Discrete Case Study PIR Exhibits

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pricing
insurance
risk
pir
Discrete scenario PIR Case Study exhibits
Author

Stephen J. Mildenhall

Published

2024-01-24

Discrete 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) Discrete estimated mean, CV, skewness and kurtosis by line and in total, gross and net. Net of 100% share of 70 xs 20 in the aggregate. applied to X2.
statistic Gross: X1 Gross: X2 Gross: Total Net: X1 Net: X2 Net: Total
Mean 4.5 22.75 27.25 4.5 5.25 9.75
CV 1.012 1.707 1.435 1.012 1.624 0.991
Skewness 0.071 1.154 1.131 0.071 1.147 0.794
Kurtosis -1.905 -0.667 -0.649 -1.905 -0.673 -0.501

Figure B

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

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

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

(C) Discrete, 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: Discrete, 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: Discrete, 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) Discrete 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 70 xs 20 in the aggregate. applied to X2.
statistic Gross: X1 Gross: X2 Gross: Benefit Gross: Sum Gross: Total Net: X1 Net: X2 Net: Benefit Net: Sum Net: Total
VaR 90.0 10 90 0.0204 100 98 10 20 0.0714 30 28
VaR 95.0 10 90 0 100 100 10 20 0 30 30
VaR 97.5 10 90 0 100 100 10 20 0 30 30
VaR 99.0 10 90 0 100 100 10 20 0 30 30
VaR 99.6 10 90 0 100 100 10 20 0 30 30
VaR 99.9 10 90 0 100 100 10 20 0 30 30
TVaR 90.0 10 90 0.00756 100 99.2 10 20 0.0256 30 29.2
TVaR 95.0 10 90 0 100 100 10 20 0 30 30
TVaR 97.5 10 90 0 100 100 10 20 0 30 30
TVaR 99.0 10 90 0 100 100 10 20 0 30 30
TVaR 99.6 10 90 0 100 100 10 20 0 30 30
TVaR 99.9 10 90 0 100 100 10 20 0 30 30
EPD 10.0 8.2 80.9 0.0658 89.1 83.6 8.2 17.9 0.231 26.1 21.2
EPD 5.0 9.1 85.5 0.0618 94.5 89 9.1 18.9 0.118 28 25.1
EPD 2.5 9.5 87.7 0.0398 97.3 93.5 9.5 19.5 0.073 29 27
EPD 1.0 9.8 89.1 0.0216 98.9 96.8 9.8 19.8 0.0411 29.6 28.4
EPD 0.4 9.9 89.6 0.0133 99.6 98.3 9.9 19.9 0.0159 29.8 29.4
EPD 0.1 10 89.9 0.00328 99.9 99.6 10 20 0.00392 30 29.8

Table F

PIR Chapter 7, Table 7.2, Pricing summary.

(F) Pricing summary for Discrete using a a 1 capital standard and 10.0% constant cost of capital for all layers.
stat Gross Net
Loss 27.25 9.75
Margin 6.614 1.841
Premium 33.864 11.591
Loss Ratio 0.805 0.841
Capital 66.136 18.409
Rate of Return 0.1 0.1
Assets 100 30
Leverage 0.512 0.63

Table H

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

(H) Classical pricing by method for Discrete. Pricing calibrated to total gross portfolio and applied to each line on a stand-alone basis. Sorted by gross premium for X2.
method Parameters: Value X1: Gross X2: Net X2: Gross Total: Net Total: Gross Total: Ceded
Net 4.5 5.2 22.8 9.8 27.2 17.5
VaR 0.750 8 1 1 11 11 0
Expected Value 0.243 5.6 6.5 28.3 12.1 33.9 21.7
Variance 0.004 4.59 5.6 29.3 10.2 33.9 23.7
Esscher 0.004 4.583 5.5 29.3 10.1 33.9 23.7
Standard Deviation 0.169 5.3 6.7 29.3 11.4 33.9 22.5
Semi-Variance 0.006 4.562 5.6 29.3 10.1 33.9 23.8
Fischer 0.196 5.1 6.7 29.3 11.3 33.9 22.6
Dutch 0.393 5.4 6.7 29.4 11.2 33.9 22.6

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 Discrete. Delta columns show the difference.
method Total: Gross Total: Net SoP: Gross SoP: Net Delta: Gross Delta: Net
Net 27.2 9.8 27.2 9.8 0 0
VaR 11 11 9 9 -2 -2
Expected Value 33.9 12.1 33.9 12.1 0 -0
Variance 33.9 10.2 33.9 10.2 0 0
Esscher 33.9 10.1 33.9 10.1 0 0
Standard Deviation 33.9 11.4 34.6 12 0.726 0.578
Semi-Variance 33.9 10.1 33.9 10.1 0.032 0.031
Fischer 33.9 11.3 34.5 11.8 0.625 0.571
Dutch 33.9 11.2 34.7 12.1 0.885 0.842

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 Discrete. Pricing calibrated to total gross portfolio and applied to each line on a stand-alone basis.
method X1: Gross X2: Net X2: Gross Total: Net Total: Gross Total: Ceded
Net 1 1 1 1 1 1
VaR 0.562 5.2 22.8 0.886 2.477 inf
Expected Value 0.805 0.805 0.805 0.805 0.805 0.805
Variance 0.98 0.943 0.777 0.96 0.805 0.738
Esscher 0.982 0.947 0.777 0.963 0.805 0.737
Standard Deviation 0.854 0.784 0.776 0.856 0.805 0.779
Semi-Variance 0.986 0.943 0.776 0.966 0.805 0.736
Fischer 0.876 0.784 0.775 0.865 0.805 0.774
Dutch 0.836 0.784 0.775 0.867 0.805 0.774

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 Discrete.
Gross SoP Gross Total Gross Redn Net SoP Net Total Net Redn
No Default: Loss 27.2 27.2 -0.0% 975.0% 975.0% 0.0%
No Default: Premium 33.9 33.9 0.0% 11.6 11.6 0.0%
No Default: Capital 66.1 66.1 0.0% 18.4 18.4 0.0%
With Default: Loss 27.2 27.2 0.0% 975.0% 975.0% 0.0%
With Default: Premium 33.9 33.9 0.0% 11.6 11.6 0.0%
With Default: Capital 66.1 66.1 0.0% 18.4 18.4 0.0%

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 Discrete, with 0.1 cost of capital and p=1. The column sop shows the sum by unit. Net of 100% share of 70 xs 20 in the aggregate. All units produce the same rate of return, by construction.
Gross: X1 Gross: X2 Gross: SoP Gross: Total Net: X1 Net: SoP Net: Total
No Default: Loss 4.5 22.8 27.2 27.2 4.5 9.8 9.8
No Default: Margin 0.5 6.1 6.6 6.6 0.5 1.841 1.841
No Default: Premium 5 28.9 33.9 33.9 5 11.6 11.6
No Default: Loss Ratio 0.9 0.788 0.805 0.805 0.9 0.841 0.841
No Default: Capital 5 61.1 66.1 66.1 5 18.4 18.4
No Default: Rate of Return 0.1 0.1 0.1 0.1 0.1 0.1 0.1
No Default: Leverage 1 0.472 0.512 0.512 1 0.63 0.63
No Default: Assets 10 90 100 100 10 30 30
With Default: Loss 4.5 22.8 27.2 27.2 4.5 9.8 9.8
With Default: Margin 0.5 6.1 6.6 6.6 0.5 1.841 1.841
With Default: Premium 5 28.9 33.9 33.9 5 11.6 11.6
With Default: Loss Ratio 0.9 0.788 0.805 0.805 0.9 0.841 0.841
With Default: Capital 5 61.1 66.1 66.1 5 18.4 18.4
With Default: Rate of Return 0.1 0.1 0.1 0.1 0.1 0.1 0.1
With Default: Leverage 1 0.472 0.512 0.512 1 0.63 0.63
With Default: Assets 10 90 100 100 10 30 30

Figure M

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

(M) Distortion envelope for Discrete, 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 Discrete, 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 33.864 66.136 0.1 0
PH 0.824 0 33.864 66.136 0.1 0
Wang 0.201 0 33.864 66.136 0.1 0
Dual 1.32 -0 33.864 66.136 0.1 0
Tvar 0.195 0 33.864 66.136 0.1 0

Figure O

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

(O) Discrete, 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) Discrete, 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) Discrete, 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) Discrete, 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 Discrete, calibrated to CCoC pricing with 0.1 cost of capital and p=1. 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 70 xs 20 in the aggregate.
Gross: X1 Gross: X2 Gross: SoP Gross: Total Net: X1 Net: SoP Net: Total
Loss: Ccoc 4.5 22.75 27.25 27.25 4.5 9.75 9.75
Margin: Ccoc 0.5 6.114 6.614 6.614 0.5 1.841 1.841
Margin: PH 0.659 6.227 6.886 6.614 0.659 2.039 1.767
Margin: Wang 0.775 6.138 6.912 6.614 0.775 2.148 1.849
Margin: Dual 0.928 5.973 6.901 6.614 0.928 2.281 1.994
Margin: TVaR 1.092 5.521 6.614 6.614 1.092 2.366 2.366
Margin: Blend 0.825 5.799 6.623 6.614 0.825 2.131 2.121
Premium: Ccoc 5 28.864 33.864 33.864 5 11.591 11.591
Premium: PH 5.159 28.977 34.136 33.864 5.159 11.789 11.517
Premium: Wang 5.275 28.888 34.162 33.864 5.275 11.898 11.599
Premium: Dual 5.428 28.723 34.151 33.864 5.428 12.031 11.744
Premium: TVaR 5.592 28.271 33.864 33.864 5.592 12.116 12.116
Premium: Blend 5.325 28.549 33.873 33.864 5.325 11.881 11.871
Loss Ratio: Ccoc 0.9 0.788 0.805 0.805 0.9 0.841 0.841
Loss Ratio: PH 0.872 0.785 0.798 0.805 0.872 0.827 0.847
Loss Ratio: Wang 0.853 0.788 0.798 0.805 0.853 0.819 0.841
Loss Ratio: Dual 0.829 0.792 0.798 0.805 0.829 0.81 0.83
Loss Ratio: TVaR 0.805 0.805 0.805 0.805 0.805 0.805 0.805
Loss Ratio: Blend 0.845 0.797 0.804 0.805 0.845 0.821 0.821
Capital: Ccoc 5 61.136 66.136 66.136 5 18.409 18.409
Capital: PH 4.841 61.023 65.864 66.136 4.841 18.211 18.483
Capital: Wang 4.725 61.112 65.838 66.136 4.725 18.102 18.401
Capital: Dual 4.572 61.277 65.849 66.136 4.572 17.969 18.256
Capital: TVaR 4.408 61.729 66.136 66.136 4.408 17.884 17.884
Capital: Blend 4.675 61.451 66.127 66.136 4.675 18.119 18.129
Rate of Return: Ccoc 0.1 0.1 0.1 0.1 0.1 0.1 0.1
Rate of Return: PH 0.136 0.102 0.105 0.1 0.136 0.112 0.096
Rate of Return: Wang 0.164 0.1 0.105 0.1 0.164 0.119 0.1
Rate of Return: Dual 0.203 0.097 0.105 0.1 0.203 0.127 0.109
Rate of Return: TVaR 0.248 0.089 0.1 0.1 0.248 0.132 0.132
Rate of Return: Blend 0.176 0.094 0.1 0.1 0.176 0.118 0.117
Leverage: Ccoc 1 0.472 0.512 0.512 1 0.63 0.63
Leverage: PH 1.066 0.475 0.518 0.512 1.066 0.647 0.623
Leverage: Wang 1.116 0.473 0.519 0.512 1.116 0.657 0.63
Leverage: Dual 1.187 0.469 0.519 0.512 1.187 0.67 0.643
Leverage: TVaR 1.269 0.458 0.512 0.512 1.269 0.678 0.678
Leverage: Blend 1.139 0.465 0.512 0.512 1.139 0.656 0.655
Assets: Ccoc 10 90 100 100 10 30 30

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 1-VaR assets.
Unit a E[Xi(a)] E[Xi ∧ ai]
X1 10 4.5 4.5
X2 90 22.75 22.75
Total 100 27.25 27.25
SoP 100 27.25 27.25

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 Discrete, classical PCP methods.
Gross: X1 Gross: X2 Gross: Total Net: X1 Net: X2 Net: Total Ceded: Diff
Loss: Expected Loss 4.5 22.75 27.25 4.5 5.25 9.75 17.5
Margin: Expected Loss 1.092 5.521 6.614 0.85 0.991 1.841 4.773
Margin: Scaled EPD 0.5 6.114 6.614 0.5 1.341 1.841 4.773
Margin: Scaled TVaR 0.5 6.114 6.614 0.5 1.341 1.841 4.773
Margin: Scaled VaR 0.5 6.114 6.614 0.5 1.341 1.841 4.773
Margin: Equal Risk EPD 0.5 6.114 6.614 0.5 1.341 1.841 4.773
Margin: Equal Risk TVaR 0.5 6.114 6.614 0.5 1.341 1.841 4.773
Margin: Equal Risk VaR 0.5 6.114 6.614 0.5 1.341 1.841 4.773
Margin: coTVaR nan nan -inf nan nan nan nan
Margin: Covar 0.09 6.524 6.614 0.409 1.432 1.841 4.773
Premium: Expected Loss 5.592 28.271 33.864 5.35 6.241 11.591 22.273
Premium: Scaled EPD 5 28.864 33.864 5 6.591 11.591 22.273
Premium: Scaled TVaR 5 28.864 33.864 5 6.591 11.591 22.273
Premium: Scaled VaR 5 28.864 33.864 5 6.591 11.591 22.273
Premium: Equal Risk EPD 5 28.864 33.864 5 6.591 11.591 22.273
Premium: Equal Risk TVaR 5 28.864 33.864 5 6.591 11.591 22.273
Premium: Equal Risk VaR 5 28.864 33.864 5 6.591 11.591 22.273
Premium: coTVaR nan nan -inf nan nan nan nan
Premium: Covar 4.59 29.274 33.864 4.909 6.682 11.591 22.273
Loss Ratio: Expected Loss 0.805 0.805 0.805 0.841 0.841 0.841 0.786
Loss Ratio: Scaled EPD 0.9 0.788 0.805 0.9 0.797 0.841 0.786
Loss Ratio: Scaled TVaR 0.9 0.788 0.805 0.9 0.797 0.841 0.786
Loss Ratio: Scaled VaR 0.9 0.788 0.805 0.9 0.797 0.841 0.786
Loss Ratio: Equal Risk EPD 0.9 0.788 0.805 0.9 0.797 0.841 0.786
Loss Ratio: Equal Risk TVaR 0.9 0.788 0.805 0.9 0.797 0.841 0.786
Loss Ratio: Equal Risk VaR 0.9 0.788 0.805 0.9 0.797 0.841 0.786
Loss Ratio: coTVaR nan nan -0 nan nan nan nan
Loss Ratio: Covar 0.98 0.777 0.805 0.917 0.786 0.841 0.786
Capital: Expected Loss 10.922 55.215 66.136 8.497 9.913 18.409 47.727
Capital: Scaled EPD 5 61.136 66.136 5 13.409 18.409 47.727
Capital: Scaled TVaR 5 61.136 66.136 5 13.409 18.409 47.727
Capital: Scaled VaR 5 61.136 66.136 5 13.409 18.409 47.727
Capital: Equal Risk EPD 5 61.136 66.136 5 13.409 18.409 47.727
Capital: Equal Risk TVaR 5 61.136 66.136 5 13.409 18.409 47.727
Capital: Equal Risk VaR 5 61.136 66.136 5 13.409 18.409 47.727
Capital: coTVaR nan nan nan nan nan nan nan
Capital: Covar 0.898 65.238 66.136 4.088 14.321 18.409 47.727
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 nan nan nan nan nan nan nan
Rate of Return: Covar 0.1 0.1 0.1 0.1 0.1 0.1 0.1
Leverage: Expected Loss 0.512 0.512 0.512 0.63 0.63 0.63 0.467
Leverage: Scaled EPD 1 0.472 0.512 1 0.492 0.63 0.467
Leverage: Scaled TVaR 1 0.472 0.512 1 0.492 0.63 0.467
Leverage: Scaled VaR 1 0.472 0.512 1 0.492 0.63 0.467
Leverage: Equal Risk EPD 1 0.472 0.512 1 0.492 0.63 0.467
Leverage: Equal Risk TVaR 1 0.472 0.512 1 0.492 0.63 0.467
Leverage: Equal Risk VaR 1 0.472 0.512 1 0.492 0.63 0.467
Leverage: coTVaR nan nan nan nan nan nan nan
Leverage: Covar 5.112 0.449 0.512 1.201 0.467 0.63 0.467
Assets: Expected Loss 16.514 83.486 100 13.846 16.154 30 70
Assets: Scaled EPD 10 90 100 10 20 30 70
Assets: Scaled TVaR 10 90 100 10 20 30 70
Assets: Scaled VaR 10 90 100 10 20 30 70
Assets: Equal Risk EPD 10 90 100 10 20 30 70
Assets: Equal Risk TVaR 10 90 100 10 20 30 70
Assets: Equal Risk VaR 10 90 100 10 20 30 70
Assets: coTVaR nan nan nan nan nan nan nan
Assets: Covar 5.488 94.512 100 8.997 21.003 30 70

Figure T_gross

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

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

Figure T_net

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

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

Figure U

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

(U) Discrete, capital density for Discrete, with ccoc gross and Tail VaR, 0.195 net distortion.

(U) Discrete, capital density for Discrete, with ccoc gross and Tail VaR, 0.195 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 Discrete, distortion, SRM methods.
Gross: X1 Gross: X2 Gross: Total Net: X1 Net: X2 Net: Total Ceded: Diff
Loss: Expected Loss 4.5 22.75 27.25 4.5 5.25 9.75 17.5
Margin: Expected Loss 1.092 5.52 6.61 0.85 0.991 1.841 4.773
Margin: Dist Ccoc 0.5 6.11 6.61 0.5 1.341 1.841 4.773
Margin: Dist PH 0.399 6.21 6.61 0.399 1.368 1.767 4.847
Margin: Dist Wang 0.495 6.12 6.61 0.495 1.353 1.849 4.765
Margin: Dist Dual 0.669 5.94 6.61 0.669 1.325 1.994 4.619
Margin: Dist Tvar 1.092 5.52 6.61 1.092 1.274 2.366 4.247
Margin: Dist Blend 0.825 5.79 6.61 0.825 1.297 2.121 4.492
Premium: Expected Loss 5.59 28.27 33.86 5.35 6.24 11.59 22.27
Premium: Dist Ccoc 5 28.86 33.86 5 6.59 11.59 22.27
Premium: Dist PH 4.899 28.96 33.86 4.899 6.62 11.52 22.35
Premium: Dist Wang 4.995 28.87 33.86 4.995 6.6 11.6 22.26
Premium: Dist Dual 5.17 28.69 33.86 5.17 6.58 11.74 22.12
Premium: Dist Tvar 5.59 28.27 33.86 5.59 6.52 12.12 21.75
Premium: Dist Blend 5.32 28.54 33.86 5.32 6.55 11.87 21.99
Loss Ratio: Expected Loss 0.805 0.805 0.805 0.841 0.841 0.841 0.786
Loss Ratio: Dist Ccoc 0.9 0.788 0.805 0.9 0.797 0.841 0.786
Loss Ratio: Dist PH 0.919 0.785 0.805 0.919 0.793 0.847 0.783
Loss Ratio: Dist Wang 0.901 0.788 0.805 0.901 0.795 0.841 0.786
Loss Ratio: Dist Dual 0.871 0.793 0.805 0.871 0.798 0.83 0.791
Loss Ratio: Dist Tvar 0.805 0.805 0.805 0.805 0.805 0.805 0.805
Loss Ratio: Dist Blend 0.845 0.797 0.805 0.845 0.802 0.821 0.796
Capital: Expected Loss 10.92 55.21 66.14 8.5 9.91 18.41 47.73
Capital: Dist Ccoc 5 61.14 66.14 5 13.41 18.41 47.73
Capital: Dist PH 4.517 61.62 66.14 4.769 13.71 18.48 47.65
Capital: Dist Wang 4.762 61.37 66.14 5.1 13.31 18.4 47.74
Capital: Dist Dual 4.701 61.44 66.14 5.23 13.03 18.26 47.88
Capital: Dist Tvar 4.763 61.37 66.14 5.33 12.55 17.88 48.25
Capital: Dist Blend 6.35 59.79 66.14 6.07 12.06 18.13 48.01
Rate of Return: Expected Loss 0.1 0.1 0.1 0.1 0.1 0.1 0.1
Rate of Return: Dist Ccoc 0.1 0.1 0.1 0.1 0.1 0.1 0.1
Rate of Return: Dist PH 0.088 0.101 0.1 0.084 0.1 0.096 0.102
Rate of Return: Dist Wang 0.104 0.1 0.1 0.097 0.102 0.1 0.1
Rate of Return: Dist Dual 0.142 0.097 0.1 0.128 0.102 0.109 0.096
Rate of Return: Dist Tvar 0.229 0.09 0.1 0.205 0.102 0.132 0.088
Rate of Return: Dist Blend 0.13 0.097 0.1 0.136 0.108 0.117 0.094
Leverage: Expected Loss 0.512 0.512 0.512 0.63 0.63 0.63 0.467
Leverage: Dist Ccoc 1 0.472 0.512 1 0.492 0.63 0.467
Leverage: Dist PH 1.085 0.47 0.512 1.027 0.483 0.623 0.469
Leverage: Dist Wang 1.049 0.47 0.512 0.98 0.496 0.63 0.466
Leverage: Dist Dual 1.1 0.467 0.512 0.989 0.505 0.643 0.462
Leverage: Dist Tvar 1.174 0.461 0.512 1.049 0.52 0.678 0.451
Leverage: Dist Blend 0.839 0.477 0.512 0.877 0.543 0.655 0.458
Assets: Expected Loss 16.51 83.49 100 13.85 16.15 30 70
Assets: Dist Ccoc 10 90 100 10 20 30 70
Assets: Dist PH 9.42 90.58 100 9.67 20.33 30 70
Assets: Dist Wang 9.76 90.24 100 10.09 19.91 30 70
Assets: Dist Dual 9.87 90.13 100 10.4 19.6 30 70
Assets: Dist Tvar 10.35 89.65 100 10.92 19.08 30 70
Assets: Dist Blend 11.67 88.33 100 11.39 18.61 30 70

Figure W

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

(W) Figure 15.11: Discrete, 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: Discrete, 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) Discrete, percentile layer of capital allocations by asset level, showing 1 capital. (Same distortions.)

(X) Discrete, percentile layer of capital allocations by asset level, showing 1 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) Discrete percentile layer of capital allocations compared to distortion allocations.
Method Gross: X1 Gross: X2 Gross: Total Net: X1 Net: X2 Net: Total Ceded: Diff
Expected Loss 16.51 83.49 100 13.85 16.15 30 70
Dist Ccoc 10 90 100 10 20 30 70
Dist PH 9.416 90.58 100 9.668 20.33 30 70
Dist Wang 9.758 90.24 100 10.09 19.91 30 70
Dist Dual 9.87 90.13 100 10.4 19.6 30 70
Dist Tvar 10.35 89.65 100 10.92 19.08 30 70
Dist Blend 11.67 88.33 100 11.39 18.61 30 70
PLC 10.35 89.65 100 10.92 19.08 30 70

Discrete Case Description

PIR Discrete Case Study (no equal points).

Distributions

# Line A (usually thinner tailed)
agg X1 1 claim dsev [0 8 10] [1/2 1/4 1/4] fixed

# Line B Gross (usually thicker tailed)
agg X2 1 claim dsev [0 1 90] [1/2 1/4 1/4] fixed

# Line B Net
agg X2 1 claim dsev [0 1 90] [1/2 1/4 1/4] fixed aggregate net of 70 xs 20

Other Parameters

  • reg_p = 1
  • roe = 0.1
  • d2tc = 0.3
  • s_values = [0.005, 0.01, 0.03]
  • gs_values = [0.029126, 0.047619, 0.074074]
  • f_discrete = True
  • log2 = 8
  • bs = 1
  • 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.