HU/Severe Convective Storm Aggregate Case Study PIR Exhibits

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HU/Severe Convective Storm with aggregate reinsurance scenario PIR Case Study exhibits
Author

Stephen J. Mildenhall

Published

2024-01-24

Hu/SCS Case 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) Hu/SCS Case estimated mean, CV, skewness and kurtosis by line and in total, gross and net. Net of 100% share of 372 xs 40 in the aggregate. applied to Hu.
statistic Gross: Hu Gross: SCS Gross: Total Net: Hu Net: SCS Net: Total
Mean 29.727 69.133 98.860 18.983 69.133 88.116
CV 10.923 0.736 3.324 16.249 0.736 3.548
Skewness 120.532 24.900 116.280 136.952 24.900 131.657
Kurtosis 26,899.193 8,713.242 25,611.757 32,391.455 8,713.242 30,705.370

Figure B

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

(B) Hu/SCS Case, gross (top) and net (bottom) densities on a nominal (left) and log (right) scale.

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

(C) Hu/SCS Case, 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: Hu/SCS Case, 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: Hu/SCS Case, 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) Hu/SCS Case 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 372 xs 40 in the aggregate. applied to Hu.
statistic Gross: Hu Gross: SCS Gross: Benefit Gross: Sum Gross: Total Net: Hu Net: SCS Net: Benefit Net: Sum Net: Total
VaR 90.0 43.5 111.0 0.0283 154.5 150.2 40.5 111.0 0.195 151.5 126.8
VaR 95.0 95.2 139.2 0.113 234.5 210.8 40.5 139.2 0.125 179.8 159.8
VaR 97.5 189.5 174.2 0.181 363.8 308.0 40.5 174.2 0.03 214.8 208.5
VaR 99.0 428.5 236.8 0.223 665.2 543.8 56.0 236.8 -0.137 292.8 339.2
VaR 99.6 899.0 326.2 0.217 1,225.2 1,006.8 526.8 326.2 0.22 853.0 699.2
VaR 99.9 2,487.5 541.2 0.173 3,028.8 2,583.0 2,115.2 541.2 0.195 2,656.5 2,223.5
TVaR 90.0 247.0 166.8 0.149 413.7 360.2 139.6 166.8 0.112 306.4 275.5
TVaR 95.0 430.2 210.5 0.175 640.8 545.5 238.7 210.5 0.0945 449.3 410.5
TVaR 97.5 727.9 267.0 0.183 995.0 841.3 437.0 267.0 0.0973 704.0 641.6
TVaR 99.0 1,403.8 369.0 0.173 1,772.8 1,510.8 1,031.4 369.0 0.146 1,400.4 1,222.4
TVaR 99.6 2,604.3 514.3 0.153 3,118.5 2,703.6 2,231.9 514.3 0.163 2,746.2 2,361.0
TVaR 99.9 6,209.2 855.6 0.122 7,064.8 6,297.9 5,836.9 855.6 0.128 6,692.4 5,932.0
EPD 10.0 3,438.0 99.6 5.8 3,537.6 522.9 5,762.0 99.6 16.2 5,861.7 341.2
EPD 5.0 8,140.4 141.5 3.9 8,281.8 1,684.1 12,594.1 141.5 6.7 12,735.6 1,646.1
EPD 2.5 16,268.4 204.2 2.7 16,472.6 4,491.2 23,183.8 204.2 3.9 23,388.0 4,820.9
EPD 1.0 32,660.4 342.5 1.6 33,002.9 12,538.1 42,070.9 342.5 2.1 42,413.3 13,623.8
EPD 0.4 53,155.8 583.0 0.99 53,738.8 27,007.8 63,044.1 583.0 1.2 63,627.0 28,894.6
EPD 0.1 82,899.8 1,278.1 0.465 84,177.9 57,442.0 90,505.4 1,278.1 0.536 91,783.6 59,749.2

Table F

PIR Chapter 7, Table 7.2, Pricing summary.

(F) Pricing summary for Hu/SCS Case using a a 0.999 capital standard and 10.0% constant cost of capital for all layers.
stat Gross Net
Loss 95.145 84.408
Margin 226.169 194.463
Premium 321.314 278.871
Loss Ratio 0.296 0.303
Capital 2,261.686 1,944.629
Rate of Return 0.100 0.100
Assets 2,583.000 2,223.500
Leverage 0.142 0.143

Table H

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

(H) Classical pricing by method for Hu/SCS Case. Pricing calibrated to total gross portfolio and applied to each line on a stand-alone basis. Sorted by gross premium for Hu.
method Parameters: Value Hu: Net Hu: Gross SCS: Gross Total: Net Total: Gross Total: Ceded
Net 19 29.7 69.1 88.1 98.9 10.7
Expected Value 2.250 61.7 96.6 224.7 286.4 321.3 34.9
Dutch 7.6 117.3 197.2 175.2 253.7 321.3 67.7
VaR 0.977 40.5 203 178.5 215.2 321.2 106
Variance 0.002 215 246.9 74.5 289.4 321.3 31.9
Semi-Variance 0.002 217.1 248.5 73.7 290.3 321.3 31
Standard Deviation 0.677 227.8 249.5 103.6 299.7 321.3 21.6
Fischer 0.681 228.9 250.3 101 300.2 321.3 21.1

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 Hu/SCS Case. Delta columns show the difference.
method Total: Gross Total: Net SoP: Gross SoP: Net Delta: Gross Delta: Net
Net 98.9 88.1 98.9 88.1 0 0
Expected Value 321.3 286.4 321.3 286.4 0 0
Dutch 321.3 253.7 372.4 292.5 51.1 38.8
VaR 321.2 215.2 381.5 219 60.2 3.75
Variance 321.3 289.4 321.4 289.4 0.04 0
Semi-Variance 321.3 290.3 322.2 290.8 0.887 0.504
Standard Deviation 321.3 299.7 353.1 331.3 31.8 31.6
Fischer 321.3 300.2 351.4 329.9 30 29.7

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 Hu/SCS Case. Pricing calibrated to total gross portfolio and applied to each line on a stand-alone basis.
method Hu: Net Hu: Gross SCS: Gross Total: Net Total: Gross Total: Ceded
Net 1 1 1 1 1 1
Expected Value 0.308 0.308 0.308 0.308 0.308 0.308
Dutch 0.162 0.151 0.395 0.347 0.308 0.159
VaR 0.469 0.146 0.387 0.409 0.308 0.101
Variance 0.088 0.12 0.928 0.304 0.308 0.337
Semi-Variance 0.087 0.12 0.938 0.304 0.308 0.346
Standard Deviation 0.083 0.119 0.668 0.294 0.308 0.498
Fischer 0.083 0.119 0.684 0.294 0.308 0.508

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 Hu/SCS Case.
Gross SoP Gross Total Gross Redn Net SoP Net Total Net Redn
No Default: Loss 98.9 98.9 -0.0% 88.1 88.1 -0.0%
No Default: Premium 365.2 324.7 -11.1% 321.6 282.2 -12.2%
No Default: Capital 2,663.5 2,258.3 -15.2% 2,334.9 1,941.3 -16.9%
With Default: Loss 94.8 95.1 0.3% 84.1 84.4 0.4%
With Default: Premium 361.5 321.3 -11.1% 317.9 278.9 -12.3%
With Default: Capital 2,667.2 2,261.7 -15.2% 2,338.6 1,944.6 -16.8%

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 Hu/SCS Case, with 0.1 cost of capital and p=0.999. The column sop shows the sum by unit. Net of 100% share of 372 xs 40 in the aggregate. All units produce the same rate of return, by construction.
Gross: Hu Gross: SCS Gross: SoP Gross: Total Net: Hu Net: SoP Net: Total
No Default: Loss 29.7 69.1 98.9 98.9 19.0 88.1 88.1
No Default: Margin 223.4 42.9 266.4 225.8 190.6 233.5 194.1
No Default: Premium 253.2 112.1 365.2 324.7 209.6 321.6 282.2
No Default: Loss Ratio 0.117 0.617 0.271 0.304 0.091 0.274 0.312
No Default: Capital 2,234.3 429.2 2,663.5 2,258.3 1,905.7 2,334.9 1,941.3
No Default: Rate of Return 0.100 0.1 0.100 0.100 0.100 0.100 0.100
No Default: Leverage 0.113 0.261 0.137 0.144 0.110 0.138 0.145
No Default: Assets 2,487.5 541.2 3,028.8 2,583.0 2,115.2 2,656.5 2,223.5
With Default: Loss 26.0 68.8 94.8 95.1 15.3 84.1 84.4
With Default: Margin 223.8 42.9 266.7 226.2 190.9 233.9 194.5
With Default: Premium 249.8 111.8 361.5 321.3 206.2 317.9 278.9
With Default: Loss Ratio 0.104 0.616 0.262 0.296 0.074 0.264 0.303
With Default: Capital 2,237.7 429.5 2,667.2 2,261.7 1,909.1 2,338.6 1,944.6
With Default: Rate of Return 0.100 0.1 0.100 0.100 0.100 0.100 0.100
With Default: Leverage 0.112 0.26 0.136 0.142 0.108 0.136 0.143
With Default: Assets 2,487.5 541.2 3,028.8 2,583.0 2,115.2 2,656.5 2,223.5

Figure M

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

(M) Distortion envelope for Hu/SCS Case, 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 Hu/SCS Case, 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 0 321.314 2261.69 0.1 0.001
PH 0.449 0 321.314 2261.69 0.1 0.001
Wang 1.19 0 321.314 2261.69 0.1 0.001
Dual 12.029 -0 321.314 2261.69 0.1 0.001
Tvar 0.899 0 321.314 2261.69 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) Hu/SCS Case, 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) Hu/SCS Case, 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) Hu/SCS Case, 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) Hu/SCS Case, 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 Hu/SCS Case, 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 372 xs 40 in the aggregate.
Gross: Hu Gross: SCS Gross: SoP Gross: Total Net: Hu Net: SoP Net: Total
Loss: Ccoc 26.006 68.819 94.824 95.145 15.262 84.081 84.408
Margin: Ccoc 223.772 42.948 266.721 226.169 190.908 233.856 194.463
Margin: PH 210.001 61.155 271.156 226.169 148.865 210.02 176.322
Margin: Wang 199.753 72.464 272.217 226.169 126.825 199.289 168.54
Margin: Dual 186.167 89.156 275.323 226.169 93.556 182.711 155.72
Margin: TVaR 182.078 94.278 276.356 226.169 86.521 180.799 152.86
Margin: Blend 92.646 31.786 124.432 103.583 61.096 92.882 78.353
Premium: Ccoc 249.778 111.767 361.545 321.314 206.17 317.937 278.871
Premium: PH 236.006 129.974 365.98 321.314 164.127 294.101 260.73
Premium: Wang 225.758 141.283 367.041 321.314 142.087 283.37 252.948
Premium: Dual 212.173 157.975 370.148 321.314 108.817 266.792 240.128
Premium: TVaR 208.083 163.097 371.18 321.314 101.783 264.88 237.268
Premium: Blend 118.651 100.605 219.256 198.728 76.358 176.963 162.761
Loss Ratio: Ccoc 0.104 0.616 0.262 0.296 0.074 0.264 0.303
Loss Ratio: PH 0.11 0.529 0.259 0.296 0.093 0.286 0.324
Loss Ratio: Wang 0.115 0.487 0.258 0.296 0.107 0.297 0.334
Loss Ratio: Dual 0.123 0.436 0.256 0.296 0.14 0.315 0.352
Loss Ratio: TVaR 0.125 0.422 0.255 0.296 0.15 0.317 0.356
Loss Ratio: Blend 0.219 0.684 0.432 0.479 0.2 0.475 0.519
Capital: Ccoc 2237.72 429.483 2667.2 2261.69 1909.08 2338.56 1944.63
Capital: PH 2251.49 411.276 2662.77 2261.69 1951.12 2362.4 1962.77
Capital: Wang 2261.74 399.967 2661.71 2261.69 1973.16 2373.13 1970.55
Capital: Dual 2275.33 383.275 2658.6 2261.69 2006.43 2389.71 1983.37
Capital: TVaR 2279.42 378.153 2657.57 2261.69 2013.47 2391.62 1986.23
Capital: Blend 2368.85 440.645 2809.49 2384.27 2038.89 2479.54 2060.74
Rate of Return: Ccoc 0.1 0.1 0.1 0.1 0.1 0.1 0.1
Rate of Return: PH 0.093 0.149 0.102 0.1 0.076 0.089 0.09
Rate of Return: Wang 0.088 0.181 0.102 0.1 0.064 0.084 0.086
Rate of Return: Dual 0.082 0.233 0.104 0.1 0.047 0.076 0.079
Rate of Return: TVaR 0.08 0.249 0.104 0.1 0.043 0.076 0.077
Rate of Return: Blend 0.039 0.072 0.044 0.043 0.03 0.037 0.038
Leverage: Ccoc 0.112 0.26 0.136 0.142 0.108 0.136 0.143
Leverage: PH 0.105 0.316 0.137 0.142 0.084 0.124 0.133
Leverage: Wang 0.1 0.353 0.138 0.142 0.072 0.119 0.128
Leverage: Dual 0.093 0.412 0.139 0.142 0.054 0.112 0.121
Leverage: TVaR 0.091 0.431 0.14 0.142 0.051 0.111 0.119
Leverage: Blend 0.05 0.228 0.078 0.083 0.037 0.071 0.079
Assets: Ccoc 2487.5 541.25 3028.75 2583 2115.25 2656.5 2223.5

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]
SCS 541.25 69.087 68.819
Hu 2487.5 26.058 26.006
Total 2583 95.145 95.145
SoP 3028.75 95.145 94.824

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 Hu/SCS Case, classical PCP methods.
Gross: Hu Gross: SCS Gross: Total Net: Hu Net: SCS Net: Total Ceded: Diff
Loss: Expected Loss 26.058 69.087 95.145 15.329 69.079 84.408 10.737
Margin: Expected Loss 61.943 164.226 226.169 35.315 159.148 194.463 31.706
Margin: Scaled EPD 229.008 -2.839 226.169 198.662 -4.199 194.463 31.706
Margin: Scaled TVaR 193.039 33.13 226.169 161.821 32.642 194.463 31.706
Margin: Scaled VaR 190.49 35.679 226.169 159.558 34.904 194.463 31.706
Margin: Equal Risk EPD 224.359 1.81 226.169 194.335 0.128 194.463 31.706
Margin: Equal Risk TVaR 190.674 35.495 226.169 158.844 35.619 194.463 31.706
Margin: Equal Risk VaR 187.563 38.606 226.169 155.72 38.743 194.463 31.706
Margin: coTVaR 219.172 7.015 226.184 181.81 12.689 194.499 31.685
Margin: Covar 220.792 5.417 226.169 189.316 5.147 194.463 31.706
Premium: Expected Loss 88.001 233.313 321.314 50.643 228.227 278.871 42.443
Premium: Scaled EPD 255.066 66.248 321.314 213.991 64.88 278.871 42.443
Premium: Scaled TVaR 219.097 102.217 321.314 177.15 101.721 278.871 42.443
Premium: Scaled VaR 216.548 104.766 321.314 174.887 103.984 278.871 42.443
Premium: Equal Risk EPD 250.417 70.897 321.314 209.664 69.207 278.871 42.443
Premium: Equal Risk TVaR 216.732 104.582 321.314 174.173 104.698 278.871 42.443
Premium: Equal Risk VaR 213.621 107.693 321.314 171.049 107.822 278.871 42.443
Premium: coTVaR 245.23 76.102 321.329 197.138 81.768 278.906 42.423
Premium: Covar 246.85 74.505 321.314 204.645 74.226 278.871 42.443
Loss Ratio: Expected Loss 0.296 0.296 0.296 0.303 0.303 0.303 0.253
Loss Ratio: Scaled EPD 0.102 1.043 0.296 0.072 1.065 0.303 0.253
Loss Ratio: Scaled TVaR 0.119 0.676 0.296 0.087 0.679 0.303 0.253
Loss Ratio: Scaled VaR 0.12 0.659 0.296 0.088 0.664 0.303 0.253
Loss Ratio: Equal Risk EPD 0.104 0.974 0.296 0.073 0.998 0.303 0.253
Loss Ratio: Equal Risk TVaR 0.12 0.661 0.296 0.088 0.66 0.303 0.253
Loss Ratio: Equal Risk VaR 0.122 0.642 0.296 0.09 0.641 0.303 0.253
Loss Ratio: coTVaR 0.106 0.908 0.296 0.078 0.845 0.303 0.253
Loss Ratio: Covar 0.106 0.927 0.296 0.075 0.931 0.303 0.253
Capital: Expected Loss 619.427 1642.26 2261.69 353.148 1591.48 1944.63 317.057
Capital: Scaled EPD 2290.08 -28.391 2261.69 1986.62 -41.994 1944.63 317.057
Capital: Scaled TVaR 1930.39 331.297 2261.69 1618.21 326.419 1944.63 317.057
Capital: Scaled VaR 1904.9 356.789 2261.69 1595.59 349.044 1944.63 317.057
Capital: Equal Risk EPD 2243.59 18.096 2261.69 1943.35 1.276 1944.63 317.057
Capital: Equal Risk TVaR 1906.74 354.945 2261.69 1588.44 356.187 1944.63 317.057
Capital: Equal Risk VaR 1875.63 386.057 2261.69 1557.2 387.428 1944.63 317.057
Capital: coTVaR 2191.72 70.147 2261.84 1818.1 126.888 1944.99 316.851
Capital: Covar 2207.92 54.175 2261.69 1893.16 51.47 1944.63 317.057
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 0.142 0.142 0.142 0.143 0.143 0.143 0.134
Leverage: Scaled EPD 0.111 -2.333 0.142 0.108 -1.545 0.143 0.134
Leverage: Scaled TVaR 0.113 0.309 0.142 0.109 0.312 0.143 0.134
Leverage: Scaled VaR 0.114 0.294 0.142 0.11 0.298 0.143 0.134
Leverage: Equal Risk EPD 0.112 3.918 0.142 0.108 54.256 0.143 0.134
Leverage: Equal Risk TVaR 0.114 0.295 0.142 0.11 0.294 0.143 0.134
Leverage: Equal Risk VaR 0.114 0.279 0.142 0.11 0.278 0.143 0.134
Leverage: coTVaR 0.112 1.085 0.142 0.108 0.644 0.143 0.134
Leverage: Covar 0.112 1.375 0.142 0.108 1.442 0.143 0.134
Assets: Expected Loss 707.428 1875.57 2583 403.791 1819.71 2223.5 359.5
Assets: Scaled EPD 2545.14 37.857 2583 2200.61 22.886 2223.5 359.5
Assets: Scaled TVaR 2149.49 433.514 2583 1795.36 428.14 2223.5 359.5
Assets: Scaled VaR 2121.45 461.555 2583 1770.47 453.028 2223.5 359.5
Assets: Equal Risk EPD 2494.01 88.993 2583 2153.02 70.482 2223.5 359.5
Assets: Equal Risk TVaR 2123.47 459.527 2583 1762.62 460.885 2223.5 359.5
Assets: Equal Risk VaR 2089.25 493.75 2583 1728.25 495.25 2223.5 359.5
Assets: coTVaR 2436.95 146.248 2583.17 2015.24 208.656 2223.89 359.274
Assets: Covar 2454.77 128.679 2583 2097.81 125.696 2223.5 359.5

Figure T_gross

(TG) Hu/SCS Case, gross twelve plot with blend distortion.

(TG) Hu/SCS Case, gross twelve plot with blend distortion.

Figure T_net

(TN) Hu/SCS Case, net twelve plot with wang distortion.

(TN) Hu/SCS Case, net twelve plot with wang distortion.

Figure U

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

(U) Hu/SCS Case, capital density for Hu/SCS Case, with blend, 0.599 gross and Wang-normal, 1.190 net distortion.

(U) Hu/SCS Case, capital density for Hu/SCS Case, with blend, 0.599 gross and Wang-normal, 1.190 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 Hu/SCS Case, distortion, SRM methods.
Gross: Hu Gross: SCS Gross: Total Net: Hu Net: SCS Net: Total Ceded: Diff
Loss: Expected Loss 26.06 69.09 95.15 15.33 69.08 84.41 10.74
Margin: Expected Loss 61.94 164.23 226.17 35.31 159.15 194.46 31.71
Margin: Dist Ccoc 232.31 -6.16 226.17 200.56 -6.12 194.46 31.71
Margin: Dist PH 203.64 22.53 226.17 142.99 33.33 176.32 49.85
Margin: Dist Wang 188.25 37.92 226.17 115.83 52.71 168.54 57.63
Margin: Dist Dual 166.49 59.68 226.17 74.71 81.01 155.72 70.45
Margin: Dist Tvar 159.5 66.67 226.17 63.86 89 152.86 73.31
Margin: Dist Blend 88.1 15.48 103.58 56.75 21.61 78.35 25.23
Premium: Expected Loss 88 233.31 321.31 50.64 228.23 278.87 42.44
Premium: Dist Ccoc 258.37 62.93 321.31 215.89 62.96 278.87 42.44
Premium: Dist PH 229.7 91.61 321.31 158.32 102.41 260.73 60.58
Premium: Dist Wang 214.31 107 321.31 131.16 121.78 252.95 68.37
Premium: Dist Dual 192.55 128.77 321.31 90.03 150.09 240.13 81.19
Premium: Dist Tvar 185.56 135.76 321.31 79.19 158.08 237.27 84.05
Premium: Dist Blend 114.16 84.57 198.73 72.08 90.68 162.76 35.97
Loss Ratio: Expected Loss 0.296 0.296 0.296 0.303 0.303 0.303 0.253
Loss Ratio: Dist Ccoc 0.101 1.098 0.296 0.071 1.097 0.303 0.253
Loss Ratio: Dist PH 0.113 0.754 0.296 0.097 0.675 0.324 0.177
Loss Ratio: Dist Wang 0.122 0.646 0.296 0.117 0.567 0.334 0.157
Loss Ratio: Dist Dual 0.135 0.537 0.296 0.17 0.46 0.352 0.132
Loss Ratio: Dist Tvar 0.14 0.509 0.296 0.194 0.437 0.356 0.128
Loss Ratio: Dist Blend 0.228 0.817 0.479 0.213 0.762 0.519 0.299
Capital: Expected Loss 619.43 1642.26 2261.69 353.15 1591.48 1944.63 317.06
Capital: Dist Ccoc 2322.63 -61.07 2261.69 2005.12 -60.69 1944.63 317.06
Capital: Dist PH 2111.4 150.29 2261.69 1757.79 204.98 1962.77 298.92
Capital: Dist Wang 2074.81 186.88 2261.69 1716.18 254.37 1970.55 291.13
Capital: Dist Dual 2049.78 211.91 2261.69 1681.66 301.71 1983.37 278.31
Capital: Dist Tvar 2051.69 210 2261.69 1684.55 301.68 1986.23 275.45
Capital: Dist Blend 2189.49 194.78 2384.27 1796.62 264.12 2060.74 323.53
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 0.101 0.1 0.1 0.101 0.1 0.1
Rate of Return: Dist PH 0.096 0.15 0.1 0.081 0.163 0.09 0.167
Rate of Return: Dist Wang 0.091 0.203 0.1 0.067 0.207 0.086 0.198
Rate of Return: Dist Dual 0.081 0.282 0.1 0.044 0.269 0.079 0.253
Rate of Return: Dist Tvar 0.078 0.317 0.1 0.038 0.295 0.077 0.266
Rate of Return: Dist Blend 0.04 0.079 0.043 0.032 0.082 0.038 0.078
Leverage: Expected Loss 0.142 0.142 0.142 0.143 0.143 0.143 0.134
Leverage: Dist Ccoc 0.111 -1.03 0.142 0.108 -1.037 0.143 0.134
Leverage: Dist PH 0.109 0.61 0.142 0.09 0.5 0.133 0.203
Leverage: Dist Wang 0.103 0.573 0.142 0.076 0.479 0.128 0.235
Leverage: Dist Dual 0.094 0.608 0.142 0.054 0.497 0.121 0.292
Leverage: Dist Tvar 0.09 0.646 0.142 0.047 0.524 0.119 0.305
Leverage: Dist Blend 0.052 0.434 0.083 0.04 0.343 0.079 0.111
Assets: Expected Loss 707.43 1875.57 2583 403.79 1819.71 2223.5 359.5
Assets: Dist Ccoc 2581.01 1.859 2583 2221.01 2.272 2223.5 359.5
Assets: Dist PH 2341.1 241.9 2583 1916.11 307.39 2223.5 359.5
Assets: Dist Wang 2289.12 293.88 2583 1847.35 376.15 2223.5 359.5
Assets: Dist Dual 2242.33 340.67 2583 1771.69 451.81 2223.5 359.5
Assets: Dist Tvar 2237.25 345.75 2583 1763.74 459.76 2223.5 359.5
Assets: Dist Blend 2303.65 279.35 2583 1868.69 354.81 2223.5 359.5

Figure W

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

(W) Figure 15.11: Hu/SCS Case, 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: Hu/SCS Case, 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) Hu/SCS Case, percentile layer of capital allocations by asset level, showing 0.999 capital. (Same distortions.)

(X) Hu/SCS Case, 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) Hu/SCS Case percentile layer of capital allocations compared to distortion allocations.
Method Gross: Hu Gross: SCS Gross: Total Net: Hu Net: SCS Net: Total Ceded: Diff
Expected Loss 707.4 1876 2583 403.8 1820 2224 359.5
Dist Ccoc 2581 1.859 2583 2221 2.272 2224 359.5
Dist PH 2341 241.9 2583 1916 307.4 2224 359.5
Dist Wang 2289 293.9 2583 1847 376.2 2224 359.5
Dist Dual 2242 340.7 2583 1772 451.8 2224 359.5
Dist Tvar 2237 345.8 2583 1764 459.8 2224 359.5
Dist Blend 2304 279.4 2583 1869 354.8 2224 359.5
PLC 2209 373.8 2583 1761 462.8 2223 359.5

Hu/SCS Case Case Description

Hu/SCS Case in the new syntax.

Distributions

# Line A (usually thinner tailed)
agg SCS 70 claims sev 0.1644744565771549 * lognorm 1.9 poisson

# Line B Gross (usually thicker tailed)
agg Hu 2 claims sev 0.6590540043511113 * lognorm 2.5 poisson

# Line B Net
agg Hu 2 claims sev 0.6590540043511113 * lognorm 2.5 poisson aggregate net of 372.4067840300255 xs 40.25298926180363

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 = 19
  • bs = 0.25
  • 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.