loss: Ccoc 19.946 79.996 99.941 99.946 17.731 97.727 97.732
Margin: Ccoc 15.031 3.841 18.872 15.205 7.992 11.833 8.168
Margin: PH 13.806 6.367 20.172 15.205 7.379 13.746 9.821
Margin: Wang 12.876 7.507 20.383 15.205 7.947 15.454 11.058
Margin: Dual 11.882 8.598 20.481 15.205 8.834 17.433 12.402
Margin: TVaR 11.208 9.171 20.38 15.205 9.151 18.322 13.152
Margin: Blend 13.652 6.308 19.96 15.043 7.304 13.612 9.726
Premium: Ccoc 34.976 83.837 118.813 115.151 25.723 109.56 105.899
Premium: PH 33.751 86.362 120.113 115.151 25.11 111.472 107.553
Premium: Wang 32.821 87.502 120.324 115.151 25.679 113.181 108.79
Premium: Dual 31.828 88.594 120.422 115.151 26.566 115.16 110.134
Premium: TVaR 31.154 89.167 120.321 115.151 26.883 116.049 110.884
Premium: Blend 33.597 86.303 119.901 114.988 25.035 111.338 107.457
Loss Ratio: Ccoc 0.57 0.954 0.841 0.868 0.689 0.892 0.923
Loss Ratio: PH 0.591 0.926 0.832 0.868 0.706 0.877 0.909
Loss Ratio: Wang 0.608 0.914 0.831 0.868 0.691 0.863 0.898
Loss Ratio: Dual 0.627 0.903 0.83 0.868 0.667 0.849 0.887
Loss Ratio: TVaR 0.64 0.897 0.831 0.868 0.66 0.842 0.881
Loss Ratio: Blend 0.594 0.927 0.834 0.869 0.708 0.878 0.909
Capital: Ccoc 150.305 38.413 188.718 152.052 79.918 118.331 81.679
Capital: PH 151.53 35.888 187.418 152.052 80.53 116.418 80.025
Capital: Wang 152.46 34.748 187.207 152.052 79.962 114.709 78.788
Capital: Dual 153.454 33.656 187.11 152.052 79.075 112.731 77.445
Capital: TVaR 154.127 33.083 187.211 152.052 78.758 111.841 76.694
Capital: Blend 151.684 35.947 187.63 152.215 80.606 116.552 80.121
Rate of Return: Ccoc 0.1 0.1 0.1 0.1 0.1 0.1 0.1
Rate of Return: PH 0.091 0.177 0.108 0.1 0.092 0.118 0.123
Rate of Return: Wang 0.084 0.216 0.109 0.1 0.099 0.135 0.14
Rate of Return: Dual 0.077 0.255 0.109 0.1 0.112 0.155 0.16
Rate of Return: TVaR 0.073 0.277 0.109 0.1 0.116 0.164 0.171
Rate of Return: Blend 0.09 0.175 0.106 0.099 0.091 0.117 0.121
Leverage: Ccoc 0.233 2.183 0.63 0.757 0.322 0.926 1.297
Leverage: PH 0.223 2.406 0.641 0.757 0.312 0.958 1.344
Leverage: Wang 0.215 2.518 0.643 0.757 0.321 0.987 1.381
Leverage: Dual 0.207 2.632 0.644 0.757 0.336 1.022 1.422
Leverage: TVaR 0.202 2.695 0.643 0.757 0.341 1.038 1.446
Leverage: Blend 0.221 2.401 0.639 0.755 0.311 0.955 1.341
Assets: Ccoc 185.281 122.25 307.531 267.203 105.641 227.891 187.578

Table R

PIR Chapter 13, Table 13.1, Comparison of gross expected losses by Case, catastrophe-prone lines.

(R) Comparison of gross expected losses by line. Second column shows allocated recovery with total assets. Third column shows stand-alone limited expected value with stand-alone 0.999-VaR assets.
Unit a E[Xi(a)] E[Xi ∧ ai]
NonCat 122.25 79.988 79.996
Cat 185.281 19.958 19.946
Total 267.203 99.946 99.946
SoP 307.531 99.946 99.941

Table S

PIR Chapter 13, Tables 13.2, 13.3, 13.4, Constant 0.10 ROE pricing for Case Study, classical PCP methods.

(S) Constant 0.10 ROE pricing for Cat/Non-Cat, classical PCP methods.
Gross: Cat Gross: NonCat Gross: Total Net: Cat Net: NonCat Net: Total Ceded: Diff
Loss: Expected Loss 19.958 79.988 99.946 17.747 79.985 97.732 2.214
Margin: Expected Loss 3.036 12.169 15.205 1.483 6.685 8.168 7.037
Margin: Scaled EPD 15.677 -0.472 15.205 9.6 -1.432 8.168 7.037
Margin: Scaled TVaR 12.846 2.36 15.205 6.166 2.002 8.168 7.037
Margin: Scaled VaR 12.82 2.385 15.205 6.292 1.876 8.168 7.037
Margin: Equal Risk EPD 13.335 1.87 15.205 6.355 1.813 8.168 7.037
Margin: Equal Risk TVaR 11.785 3.42 15.205 4.767 3.4 8.168 7.037
Margin: Equal Risk VaR 11.735 3.47 15.205 4.698 3.47 8.168 7.037
Margin: coTVaR 14.84 0.37 15.207 6.845 1.326 8.171 7.036
Margin: Covar 11.181 4.027 15.205 4.443 3.725 8.168 7.037
Premium: Expected Loss 22.994 92.157 115.151 19.23 86.669 105.899 9.252
Premium: Scaled EPD 35.635 79.516 115.151 27.347 78.552 105.899 9.252
Premium: Scaled TVaR 32.803 82.348 115.151 23.913 81.987 105.899 9.252
Premium: Scaled VaR 32.778 82.373 115.151 24.039 81.861 105.899 9.252
Premium: Equal Risk EPD 33.293 81.858 115.151 24.102 81.797 105.899 9.252
Premium: Equal Risk TVaR 31.743 83.408 115.151 22.515 83.385 105.899 9.252
Premium: Equal Risk VaR 31.693 83.458 115.151 22.445 83.455 105.899 9.252
Premium: coTVaR 34.798 80.358 115.153 24.592 81.311 105.902 9.25
Premium: Covar 31.138 84.015 115.151 22.19 83.709 105.899 9.252
Loss Ratio: Expected Loss 0.868 0.868 0.868 0.923 0.923 0.923 0.239
Loss Ratio: Scaled EPD 0.56 1.006 0.868 0.649 1.018 0.923 0.239
Loss Ratio: Scaled TVaR 0.608 0.971 0.868 0.742 0.976 0.923 0.239
Loss Ratio: Scaled VaR 0.609 0.971 0.868 0.738 0.977 0.923 0.239
Loss Ratio: Equal Risk EPD 0.599 0.977 0.868 0.736 0.978 0.923 0.239
Loss Ratio: Equal Risk TVaR 0.629 0.959 0.868 0.788 0.959 0.923 0.239
Loss Ratio: Equal Risk VaR 0.63 0.958 0.868 0.791 0.958 0.923 0.239
Loss Ratio: coTVaR 0.574 0.995 0.868 0.722 0.984 0.923 0.239
Loss Ratio: Covar 0.641 0.952 0.868 0.8 0.956 0.923 0.239
Capital: Expected Loss 30.363 121.689 152.052 14.832 66.847 81.679 70.373
Capital: Scaled EPD 156.767 -4.715 152.052 96 -14.321 81.679 70.373
Capital: Scaled TVaR 128.455 23.597 152.052 61.657 20.022 81.679 70.373
Capital: Scaled VaR 128.203 23.849 152.052 62.916 18.763 81.679 70.373
Capital: Equal Risk EPD 133.349 18.703 152.052 63.551 18.127 81.679 70.373
Capital: Equal Risk TVaR 117.848 34.205 152.052 47.674 34.004 81.679 70.373
Capital: Equal Risk VaR 117.354 34.698 152.052 46.977 34.702 81.679 70.373
Capital: coTVaR 148.399 3.698 152.067 68.445 13.262 81.706 70.361
Capital: Covar 111.806 40.27 152.052 44.433 37.246 81.679 70.373
Rate of Return: Expected Loss 0.1 0.1 0.1 0.1 0.1 0.1 0.1
Rate of Return: Scaled EPD 0.1 0.1 0.1 0.1 0.1 0.1 0.1
Rate of Return: Scaled TVaR 0.1 0.1 0.1 0.1 0.1 0.1 0.1
Rate of Return: Scaled VaR 0.1 0.1 0.1 0.1 0.1 0.1 0.1
Rate of Return: Equal Risk EPD 0.1 0.1 0.1 0.1 0.1 0.1 0.1
Rate of Return: Equal Risk TVaR 0.1 0.1 0.1 0.1 0.1 0.1 0.1
Rate of Return: Equal Risk VaR 0.1 0.1 0.1 0.1 0.1 0.1 0.1
Rate of Return: coTVaR 0.1 0.1 0.1 0.1 0.1 0.1 0.1
Rate of Return: Covar 0.1 0.1 0.1 0.1 0.1 0.1 0.1
Leverage: Expected Loss 0.757 0.757 0.757 1.297 1.297 1.297 0.131
Leverage: Scaled EPD 0.227 -16.863 0.757 0.285 -5.485 1.297 0.131
Leverage: Scaled TVaR 0.255 3.49 0.757 0.388 4.095 1.297 0.131
Leverage: Scaled VaR 0.256 3.454 0.757 0.382 4.363 1.297 0.131
Leverage: Equal Risk EPD 0.25 4.377 0.757 0.379 4.512 1.297 0.131
Leverage: Equal Risk TVaR 0.269 2.439 0.757 0.472 2.452 1.297 0.131
Leverage: Equal Risk VaR 0.27 2.405 0.757 0.478 2.405 1.297 0.131
Leverage: coTVaR 0.234 21.728 0.757 0.359 6.131 1.296 0.131
Leverage: Covar 0.279 2.086 0.757 0.499 2.247 1.297 0.131
Assets: Expected Loss 53.357 213.846 267.203 34.062 153.516 187.578 79.625
Assets: Scaled EPD 192.402 74.801 267.203 123.347 64.231 187.578 79.625
Assets: Scaled TVaR 161.259 105.944 267.203 85.57 102.009 187.578 79.625
Assets: Scaled VaR 160.982 106.222 267.203 86.954 100.624 187.578 79.625
Assets: Equal Risk EPD 166.642 100.561 267.203 87.654 99.925 187.578 79.625
Assets: Equal Risk TVaR 149.59 117.613 267.203 70.189 117.389 187.578 79.625
Assets: Equal Risk VaR 149.047 118.156 267.203 69.422 118.156 187.578 79.625
Assets: coTVaR 183.197 84.056 267.22 93.037 94.573 187.608 79.612
Assets: Covar 142.944 124.285 267.203 66.624 120.956 187.578 79.625

Figure T_gross

(TG) Cat/Non-Cat, gross twelve plot with dual distortion.

(TG) Cat/Non-Cat, gross twelve plot with dual distortion.

Figure T_net

(TN) Cat/Non-Cat, net twelve plot with ph distortion.

(TN) Cat/Non-Cat, net twelve plot with ph distortion.

Figure U

PIR Chapter 15, Figures 15.8, 15.9, 15.10, Capital density by layer.

(U) Cat/Non-Cat, capital density for Cat/Non-Cat, with Dual Moment, 2.463 gross and Proportional Hazard, 0.596 net distortion.

(U) Cat/Non-Cat, capital density for Cat/Non-Cat, with Dual Moment, 2.463 gross and Proportional Hazard, 0.596 net distortion.

Table V

PIR Chapter 15, Tables 15.35, 15.36, 15.37, Constant 0.10 ROE pricing for Cat/Non-Cat Case Study, distortion, SRM methods.

(V) Constant 0.10 ROE pricing for Cat/Non-Cat, distortion, SRM methods.
Gross: Cat Gross: NonCat Gross: Total Net: Cat Net: NonCat Net: Total Ceded: Diff
Loss: Expected Loss 19.96 79.99 99.95 17.75 79.98 97.73 2.214
Margin: Expected Loss 3.036 12.17 15.21 1.483 6.68 8.17 7.04
Margin: Dist Ccoc 20.55 -5.35 15.21 13.94 -5.77 8.17 7.04
Margin: Dist PH 12.94 2.269 15.21 6.32 3.506 9.82 5.38
Margin: Dist Wang 11.31 3.898 15.21 6.15 4.91 11.06 4.147
Margin: Dist Dual 9.72 5.49 15.21 6.47 5.94 12.4 2.803
Margin: Dist Tvar 8.74 6.47 15.21 6.65 6.5 13.15 2.053
Margin: Dist Blend 12.78 2.258 15.04 6.24 3.482 9.73 5.32
Premium: Expected Loss 22.99 92.16 115.15 19.23 86.67 105.9 9.25
Premium: Dist Ccoc 40.51 74.64 115.15 31.68 74.21 105.9 9.25
Premium: Dist PH 32.89 82.26 115.15 24.06 83.49 107.55 7.6
Premium: Dist Wang 31.26 83.89 115.15 23.89 84.9 108.79 6.36
Premium: Dist Dual 29.68 85.48 115.15 24.21 85.92 110.13 5.02
Premium: Dist Tvar 28.69 86.46 115.15 24.39 86.49 110.88 4.267
Premium: Dist Blend 32.74 82.25 114.99 23.99 83.47 107.46 7.53
Loss Ratio: Expected Loss 0.868 0.868 0.868 0.923 0.923 0.923 0.239
Loss Ratio: Dist Ccoc 0.493 1.072 0.868 0.56 1.078 0.923 0.239
Loss Ratio: Dist PH 0.607 0.972 0.868 0.738 0.958 0.909 0.291
Loss Ratio: Dist Wang 0.638 0.954 0.868 0.743 0.942 0.898 0.348
Loss Ratio: Dist Dual 0.673 0.936 0.868 0.733 0.931 0.887 0.441
Loss Ratio: Dist Tvar 0.696 0.925 0.868 0.728 0.925 0.881 0.519
Loss Ratio: Dist Blend 0.61 0.973 0.869 0.74 0.958 0.909 0.294
Capital: Expected Loss 30.36 121.69 152.05 14.83 66.85 81.68 70.37
Capital: Dist Ccoc 193.41 -41.35 152.05 127.25 -45.57 81.68 70.37
Capital: Dist PH 103.52 48.54 152.05 43.14 36.88 80.03 72.03
Capital: Dist Wang 92.18 59.87 152.05 35.78 43.01 78.79 73.26
Capital: Dist Dual 89.01 63.04 152.05 33.53 43.92 77.44 74.61
Capital: Dist Tvar 88.87 63.19 152.05 33.27 43.43 76.69 75.36
Capital: Dist Blend 103.52 48.7 152.21 43.12 37 80.12 72.09
Rate of Return: Expected Loss 0.1 0.1 0.1 0.1 0.1 0.1 0.1
Rate of Return: Dist Ccoc 0.106 0.129 0.1 0.11 0.127 0.1 0.1
Rate of Return: Dist PH 0.125 0.047 0.1 0.146 0.095 0.123 0.075
Rate of Return: Dist Wang 0.123 0.065 0.1 0.172 0.114 0.14 0.057
Rate of Return: Dist Dual 0.109 0.087 0.1 0.193 0.135 0.16 0.038
Rate of Return: Dist Tvar 0.098 0.102 0.1 0.2 0.15 0.171 0.027
Rate of Return: Dist Blend 0.123 0.046 0.099 0.145 0.094 0.121 0.074
Leverage: Expected Loss 0.757 0.757 0.757 1.297 1.297 1.297 0.131
Leverage: Dist Ccoc 0.209 -1.805 0.757 0.249 -1.628 1.297 0.131
Leverage: Dist PH 0.318 1.695 0.757 0.558 2.264 1.344 0.105
Leverage: Dist Wang 0.339 1.401 0.757 0.668 1.974 1.381 0.087
Leverage: Dist Dual 0.333 1.356 0.757 0.722 1.956 1.422 0.067
Leverage: Dist Tvar 0.323 1.368 0.757 0.733 1.991 1.446 0.057
Leverage: Dist Blend 0.316 1.689 0.755 0.556 2.256 1.341 0.104
Assets: Expected Loss 53.36 213.85 267.2 34.06 153.52 187.58 79.62
Assets: Dist Ccoc 233.92 33.29 267.2 158.94 28.64 187.58 79.62
Assets: Dist PH 136.41 130.79 267.2 67.2 120.37 187.58 79.62
Assets: Dist Wang 123.44 143.76 267.2 59.67 127.91 187.58 79.62
Assets: Dist Dual 118.69 148.52 267.2 57.74 129.84 187.58 79.62
Assets: Dist Tvar 117.57 149.64 267.2 57.66 129.92 187.58 79.62
Assets: Dist Blend 136.26 130.95 267.2 67.11 120.47 187.58 79.62

Figure W

PIR Chapter 15, Figure 15.11, Loss and loss spectrums.

(W) Figure 15.11: Cat/Non-Cat, loss spectrum (gross/net top row). Rows 2 and show VaR weights by distortion. In the second row, the CCoC distortion includes a mass putting weight 𝑑 = 0.1∕1.1 at the maximum loss, corresponding to an infinite density. The lower right-hand plot compares all five distortions on a log-log scale.

(W) Figure 15.11: Cat/Non-Cat, loss spectrum (gross/net top row). Rows 2 and show VaR weights by distortion. In the second row, the CCoC distortion includes a mass putting weight 𝑑 = 0.1∕1.1 at the maximum loss, corresponding to an infinite density. The lower right-hand plot compares all five distortions on a log-log scale.

Figure X

PIR Chapter 15, Figures 15.12, 15.13, 15.14, Percentile layer of capital allocations by asset level.

(X) Cat/Non-Cat, percentile layer of capital allocations by asset level, showing 0.999 capital. (Same distortions.)

(X) Cat/Non-Cat, percentile layer of capital allocations by asset level, showing 0.999 capital. (Same distortions.)

Table Y

PIR Chapter 15, Tables 15.38, 15.39, 15.40, Percentile layer of capital allocations compared to distortion allocations.

(Y) Cat/Non-Cat percentile layer of capital allocations compared to distortion allocations.
Method Gross: Cat Gross: NonCat Gross: Total Net: Cat Net: NonCat Net: Total Ceded: Diff
Expected Loss 53.36 213.8 267.2 34.06 153.5 187.6 79.62
Dist Ccoc 233.9 33.29 267.2 158.9 28.64 187.6 79.62
Dist PH 136.4 130.8 267.2 67.2 120.4 187.6 79.62
Dist Wang 123.4 143.8 267.2 59.67 127.9 187.6 79.62
Dist Dual 118.7 148.5 267.2 57.74 129.8 187.6 79.62
Dist Tvar 117.6 149.6 267.2 57.66 129.9 187.6 79.62
Dist Blend 136.3 130.9 267.2 67.11 120.5 187.6 79.62
PLC 113.2 154 267.2 54 133.6 187.6 79.62

Cat/Non-Cat Case Description

Cat/Non-Cat in the new syntax.

Distributions

# Line A (usually thinner tailed)
agg NonCat 1 claim sev gamma 80 cv 0.15 fixed

# Line B Gross (usually thicker tailed)
agg Cat 1 claim sev lognorm 20 cv 1.00 fixed

# Line B Net
agg Cat 1 claim sev lognorm 20 cv 1.00 fixed aggregate net of 79.640625 xs 41.109375

Other Parameters

  • reg_p = 0.999
  • roe = 0.1
  • d2tc = 0.3
  • s_values = [0.005, 0.01, 0.03]
  • gs_values = [0.029126, 0.047619, 0.074074]
  • f_discrete = False
  • log2 = 16
  • bs = 0.015625
  • padding = 1

Description of Tables and Figures

Ref. Kind Chapter Number(s) Description
A Table 2 2.3, 2.5, 2.6, 2.7 Estimated mean, CV, skewness and kurtosis by line and in total, gross and net.
B Figure 2 2.2, 2.4, 2.6 Gross and net densities on a linear and log scale.
C Figure 2 2.3, 2.5, 2.7 Bivariate densities: gross and net with gross sample.
D Figure 4 4.9, 4.10, 4.11, 4.12 TVaR, and VaR for unlimited and limited variables, gross and net.
E Table 4 4.6, 4.7, 4.8 Estimated VaR, TVaR, and EPD by line and in total, gross, and net.
F Table 7 7.2 Pricing summary.
G Table 7 7.3 Details of reinsurance.
H Table 9 9.2, 9.5, 9.8 Classical pricing by method.
I Table 9 9.3, 9.6, 9.9 Sum of parts (SoP) stand-alone vs. diversified classical pricing by method.
J Table 9 9.4, 9.7, 9.10 Implied loss ratios from classical pricing by method.
K Table 9 9.11 Comparison of stand-alone and sum of parts premium.
L Table 9 9.12, 9.13, 9.14 Constant CoC pricing by unit for Case Study.
M Figure 11 11.2, 11.3, 11.4,11.5 Distortion envelope for Case Study, gross.
N Table 11 11.5 Parameters for the six SRMs and associated distortions.
O Figure 11 11.6, 11.7, 11.8 Variation in insurance statistics for six distortions as \(s\) varies.
P Figure 11 11.9, 11.10, 11.11 Variation in insurance statistics as the asset limit is varied.
Q Table 11 11.7, 11.8, 11.9 Pricing by unit and distortion for Case Study.
R Table 13 13.1 missing Comparison of gross expected losses by Case, catastrophe-prone lines.
S Table 13 13.2, 13.3, 13.4 Constant 0.10 ROE pricing for Case Study, classical PCP methods.
T Figure 15 15.2 - 15.7 (G/N) Twelve plot.
U Figure 15 15.8, 15.9, 15.10 Capital density by layer.
V Table 15 15.35, 15.36, 15.37 Constant 0.10 ROE pricing for Cat/Non-Cat Case Study, distortion, SRM methods.
W Figure 15 15.11 Loss and loss spectrums.
X Figure 15 15.12, 15.13, 15.14 Percentile layer of capital allocations by asset level.
Y Table 15 15.38, 15.39, 15.40 Percentile layer of capital allocations compared to distortion allocations.