Hurricane/Severe Convective Storm Case Study Results

Case Study Description

Book Hu/SCS Case, two thick-tailed units.

Contents

Chapter 2

The Insurance Market and Our Case Studies.

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

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(A) Table 2.7: Hu/SCS Case Study estimated mean, CV, skewness and kurtosis by line and in total, gross and net. Aggregate* reinsurance applied to HU with an attachment probability 0.05 (¤ 0) and detachment probability 0.005 (¤ 0).
view Gross Net
line Hu SCS Total Hu SCS Total
statistic            
Mean 29.727 69.133 98.860 29.727 69.133 98.860
CV 10.923 0.736 3.324 10.923 0.736 3.324
Skewness 121 24.900 116 121 24.900 116
Kurtosis 27 9 26 27 9 26
Figure Figure(900x650)
(B) Figure 2.2: Hu/SCS Case Study, gross (top) and net (bottom) densities on a nominal (left) and log (right) scale.
Figure Figure(900x300)
(C) Figure 2.3: Hu/SCS 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: Hu/SCS 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 4.8: Hu/SCS 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 HU with an attachment probability 0.05 (¤ 0) and detachment probability 0.005 (¤ 0).
view Gross Net
line Hu SCS Benefit Sum Total Hu SCS Benefit Sum Total
statistic                    
VaR 90.0 44 111 0.0283 154 150 44 111 0.0283 154 150
VaR 95.0 95 139 0.113 234 211 95 139 0.113 234 211
VaR 97.5 190 174 0.181 364 308 190 174 0.181 364 308
VaR 99.0 428 237 0.223 665 544 428 237 0.223 665 544
VaR 99.6 899 326 0.217 1,225 1,007 899 326 0.217 1,225 1,007
VaR 99.9 2,488 541 0.173 3,029 2,583 2,488 541 0.173 3,029 2,583
TVaR 90.0 247 167 0.149 414 360 247 167 0.149 414 360
TVaR 95.0 430 211 0.175 641 546 430 211 0.175 641 546
TVaR 97.5 728 267 0.183 995 841 728 267 0.183 995 841
TVaR 99.0 1,404 369 0.173 1,773 1,511 1,404 369 0.173 1,773 1,511
TVaR 99.6 2,604 514 0.153 3,119 2,704 2,604 514 0.153 3,119 2,704
TVaR 99.9 6,209 856 0.122 7,065 6,298 6,209 856 0.122 7,065 6,298
EPD 10.0 3,438 100 6 3,538 523 3,438 100 6 3,538 523
EPD 5.0 8,140 141 4 8,282 1,684 8,140 141 4 8,282 1,684
EPD 2.5 16,268 204 3 16,473 4,491 16,268 204 3 16,473 4,491
EPD 1.0 32,660 342 2 33,003 12,538 32,660 342 2 33,003 12,538
EPD 0.4 53,156 583 0.99 53,739 27,010 53,156 583 0.99 53,739 27,010
EPD 0.1 82,900 1,278 0.465 84,178 57,464 82,900 1,278 0.465 84,178 57,464

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 Hu/SCS Case Study. Pricing summary by Case. Tame Case uses a 0.9999 capital standard; Cat/Non-Cat and HU/SCS use 0.999. Cost of capital is 0.10.
portfolio Gross Net
stat    
Loss 95.15 95.15
Margin 226.2 226.2
Premium 321.3 321.3
Loss Ratio 0.296 0.296
Capital 2,262 2,262
Rate of Return 0.1 0.1
Assets 2,583 2,583
Leverage 0.142 0.142
(G) Table 7.3: Reinsurance summary for Hu/SCS Case Study.
  HS
item  
Reinsured Line HU
Reinsurance Type Aggregate*
Attachment Probability 0.05
Attachment 0
Exhaustion Probability 0.005
Limit 0

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 9.8: Classical pricing by method for Hu/SCS Case Study. Pricing calibrated to total gross portfolio and applied to each line on a stand-alone basis. Sorted by gross premium for HU.
  Parameters Hu SCS Total
  Value Net Gross Gross Net Gross Ceded
method              
Net 29.727 29.727 69.133 98.860 98.860 -0.000
Expected Value 2.250 96.619 96.619 224.695 321.314 321.314 -0.000
Dutch 7.635 197.224 197.224 175.181 321.314 321.314 -0.000
VaR 0.977 203.000 203.000 178.500 321.500 321.250 -0.250
Variance 0.002 246.893 246.893 74.462 321.314 321.314 -0.000
Semi-Variance 0.002 248.496 248.496 73.705 321.314 321.314 -0.000
Standard Deviation 0.677 249.521 249.521 103.562 321.314 321.314 -0.000
Fischer 0.681 250.331 250.331 101.024 321.314 321.314 -0.000
(I) Table 9.9: Sum of parts (SoP) stand-alone vs. diversified classical pricing by method for Hu/SCS Case Study. Delta columns show the difference.
  Total SoP Delta
  Gross Net Gross Net Gross Net
method            
Net 98.860 98.860 98.860 98.860 0.000 0.000
Expected Value 321.314 321.314 321.314 321.314 0.000 0.000
Dutch 321.314 321.314 372.405 372.405 51.091 51.091
VaR 321.250 321.500 381.500 381.500 60.250 60.000
Variance 321.314 321.314 321.354 321.354 0.040 0.040
Semi-Variance 321.314 321.314 322.201 322.201 0.887 0.887
Standard Deviation 321.314 321.314 353.083 353.083 31.769 31.769
Fischer 321.314 321.314 351.355 351.355 30.041 30.041
(J) Table 9.1: Implied loss ratios from classical pricing by method for Hu/SCS Case Study. Pricing calibrated to total gross portfolio and applied to each line on a stand-alone basis.
  Hu SCS Total
  Net Gross Gross Net Gross Ceded
method            
Net 1 1 1 1 1 1
Expected Value 0.308 0.308 0.308 0.308 0.308 0.308
Dutch 0.151 0.151 0.395 0.308 0.308 0.149
VaR 0.146 0.146 0.387 0.307 0.308 1.49e-09
Variance 0.12 0.12 0.928 0.308 0.308 0.00505
Semi-Variance 0.12 0.12 0.938 0.308 0.308 0.00499
Standard Deviation 0.119 0.119 0.668 0.308 0.308 0.0101
Fischer 0.119 0.119 0.684 0.308 0.308 0.00993
(K) Table 9.11: Comparison of stand-alone and sum of parts (SoP) premium for Hu/SCS Case Study.
    Gross SoP Gross Total Gross Redn Net SoP Net Total Net Redn
method statistic            
No Default Loss 98.86 98.86 -0.0% 98.86 98.86 -0.0%
Premium 365.2 324.7 -11.1% 365.2 324.7 -11.1%
Capital 2664 2258 -15.2% 2664 2258 -15.2%
With Default Loss 94.82 95.15 0.3% 94.82 95.15 0.3%
Premium 361.5 321.3 -11.1% 361.5 321.3 -11.1%
Capital 2667 2262 -15.2% 2667 2262 -15.2%
(L) Table 9.14: Constant CoC pricing by unit for Hu/SCS Case Study, with 0.1 cost of capital and $p=0.999$. The column sop shows the sum by unit. ¤0.0 excess ¤0.0 aggregate* reinsurance applied to HU. All units produce the same rate of return, by construction.
  portfolio Gross Net
  line Hu SCS SoP Total Hu SoP Total
method statistic              
No Default Loss 29.73 69.13 98.86 98.86 29.73 98.86 98.86
Margin 223.4 42.92 266.4 225.8 223.4 266.4 225.8
Premium 253.2 112.1 365.2 324.7 253.2 365.2 324.7
Loss Ratio 0.117 0.617 0.271 0.304 0.117 0.271 0.304
Capital 2,234 429.2 2,664 2,258 2,234 2,664 2,258
Rate of Return 0.1 0.1 0.1 0.1 0.1 0.1 0.1
Leverage 0.113 0.261 0.137 0.144 0.113 0.137 0.144
Assets 2,488 541.2 3,029 2,583 2,488 3,029 2,583
With Default Loss 26.01 68.82 94.82 95.15 26.01 94.82 95.15
Margin 223.8 42.95 266.7 226.2 223.8 266.7 226.2
Premium 249.8 111.8 361.5 321.3 249.8 361.5 321.3
Loss Ratio 0.104 0.616 0.262 0.296 0.104 0.262 0.296
Capital 2,238 429.5 2,667 2,262 2,238 2,667 2,262
Rate of Return 0.1 0.1 0.1 0.1 0.1 0.1 0.1
Leverage 0.112 0.26 0.136 0.142 0.112 0.136 0.142
Assets 2,488 541.2 3,029 2,583 2,488 3,029 2,583

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 Hu/SCS 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 0 321.3 2,262 0.1 999.907u
PH 0.449 5.417u 321.3 2,262 0.1 999.907u
Wang 1.19 28.820p 321.3 2,262 0.1 999.907u
Dual 12.03 -1.197u 321.3 2,262 0.1 999.907u
Tvar 0.899 6.497u 321.3 2,262 0.1 999.907u
Figure Figure(750x1000)
(O) Figure 11.6: Hu/SCS 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(1000x1000)
(P) Figure 11.9: Hu/SCS 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, Dual, and TVaR.
(Q) Table 11.9: Traditional and stand alone Pricing by distortion. Pricing by unit and distortion for Hu/SCS 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. ¤0.0 excess ¤0.0 aggregate* reinsurance applied to HU.
  portfolio Gross Net
  line Hu SCS SoP Total Hu SoP Total
statistic distortion              
Loss CCoC 26.01 68.82 94.82 95.15 26.01 94.82 95.15
Margin CCoC 223.8 42.95 266.7 226.2 223.8 266.7 226.2
PH 210.0 61.16 271.2 226.2 210.0 271.2 226.2
Wang 199.8 72.46 272.2 226.2 199.8 272.2 226.2
Dual 186.2 89.16 275.3 226.2 186.2 275.3 226.2
TVaR 182.1 94.28 276.4 226.2 182.1 276.4 226.2
Blend 63.72 16.28 80.01 67.2 63.72 80.01 67.2
Premium CCoC 249.8 111.8 361.5 321.3 249.8 361.5 321.3
PH 236.0 130.0 366.0 321.3 236.0 366.0 321.3
Wang 225.8 141.3 367.0 321.3 225.8 367.0 321.3
Dual 212.2 158.0 370.1 321.3 212.2 370.1 321.3
TVaR 208.1 163.1 371.2 321.3 208.1 371.2 321.3
Blend 89.73 85.1 174.8 162.3 89.73 174.8 162.3
Loss Ratio CCoC 0.104 0.616 0.262 0.296 0.104 0.262 0.296
PH 0.11 0.529 0.259 0.296 0.11 0.259 0.296
Wang 0.115 0.487 0.258 0.296 0.115 0.258 0.296
Dual 0.123 0.436 0.256 0.296 0.123 0.256 0.296
TVaR 0.125 0.422 0.255 0.296 0.125 0.255 0.296
Blend 0.29 0.809 0.542 0.586 0.29 0.542 0.586
Capital CCoC 2,238 429.5 2,667 2,262 2,238 2,667 2,262
PH 2,251 411.3 2,663 2,262 2,251 2,663 2,262
Wang 2,262 400.0 2,662 2,262 2,262 2,662 2,262
Dual 2,275 383.3 2,659 2,262 2,275 2,659 2,262
TVaR 2,279 378.2 2,658 2,262 2,279 2,658 2,262
Blend 2,398 456.1 2,854 2,421 2,398 2,854 2,421
Rate of Return CCoC 0.1 0.1 0.1 0.1 0.1 0.1 0.1
PH 0.0933 0.149 0.102 0.1 0.0933 0.102 0.1
Wang 0.0883 0.181 0.102 0.1 0.0883 0.102 0.1
Dual 0.0818 0.233 0.104 0.1 0.0818 0.104 0.1
TVaR 0.0799 0.249 0.104 0.1 0.0799 0.104 0.1
Blend 0.0266 0.0357 0.028 0.0278 0.0266 0.028 0.0278
Leverage CCoC 0.112 0.26 0.136 0.142 0.112 0.136 0.142
PH 0.105 0.316 0.137 0.142 0.105 0.137 0.142
Wang 0.0998 0.353 0.138 0.142 0.0998 0.138 0.142
Dual 0.0932 0.412 0.139 0.142 0.0932 0.139 0.142
TVaR 0.0913 0.431 0.14 0.142 0.0913 0.14 0.142
Blend 0.0374 0.187 0.0613 0.0671 0.0374 0.0613 0.0671
Assets CCoC 2,488 541.2 3,029 2,583 2,488 3,029 2,583

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      
SCS 541.2 69.09 68.82
HU 2,488 26.06 26.01
Total 2,583 95.15 95.15
SoP 3,029 95.15 94.82
(S) Table 13.4: Constant 0.10 ROE pricing for Hu/SCS Case Study, classical PCP methods.
    Gross Net Ceded
  line Hu SCS Total Hu SCS Total Diff
stat Method              
Loss Expected Loss 26.06 69.09 95.15 26.06 69.09 95.15 -15.163p
Margin Expected Loss 61.94 164.2 226.2 61.94 164.2 226.2 14.694p
Scaled EPD 229.0 -2.839 226.2 229.0 -2.839 226.2 14.694p
Scaled TVaR 193.0 33.13 226.2 193.0 33.13 226.2 14.751p
Scaled VaR 190.5 35.68 226.2 190.5 35.68 226.2 14.694p
Equal Risk EPD 224.4 1.81 226.2 224.4 1.81 226.2 14.694p
Equal Risk TVaR 190.7 35.49 226.2 190.7 35.49 226.2 14.751p
Equal Risk VaR 187.6 38.61 226.2 187.6 38.61 226.2 14.694p
coTVaR 219.2 7.015 226.2 219.2 7.015 226.2 -7.561n
Covar 220.8 5.417 226.2 220.8 5.417 226.2 14.637p
Premium Expected Loss 88 233.3 321.3 88 233.3 321.3 -454.747f
Scaled EPD 255.1 66.25 321.3 255.1 66.25 321.3 -454.747f
Scaled TVaR 219.1 102.2 321.3 219.1 102.2 321.3 -397.904f
Scaled VaR 216.5 104.8 321.3 216.5 104.8 321.3 -454.747f
Equal Risk EPD 250.4 70.9 321.3 250.4 70.9 321.3 -454.747f
Equal Risk TVaR 216.7 104.6 321.3 216.7 104.6 321.3 -397.904f
Equal Risk VaR 213.6 107.7 321.3 213.6 107.7 321.3 -454.747f
coTVaR 245.2 76.1 321.3 245.2 76.1 321.3 -7.576n
Covar 246.9 74.5 321.3 246.9 74.5 321.3 -511.591f
Loss Ratio Expected Loss 0.296 0.296 0.296 0.296 0.296 0.296 33.34
Scaled EPD 0.102 1.043 0.296 0.102 1.043 0.296 33.34
Scaled TVaR 0.119 0.676 0.296 0.119 0.676 0.296 38.11
Scaled VaR 0.12 0.659 0.296 0.12 0.659 0.296 33.34
Equal Risk EPD 0.104 0.974 0.296 0.104 0.974 0.296 33.34
Equal Risk TVaR 0.12 0.661 0.296 0.12 0.661 0.296 38.11
Equal Risk VaR 0.122 0.642 0.296 0.122 0.642 0.296 33.34
coTVaR 0.106 0.908 0.296 0.106 0.908 0.296 0.002
Covar 0.106 0.927 0.296 0.106 0.927 0.296 29.64
Capital Expected Loss 619.4 1,642 2,262 619.4 1,642 2,262 454.747f
Scaled EPD 2,290 -28.39 2,262 2,290 -28.39 2,262 454.747f
Scaled TVaR 1,930 331.3 2,262 1,930 331.3 2,262 1.819p
Scaled VaR 1,905 356.8 2,262 1,905 356.8 2,262 454.747f
Equal Risk EPD 2,244 18.1 2,262 2,244 18.1 2,262 454.747f
Equal Risk TVaR 1,907 354.9 2,262 1,907 354.9 2,262 1.819p
Equal Risk VaR 1,876 386.1 2,262 1,876 386.1 2,262 454.747f
coTVaR 2,192 70.15 2,262 2,192 70.15 2,262 -75.758n
Covar 2,208 54.17 2,262 2,208 54.17 2,262 0
Rate of Return Expected Loss 0.1 0.1 0.1 0.1 0.1 0.1 32.31
Scaled EPD 0.1 0.1 0.1 0.1 0.1 0.1 32.31
Scaled TVaR 0.1 0.1 0.1 0.1 0.1 0.1 8.109
Scaled VaR 0.1 0.1 0.1 0.1 0.1 0.1 32.31
Equal Risk EPD 0.1 0.1 0.1 0.1 0.1 0.1 32.31
Equal Risk TVaR 0.1 0.1 0.1 0.1 0.1 0.1 8.109
Equal Risk VaR 0.1 0.1 0.1 0.1 0.1 0.1 32.31
coTVaR 0.1 0.1 0.1 0.1 0.1 0.1 0.0998
Covar 0.1 0.1 0.1 0.1 0.1 0.1 inf
Leverage Expected Loss 0.142 0.142 0.142 0.142 0.142 0.142 -1
Scaled EPD 0.111 -2.333 0.142 0.111 -2.333 0.142 -1
Scaled TVaR 0.113 0.309 0.142 0.113 0.309 0.142 -0.219
Scaled VaR 0.114 0.294 0.142 0.114 0.294 0.142 -1
Equal Risk EPD 0.112 3.918 0.142 0.112 3.918 0.142 -1
Equal Risk TVaR 0.114 0.295 0.142 0.114 0.295 0.142 -0.219
Equal Risk VaR 0.114 0.279 0.142 0.114 0.279 0.142 -1
coTVaR 0.112 1.085 0.142 0.112 1.085 0.142 0.1
Covar 0.112 1.375 0.142 0.112 1.375 0.142 inf
Assets Expected Loss 707.4 1,876 2,583 707.4 1,876 2,583 0
Scaled EPD 2,545 37.86 2,583 2,545 37.86 2,583 0
Scaled TVaR 2,149 433.5 2,583 2,149 433.5 2,583 1.364p
Scaled VaR 2,121 461.6 2,583 2,121 461.6 2,583 0
Equal Risk EPD 2,494 88.99 2,583 2,494 88.99 2,583 0
Equal Risk TVaR 2,123 459.5 2,583 2,123 459.5 2,583 1.364p
Equal Risk VaR 2,089 493.8 2,583 2,089 493.8 2,583 0
coTVaR 2,437 146.2 2,583 2,437 146.2 2,583 -83.334n
Covar 2,455 128.7 2,583 2,455 128.7 2,583 -454.747f

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: Hu/SCS Case Study, gross twelve plot with blend distortion.
Figure Figure(1080x1200)
(TN) Figure 15.3: Hu/SCS Case Study, net twelve plot with wang distortion.
Figure Figure(900x325)
(U) Figure 15.8: Hu/SCS Case Study, capital density for Hu/SCS Case Study, with blend gross and Wang-normal, 1.190 net distortion.
(V) Table 13.37: Constant 0.10 ROE pricing for Hu/SCS Case Study, distortion, SRM methods.
    Gross Net Ceded
  line Hu SCS Total Hu SCS Total Diff
stat Method              
Loss Expected Loss 26.06 69.09 95.15 26.06 69.09 95.15 -0.00
Margin Expected Loss 61.94 164.23 226.17 61.94 164.23 226.17 0.00
Dist ROE 232.31 -6.16 226.17 232.31 -6.16 226.17 0.00
Dist PH 203.64 22.53 226.17 203.64 22.53 226.17 0.00
Dist Wang 188.25 37.92 226.17 188.25 37.92 226.17 0.00
Dist Dual 166.49 59.68 226.17 166.49 59.68 226.17 0.00
Dist Tvar 159.50 66.67 226.17 159.50 66.67 226.17 0.00
Dist Blend 64.00 3.20 67.20 64.00 3.20 67.20 0.00
Premium Expected Loss 88.00 233.31 321.31 88.00 233.31 321.31 -0.00
Dist ROE 258.37 62.93 321.31 258.37 62.93 321.31 -0.00
Dist PH 229.70 91.61 321.31 229.70 91.61 321.31 -0.00
Dist Wang 214.31 107.00 321.31 214.31 107.00 321.31 -0.00
Dist Dual 192.55 128.77 321.31 192.55 128.77 321.31 -0.00
Dist Tvar 185.56 135.76 321.31 185.56 135.76 321.31 -0.00
Dist Blend 90.06 72.29 162.35 90.06 72.29 162.35 -0.00
Loss Ratio Expected Loss 0.30 0.30 0.30 0.30 0.30 0.30 33.34
Dist ROE 0.10 1.10 0.30 0.10 1.10 0.30 29.64
Dist PH 0.11 0.75 0.30 0.11 0.75 0.30 3.51
Dist Wang 0.12 0.65 0.30 0.12 0.65 0.30 3.25
Dist Dual 0.14 0.54 0.30 0.14 0.54 0.30 4.30
Dist Tvar 0.14 0.51 0.30 0.14 0.51 0.30 6.06
Dist Blend 0.29 0.96 0.59 0.29 0.96 0.59 33.34
Capital Expected Loss 619.43 1642.26 2261.69 619.43 1642.26 2261.69 0.00
Dist ROE 2322.94 -61.38 2261.69 2322.94 -61.38 2261.69 0.00
Dist PH 2111.29 150.39 2261.69 2111.29 150.39 2261.69 0.00
Dist Wang 2074.81 186.88 2261.69 2074.81 186.88 2261.69 0.00
Dist Dual 2049.78 211.91 2261.69 2049.78 211.91 2261.69 0.00
Dist Tvar 2051.69 210.00 2261.69 2051.69 210.00 2261.69 0.00
Dist Blend 2344.33 76.32 2420.65 2344.33 76.32 2420.65 0.00
Rate of Return Expected Loss 0.10 0.10 0.10 0.10 0.10 0.10 32.31
Dist ROE 0.10 0.10 0.10 0.10 0.10 0.10 32.19
Dist PH 0.10 0.15 0.10 0.10 0.15 0.10 2.38
Dist Wang 0.09 0.20 0.10 0.09 0.20 0.10 2.31
Dist Dual 0.08 0.28 0.10 0.08 0.28 0.10 3.65
Dist Tvar 0.08 0.32 0.10 0.08 0.32 0.10 4.64
Dist Blend 0.03 0.04 0.03 0.03 0.04 0.03 32.34
Leverage Expected Loss 0.14 0.14 0.14 0.14 0.14 0.14 -1.00
Dist ROE 0.11 -1.03 0.14 0.11 -1.03 0.14 -1.12
Dist PH 0.11 0.61 0.14 0.11 0.61 0.14 -0.95
Dist Wang 0.10 0.57 0.14 0.10 0.57 0.14 -1.02
Dist Dual 0.09 0.61 0.14 0.09 0.61 0.14 -1.11
Dist Tvar 0.09 0.65 0.14 0.09 0.65 0.14 -0.92
Dist Blend 0.04 0.95 0.07 0.04 0.95 0.07 -1.00
Assets Expected Loss 707.43 1875.57 2583.00 707.43 1875.57 2583.00 0.00
Dist ROE 2581.31 1.55 2583.00 2581.31 1.55 2583.00 0.00
Dist PH 2340.99 242.01 2583.00 2340.99 242.01 2583.00 0.00
Dist Wang 2289.12 293.88 2583.00 2289.12 293.88 2583.00 0.00
Dist Dual 2242.33 340.67 2583.00 2242.33 340.67 2583.00 0.00
Dist Tvar 2237.25 345.75 2583.00 2237.25 345.75 2583.00 0.00
Dist Blend 2434.39 148.61 2583.00 2434.39 148.61 2583.00 0.00
Figure Figure(900x975)
(W) Figure 15.11: Hu/SCS 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: Hu/SCS Case Study, percentile layer of capital allocations by asset level, showing 0.999 capital. (Same distortions.)
(Y) Table 15.4: Hu/SCS Case Study percentile layer of capital allocations compared to distortion allocations.
  Gross Net Ceded
line Hu SCS Total Hu SCS Total Diff
Method              
Expected Loss 707.4 1876 2583 707.4 1876 2583 0
Dist ROE 2581 1.553 2583 2581 1.553 2583 0
Dist PH 2341 242 2583 2341 242 2583 0
Dist Wang 2289 293.9 2583 2289 293.9 2583 0
Dist Dual 2242 340.7 2583 2242 340.7 2583 0
Dist Tvar 2237 345.8 2583 2237 345.8 2583 0
Dist Blend 2434 148.6 2583 2434 148.6 2583 0
PLC 2209 373.8 2583 2209 373.8 2583 -1.847e-08

Created 2024-09-15 04:32:27.684515

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.