Cat/Noncat Case Study Results

Case Study Description

Book Cat/NonCat Case, one thin- and one thick-tailed unit.

Contents

Chapter 2

The Insurance Market and Our Case Studies.

  • Basic statistics by line.
  • Density plots.
  • Bivariate distributions.

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(A) Table 2.6: Cat/NonCat Case Study estimated mean, CV, skewness and kurtosis by line and in total, gross and net. Aggregate reinsurance applied to Cat with an attachment probability 0.1 (¤ 41) and detachment probability 0.005 (¤ 121).
view Gross Net
line Cat NonCat Total Cat NonCat Total
statistic            
Mean 20.000 80.000 100.000 17.786 80.000 97.786
CV 1.000 0.150 0.233 0.737 0.150 0.182
Skewness 3.972 0.300 2.539 3.139 0.300 1.351
Kurtosis 35.933 0.135 19.173 55.220 0.135 16.336
Figure Figure(900x650)
(B) Figure 2.2: Cat/NonCat Case Study, gross (top) and net (bottom) densities on a nominal (left) and log (right) scale.
Figure Figure(900x300)
(C) Figure 2.3: Cat/NonCat Case Study, bivariate densities: gross (left), net (center), and a sample from gross (right). Impact of reinsurance is clear in net plot.

Chapter 4

Measuring Risk with Quantiles, VaR, and TVaR.

  • VaR, TVaR, and EPD plots and statistics.

Home.

Figure Figure(900x650)
(D) Figure 4.10: Cat/NonCat Case Study, TVaR, and VaR for unlimited and limited variables, gross (left) and net (right). Lower view uses a log return period horizontal axis.
(E) Table 4.7: Cat/NonCat Case Study estimated VaR, TVaR and EPD by line and in total, gross and net. EPD shows assets required for indicated EPD percentage. Sum shows sum of parts by line with no diversification and benefit shows percentage reduction compared to total. Aggregate reinsurance applied to Cat with an attachment probability 0.1 (¤ 41) and detachment probability 0.005 (¤ 121).
view Gross Net
line Cat NonCat Benefit Sum Total Cat NonCat Benefit Sum Total
statistic                    
VaR 90.0 41 96 0.0857 137 126 41 96 0.132 137 121
VaR 95.0 56 101 0.117 156 140 41 101 0.108 142 128
VaR 97.5 72 105 0.138 177 156 41 105 0.0901 146 134
VaR 99.0 98 111 0.152 209 181 41 111 0.0686 152 142
VaR 99.6 129 115 0.156 244 211 49 115 0.0919 164 151
VaR 99.9 185 122 0.151 308 267 106 122 0.215 228 188
TVaR 90.0 65 102 0.122 168 149 43 102 0.108 146 131
TVaR 95.0 83 107 0.139 190 167 64 107 0.233 171 139
TVaR 97.5 104 111 0.149 215 187 75 111 0.267 185 146
TVaR 99.0 135 116 0.153 251 218 81 116 0.232 197 160
TVaR 99.6 172 120 0.15 292 254 92 120 0.177 213 181
TVaR 99.9 240 127 0.141 366 321 160 127 0.187 287 242
EPD 10.0 46 74 0.253 120 96 30 74 0.128 104 92
EPD 5.0 65 82 0.338 147 110 35 82 0.139 117 103
EPD 2.5 87 88 0.407 175 124 39 88 0.135 127 112
EPD 1.0 123 94 0.464 217 148 48 94 0.172 142 122
EPD 0.4 165 100 0.491 265 178 91 100 0.46 192 131
EPD 0.1 245 108 0.496 353 236 173 108 0.776 281 158

Chapter 7

Guide to the Practice Chapters.

  • Summary of pricing by unit.
  • Specification of ceded reinsurance.

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(F) Table 7.2: Pricing summary for Cat/NonCat Case Study. Pricing summary by Case. The Case uses a 0.999 capital standard. Cost of capital is 10.0%.
portfolio Gross Net
stat    
Loss 99.95 97.73
Margin 15.21 8.168
Premium 115.2 105.9
Loss Ratio 0.868 0.923
Capital 152.1 81.68
Rate of Return 0.1 0.1
Assets 267.2 187.6
Leverage 0.757 1.297
(G) Table 7.3: Reinsurance summary for Cat/NonCat Case Study.
  CNC
item  
Reinsured Line Cat
Reinsurance Type Aggregate
Attachment Probability 0.1
Attachment 41.11
Exhaustion Probability 0.005
Limit 79.64

Chapter 9

Classical Portfolio Pricing Practice.

  • Classical pricing stand-alone by unit and in total, with parameters.
  • Impact of diversification: sum of stand-alone premiums compared to portfolio premium.
  • Stand-alone vs. diversified loss ratios.
  • Stand-alone vs. diversified loss, premium, and capital for CCoC pricing.
  • Stand-alone vs. diversified all insurance statistics for CCoC pricing.

Home.

(H) Table 9.5: Classical pricing by method for Cat/NonCat Case Study. Pricing calibrated to total gross portfolio and applied to each line on a stand-alone basis. Sorted by gross premium for Cat.
  Parameters Cat NonCat Total
  Value Net Gross Gross Net Gross Ceded
method              
Net 17.786 20.000 80.000 97.786 100.000 2.214
Expected Value 0.152 20.480 23.030 92.121 112.601 115.151 2.550
VaR 0.818 30.109 30.109 90.734 113.391 115.141 1.750
Variance 0.028 22.573 31.141 84.013 106.585 115.151 8.566
Dutch 1.859 27.170 32.000 88.882 110.638 115.151 4.513
Semi-Variance 0.040 22.513 32.785 83.092 105.194 115.151 9.957
Standard Deviation 0.650 26.302 32.992 87.797 109.332 115.151 5.819
Fischer 0.776 26.249 33.918 86.844 108.380 115.151 6.771
Esscher 0.011 23.575 35.944 81.561 105.054 115.151 10.098
Exponential 0.014 24.760 39.697 81.059 105.619 115.151 9.532
(I) Table 9.6: Sum of parts (SoP) stand-alone vs. diversified classical pricing by method for Cat/NonCat Case Study. Delta columns show the difference.
  Total SoP Delta
  Gross Net Gross Net Gross Net
method            
Net 100.000 97.786 100.000 97.786 0.000 0.000
Expected Value 115.151 112.601 115.151 112.601 0.000 0.000
VaR 115.141 113.391 120.844 120.844 5.703 7.453
Variance 115.151 106.585 115.154 106.585 0.002 0.000
Dutch 115.151 110.638 120.883 116.052 5.732 5.414
Semi-Variance 115.151 105.194 115.877 105.605 0.726 0.410
Standard Deviation 115.151 109.332 120.789 114.099 5.638 4.767
Fischer 115.151 108.380 120.762 113.093 5.611 4.712
Esscher 115.151 105.054 117.505 105.136 2.354 0.083
Exponential 115.151 105.619 120.756 105.819 5.604 0.200
(J) Table 9.7: Implied loss ratios from classical pricing by method for Cat/NonCat Case Study. Pricing calibrated to total gross portfolio and applied to each line on a stand-alone basis.
  Cat NonCat Total
  Net Gross Gross Net Gross Ceded
method            
Net 1 1 1 1 1 1
Expected Value 0.868 0.868 0.868 0.868 0.868 0.868
VaR 0.591 0.664 0.882 0.862 0.869 1.27
Variance 0.788 0.642 0.952 0.917 0.868 0.258
Dutch 0.655 0.625 0.9 0.884 0.868 0.491
Semi-Variance 0.79 0.61 0.963 0.93 0.868 0.222
Standard Deviation 0.676 0.606 0.911 0.894 0.868 0.381
Fischer 0.678 0.59 0.921 0.902 0.868 0.327
Esscher 0.754 0.556 0.981 0.931 0.868 0.219
Exponential 0.718 0.504 0.987 0.926 0.868 0.232
(K) Table 9.11: Comparison of stand-alone and sum of parts (SoP) premium for Cat/NonCat Case Study.
    Gross SoP Gross Total Gross Redn Net SoP Net Total Net Redn
method statistic            
No Default Loss 100 100 -0.0% 97.79 97.79 -0.0%
Premium 118.9 115.2 -3.1% 109.6 105.9 -3.3%
Capital 188.7 152 -19.4% 118.3 81.63 -31.0%
With Default Loss 99.94 99.95 0.0% 97.73 97.73 0.0%
Premium 118.8 115.2 -3.1% 109.6 105.9 -3.3%
Capital 188.7 152.1 -19.4% 118.3 81.68 -31.0%
(L) Table 9.13: Constant CoC pricing by unit for Cat/NonCat Case Study, with 0.1 cost of capital and $p=0.999$. The column sop shows the sum by unit. ¤79.6 excess ¤41.1 aggregate reinsurance applied to Cat. All units produce the same rate of return, by construction.
  portfolio Gross Net
  line Cat NonCat SoP Total Cat SoP Total
method statistic              
No Default Loss 20 80 100 100 17.79 97.79 97.79
Margin 15.03 3.841 18.87 15.2 7.987 11.83 8.163
Premium 35.03 83.84 118.9 115.2 25.77 109.6 105.9
Loss Ratio 0.571 0.954 0.841 0.868 0.69 0.892 0.923
Capital 150.3 38.41 188.7 152.0 79.87 118.3 81.63
Rate of Return 0.1 0.1 0.1 0.1 0.1 0.1 0.1
Leverage 0.233 2.183 0.63 0.758 0.323 0.927 1.298
Assets 185.3 122.2 307.5 267.2 105.6 227.9 187.6
With Default Loss 19.95 80 99.94 99.95 17.73 97.73 97.73
Margin 15.03 3.841 18.87 15.21 7.992 11.83 8.168
Premium 34.98 83.84 118.8 115.2 25.72 109.6 105.9
Loss Ratio 0.57 0.954 0.841 0.868 0.689 0.892 0.923
Capital 150.3 38.41 188.7 152.1 79.92 118.3 81.68
Rate of Return 0.1 0.1 0.1 0.1 0.1 0.1 0.1
Leverage 0.233 2.183 0.63 0.757 0.322 0.926 1.297
Assets 185.3 122.2 307.5 267.2 105.6 227.9 187.6

Chapter 11

Modern Portfolio Pricing Practice.

  • Distortion envelopes based on gross pricing.
  • Distortion parameter estimates calibrated to gross pricing.
  • Distortion, loss ratio, markup, margin, discount and premium leverage for PH, Wang, Dual, TVaR and CCoC.
  • Distortion, loss ratio, markup, margin, discount and premium leverage for PH, Wang, Dual, TVaR, CCoC, and Blend.
  • Insurance statistics by asset level for CCoC, PH, Dual, and TVaR distortions.
  • Stand-alone pricing and insurance statistics, gross and net, by unit by distortion.

Home.

Figure Figure(900x300)
(M) Figure 11.2: Distortion envelope for Cat/NonCat Case Study, gross. Left plot shows the distortion envelope, middle overlays the CCoC and TVaR distortions, right overlays proportional hazard, Wang, and dual moment distortions.
(N) Table 11.5: Parameter estimates for the five base SRMs
  Param Error $P$ $K$ Rate of Return $S$
method            
ROE 0.1 0 115.2 152.1 0.1 999.586u
PH 0.596 395.809n 115.2 152.1 0.1 999.586u
Wang 0.611 384.361n 115.2 152.1 0.1 999.586u
Dual 2.463 -163.814n 115.2 152.1 0.1 999.586u
Tvar 0.482 8.936u 115.2 152.1 0.1 999.586u
Figure Figure(750x1000)
(O) Figure 11.6: Cat/NonCat Case Study, 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 Figure(1000x1500)
(P) Figure 11.9: Cat/NonCat Case Study, 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.
(Q) Table 11.8: Traditional and stand alone Pricing by distortion. Pricing by unit and distortion for Cat/NonCat Case Study, 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 by unit, the different with total shows the impact of diversification. ¤79.6 excess ¤41.1 aggregate reinsurance applied to Cat.
  portfolio Gross Net
  line Cat NonCat SoP Total Cat SoP Total
statistic distortion              
Loss CCoC 19.95 80 99.94 99.95 17.73 97.73 97.73
Margin CCoC 15.03 3.841 18.87 15.21 7.992 11.83 8.168
PH 13.81 6.367 20.17 15.21 7.379 13.75 9.821
Wang 12.88 7.507 20.38 15.21 7.947 15.45 11.06
Dual 11.88 8.598 20.48 15.21 8.834 17.43 12.4
TVaR 11.21 9.171 20.38 15.21 9.151 18.32 13.15
Blend 14.53 5.329 19.86 15.22 7.75 13.08 9.137
Premium CCoC 34.98 83.84 118.8 115.2 25.72 109.6 105.9
PH 33.75 86.36 120.1 115.2 25.11 111.5 107.6
Wang 32.82 87.5 120.3 115.2 25.68 113.2 108.8
Dual 31.83 88.59 120.4 115.2 26.57 115.2 110.1
TVaR 31.15 89.17 120.3 115.2 26.88 116.0 110.9
Blend 34.48 85.32 119.8 115.2 25.48 110.8 106.9
Loss Ratio CCoC 0.57 0.954 0.841 0.868 0.689 0.892 0.923
PH 0.591 0.926 0.832 0.868 0.706 0.877 0.909
Wang 0.608 0.914 0.831 0.868 0.691 0.863 0.898
Dual 0.627 0.903 0.83 0.868 0.667 0.849 0.887
TVaR 0.64 0.897 0.831 0.868 0.66 0.842 0.881
Blend 0.578 0.938 0.834 0.868 0.696 0.882 0.914
Capital CCoC 150.3 38.41 188.7 152.1 79.92 118.3 81.68
PH 151.5 35.89 187.4 152.1 80.53 116.4 80.03
Wang 152.5 34.75 187.2 152.1 79.96 114.7 78.79
Dual 153.5 33.66 187.1 152.1 79.08 112.7 77.44
TVaR 154.1 33.08 187.2 152.1 78.76 111.8 76.69
Blend 150.8 36.93 187.7 152.0 80.16 117.1 80.71
Rate of Return CCoC 0.1 0.1 0.1 0.1 0.1 0.1 0.1
PH 0.0911 0.177 0.108 0.1 0.0916 0.118 0.123
Wang 0.0845 0.216 0.109 0.1 0.0994 0.135 0.14
Dual 0.0774 0.255 0.109 0.1 0.112 0.155 0.16
TVaR 0.0727 0.277 0.109 0.1 0.116 0.164 0.171
Blend 0.0964 0.144 0.106 0.1 0.0967 0.112 0.113
Leverage CCoC 0.233 2.183 0.63 0.757 0.322 0.926 1.297
PH 0.223 2.406 0.641 0.757 0.312 0.958 1.344
Wang 0.215 2.518 0.643 0.757 0.321 0.987 1.381
Dual 0.207 2.632 0.644 0.757 0.336 1.022 1.422
TVaR 0.202 2.695 0.643 0.757 0.341 1.038 1.446
Blend 0.229 2.311 0.638 0.757 0.318 0.946 1.324
Assets CCoC 185.3 122.2 307.5 267.2 105.6 227.9 187.6

Chapter 13

Classical Price Allocation Practice.

  • Comparison of stand-alone and equal priority loss recoveries by unit and in total.
  • Allocated pricing and insurance statistics, gross and net, by unit by classical pricing method. including scaled VaR, EPD, TVaR, equal risk VaR, EPD, TVaR, coTVaR, and covariance.

Home.

(R) Table 13.1: 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.
  a E[Xi(a)] E[Xi ∧ ai]
Unit      
NonCat 122.2 79.99 80
Cat 185.3 19.96 19.95
Total 267.2 99.95 99.95
SoP 307.5 99.95 99.94
(S) Table 13.3: Constant 0.10 ROE pricing for Cat/NonCat Case Study, classical PCP methods.
    Gross Net Ceded
  line Cat NonCat Total Cat NonCat Total Diff
stat Method              
Loss Expected Loss 19.96 79.99 99.95 17.75 79.98 97.73 2.214
Margin Expected Loss 3.036 12.17 15.21 1.483 6.685 8.168 7.037
Scaled EPD 15.68 -0.471 15.21 9.6 -1.432 8.168 7.037
Scaled TVaR 12.85 2.36 15.21 6.166 2.002 8.168 7.037
Scaled VaR 12.82 2.385 15.21 6.292 1.876 8.168 7.037
Equal Risk EPD 13.33 1.87 15.21 6.355 1.813 8.168 7.037
Equal Risk TVaR 11.78 3.42 15.21 5.294 2.874 8.168 7.037
Equal Risk VaR 11.74 3.47 15.21 4.698 3.47 8.168 7.037
coTVaR 14.84 0.37 15.21 6.845 1.326 8.171 7.035
Covar 11.18 4.027 15.21 4.443 3.725 8.168 7.037
Premium Expected Loss 22.99 92.16 115.2 19.23 86.67 105.9 9.252
Scaled EPD 35.63 79.52 115.2 27.35 78.55 105.9 9.252
Scaled TVaR 32.8 82.35 115.2 23.91 81.99 105.9 9.252
Scaled VaR 32.78 82.37 115.2 24.04 81.86 105.9 9.252
Equal Risk EPD 33.29 81.86 115.2 24.1 81.8 105.9 9.252
Equal Risk TVaR 31.74 83.41 115.2 23.04 82.86 105.9 9.252
Equal Risk VaR 31.69 83.46 115.2 22.44 83.45 105.9 9.252
coTVaR 34.8 80.36 115.2 24.59 81.31 105.9 9.249
Covar 31.14 84.01 115.2 22.19 83.71 105.9 9.252
Loss Ratio Expected Loss 0.868 0.868 0.868 0.923 0.923 0.923 0.239
Scaled EPD 0.56 1.006 0.868 0.649 1.018 0.923 0.239
Scaled TVaR 0.608 0.971 0.868 0.742 0.976 0.923 0.239
Scaled VaR 0.609 0.971 0.868 0.738 0.977 0.923 0.239
Equal Risk EPD 0.599 0.977 0.868 0.736 0.978 0.923 0.239
Equal Risk TVaR 0.629 0.959 0.868 0.77 0.965 0.923 0.239
Equal Risk VaR 0.63 0.958 0.868 0.791 0.958 0.923 0.239
coTVaR 0.574 0.995 0.868 0.722 0.984 0.923 0.239
Covar 0.641 0.952 0.868 0.8 0.956 0.923 0.239
Capital Expected Loss 30.36 121.7 152.1 14.83 66.85 81.68 70.37
Scaled EPD 156.8 -4.711 152.1 96 -14.32 81.68 70.37
Scaled TVaR 128.5 23.6 152.1 61.66 20.02 81.68 70.37
Scaled VaR 128.2 23.85 152.1 62.92 18.76 81.68 70.37
Equal Risk EPD 133.3 18.7 152.1 63.55 18.13 81.68 70.37
Equal Risk TVaR 117.8 34.2 152.1 52.94 28.74 81.68 70.37
Equal Risk VaR 117.4 34.7 152.1 46.98 34.7 81.68 70.37
coTVaR 148.4 3.701 152.1 68.45 13.26 81.71 70.35
Covar 111.8 40.27 152.1 44.43 37.25 81.68 70.37
Rate of Return Expected Loss 0.1 0.1 0.1 0.1 0.1 0.1 0.1
Scaled EPD 0.1 0.1 0.1 0.1 0.1 0.1 0.1
Scaled TVaR 0.1 0.1 0.1 0.1 0.1 0.1 0.1
Scaled VaR 0.1 0.1 0.1 0.1 0.1 0.1 0.1
Equal Risk EPD 0.1 0.1 0.1 0.1 0.1 0.1 0.1
Equal Risk TVaR 0.1 0.1 0.1 0.1 0.1 0.1 0.1
Equal Risk VaR 0.1 0.1 0.1 0.1 0.1 0.1 0.1
coTVaR 0.1 0.1 0.1 0.1 0.1 0.1 0.1
Covar 0.1 0.1 0.1 0.1 0.1 0.1 0.1
Leverage Expected Loss 0.757 0.757 0.757 1.297 1.297 1.297 0.131
Scaled EPD 0.227 -16.88 0.757 0.285 -5.485 1.297 0.131
Scaled TVaR 0.255 3.489 0.757 0.388 4.095 1.297 0.131
Scaled VaR 0.256 3.454 0.757 0.382 4.363 1.297 0.131
Equal Risk EPD 0.25 4.377 0.757 0.379 4.512 1.297 0.131
Equal Risk TVaR 0.269 2.439 0.757 0.435 2.883 1.297 0.131
Equal Risk VaR 0.27 2.405 0.757 0.478 2.405 1.297 0.131
coTVaR 0.234 21.71 0.757 0.359 6.131 1.296 0.131
Covar 0.279 2.086 0.757 0.499 2.247 1.297 0.131
Assets Expected Loss 53.36 213.8 267.2 34.06 153.5 187.6 79.62
Scaled EPD 192.4 74.81 267.2 123.3 64.23 187.6 79.62
Scaled TVaR 161.3 105.9 267.2 85.57 102.0 187.6 79.62
Scaled VaR 161.0 106.2 267.2 86.95 100.6 187.6 79.62
Equal Risk EPD 166.6 100.6 267.2 87.65 99.92 187.6 79.62
Equal Risk TVaR 149.6 117.6 267.2 75.98 111.6 187.6 79.62
Equal Risk VaR 149.0 118.2 267.2 69.42 118.2 187.6 79.62
coTVaR 183.2 84.06 267.2 93.04 94.57 187.6 79.6
Covar 142.9 124.3 267.2 66.62 121.0 187.6 79.62

Chapter 15

Modern Price Allocation Practice.

  • Twelve-plot recapping densities and plotting κ. α, and β; premium and margin density by unit, cumulative margin by unit, and comparing the lifted natural allocation with stand-alone margins. Shown for gross and net losses, with different distortions.
  • Gross and net capital densities (marginal capital) as a function of assets.
  • Allocated pricing and insurance statistics, gross and net, by unit by distortion, shown for CCoC, PH, Wang, Dual, TVaR, and the blend distortion.
  • Conditional gross and net loss densities, κ, and distortion spectra by loss return period.
  • Percentile layer of capital (PLC) allocated capital by asset level.
  • Comparison of PLC and distortion pricing.

Home.

Figure Figure(1080x1200)
(TG) Figure 12.2: Cat/NonCat Case Study, gross twelve plot with dual distortion.
Figure Figure(1080x1200)
(TN) Figure 15.3: Cat/NonCat Case Study, net twelve plot with ph distortion.
Figure Figure(900x325)
(U) Figure 15.8: Cat/NonCat Case Study, capital density for Cat/NonCat Case Study, with Dual Moment, 2.463 gross and Proportional Hazard, 0.596 net distortion.
(V) Table 13.36: Constant 0.10 ROE pricing for Cat/NonCat Case Study, distortion, SRM methods.
    Gross Net Ceded
  line Cat NonCat Total Cat NonCat Total Diff
stat Method              
Loss Expected Loss 19.96 79.99 99.95 17.75 79.98 97.73 2.21
Margin Expected Loss 3.04 12.17 15.21 1.48 6.68 8.17 7.04
Dist ROE 20.55 -5.35 15.21 13.94 -5.77 8.17 7.04
Dist PH 12.94 2.27 15.21 6.32 3.51 9.82 5.38
Dist Wang 11.31 3.90 15.21 6.15 4.91 11.06 4.15
Dist Dual 9.72 5.49 15.21 6.47 5.94 12.40 2.80
Dist Tvar 8.74 6.47 15.21 6.65 6.50 13.15 2.05
Dist Blend 15.62 -0.40 15.22 8.93 0.21 9.14 6.08
Premium Expected Loss 22.99 92.16 115.15 19.23 86.67 105.90 9.25
Dist ROE 40.51 74.64 115.15 31.68 74.21 105.90 9.25
Dist PH 32.89 82.26 115.15 24.06 83.49 107.55 7.60
Dist Wang 31.26 83.89 115.15 23.89 84.89 108.79 6.36
Dist Dual 29.68 85.48 115.15 24.21 85.92 110.13 5.02
Dist Tvar 28.69 86.46 115.15 24.39 86.49 110.88 4.27
Dist Blend 35.58 79.59 115.17 26.68 80.19 106.87 8.30
Loss Ratio Expected Loss 0.87 0.87 0.87 0.92 0.92 0.92 0.24
Dist ROE 0.49 1.07 0.87 0.56 1.08 0.92 0.24
Dist PH 0.61 0.97 0.87 0.74 0.96 0.91 0.29
Dist Wang 0.64 0.95 0.87 0.74 0.94 0.90 0.35
Dist Dual 0.67 0.94 0.87 0.73 0.93 0.89 0.44
Dist Tvar 0.70 0.93 0.87 0.73 0.92 0.88 0.52
Dist Blend 0.56 1.01 0.87 0.67 1.00 0.91 0.27
Capital Expected Loss 30.36 121.69 152.05 14.83 66.85 81.68 70.37
Dist ROE 205.55 -53.50 152.05 139.40 -57.72 81.68 70.37
Dist PH 103.25 48.80 152.05 42.98 37.04 80.03 72.03
Dist Wang 92.20 59.85 152.05 35.79 43.00 78.79 73.26
Dist Dual 89.01 63.04 152.05 33.53 43.92 77.44 74.61
Dist Tvar 88.87 63.19 152.05 33.27 43.43 76.69 75.36
Dist Blend 134.52 17.52 152.04 68.55 12.16 80.71 71.33
Rate of Return Expected Loss 0.10 0.10 0.10 0.10 0.10 0.10 0.10
Dist ROE 0.10 0.10 0.10 0.10 0.10 0.10 0.10
Dist PH 0.13 0.05 0.10 0.15 0.09 0.12 0.07
Dist Wang 0.12 0.07 0.10 0.17 0.11 0.14 0.06
Dist Dual 0.11 0.09 0.10 0.19 0.14 0.16 0.04
Dist Tvar 0.10 0.10 0.10 0.20 0.15 0.17 0.03
Dist Blend 0.12 -0.02 0.10 0.13 0.02 0.11 0.09
Leverage Expected Loss 0.76 0.76 0.76 1.30 1.30 1.30 0.13
Dist ROE 0.20 -1.40 0.76 0.23 -1.29 1.30 0.13
Dist PH 0.32 1.69 0.76 0.56 2.25 1.34 0.11
Dist Wang 0.34 1.40 0.76 0.67 1.97 1.38 0.09
Dist Dual 0.33 1.36 0.76 0.72 1.96 1.42 0.07
Dist Tvar 0.32 1.37 0.76 0.73 1.99 1.45 0.06
Dist Blend 0.26 4.54 0.76 0.39 6.60 1.32 0.12
Assets Expected Loss 53.36 213.85 267.20 34.06 153.52 187.58 79.62
Dist ROE 246.06 21.14 267.20 171.08 16.50 187.58 79.62
Dist PH 136.14 131.06 267.20 67.05 120.53 187.58 79.62
Dist Wang 123.46 143.74 267.20 59.68 127.89 187.58 79.62
Dist Dual 118.69 148.52 267.20 57.74 129.84 187.58 79.62
Dist Tvar 117.57 149.64 267.20 57.66 129.92 187.58 79.62
Dist Blend 170.10 97.10 267.20 95.23 92.35 187.58 79.62
Figure Figure(900x975)
(W) Figure 15.11: Cat/NonCat Case Study, 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 Figure(900x325)
(X) Figure 15.12: Cat/NonCat Case Study, percentile layer of capital allocations by asset level, showing 0.999 capital. (Same distortions.)
(Y) Table 15.39: Cat/NonCat Case Study percentile layer of capital allocations compared to distortion allocations.
  Gross Net Ceded
line Cat NonCat Total Cat NonCat Total Diff
Method              
Expected Loss 53.36 213.8 267.2 34.06 153.5 187.6 79.62
Dist ROE 246.1 21.14 267.2 171.1 16.5 187.6 79.62
Dist PH 136.1 131.1 267.2 67.05 120.5 187.6 79.62
Dist Wang 123.5 143.7 267.2 59.68 127.9 187.6 79.62
Dist Dual 118.7 148.5 267.2 57.74 129.8 187.6 79.62
Dist Tvar 117.6 149.6 267.2 57.66 129.9 187.6 79.62
Dist Blend 170.1 97.1 267.2 95.23 92.35 187.6 79.62
PLC 113.2 154 267.2 54 133.6 187.6 79.62

Created 2024-05-19 02:50:53.535507

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.