Tame Case Study Results

Case Study Description

Book Tame Case, two thin-tailed units.

Contents

Chapter 2

The Insurance Market and Our Case Studies.

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

Home.

(A) Table 2.5: Tame Case Study estimated mean, CV, skewness and kurtosis by line and in total, gross and net. Aggregate reinsurance applied to B with an attachment probability 0.2 (¤ 56) and detachment probability 0.01 (¤ 69).
view Gross Net
line A B Total A B Total
statistic            
Mean 50.000 50.000 100.000 50.000 49.084 99.084
CV 0.100 0.150 0.090 0.100 0.123 0.079
Skewness 0.200 0.300 0.207 0.200 -0.484 -0.169
Kurtosis 0.060 0.135 0.070 0.060 -0.474 -0.157
Figure Figure(900x650)
(B) Figure 2.2: Tame Case Study, gross (top) and net (bottom) densities on a nominal (left) and log (right) scale.
Figure Figure(900x300)
(C) Figure 2.3: Tame 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: Tame 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 4.6: Tame 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 B with an attachment probability 0.2 (¤ 56) and detachment probability 0.01 (¤ 69).
view Gross Net
line A B Benefit Sum Total A B Benefit Sum Total
statistic                    
VaR 90.0 56 60 0.041 116 112 56 56 0.0344 113 109
VaR 95.0 58 63 0.053 121 115 58 56 0.0296 115 111
VaR 97.5 60 66 0.063 126 119 60 56 0.026 116 113
VaR 99.0 62 69 0.0743 131 122 62 56 0.0224 119 116
VaR 99.6 64 72 0.0843 136 126 64 59 0.0452 124 118
VaR 99.9 67 76 0.0974 143 131 67 64 0.0754 130 121
TVaR 90.0 59 64 0.0568 123 117 59 58 0.0444 117 112
TVaR 95.0 61 67 0.0664 128 120 61 59 0.0477 120 114
TVaR 97.5 62 69 0.075 132 123 62 59 0.048 122 116
TVaR 99.0 64 72 0.0849 137 126 64 59 0.0464 124 118
TVaR 99.6 66 75 0.0938 141 129 66 62 0.0666 128 120
TVaR 99.9 69 79 0.106 148 134 69 66 0.0931 135 123
EPD 10.0 45 46 0.0146 92 91 45 45 0.0122 91 90
EPD 5.0 49 51 0.0277 100 97 49 49 0.0232 98 96
EPD 2.5 52 55 0.0397 107 103 52 52 0.0323 104 101
EPD 1.0 55 59 0.0536 114 108 55 54 0.0392 109 105
EPD 0.4 57 63 0.0656 120 113 57 55 0.0407 113 108
EPD 0.1 60 68 0.081 128 119 60 56 0.0374 117 112

Chapter 7

Guide to the Practice Chapters.

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

Home.

(F) Table 7.2: Pricing summary for Tame Case Study. Pricing summary by Case. The Case uses a 0.9999 capital standard. Cost of capital is 10.0%.
portfolio Gross Net
stat    
Loss 100 99.08
Margin 3.423 2.464
Premium 103.4 101.5
Loss Ratio 0.967 0.976
Capital 34.23 24.64
Rate of Return 0.1 0.1
Assets 137.7 126.2
Leverage 3.021 4.121
(G) Table 7.3: Reinsurance summary for Tame Case Study.
  Tame
item  
Reinsured Line B
Reinsurance Type Aggregate
Attachment Probability 0.2
Attachment 56.17
Exhaustion Probability 0.01
Limit 12.91

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 9.2: Classical pricing by method for Tame Case Study. Pricing calibrated to total gross portfolio and applied to each line on a stand-alone basis. Sorted by gross premium for B.
  Parameters A B Total
  Value Gross Net Gross Net Gross Ceded
method              
Net 50.000 49.084 50.000 99.084 100.000 0.916
Expected Value 0.034 51.712 50.765 51.712 102.476 103.423 0.947
Variance 0.042 51.053 50.620 52.370 101.673 103.423 1.750
Esscher 0.041 51.034 50.468 52.389 101.503 103.423 1.920
Exponential 0.080 51.028 50.420 52.395 101.447 103.423 1.976
Semi-Variance 0.080 51.051 50.330 52.425 101.411 103.423 2.012
VaR 0.659 51.906 52.750 52.750 102.672 103.422 0.750
Dutch 0.953 51.899 51.498 52.846 102.096 103.423 1.327
Standard Deviation 0.380 51.899 51.377 52.848 102.061 103.423 1.362
Fischer 0.523 51.897 51.149 52.881 101.907 103.423 1.516
(I) Table 9.3: Sum of parts (SoP) stand-alone vs. diversified classical pricing by method for Tame Case Study. Delta columns show the difference.
  Total SoP Delta
  Gross Net Gross Net Gross Net
method            
Net 100.000 99.084 100.000 99.084 0.000 0.000
Expected Value 103.423 102.476 103.423 102.476 0.000 0.000
Variance 103.423 101.673 103.423 101.673 -0.000 -0.000
Esscher 103.423 101.503 103.423 101.503 0.000 0.000
Exponential 103.423 101.447 103.423 101.447 0.000 0.000
Semi-Variance 103.423 101.411 103.477 101.382 0.054 -0.029
VaR 103.422 102.672 104.656 104.656 1.234 1.984
Dutch 103.423 102.096 104.745 103.397 1.322 1.301
Standard Deviation 103.423 102.061 104.747 103.276 1.324 1.215
Fischer 103.423 101.907 104.778 103.047 1.355 1.140
(J) Table 9.4: Implied loss ratios from classical pricing by method for Tame Case Study. Pricing calibrated to total gross portfolio and applied to each line on a stand-alone basis.
  A B Total
  Gross Net Gross Net Gross Ceded
method            
Net 1 1 1 1 1 1
Expected Value 0.967 0.967 0.967 0.967 0.967 0.967
Variance 0.979 0.97 0.955 0.975 0.967 0.523
Esscher 0.98 0.973 0.954 0.976 0.967 0.477
Exponential 0.98 0.974 0.954 0.977 0.967 0.463
Semi-Variance 0.979 0.975 0.954 0.977 0.967 0.455
VaR 0.963 0.931 0.948 0.965 0.967 1.22
Dutch 0.963 0.953 0.946 0.97 0.967 0.69
Standard Deviation 0.963 0.955 0.946 0.971 0.967 0.672
Fischer 0.963 0.96 0.946 0.972 0.967 0.604
(K) Table 9.11: Comparison of stand-alone and sum of parts (SoP) premium for Tame Case Study.
    Gross SoP Gross Total Gross Redn Net SoP Net Total Net Redn
method statistic            
No Default Loss 100 100 -0.0% 99.08 99.08 -0.0%
Premium 104.9 103.4 -1.4% 102.9 101.5 -1.3%
Capital 48.69 34.23 -29.7% 37.79 24.64 -34.8%
With Default Loss 100 100 0.0% 99.08 99.08 0.0%
Premium 104.9 103.4 -1.4% 102.9 101.5 -1.3%
Capital 48.69 34.23 -29.7% 37.79 24.64 -34.8%
(L) Table 9.12: Constant CoC pricing by unit for Tame Case Study, with 0.1 cost of capital and $p=0.9999$. The column sop shows the sum by unit. ¤12.9 excess ¤56.2 aggregate reinsurance applied to B. All units produce the same rate of return, by construction.
  portfolio Gross Net
  line A B SoP Total A SoP Total
method statistic              
No Default Loss 50 50 100 100 50 99.08 99.08
Margin 1.888 2.982 4.869 3.423 1.888 3.779 2.464
Premium 51.89 52.98 104.9 103.4 51.89 102.9 101.5
Loss Ratio 0.964 0.944 0.954 0.967 0.964 0.963 0.976
Capital 18.88 29.82 48.69 34.23 18.88 37.79 24.64
Rate of Return 0.1 0.1 0.1 0.1 0.1 0.1 0.1
Leverage 2.749 1.777 2.154 3.021 2.749 2.722 4.121
Assets 70.77 82.8 153.6 137.7 70.77 140.7 126.2
With Default Loss 50 50 100 100 50 99.08 99.08
Margin 1.888 2.982 4.869 3.423 1.888 3.779 2.464
Premium 51.89 52.98 104.9 103.4 51.89 102.9 101.5
Loss Ratio 0.964 0.944 0.954 0.967 0.964 0.963 0.976
Capital 18.88 29.82 48.69 34.23 18.88 37.79 24.64
Rate of Return 0.1 0.1 0.1 0.1 0.1 0.1 0.1
Leverage 2.749 1.777 2.154 3.021 2.749 2.722 4.121
Assets 70.77 82.8 153.6 137.7 70.77 140.7 126.2

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 Tame 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 0 103.4 34.23 0.1 99.957u
PH 0.683 7.920n 103.4 34.23 0.1 99.957u
Wang 0.375 3.396p 103.4 34.23 0.1 99.957u
Dual 1.576 -5.063u 103.4 34.23 0.1 99.957u
Tvar 0.227 9.164u 103.4 34.23 0.1 99.957u
Figure Figure(750x1000)
(O) Figure 11.6: Tame 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: Tame 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 11.7: Traditional and stand alone Pricing by distortion. Pricing by unit and distortion for Tame Case Study, calibrated to CCoC pricing with 0.1 cost of capital and $p=0.9999$. Losses and assets are the same for all distortions. The column sop shows sum by unit, the different with total shows the impact of diversification. ¤12.9 excess ¤56.2 aggregate reinsurance applied to B.
  portfolio Gross Net
  line A B SoP Total B SoP Total
statistic distortion              
Loss CCoC 50 50 100 100 49.08 99.08 99.08
Margin CCoC 1.888 2.982 4.869 3.423 1.891 3.779 2.464
PH 1.896 2.891 4.787 3.423 1.822 3.718 2.765
Wang 1.898 2.861 4.759 3.423 2.09 3.988 2.904
Dual 1.9 2.835 4.735 3.423 2.355 4.255 3.03
TVaR 1.901 2.81 4.711 3.423 2.541 4.442 3.155
Blend 1.9 2.959 4.859 3.438 1.621 3.521 2.562
Premium CCoC 51.89 52.98 104.9 103.4 50.98 102.9 101.5
PH 51.9 52.89 104.8 103.4 50.91 102.8 101.8
Wang 51.9 52.86 104.8 103.4 51.17 103.1 102.0
Dual 51.9 52.83 104.7 103.4 51.44 103.3 102.1
TVaR 51.9 52.81 104.7 103.4 51.63 103.5 102.2
Blend 51.9 52.96 104.9 103.4 50.71 102.6 101.6
Loss Ratio CCoC 0.964 0.944 0.954 0.967 0.963 0.963 0.976
PH 0.963 0.945 0.954 0.967 0.964 0.964 0.973
Wang 0.963 0.946 0.955 0.967 0.959 0.961 0.972
Dual 0.963 0.946 0.955 0.967 0.954 0.959 0.97
TVaR 0.963 0.947 0.955 0.967 0.951 0.957 0.969
Blend 0.963 0.944 0.954 0.967 0.968 0.966 0.975
Capital CCoC 18.88 29.82 48.69 34.23 18.91 37.79 24.64
PH 18.87 29.91 48.78 34.23 18.98 37.85 24.34
Wang 18.87 29.94 48.8 34.23 18.72 37.58 24.2
Dual 18.87 29.96 48.83 34.23 18.45 37.32 24.07
TVaR 18.86 29.99 48.85 34.23 18.27 37.13 23.95
Blend 18.87 29.84 48.7 34.22 19.19 38.05 24.54
Rate of Return CCoC 0.1 0.1 0.1 0.1 0.1 0.1 0.1
PH 0.1 0.0967 0.0981 0.1 0.096 0.0982 0.114
Wang 0.101 0.0956 0.0975 0.1 0.112 0.106 0.12
Dual 0.101 0.0946 0.097 0.1 0.128 0.114 0.126
TVaR 0.101 0.0937 0.0964 0.1 0.139 0.12 0.132
Blend 0.101 0.0992 0.0998 0.1 0.0845 0.0925 0.104
Leverage CCoC 2.749 1.777 2.154 3.021 2.695 2.722 4.121
PH 2.75 1.769 2.148 3.021 2.681 2.716 4.185
Wang 2.751 1.766 2.147 3.021 2.734 2.742 4.214
Dual 2.751 1.763 2.145 3.021 2.788 2.769 4.242
TVaR 2.751 1.761 2.143 3.021 2.826 2.788 4.269
Blend 2.751 1.775 2.153 3.023 2.643 2.696 4.142
Assets CCoC 70.77 82.8 153.6 137.7 69.89 140.7 126.2

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 0.9999-VaR assets.
  a E[Xi(a)] E[Xi ∧ ai]
Unit      
A 70.77 50 50
B 82.8 50 50
Total 137.7 100 100
SoP 153.6 100 100
(S) Table 13.2: Constant 0.10 ROE pricing for Tame Case Study, classical PCP methods.
    Gross Net Ceded
  line A B Total A B Total Diff
stat Method              
Loss Expected Loss 50 50 100 50 49.08 99.08 0.916
Margin Expected Loss 1.712 1.712 3.423 1.243 1.221 2.464 0.959
Scaled EPD 1.171 2.252 3.423 1.16 1.304 2.464 0.959
Scaled TVaR 1.221 2.202 3.423 1.227 1.237 2.464 0.959
Scaled VaR 1.221 2.202 3.423 1.225 1.239 2.464 0.959
Equal Risk EPD 1.289 2.134 3.423 1.338 1.126 2.464 0.959
Equal Risk TVaR 1.34 2.084 3.423 1.389 1.075 2.464 0.959
Equal Risk VaR 1.339 2.084 3.423 1.389 1.075 2.464 0.959
coTVaR 0.883 2.542 3.425 1.39 1.075 2.465 0.959
Covar 1.053 2.37 3.423 1.003 1.461 2.464 0.959
Premium Expected Loss 51.71 51.71 103.4 51.24 50.3 101.5 1.875
Scaled EPD 51.17 52.25 103.4 51.16 50.39 101.5 1.875
Scaled TVaR 51.22 52.2 103.4 51.23 50.32 101.5 1.875
Scaled VaR 51.22 52.2 103.4 51.23 50.32 101.5 1.875
Equal Risk EPD 51.29 52.13 103.4 51.34 50.21 101.5 1.875
Equal Risk TVaR 51.34 52.08 103.4 51.39 50.16 101.5 1.875
Equal Risk VaR 51.34 52.08 103.4 51.39 50.16 101.5 1.875
coTVaR 50.88 52.54 103.4 51.39 50.16 101.5 1.875
Covar 51.05 52.37 103.4 51 50.55 101.5 1.875
Loss Ratio Expected Loss 0.967 0.967 0.967 0.976 0.976 0.976 0.488
Scaled EPD 0.977 0.957 0.967 0.977 0.974 0.976 0.488
Scaled TVaR 0.976 0.958 0.967 0.976 0.975 0.976 0.488
Scaled VaR 0.976 0.958 0.967 0.976 0.975 0.976 0.488
Equal Risk EPD 0.975 0.959 0.967 0.974 0.978 0.976 0.488
Equal Risk TVaR 0.974 0.96 0.967 0.973 0.979 0.976 0.488
Equal Risk VaR 0.974 0.96 0.967 0.973 0.979 0.976 0.488
coTVaR 0.983 0.952 0.967 0.973 0.979 0.976 0.488
Covar 0.979 0.955 0.967 0.98 0.971 0.976 0.488
Capital Expected Loss 17.12 17.12 34.23 12.43 12.21 24.64 9.594
Scaled EPD 11.71 22.52 34.23 11.6 13.04 24.64 9.594
Scaled TVaR 12.21 22.02 34.23 12.27 12.37 24.64 9.594
Scaled VaR 12.21 22.02 34.23 12.25 12.39 24.64 9.594
Equal Risk EPD 12.89 21.34 34.23 13.38 11.26 24.64 9.594
Equal Risk TVaR 13.4 20.84 34.23 13.89 10.75 24.64 9.594
Equal Risk VaR 13.39 20.84 34.23 13.89 10.75 24.64 9.594
coTVaR 8.827 25.42 34.25 13.9 10.75 24.65 9.593
Covar 10.53 23.7 34.23 10.03 14.61 24.64 9.594
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 3.021 3.021 3.021 4.121 4.121 4.121 0.195
Scaled EPD 4.37 2.32 3.021 4.409 3.865 4.121 0.195
Scaled TVaR 4.195 2.37 3.021 4.175 4.069 4.121 0.195
Scaled VaR 4.196 2.37 3.021 4.18 4.063 4.121 0.195
Equal Risk EPD 3.978 2.443 3.021 3.837 4.459 4.121 0.195
Equal Risk TVaR 3.833 2.5 3.021 3.699 4.668 4.121 0.195
Equal Risk VaR 3.833 2.499 3.021 3.699 4.667 4.121 0.195
coTVaR 5.764 2.067 3.02 3.696 4.667 4.119 0.195
Covar 4.847 2.21 3.021 5.087 3.459 4.121 0.195
Assets Expected Loss 68.83 68.83 137.7 63.68 62.51 126.2 11.47
Scaled EPD 62.88 74.78 137.7 62.76 63.42 126.2 11.47
Scaled TVaR 63.43 74.23 137.7 63.5 62.69 126.2 11.47
Scaled VaR 63.43 74.23 137.7 63.48 62.71 126.2 11.47
Equal Risk EPD 64.18 73.47 137.7 64.72 61.47 126.2 11.47
Equal Risk TVaR 64.74 72.92 137.7 65.28 60.9 126.2 11.47
Equal Risk VaR 64.73 72.92 137.7 65.28 60.91 126.2 11.47
coTVaR 59.71 77.96 137.7 65.29 60.91 126.2 11.47
Covar 61.59 76.07 137.7 61.03 65.16 126.2 11.47

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: Tame Case Study, gross twelve plot with roe distortion.
Figure Figure(1080x1200)
(TN) Figure 15.3: Tame Case Study, net twelve plot with tvar distortion.
Figure Figure(900x325)
(U) Figure 15.8: Tame Case Study, capital density for Tame Case Study, with roe gross and Tail VaR, 0.227 net distortion.
(V) Table 13.35: Constant 0.10 ROE pricing for Tame Case Study, distortion, SRM methods.
    Gross Net Ceded
  line A B Total A B Total Diff
stat Method              
Loss Expected Loss 50.00 50.00 100.00 50.00 49.08 99.08 0.92
Margin Expected Loss 1.71 1.71 3.42 1.24 1.22 2.46 0.96
Dist ROE 0.23 3.20 3.42 0.18 2.28 2.46 0.96
Dist PH 1.00 2.42 3.42 1.37 1.39 2.76 0.66
Dist Wang 1.04 2.39 3.42 1.28 1.63 2.90 0.52
Dist Dual 1.07 2.36 3.42 1.18 1.85 3.03 0.39
Dist Tvar 1.10 2.33 3.42 1.10 2.05 3.15 0.27
Dist Blend 0.88 2.55 3.44 1.33 1.23 2.56 0.88
Premium Expected Loss 51.71 51.71 103.42 51.24 50.30 101.55 1.87
Dist ROE 50.23 53.20 103.42 50.18 51.37 101.55 1.87
Dist PH 51.00 52.42 103.42 51.37 50.48 101.85 1.57
Dist Wang 51.04 52.39 103.42 51.28 50.71 101.99 1.44
Dist Dual 51.07 52.35 103.42 51.18 50.93 102.11 1.31
Dist Tvar 51.10 52.33 103.42 51.10 51.14 102.24 1.18
Dist Blend 50.88 52.55 103.44 51.33 50.31 101.65 1.79
Loss Ratio Expected Loss 0.97 0.97 0.97 0.98 0.98 0.98 0.49
Dist ROE 1.00 0.94 0.97 1.00 0.96 0.98 0.49
Dist PH 0.98 0.95 0.97 0.97 0.97 0.97 0.58
Dist Wang 0.98 0.95 0.97 0.98 0.97 0.97 0.64
Dist Dual 0.98 0.96 0.97 0.98 0.96 0.97 0.70
Dist Tvar 0.98 0.96 0.97 0.98 0.96 0.97 0.77
Dist Blend 0.98 0.95 0.97 0.97 0.98 0.97 0.51
Capital Expected Loss 17.12 17.12 34.23 12.43 12.21 24.64 9.59
Dist ROE 2.25 31.98 34.23 1.82 22.82 24.64 9.59
Dist PH 14.19 20.04 34.23 12.46 11.88 24.34 9.89
Dist Wang 15.46 18.78 34.23 12.43 11.77 24.20 10.03
Dist Dual 15.63 18.60 34.23 12.37 11.70 24.07 10.16
Dist Tvar 15.66 18.57 34.23 12.31 11.64 23.95 10.28
Dist Blend 10.90 23.32 34.22 12.58 11.96 24.54 9.68
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 0.07 0.12 0.10 0.11 0.12 0.11 0.07
Dist Wang 0.07 0.13 0.10 0.10 0.14 0.12 0.05
Dist Dual 0.07 0.13 0.10 0.10 0.16 0.13 0.04
Dist Tvar 0.07 0.13 0.10 0.09 0.18 0.13 0.03
Dist Blend 0.08 0.11 0.10 0.11 0.10 0.10 0.09
Leverage Expected Loss 3.02 3.02 3.02 4.12 4.12 4.12 0.20
Dist ROE 22.34 1.66 3.02 27.64 2.25 4.12 0.20
Dist PH 3.59 2.62 3.02 4.12 4.25 4.18 0.16
Dist Wang 3.30 2.79 3.02 4.12 4.31 4.21 0.14
Dist Dual 3.27 2.81 3.02 4.14 4.35 4.24 0.13
Dist Tvar 3.26 2.82 3.02 4.15 4.39 4.27 0.12
Dist Blend 4.67 2.25 3.02 4.08 4.21 4.14 0.19
Assets Expected Loss 68.83 68.83 137.66 63.68 62.51 126.19 11.47
Dist ROE 52.47 85.18 137.66 52.00 74.18 126.19 11.47
Dist PH 65.19 72.46 137.66 63.83 62.35 126.19 11.47
Dist Wang 66.49 71.16 137.66 63.71 62.48 126.19 11.47
Dist Dual 66.70 70.96 137.66 63.55 62.63 126.19 11.47
Dist Tvar 66.76 70.90 137.66 63.41 62.77 126.19 11.47
Dist Blend 61.78 75.87 137.66 63.92 62.27 126.19 11.47
Figure Figure(900x975)
(W) Figure 15.11: Tame 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: Tame Case Study, percentile layer of capital allocations by asset level, showing 0.9999 capital. (Same distortions.)
(Y) Table 15.38: Tame Case Study percentile layer of capital allocations compared to distortion allocations.
  Gross Net Ceded
line A B Total A B Total Diff
Method              
Expected Loss 68.83 68.83 137.7 63.68 62.51 126.2 11.47
Dist ROE 52.47 85.18 137.7 52 74.18 126.2 11.47
Dist PH 65.19 72.46 137.7 63.83 62.35 126.2 11.47
Dist Wang 66.49 71.16 137.7 63.71 62.48 126.2 11.47
Dist Dual 66.7 70.96 137.7 63.55 62.63 126.2 11.47
Dist Tvar 66.76 70.9 137.7 63.41 62.77 126.2 11.47
Dist Blend 61.78 75.87 137.7 63.92 62.27 126.2 11.47
PLC 67.34 70.32 137.7 63.89 62.3 126.2 11.47

Created 2023-03-23 15:26:03.395913

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.