Discrete Case Study PIR Exhibits

publications
pricing
insurance
risk
pir
Discrete scenario PIR Case Study exhibits
Author

Stephen J. Mildenhall

Published

2024-01-24

Discrete (equal points) 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 (equal points) 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 4.75 22.75 27.5 4.75 5.25 10
CV 1.003 1.707 1.423 1.003 1.624 0.977
Skewness 0.016 1.154 1.129 0.016 1.147 0.765
Kurtosis -1.978 -0.667 -0.648 -1.978 -0.673 -0.503

Figure B

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

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

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

(C) Discrete (equal points), 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 (equal points), 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 (equal points), 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 (equal points) 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 0.0101 100 99 10 20 0.0345 30 29
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 0.00376 100 99.6 10 20 0.0127 30 29.6
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 8.6 80.9 0.0681 89.5 83.8 8.6 17.9 0.23 26.4 21.5
EPD 5.0 9 85.5 0.0588 94.5 89.2 9 18.9 0.098 28 25.5
EPD 2.5 9.5 87.7 0.0346 97.2 94 9.5 19.5 0.0545 29 27.5
EPD 1.0 9.8 89.1 0.0164 98.9 97.3 9.8 19.8 0.0314 29.6 28.7
EPD 0.4 9.9 89.6 0.00953 99.6 98.6 9.9 19.9 0.0163 29.8 29.4
EPD 0.1 10 89.9 0.00331 99.9 99.6 10 20 0.00402 30 29.8

Table F

PIR Chapter 7, Table 7.2, Pricing summary.

(F) Pricing summary for Discrete (equal points) using a a 1 capital standard and 10.0% constant cost of capital for all layers.
stat Gross Net
Loss 27.5 10
Margin 6.591 1.818
Premium 34.091 11.818
Loss Ratio 0.807 0.846
Capital 65.909 18.182
Rate of Return 0.1 0.1
Assets 100 30
Leverage 0.517 0.65

Table H

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

(H) Classical pricing by method for Discrete (equal points). 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: Net X1: Gross X2: Net X2: Gross Total: Net Total: Gross Total: Ceded
Net 4.750 4.75 5.2 22.8 10 27.5 17.5
VaR 0.750 9 1 1 11 11 0
Expected Value 0.240 5.9 6.5 28.2 12.4 34.1 21.7
Variance 0.004 4.848 5.6 29.2 10.4 34.1 23.7
Esscher 0.004 4.84 5.5 29.3 10.4 34.1 23.7
Standard Deviation 0.168 5.6 6.7 29.3 11.6 34.1 22.4
Semi-Variance 0.006 4.816 5.6 29.3 10.3 34.1 23.7
Fischer 0.196 5.4 6.7 29.3 11.5 34.1 22.6
Dutch 0.392 5.7 6.7 29.3 11.5 34.1 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 (equal points). Delta columns show the difference.
method Total: Gross Total: Net SoP: Gross SoP: Net Delta: Gross Delta: Net
Net 27.5 10 27.5 10 0 0
VaR 11 11 10 10 -1 -1
Expected Value 34.1 12.4 34.1 12.4 0 0
Variance 34.1 10.4 34.1 10.4 0 0
Esscher 34.1 10.4 34.1 10.4 0 -0
Standard Deviation 34.1 11.6 34.8 12.2 0.753 0.594
Semi-Variance 34.1 10.3 34.1 10.4 0.033 0.033
Fischer 34.1 11.5 34.7 12.1 0.644 0.586
Dutch 34.1 11.5 35 12.4 0.931 0.907

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 (equal points). Pricing calibrated to total gross portfolio and applied to each line on a stand-alone basis.
method X1: Net X1: Gross X2: Net X2: Gross Total: Net Total: Gross Total: Ceded
Net 1.000 1 1 1 1 1 1
VaR 0.528 5.2 22.8 0.909 2.5 inf
Expected Value 0.807 0.807 0.807 0.807 0.807 0.807
Variance 0.98 0.944 0.778 0.961 0.807 0.739
Esscher 0.981 0.947 0.778 0.963 0.807 0.738
Standard Deviation 0.855 0.785 0.777 0.859 0.807 0.78
Semi-Variance 0.986 0.943 0.776 0.966 0.807 0.737
Fischer 0.878 0.785 0.776 0.868 0.807 0.775
Dutch 0.836 0.784 0.775 0.872 0.807 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 (equal points).
Gross SoP Gross Total Gross Redn Net SoP Net Total Net Redn
No Default: Loss 27.5 27.5 0.0% 1000.0% 1000.0% 0.0%
No Default: Premium 34.1 34.1 0.0% 11.8 11.8 -0.0%
No Default: Capital 65.9 65.9 0.0% 18.2 18.2 0.0%
With Default: Loss 27.5 27.5 0.0% 1000.0% 1000.0% 0.0%
With Default: Premium 34.1 34.1 0.0% 11.8 11.8 -0.0%
With Default: Capital 65.9 65.9 0.0% 18.2 18.2 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 (equal points), 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 4.75 22.8 27.5 27.5 4.75 10 10
No Default: Margin 0.477 6.1 6.6 6.6 0.477 1.818 1.818
No Default: Premium 5.2 28.9 34.1 34.1 5.2 11.8 11.8
No Default: Loss Ratio 0.909 0.788 0.807 0.807 0.909 0.846 0.846
No Default: Capital 4.773 61.1 65.9 65.9 4.773 18.2 18.2
No Default: Rate of Return 0.1 0.1 0.1 0.1 0.1 0.1 0.1
No Default: Leverage 1.095 0.472 0.517 0.517 1.095 0.65 0.65
No Default: Assets 10 90 100 100 10 30 30
With Default: Loss 4.75 22.8 27.5 27.5 4.75 10 10
With Default: Margin 0.477 6.1 6.6 6.6 0.477 1.818 1.818
With Default: Premium 5.2 28.9 34.1 34.1 5.2 11.8 11.8
With Default: Loss Ratio 0.909 0.788 0.807 0.807 0.909 0.846 0.846
With Default: Capital 4.773 61.1 65.9 65.9 4.773 18.2 18.2
With Default: Rate of Return 0.1 0.1 0.1 0.1 0.1 0.1 0.1
With Default: Leverage 1.095 0.472 0.517 0.517 1.095 0.65 0.65
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 (equal points), 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 (equal points), 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 34.091 65.909 0.1 0
PH 0.824 0 34.091 65.909 0.1 0
Wang 0.2 0 34.091 65.909 0.1 0
Dual 1.318 -0 34.091 65.909 0.1 0
Tvar 0.193 0 34.091 65.909 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 (equal points), 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 (equal points), 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 (equal points), 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 (equal points), 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 (equal points), 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 4.75 22.75 27.5 27.5 4.75 10 10
Margin: Ccoc 0.477 6.114 6.591 6.591 0.477 1.818 1.818
Margin: PH 0.652 6.205 6.857 6.591 0.652 2.028 1.762
Margin: Wang 0.782 6.107 6.889 6.591 0.782 2.149 1.85
Margin: Dual 0.955 5.93 6.885 6.591 0.955 2.299 2.004
Margin: TVaR 1.138 5.452 6.591 6.591 1.138 2.397 2.397
Margin: Blend 0.839 5.762 6.601 6.591 0.839 2.137 2.127
Premium: Ccoc 5.227 28.864 34.091 34.091 5.227 11.818 11.818
Premium: PH 5.402 28.955 34.357 34.091 5.402 12.028 11.762
Premium: Wang 5.532 28.857 34.389 34.091 5.532 12.149 11.85
Premium: Dual 5.705 28.68 34.385 34.091 5.705 12.299 12.004
Premium: TVaR 5.888 28.202 34.091 34.091 5.888 12.397 12.397
Premium: Blend 5.589 28.512 34.101 34.091 5.589 12.137 12.127
Loss Ratio: Ccoc 0.909 0.788 0.807 0.807 0.909 0.846 0.846
Loss Ratio: PH 0.879 0.786 0.8 0.807 0.879 0.831 0.85
Loss Ratio: Wang 0.859 0.788 0.8 0.807 0.859 0.823 0.844
Loss Ratio: Dual 0.833 0.793 0.8 0.807 0.833 0.813 0.833
Loss Ratio: TVaR 0.807 0.807 0.807 0.807 0.807 0.807 0.807
Loss Ratio: Blend 0.85 0.798 0.806 0.807 0.85 0.824 0.825
Capital: Ccoc 4.773 61.136 65.909 65.909 4.773 18.182 18.182
Capital: PH 4.598 61.045 65.643 65.909 4.598 17.972 18.238
Capital: Wang 4.468 61.143 65.611 65.909 4.468 17.851 18.15
Capital: Dual 4.295 61.32 65.615 65.909 4.295 17.701 17.996
Capital: TVaR 4.112 61.798 65.909 65.909 4.112 17.603 17.603
Capital: Blend 4.411 61.488 65.899 65.909 4.411 17.863 17.873
Rate of Return: Ccoc 0.1 0.1 0.1 0.1 0.1 0.1 0.1
Rate of Return: PH 0.142 0.102 0.104 0.1 0.142 0.113 0.097
Rate of Return: Wang 0.175 0.1 0.105 0.1 0.175 0.12 0.102
Rate of Return: Dual 0.222 0.097 0.105 0.1 0.222 0.13 0.111
Rate of Return: TVaR 0.277 0.088 0.1 0.1 0.277 0.136 0.136
Rate of Return: Blend 0.19 0.094 0.1 0.1 0.19 0.12 0.119
Leverage: Ccoc 1.095 0.472 0.517 0.517 1.095 0.65 0.65
Leverage: PH 1.175 0.474 0.523 0.517 1.175 0.669 0.645
Leverage: Wang 1.238 0.472 0.524 0.517 1.238 0.681 0.653
Leverage: Dual 1.328 0.468 0.524 0.517 1.328 0.695 0.667
Leverage: TVaR 1.432 0.456 0.517 0.517 1.432 0.704 0.704
Leverage: Blend 1.267 0.464 0.517 0.517 1.267 0.679 0.679
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 4.75 4.75
X2 90 22.75 22.75
Total 100 27.5 27.5
SoP 100 27.5 27.5

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 (equal points), classical PCP methods.
Gross: X1 Gross: X2 Gross: Total Net: X1 Net: X2 Net: Total Ceded: Diff
Loss: Expected Loss 4.75 22.75 27.5 4.75 5.25 10 17.5
Margin: Expected Loss 1.138 5.452 6.591 0.864 0.955 1.818 4.773
Margin: Scaled EPD 0.477 6.114 6.591 0.477 1.341 1.818 4.773
Margin: Scaled TVaR 0.477 6.114 6.591 0.477 1.341 1.818 4.773
Margin: Scaled VaR 0.477 6.114 6.591 0.477 1.341 1.818 4.773
Margin: Equal Risk EPD 0.477 6.114 6.591 0.477 1.341 1.818 4.773
Margin: Equal Risk TVaR 0.477 6.114 6.591 0.477 1.341 1.818 4.773
Margin: Equal Risk VaR 0.477 6.114 6.591 0.477 1.341 1.818 4.773
Margin: coTVaR nan nan nan nan nan nan nan
Margin: Covar 0.098 6.493 6.591 0.433 1.386 1.818 4.773
Premium: Expected Loss 5.888 28.202 34.091 5.614 6.205 11.818 22.273
Premium: Scaled EPD 5.227 28.864 34.091 5.227 6.591 11.818 22.273
Premium: Scaled TVaR 5.227 28.864 34.091 5.227 6.591 11.818 22.273
Premium: Scaled VaR 5.227 28.864 34.091 5.227 6.591 11.818 22.273
Premium: Equal Risk EPD 5.227 28.864 34.091 5.227 6.591 11.818 22.273
Premium: Equal Risk TVaR 5.227 28.864 34.091 5.227 6.591 11.818 22.273
Premium: Equal Risk VaR 5.227 28.864 34.091 5.227 6.591 11.818 22.273
Premium: coTVaR nan nan nan nan nan nan nan
Premium: Covar 4.848 29.243 34.091 5.183 6.636 11.818 22.273
Loss Ratio: Expected Loss 0.807 0.807 0.807 0.846 0.846 0.846 0.786
Loss Ratio: Scaled EPD 0.909 0.788 0.807 0.909 0.797 0.846 0.786
Loss Ratio: Scaled TVaR 0.909 0.788 0.807 0.909 0.797 0.846 0.786
Loss Ratio: Scaled VaR 0.909 0.788 0.807 0.909 0.797 0.846 0.786
Loss Ratio: Equal Risk EPD 0.909 0.788 0.807 0.909 0.797 0.846 0.786
Loss Ratio: Equal Risk TVaR 0.909 0.788 0.807 0.909 0.797 0.846 0.786
Loss Ratio: Equal Risk VaR 0.909 0.788 0.807 0.909 0.797 0.846 0.786
Loss Ratio: coTVaR nan nan nan nan nan nan nan
Loss Ratio: Covar 0.98 0.778 0.807 0.917 0.791 0.846 0.786
Capital: Expected Loss 11.384 54.525 65.909 8.636 9.545 18.182 47.727
Capital: Scaled EPD 4.773 61.136 65.909 4.773 13.409 18.182 47.727
Capital: Scaled TVaR 4.773 61.136 65.909 4.773 13.409 18.182 47.727
Capital: Scaled VaR 4.773 61.136 65.909 4.773 13.409 18.182 47.727
Capital: Equal Risk EPD 4.773 61.136 65.909 4.773 13.409 18.182 47.727
Capital: Equal Risk TVaR 4.773 61.136 65.909 4.773 13.409 18.182 47.727
Capital: Equal Risk VaR 4.773 61.136 65.909 4.773 13.409 18.182 47.727
Capital: coTVaR nan nan nan nan nan nan nan
Capital: Covar 0.977 64.932 65.909 4.325 13.857 18.182 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 0.517 0.517 0.517 0.65 0.65 0.65 0.467
Leverage: Scaled EPD 1.095 0.472 0.517 1.095 0.492 0.65 0.467
Leverage: Scaled TVaR 1.095 0.472 0.517 1.095 0.492 0.65 0.467
Leverage: Scaled VaR 1.095 0.472 0.517 1.095 0.492 0.65 0.467
Leverage: Equal Risk EPD 1.095 0.472 0.517 1.095 0.492 0.65 0.467
Leverage: Equal Risk TVaR 1.095 0.472 0.517 1.095 0.492 0.65 0.467
Leverage: Equal Risk VaR 1.095 0.472 0.517 1.095 0.492 0.65 0.467
Leverage: coTVaR nan nan nan nan nan nan nan
Leverage: Covar 4.961 0.45 0.517 1.198 0.479 0.65 0.467
Assets: Expected Loss 17.273 82.727 100 14.25 15.75 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 5.825 94.175 100 9.508 20.492 30 70

Figure T_gross

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

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

Figure T_net

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

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

Figure U

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

(U) Discrete (equal points), capital density for Discrete (equal points), with ccoc gross and Tail VaR, 0.193 net distortion.

(U) Discrete (equal points), capital density for Discrete (equal points), with ccoc gross and Tail VaR, 0.193 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 (equal points), distortion, SRM methods.
Gross: X1 Gross: X2 Gross: Total Net: X1 Net: X2 Net: Total Ceded: Diff
Loss: Expected Loss 4.75 22.75 27.5 4.75 5.25 10 17.5
Margin: Expected Loss 1.138 5.45 6.59 0.864 0.955 1.818 4.773
Margin: Dist Ccoc 0.477 6.11 6.59 0.477 1.341 1.818 4.773
Margin: Dist PH 0.397 6.19 6.59 0.397 1.365 1.762 4.829
Margin: Dist Wang 0.502 6.09 6.59 0.502 1.349 1.85 4.74
Margin: Dist Dual 0.686 5.9 6.59 0.686 1.318 2.004 4.587
Margin: Dist Tvar 1.138 5.45 6.59 1.138 1.258 2.397 4.194
Margin: Dist Blend 0.839 5.75 6.59 0.839 1.288 2.127 4.464
Premium: Expected Loss 5.89 28.2 34.09 5.61 6.2 11.82 22.27
Premium: Dist Ccoc 5.23 28.86 34.09 5.23 6.59 11.82 22.27
Premium: Dist PH 5.15 28.94 34.09 5.15 6.61 11.76 22.33
Premium: Dist Wang 5.25 28.84 34.09 5.25 6.6 11.85 22.24
Premium: Dist Dual 5.44 28.65 34.09 5.44 6.57 12 22.09
Premium: Dist Tvar 5.89 28.2 34.09 5.89 6.51 12.4 21.69
Premium: Dist Blend 5.59 28.5 34.09 5.59 6.54 12.13 21.96
Loss Ratio: Expected Loss 0.807 0.807 0.807 0.846 0.846 0.846 0.786
Loss Ratio: Dist Ccoc 0.909 0.788 0.807 0.909 0.797 0.846 0.786
Loss Ratio: Dist PH 0.923 0.786 0.807 0.923 0.794 0.85 0.784
Loss Ratio: Dist Wang 0.904 0.789 0.807 0.904 0.796 0.844 0.787
Loss Ratio: Dist Dual 0.874 0.794 0.807 0.874 0.799 0.833 0.792
Loss Ratio: Dist Tvar 0.807 0.807 0.807 0.807 0.807 0.807 0.807
Loss Ratio: Dist Blend 0.85 0.798 0.807 0.85 0.803 0.825 0.797
Capital: Expected Loss 11.38 54.52 65.91 8.64 9.55 18.18 47.73
Capital: Dist Ccoc 4.773 61.14 65.91 4.773 13.41 18.18 47.73
Capital: Dist PH 4.549 61.36 65.91 4.73 13.51 18.24 47.67
Capital: Dist Wang 4.848 61.06 65.91 5.1 13.05 18.15 47.76
Capital: Dist Dual 4.804 61.11 65.91 5.25 12.74 18 47.91
Capital: Dist Tvar 4.861 61.05 65.91 5.36 12.25 17.6 48.31
Capital: Dist Blend 6.39 59.52 65.91 6.03 11.84 17.87 48.04
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 0.087 0.101 0.1 0.084 0.101 0.097 0.101
Rate of Return: Dist Wang 0.104 0.1 0.1 0.098 0.103 0.102 0.099
Rate of Return: Dist Dual 0.143 0.097 0.1 0.131 0.103 0.111 0.096
Rate of Return: Dist Tvar 0.234 0.089 0.1 0.213 0.103 0.136 0.087
Rate of Return: Dist Blend 0.131 0.097 0.1 0.139 0.109 0.119 0.093
Leverage: Expected Loss 0.517 0.517 0.517 0.65 0.65 0.65 0.467
Leverage: Dist Ccoc 1.095 0.472 0.517 1.095 0.492 0.65 0.467
Leverage: Dist PH 1.132 0.472 0.517 1.088 0.49 0.645 0.468
Leverage: Dist Wang 1.083 0.472 0.517 1.03 0.506 0.653 0.466
Leverage: Dist Dual 1.132 0.469 0.517 1.035 0.516 0.667 0.461
Leverage: Dist Tvar 1.211 0.462 0.517 1.099 0.531 0.704 0.449
Leverage: Dist Blend 0.875 0.479 0.517 0.927 0.552 0.679 0.457
Assets: Expected Loss 17.27 82.73 100 14.25 15.75 30 70
Assets: Dist Ccoc 10 90 100 10 20 30 70
Assets: Dist PH 9.7 90.3 100 9.88 20.12 30 70
Assets: Dist Wang 10.1 89.9 100 10.35 19.65 30 70
Assets: Dist Dual 10.24 89.76 100 10.69 19.31 30 70
Assets: Dist Tvar 10.75 89.25 100 11.25 18.75 30 70
Assets: Dist Blend 11.98 88.02 100 11.62 18.38 30 70

Figure W

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

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

(X) Discrete (equal points), 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 (equal points) 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 17.27 82.73 100 14.25 15.75 30 70
Dist Ccoc 10 90 100 10 20 30 70
Dist PH 9.696 90.3 100 9.877 20.12 30 70
Dist Wang 10.1 89.9 100 10.35 19.65 30 70
Dist Dual 10.24 89.76 100 10.69 19.31 30 70
Dist Tvar 10.75 89.25 100 11.25 18.75 30 70
Dist Blend 11.98 88.02 100 11.62 18.38 30 70
PLC 10.75 89.25 100 11.25 18.75 30 70

Discrete (equal points) Case Description

PIR Discrete Case Study with equal points.

Distributions

# Line A (usually thinner tailed)
agg X1 1 claim dsev [0 9 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.