Tame Case Study PIR Exhibits

publications
pricing
insurance
risk
pir
Tame Scenario PIR Case Study Exhibits
Author

Stephen J. Mildenhall

Published

2024-06-24

Tame 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) Tame estimated mean, CV, skewness and kurtosis by line and in total, gross and net. Net of 100% share of 13 xs 56 in the aggregate. applied to B.
statistic Gross: A Gross: B Gross: Total Net: A Net: B Net: Total
Mean 50 50 100 50 49.084 99.084
CV 0.1 0.15 0.09 0.1 0.123 0.079
Skewness 0.2 0.3 0.207 0.2 -0.484 -0.169
Kurtosis 0.06 0.135 0.07 0.06 -0.474 -0.157

Figure B

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

(B) Tame, gross (top) and net (bottom) densities on a nominal (left) and log (right) scale.

(B) Tame, 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) Tame, bivariate densities: gross (left), net (center), and a sample from gross (right). Impact of reinsurance is clear in net plot.

(C) Tame, 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: Tame, 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: Tame, 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) Tame 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 13 xs 56 in the aggregate. applied to B.
statistic Gross: A Gross: B Gross: Benefit Gross: Sum Gross: Total Net: A Net: B Net: Benefit Net: Sum Net: Total
VaR 90.0 56.5 59.8 0.041 116.3 111.7 56.5 56.2 0.0344 112.7 108.9
VaR 95.0 58.5 62.9 0.053 121.4 115.3 58.5 56.2 0.0296 114.7 111.4
VaR 97.5 60.3 65.7 0.063 126 118.5 60.3 56.2 0.026 116.4 113.5
VaR 99.0 62.4 69.1 0.0743 131.4 122.3 62.4 56.2 0.0224 118.5 115.9
VaR 99.6 64.3 72.1 0.0843 136.4 125.8 64.3 59.2 0.0452 123.5 118.2
VaR 99.9 66.9 76.4 0.0974 143.3 130.6 66.9 63.5 0.0754 130.4 121.2
TVaR 90.0 59.1 64 0.0568 123.1 116.5 59.1 56.5 0.0314 115.6 112.1
TVaR 95.0 60.9 66.7 0.0664 127.6 119.6 60.9 56.8 0.0307 117.7 114.2
TVaR 97.5 62.4 69.2 0.075 131.7 122.5 62.4 57.5 0.0333 119.9 116
TVaR 99.0 64.4 72.3 0.0849 136.7 126 64.4 59.4 0.0463 123.8 118.3
TVaR 99.6 66.1 75.2 0.0938 141.3 129.2 66.1 62.3 0.0666 128.4 120.4
TVaR 99.9 68.6 79.2 0.106 147.8 133.7 68.6 66.3 0.0931 134.9 123.4
EPD 10.0 45.5 46.5 0.0146 91.9 90.6 45.5 45.3 0.0122 90.8 89.7
EPD 5.0 49 51.1 0.0277 100.1 97.4 49 49.2 0.0232 98.3 96
EPD 2.5 51.8 54.8 0.0397 106.6 102.5 51.8 52 0.0323 103.7 100.5
EPD 1.0 54.7 59 0.0536 113.8 108 54.7 54.2 0.0392 109 104.8
EPD 0.4 57.2 62.7 0.0656 119.9 112.6 57.2 55.4 0.0407 112.6 108.2
EPD 0.1 60.5 67.6 0.081 128.1 118.5 60.5 56.1 0.0374 116.5 112.3

Table F

PIR Chapter 7, Table 7.2, Pricing summary.

(F) Pricing summary for Tame using a a 0.9999 capital standard and 10.0% constant cost of capital for all layers.
stat Gross Net
Loss 100 99.084
Margin 3.423 2.464
Premium 103.423 101.548
Loss Ratio 0.967 0.976
Capital 34.233 24.639
Rate of Return 0.1 0.1
Assets 137.656 126.188
Leverage 3.021 4.121

Table H

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

(H) Classical pricing by method for Tame. Pricing calibrated to total gross portfolio and applied to each line on a stand-alone basis. Sorted by gross premium for B.
method Parameters: Value A: Gross B: Net B: Gross Total: Net Total: Gross Total: Ceded
Net 50 49.1 50 99.1 100 0.916
Expected Value 0.034 51.7 50.8 51.7 102.5 103.4 0.947
Variance 0.042 51.1 50.6 52.4 101.7 103.4 1.75
Esscher 0.041 51 50.5 52.4 101.5 103.4 1.92
Exponential 0.080 51 50.4 52.4 101.4 103.4 1.976
Semi-Variance 0.080 51.1 50.3 52.4 101.4 103.4 2.012
VaR 0.659 51.9 52.8 52.8 102.7 103.4 0.75
Dutch 0.953 51.9 51.5 52.8 102.1 103.4 1.327
Standard Deviation 0.380 51.9 51.4 52.8 102.1 103.4 1.362
Fischer 0.523 51.9 51.1 52.9 101.9 103.4 1.516

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 Tame. Delta columns show the difference.
method Total: Gross Total: Net SoP: Gross SoP: Net Delta: Gross Delta: Net
Net 100 99.1 100 99.1 0 0
Expected Value 103.4 102.5 103.4 102.5 0 0
Variance 103.4 101.7 103.4 101.7 -0 -0
Esscher 103.4 101.5 103.4 101.5 0 0
Exponential 103.4 101.4 103.4 101.4 0 0
Semi-Variance 103.4 101.4 103.5 101.4 0.054 -0.029
VaR 103.4 102.7 104.7 104.7 1.234 1.984
Dutch 103.4 102.1 104.7 103.4 1.322 1.301
Standard Deviation 103.4 102.1 104.7 103.3 1.324 1.215
Fischer 103.4 101.9 104.8 103 1.355 1.14

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 Tame. Pricing calibrated to total gross portfolio and applied to each line on a stand-alone basis.
method A: Gross B: Net B: Gross Total: Net Total: Gross Total: Ceded
Net 1 1 1 1 1 1
Expected Value 0.967 0.967 0.967 0.967 0.967 0.967
Variance 0.979 0.97 0.955 0.975 0.967 0.523
Esscher 0.98 0.973 0.954 0.976 0.967 0.477
Exponential 0.98 0.974 0.954 0.977 0.967 0.463
Semi-Variance 0.979 0.975 0.954 0.977 0.967 0.455
VaR 0.963 0.931 0.948 0.965 0.967 1.221
Dutch 0.963 0.953 0.946 0.97 0.967 0.69
Standard Deviation 0.963 0.955 0.946 0.971 0.967 0.672
Fischer 0.963 0.96 0.946 0.972 0.967 0.604

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 Tame.
Gross SoP Gross Total Gross Redn Net SoP Net Total Net Redn
No Default: Loss 100 100 -0.0% 99.1 99.1 -0.0%
No Default: Premium 104.9 103.4 -1.4% 102.9 101.5 -1.3%
No Default: Capital 48.7 34.2 -29.7% 37.8 24.6 -34.8%
With Default: Loss 100 100 0.0% 99.1 99.1 0.0%
With Default: Premium 104.9 103.4 -1.4% 102.9 101.5 -1.3%
With Default: Capital 48.7 34.2 -29.7% 37.8 24.6 -34.8%

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 Tame, with 0.1 cost of capital and p=0.9999. The column sop shows the sum by unit. Net of 100% share of 13 xs 56 in the aggregate. All units produce the same rate of return, by construction.
Gross: A Gross: B Gross: SoP Gross: Total Net: A Net: SoP Net: Total
No Default: Loss 50 50 100 100 50 99.1 99.1
No Default: Margin 1.888 2.982 4.869 3.423 1.888 3.779 2.464
No Default: Premium 51.9 53 104.9 103.4 51.9 102.9 101.5
No Default: Loss Ratio 0.964 0.944 0.954 0.967 0.964 0.963 0.976
No Default: Capital 18.9 29.8 48.7 34.2 18.9 37.8 24.6
No Default: Rate of Return 0.1 0.1 0.1 0.1 0.1 0.1 0.1
No Default: Leverage 2.749 1.777 2.154 3.021 2.749 2.722 4.121
No Default: Assets 70.8 82.8 153.6 137.7 70.8 140.7 126.2
With Default: Loss 50 50 100 100 50 99.1 99.1
With Default: Margin 1.888 2.982 4.869 3.423 1.888 3.779 2.464
With Default: Premium 51.9 53 104.9 103.4 51.9 102.9 101.5
With Default: Loss Ratio 0.964 0.944 0.954 0.967 0.964 0.963 0.976
With Default: Capital 18.9 29.8 48.7 34.2 18.9 37.8 24.6
With Default: Rate of Return 0.1 0.1 0.1 0.1 0.1 0.1 0.1
With Default: Leverage 2.749 1.777 2.154 3.021 2.749 2.722 4.121
With Default: Assets 70.8 82.8 153.6 137.7 70.8 140.7 126.2

Figure M

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

(M) Distortion envelope for Tame, 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 Tame, 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 0 103.423 34.233 0.1 9.9957e-05
PH 0.683 0 103.423 34.233 0.1 9.9957e-05
Wang 0.375 0 103.423 34.233 0.1 9.9957e-05
Dual 1.576 -0 103.423 34.233 0.1 9.9957e-05
Tvar 0.227 0 103.423 34.233 0.1 9.9957e-05

Figure O

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

(O) Tame, 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) Tame, 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) Tame, 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) Tame, 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 Tame, calibrated to CCoC pricing with 0.1 cost of capital and p=0.9999. 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 13 xs 56 in the aggregate.
Gross: A Gross: B Gross: SoP Gross: Total Net: B Net: SoP Net: Total
Loss: Ccoc 50 50 100 100 49.084 99.084 99.084
Margin: Ccoc 1.888 2.982 4.869 3.423 1.891 3.779 2.464
Margin: PH 1.896 2.891 4.787 3.423 1.822 3.718 2.765
Margin: Wang 1.898 2.861 4.759 3.423 2.09 3.988 2.904
Margin: Dual 1.9 2.835 4.735 3.423 2.355 4.255 3.03
Margin: TVaR 1.901 2.81 4.711 3.423 2.541 4.442 3.155
Margin: Blend 1.898 2.853 4.752 3.423 2.193 4.091 2.956
Premium: Ccoc 51.888 52.981 104.869 103.423 50.976 102.863 101.548
Premium: PH 51.896 52.891 104.787 103.423 50.906 102.802 101.849
Premium: Wang 51.898 52.861 104.759 103.423 51.174 103.072 101.988
Premium: Dual 51.9 52.835 104.734 103.423 51.44 103.339 102.114
Premium: TVaR 51.901 52.81 104.711 103.423 51.625 103.526 102.239
Premium: Blend 51.898 52.853 104.751 103.423 51.277 103.175 102.04
Loss Ratio: Ccoc 0.964 0.944 0.954 0.967 0.963 0.963 0.976
Loss Ratio: PH 0.963 0.945 0.954 0.967 0.964 0.964 0.973
Loss Ratio: Wang 0.963 0.946 0.955 0.967 0.959 0.961 0.972
Loss Ratio: Dual 0.963 0.946 0.955 0.967 0.954 0.959 0.97
Loss Ratio: TVaR 0.963 0.947 0.955 0.967 0.951 0.957 0.969
Loss Ratio: Blend 0.963 0.946 0.955 0.967 0.957 0.96 0.971
Capital: Ccoc 18.878 29.816 48.694 34.233 18.915 37.793 24.639
Capital: PH 18.869 29.906 48.776 34.233 18.985 37.854 24.339
Capital: Wang 18.868 29.936 48.804 34.233 18.717 37.584 24.2
Capital: Dual 18.866 29.962 48.828 34.233 18.451 37.317 24.073
Capital: TVaR 18.865 29.987 48.852 34.233 18.265 37.13 23.949
Capital: Blend 18.868 29.944 48.811 34.233 18.613 37.481 24.148
Rate of Return: Ccoc 0.1 0.1 0.1 0.1 0.1 0.1 0.1
Rate of Return: PH 0.1 0.097 0.098 0.1 0.096 0.098 0.114
Rate of Return: Wang 0.101 0.096 0.098 0.1 0.112 0.106 0.12
Rate of Return: Dual 0.101 0.095 0.097 0.1 0.128 0.114 0.126
Rate of Return: TVaR 0.101 0.094 0.096 0.1 0.139 0.12 0.132
Rate of Return: Blend 0.101 0.095 0.097 0.1 0.118 0.109 0.122
Leverage: Ccoc 2.749 1.777 2.154 3.021 2.695 2.722 4.121
Leverage: PH 2.75 1.769 2.148 3.021 2.681 2.716 4.185
Leverage: Wang 2.751 1.766 2.147 3.021 2.734 2.742 4.214
Leverage: Dual 2.751 1.763 2.145 3.021 2.788 2.769 4.242
Leverage: TVaR 2.751 1.761 2.143 3.021 2.826 2.788 4.269
Leverage: Blend 2.751 1.765 2.146 3.021 2.755 2.753 4.226
Assets: Ccoc 70.766 82.797 153.562 137.656 69.891 140.656 126.188

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 0.9999-VaR assets.
Unit a E[Xi(a)] E[Xi ∧ ai]
A 70.766 50 50
B 82.797 50 50
Total 137.656 100 100
SoP 153.562 100 100

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 Tame, classical PCP methods.
Gross: A Gross: B Gross: Total Net: A Net: B Net: Total Ceded: Diff
Loss: Expected Loss 50 50 100 50 49.084 99.084 0.916
Margin: Expected Loss 1.712 1.712 3.423 1.243 1.221 2.464 0.959
Margin: Scaled EPD nan nan nan 1.16 1.304 2.464 nan
Margin: Scaled TVaR nan nan nan 1.227 1.237 2.464 nan
Margin: Scaled VaR nan nan nan 1.225 1.239 2.464 nan
Margin: Equal Risk EPD nan nan nan 1.338 1.126 2.464 nan
Margin: Equal Risk TVaR nan nan nan 1.389 1.075 2.464 nan
Margin: Equal Risk VaR nan nan nan 1.391 1.075 2.464 nan
Margin: coTVaR nan nan nan 1.39 1.075 2.465 nan
Margin: Covar nan nan nan 1.003 1.461 2.464 nan
Premium: Expected Loss 51.712 51.711 103.423 51.243 50.305 101.548 1.875
Premium: Scaled EPD nan nan nan 51.16 50.388 101.548 nan
Premium: Scaled TVaR nan nan nan 51.227 50.321 101.548 nan
Premium: Scaled VaR nan nan nan 51.225 50.323 101.548 nan
Premium: Equal Risk EPD nan nan nan 51.338 50.21 101.548 nan
Premium: Equal Risk TVaR nan nan nan 51.389 50.159 101.548 nan
Premium: Equal Risk VaR nan nan nan 51.391 50.159 101.548 nan
Premium: coTVaR nan nan nan 51.39 50.159 101.549 nan
Premium: Covar nan nan nan 51.002 50.546 101.548 nan
Loss Ratio: Expected Loss 0.967 0.967 0.967 0.976 0.976 0.976 0.488
Loss Ratio: Scaled EPD nan nan nan 0.977 0.974 0.976 nan
Loss Ratio: Scaled TVaR nan nan nan 0.976 0.975 0.976 nan
Loss Ratio: Scaled VaR nan nan nan 0.976 0.975 0.976 nan
Loss Ratio: Equal Risk EPD nan nan nan 0.974 0.978 0.976 nan
Loss Ratio: Equal Risk TVaR nan nan nan 0.973 0.979 0.976 nan
Loss Ratio: Equal Risk VaR nan nan nan 0.973 0.979 0.976 nan
Loss Ratio: coTVaR nan nan nan 0.973 0.979 0.976 nan
Loss Ratio: Covar nan nan nan 0.98 0.971 0.976 nan
Capital: Expected Loss 17.117 17.117 34.233 12.434 12.206 24.639 9.594
Capital: Scaled EPD nan nan nan 11.603 13.036 24.639 nan
Capital: Scaled TVaR nan nan nan 12.271 12.368 24.639 nan
Capital: Scaled VaR nan nan nan 12.254 12.385 24.639 nan
Capital: Equal Risk EPD nan nan nan 13.379 11.261 24.639 nan
Capital: Equal Risk TVaR nan nan nan 13.894 10.745 24.639 nan
Capital: Equal Risk VaR nan nan nan 13.906 10.747 24.639 nan
Capital: coTVaR nan nan nan 13.904 10.749 24.653 nan
Capital: Covar nan nan nan 10.026 14.613 24.639 nan
Rate of Return: Expected Loss 0.1 0.1 0.1 0.1 0.1 0.1 0.1
Rate of Return: Scaled EPD nan nan nan 0.1 0.1 0.1 nan
Rate of Return: Scaled TVaR nan nan nan 0.1 0.1 0.1 nan
Rate of Return: Scaled VaR nan nan nan 0.1 0.1 0.1 nan
Rate of Return: Equal Risk EPD nan nan nan 0.1 0.1 0.1 nan
Rate of Return: Equal Risk TVaR nan nan nan 0.1 0.1 0.1 nan
Rate of Return: Equal Risk VaR nan nan nan 0.1 0.1 0.1 nan
Rate of Return: coTVaR nan nan nan 0.1 0.1 0.1 nan
Rate of Return: Covar nan nan nan 0.1 0.1 0.1 nan
Leverage: Expected Loss 3.021 3.021 3.021 4.121 4.121 4.121 0.195
Leverage: Scaled EPD nan nan nan 4.409 3.865 4.121 nan
Leverage: Scaled TVaR nan nan nan 4.175 4.069 4.121 nan
Leverage: Scaled VaR nan nan nan 4.18 4.063 4.121 nan
Leverage: Equal Risk EPD nan nan nan 3.837 4.459 4.121 nan
Leverage: Equal Risk TVaR nan nan nan 3.699 4.668 4.121 nan
Leverage: Equal Risk VaR nan nan nan 3.695 4.667 4.121 nan
Leverage: coTVaR nan nan nan 3.696 4.667 4.119 nan
Leverage: Covar nan nan nan 5.087 3.459 4.121 nan
Assets: Expected Loss 68.828 68.828 137.656 63.677 62.511 126.188 11.469
Assets: Scaled EPD nan nan nan 62.764 63.424 126.188 nan
Assets: Scaled TVaR nan nan nan 63.498 62.689 126.188 nan
Assets: Scaled VaR nan nan nan 63.479 62.708 126.188 nan
Assets: Equal Risk EPD nan nan nan 64.716 61.471 126.188 nan
Assets: Equal Risk TVaR nan nan nan 65.283 60.904 126.188 nan
Assets: Equal Risk VaR nan nan nan 65.297 60.906 126.188 nan
Assets: coTVaR nan nan nan 65.294 60.908 126.202 nan
Assets: Covar nan nan nan 61.029 65.159 126.188 nan

Figure T_gross

(TG) Tame, gross twelve plot with ccoc distortion.

(TG) Tame, gross twelve plot with ccoc distortion.

Figure T_net

(TN) Tame, net twelve plot with tvar distortion.

(TN) Tame, net twelve plot with tvar distortion.

Figure U

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

(U) Tame, capital density for Tame, with ccoc gross and Tail VaR, 0.227 net distortion.

(U) Tame, capital density for Tame, with ccoc gross and Tail VaR, 0.227 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 Tame, distortion, SRM methods.
Gross: A Gross: B Gross: Total Net: A Net: B Net: Total Ceded: Diff
Loss: Expected Loss 50 50 100 50 49.08 99.08 0.916
Margin: Expected Loss 1.712 1.712 3.423 1.243 1.221 2.464 0.959
Margin: Dist Ccoc 0.224 3.199 3.423 0.177 2.287 2.464 0.959
Margin: Dist PH 1 2.423 3.423 1.374 1.391 2.765 0.659
Margin: Dist Wang 1.038 2.386 3.423 1.276 1.627 2.904 0.52
Margin: Dist Dual 1.068 2.355 3.423 1.18 1.85 3.03 0.393
Margin: Dist Tvar 1.096 2.327 3.423 1.102 2.053 3.155 0.269
Margin: Dist Blend 1.045 2.378 3.423 1.237 1.718 2.956 0.468
Premium: Expected Loss 51.71 51.71 103.42 51.24 50.3 101.55 1.875
Premium: Dist Ccoc 50.22 53.2 103.42 50.18 51.37 101.55 1.875
Premium: Dist PH 51 52.42 103.42 51.37 50.48 101.85 1.574
Premium: Dist Wang 51.04 52.39 103.42 51.28 50.71 101.99 1.435
Premium: Dist Dual 51.07 52.35 103.42 51.18 50.93 102.11 1.309
Premium: Dist Tvar 51.1 52.33 103.42 51.1 51.14 102.24 1.184
Premium: Dist Blend 51.04 52.38 103.42 51.24 50.8 102.04 1.383
Loss Ratio: Expected Loss 0.967 0.967 0.967 0.976 0.976 0.976 0.488
Loss Ratio: Dist Ccoc 0.996 0.94 0.967 0.996 0.955 0.976 0.488
Loss Ratio: Dist PH 0.98 0.954 0.967 0.973 0.972 0.973 0.582
Loss Ratio: Dist Wang 0.98 0.954 0.967 0.975 0.968 0.972 0.638
Loss Ratio: Dist Dual 0.979 0.955 0.967 0.977 0.964 0.97 0.699
Loss Ratio: Dist Tvar 0.979 0.956 0.967 0.978 0.96 0.969 0.773
Loss Ratio: Dist Blend 0.98 0.955 0.967 0.976 0.966 0.971 0.662
Capital: Expected Loss 17.12 17.12 34.23 12.43 12.21 24.64 9.59
Capital: Dist Ccoc 6.24 27.99 34.23 4.882 19.76 24.64 9.59
Capital: Dist PH 14.4 19.83 34.23 12.48 11.86 24.34 9.89
Capital: Dist Wang 15.48 18.75 34.23 12.44 11.76 24.2 10.03
Capital: Dist Dual 15.63 18.6 34.23 12.37 11.7 24.07 10.16
Capital: Dist Tvar 15.66 18.57 34.23 12.31 11.64 23.95 10.28
Capital: Dist Blend 14.84 19.39 34.23 12.21 11.94 24.15 10.09
Rate of Return: Expected Loss 0.1 0.1 0.1 0.1 0.1 0.1 0.1
Rate of Return: Dist Ccoc 0.036 0.114 0.1 0.036 0.116 0.1 0.1
Rate of Return: Dist PH 0.069 0.122 0.1 0.11 0.117 0.114 0.067
Rate of Return: Dist Wang 0.067 0.127 0.1 0.103 0.138 0.12 0.052
Rate of Return: Dist Dual 0.068 0.127 0.1 0.095 0.158 0.126 0.039
Rate of Return: Dist Tvar 0.07 0.125 0.1 0.089 0.176 0.132 0.026
Rate of Return: Dist Blend 0.07 0.123 0.1 0.101 0.144 0.122 0.046
Leverage: Expected Loss 3.021 3.021 3.021 4.121 4.121 4.121 0.195
Leverage: Dist Ccoc 8.05 1.901 3.021 10.28 2.6 4.121 0.195
Leverage: Dist PH 3.542 2.643 3.021 4.117 4.255 4.185 0.159
Leverage: Dist Wang 3.296 2.794 3.021 4.121 4.313 4.214 0.143
Leverage: Dist Dual 3.267 2.814 3.021 4.136 4.353 4.242 0.129
Leverage: Dist Tvar 3.263 2.817 3.021 4.15 4.395 4.269 0.115
Leverage: Dist Blend 3.439 2.701 3.021 4.198 4.254 4.226 0.137
Assets: Expected Loss 68.83 68.83 137.66 63.68 62.51 126.19 11.47
Assets: Dist Ccoc 56.47 81.19 137.66 55.06 71.13 126.19 11.47
Assets: Dist PH 65.4 72.26 137.66 63.85 62.34 126.19 11.47
Assets: Dist Wang 66.52 71.14 137.66 63.72 62.47 126.19 11.47
Assets: Dist Dual 66.7 70.96 137.66 63.55 62.63 126.19 11.47
Assets: Dist Tvar 66.76 70.9 137.66 63.41 62.77 126.19 11.47
Assets: Dist Blend 65.89 71.77 137.66 63.44 62.74 126.19 11.47

Figure W

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

(W) Figure 15.11: Tame, 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: Tame, 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) Tame, percentile layer of capital allocations by asset level, showing 0.9999 capital. (Same distortions.)

(X) Tame, percentile layer of capital allocations by asset level, showing 0.9999 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) Tame percentile layer of capital allocations compared to distortion allocations.
Method Gross: A Gross: B Gross: Total Net: A Net: B Net: Total Ceded: Diff
Expected Loss 68.83 68.83 137.7 63.68 62.51 126.2 11.47
Dist Ccoc 56.47 81.19 137.7 55.06 71.13 126.2 11.47
Dist PH 65.4 72.26 137.7 63.85 62.34 126.2 11.47
Dist Wang 66.52 71.14 137.7 63.72 62.47 126.2 11.47
Dist Dual 66.7 70.96 137.7 63.55 62.63 126.2 11.47
Dist Tvar 66.76 70.9 137.7 63.41 62.77 126.2 11.47
Dist Blend 65.89 71.77 137.7 63.44 62.74 126.2 11.47
PLC 67.34 70.32 137.7 63.89 62.3 126.2 11.47

Tame Case Description

Tame Case in the new syntax.

Distributions

# Line A (usually thinner tailed)
agg A 1 claim sev gamma  50 cv 0.10 fixed

# Line B Gross (usually thicker tailed)
agg B 1 claim sev gamma  50 cv 0.15 fixed

# Line B Net
agg B 1 claim sev gamma  50 cv 0.15 fixed aggregate net of 12.90625 xs 56.171875

Other Parameters

  • reg_p = 0.9999
  • 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.