Discrete Case Study Results

Case Study Description

Book simple discrete example.

Contents

Chapter 2

The Insurance Market and Our Case Studies.

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

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(A) Table 2.5: Discrete Example estimated mean, CV, skewness and kurtosis by line and in total, gross and net. Aggregate reinsurance applied to X2 with an attachment probability 0.75 (¤ 20) and detachment probability 1.0 (¤ 100).
view Gross Net
line X1 X2 Total X1 X2 Total
statistic            
Mean 4.500 22.750 27.250 4.500 5.250 9.750
CV 1.012 1.707 1.435 1.012 1.624 0.991
Skewness 0.071 1.154 1.131 0.071 1.147 0.794
Kurtosis -1.905 -0.667 -0.649 -1.905 -0.673 -0.501
Figure Figure(900x650)
('(B) Figure (suppl.) 2.2: Discrete Example, gross and net densities by line and combined gross.',)
Figure Figure(900x300)
(C) Figure 2.3: Discrete Example, 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: Discrete Example, 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: Discrete Example 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 X2 with an attachment probability 0.75 (¤ 20) and detachment probability 1.0 (¤ 100).
view Gross Net
line X1 X2 Benefit Sum Total X1 X2 Benefit Sum Total
statistic                    
VaR 90.0 10 90 0.0204 100 98 10 20 0.0714 30 28
VaR 95.0 10 90 0 100 100 10 20 0 30 30
VaR 97.5 10 90 0 100 100 10 20 0 30 30
VaR 99.0 10 90 0 100 100 10 20 0 30 30
VaR 99.6 10 90 0 100 100 10 20 0 30 30
VaR 99.9 10 90 0 100 100 10 20 0 30 30
TVaR 90.0 10 90 0.00604 100 99 10 20 0.0204 30 29
TVaR 95.0 10 90 0 100 100 10 20 0 30 30
TVaR 97.5 10 90 0 100 100 10 20 0 30 30
TVaR 99.0 10 90 0 100 100 10 20 0 30 30
TVaR 99.6 10 90 0 100 100 10 20 0 30 30
TVaR 99.9 10 90 0 100 100 10 20 0 30 30
EPD 10.0 8 81 0.0658 89 84 8 18 0.231 26 21
EPD 5.0 9 85 0.0618 95 89 9 19 0.118 28 25
EPD 2.5 10 88 0.0398 97 94 10 19 0.073 29 27
EPD 1.0 10 89 0.0216 99 97 10 20 0.0411 30 28
EPD 0.4 10 90 0.0133 100 98 10 20 0.0159 30 29
EPD 0.1 10 90 0.00328 100 100 10 20 0.00392 30 30

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 Discrete Example. 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 27.25 9.75
Margin 23.89 6.65
Premium 51.14 16.4
Loss Ratio 0.533 0.595
Capital 48.86 13.6
Rate of Return 0.489 0.489
Assets 100 30
Leverage 1.047 1.206
(G) Table 7.3: Reinsurance summary for Discrete Example.
  Discrete
item  
Reinsured Line X2
Reinsurance Type Aggregate
Attachment Probability 0.75
Attachment 20
Exhaustion Probability 1
Limit 80

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 Discrete Example. Pricing calibrated to total gross portfolio and applied to each line on a stand-alone basis. Sorted by gross premium for X2.
  Parameters X1 X2 Total
  Value Gross Net Gross Net Gross Ceded
method              
Net 4.500 5.250 22.750 9.750 27.250 17.500
VaR 0.750 8.000 1.000 1.000 11.000 11.000 0.000
Expected Value 0.877 8.445 9.853 42.696 18.298 51.142 32.843
Variance 0.016 4.824 6.386 46.317 11.211 51.142 39.931
Esscher 0.013 4.767 6.241 46.375 11.008 51.142 40.133
Standard Deviation 0.611 7.284 10.460 46.479 15.657 51.142 35.484
Semi-Variance 0.021 4.723 6.394 46.533 11.005 51.142 40.136
Fischer 0.709 6.811 10.478 46.587 15.227 51.142 35.915
Dutch 1.421 7.697 10.490 46.642 15.146 51.142 35.996
(I) Table 9.3: Sum of parts (SoP) stand-alone vs. diversified classical pricing by method for Discrete Example. Delta columns show the difference.
  Total SoP Delta
  Gross Net Gross Net Gross Net
method            
Net 27.250 9.750 27.250 9.750 0.000 0.000
VaR 11.000 11.000 9.000 9.000 -2.000 -2.000
Expected Value 51.142 18.298 51.142 18.298 0.000 0.000
Variance 51.142 11.211 51.142 11.211 0.000 -0.000
Esscher 51.142 11.008 51.142 11.008 0.000 0.000
Standard Deviation 51.142 15.657 53.763 17.744 2.621 2.087
Semi-Variance 51.142 11.005 51.256 11.118 0.114 0.112
Fischer 51.142 15.227 53.398 17.289 2.256 2.062
Dutch 51.142 15.146 54.339 18.188 3.197 3.042
(J) Table 9.4: Implied loss ratios from classical pricing by method for Discrete Example. Pricing calibrated to total gross portfolio and applied to each line on a stand-alone basis.
  X1 X2 Total
  Gross Net Gross Net Gross Ceded
method            
Net 1 1 1 1 1 1
VaR 0.562 5.25 22.8 0.886 2.48 inf
Expected Value 0.533 0.533 0.533 0.533 0.533 0.533
Variance 0.933 0.822 0.491 0.87 0.533 0.438
Esscher 0.944 0.841 0.491 0.886 0.533 0.436
Standard Deviation 0.618 0.502 0.489 0.623 0.533 0.493
Semi-Variance 0.953 0.821 0.489 0.886 0.533 0.436
Fischer 0.661 0.501 0.488 0.64 0.533 0.487
Dutch 0.585 0.5 0.488 0.644 0.533 0.486
(K) Table 9.11: Comparison of stand-alone and sum of parts (SoP) premium for Discrete Example.
    Gross SoP Gross Total Gross Redn Net SoP Net Total Net Redn
method statistic            
No Default Loss 27.25 27.25 0.0% 975.0% 975.0% 0.0%
Premium 51.14 51.14 -0.0% 16.4 16.4 0.0%
Capital 48.86 48.86 0.0% 13.6 13.6 0.0%
With Default Loss 27.25 27.25 0.0% 975.0% 975.0% 0.0%
Premium 51.14 51.14 -0.0% 16.4 16.4 0.0%
Capital 48.86 48.86 0.0% 13.6 13.6 0.0%
(L) Table 9.12: Constant CoC pricing by unit for Discrete Example, with 0.489 cost of capital and $p=1.0$. The column sop shows the sum by unit. ¤80.0 excess ¤20.0 aggregate reinsurance applied to X2. All units produce the same rate of return, by construction.
  portfolio Gross Net
  line X1 X2 SoP Total X1 SoP Total
method statistic              
No Default Loss 4.5 22.75 27.25 27.25 4.5 9.75 9.75
Margin 1.806 22.09 23.89 23.89 1.806 6.65 6.65
Premium 6.306 44.84 51.14 51.14 6.306 16.4 16.4
Loss Ratio 0.714 0.507 0.533 0.533 0.714 0.595 0.595
Capital 3.694 45.16 48.86 48.86 3.694 13.6 13.6
Rate of Return 0.489 0.489 0.489 0.489 0.489 0.489 0.489
Leverage 1.707 0.993 1.047 1.047 1.707 1.206 1.206
Assets 10 90 100 100 10 30 30
With Default Loss 4.5 22.75 27.25 27.25 4.5 9.75 9.75
Margin 1.806 22.09 23.89 23.89 1.806 6.65 6.65
Premium 6.306 44.84 51.14 51.14 6.306 16.4 16.4
Loss Ratio 0.714 0.507 0.533 0.533 0.714 0.595 0.595
Capital 3.694 45.16 48.86 48.86 3.694 13.6 13.6
Rate of Return 0.489 0.489 0.489 0.489 0.489 0.489 0.489
Leverage 1.707 0.993 1.047 1.047 1.707 1.206 1.206
Assets 10 90 100 100 10 30 30

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 Discrete Example, 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 0.489 0 51.14 48.86 0.489 0
PH 0.5 4.016u 51.14 48.86 0.489 0
Wang 0.672 -7.568u 51.14 48.86 0.489 0
Dual 2.386 -18.071n 51.14 48.86 0.489 0
Tvar 0.487 -5.700u 51.14 48.86 0.489 0
Figure Figure(750x1000)
(O) Figure 11.6: Discrete Example, 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: Discrete Example, 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 11.7: Traditional and stand alone Pricing by distortion. Pricing by unit and distortion for Discrete Example, calibrated to CCoC pricing with 0.489 cost of capital and $p=1.0$. 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. ¤80.0 excess ¤20.0 aggregate reinsurance applied to X2.
  portfolio Gross Net
  line X1 X2 SoP Total X1 SoP Total
statistic distortion              
Loss CCoC 4.5 22.75 27.25 27.25 4.5 9.75 9.75
Margin CCoC 1.806 22.09 23.89 23.89 1.806 6.65 6.65
PH 2.156 22.44 24.6 23.89 2.156 7.11 6.402
Wang 2.492 22.41 24.9 23.89 2.492 7.472 6.462
Dual 2.963 22.26 25.23 23.89 2.963 7.959 6.625
TVaR 4.275 21.61 25.88 23.89 4.275 9.262 7.268
Blend 1.806 22.09 23.89 22.54 1.806 6.65 5.299
Premium CCoC 6.306 44.84 51.14 51.14 6.306 16.4 16.4
PH 6.656 45.19 51.85 51.14 6.656 16.86 16.15
Wang 6.992 45.16 52.15 51.14 6.992 17.22 16.21
Dual 7.463 45.01 52.48 51.14 7.463 17.71 16.37
TVaR 8.775 44.36 53.13 51.14 8.775 19.01 17.02
Blend 6.306 44.84 51.14 49.79 6.306 16.4 15.05
Loss Ratio CCoC 0.714 0.507 0.533 0.533 0.714 0.595 0.595
PH 0.676 0.503 0.526 0.533 0.676 0.578 0.604
Wang 0.644 0.504 0.523 0.533 0.644 0.566 0.601
Dual 0.603 0.505 0.519 0.533 0.603 0.551 0.595
TVaR 0.513 0.513 0.513 0.533 0.513 0.513 0.573
Blend 0.714 0.507 0.533 0.547 0.714 0.595 0.648
Capital CCoC 3.694 45.16 48.86 48.86 3.694 13.6 13.6
PH 3.344 44.81 48.15 48.86 3.344 13.14 13.85
Wang 3.008 44.84 47.85 48.86 3.008 12.78 13.79
Dual 2.537 44.99 47.52 48.86 2.537 12.29 13.63
TVaR 1.225 45.64 46.87 48.86 1.225 10.99 12.98
Blend 3.694 45.16 48.86 50.21 3.694 13.6 14.95
Rate of Return CCoC 0.489 0.489 0.489 0.489 0.489 0.489 0.489
PH 0.645 0.501 0.511 0.489 0.645 0.541 0.462
Wang 0.828 0.5 0.52 0.489 0.828 0.585 0.469
Dual 1.168 0.495 0.531 0.489 1.168 0.648 0.486
TVaR 3.488 0.474 0.552 0.489 3.488 0.843 0.56
Blend 0.489 0.489 0.489 0.449 0.489 0.489 0.354
Leverage CCoC 1.707 0.993 1.047 1.047 1.707 1.206 1.206
PH 1.99 1.009 1.077 1.047 1.99 1.283 1.166
Wang 2.324 1.007 1.09 1.047 2.324 1.348 1.176
Dual 2.942 1.001 1.104 1.047 2.942 1.441 1.202
TVaR 7.16 0.972 1.134 1.047 7.16 1.73 1.311
Blend 1.707 0.993 1.047 0.992 1.707 1.206 1.007
Assets CCoC 10 90 100 100 10 30 30

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 1.0-VaR assets.
  a E[Xi(a)] E[Xi ∧ ai]
Unit      
X1 10 4.5 4.5
X2 90 22.75 22.75
Total 100 27.25 27.25
SoP 100 27.25 27.25
(S) Table 13.2: Constant 0.49 ROE pricing for Discrete Example, classical PCP methods.
    Gross Net Ceded
  line X1 X2 Total X1 X2 Total Diff
stat Method              
Loss Expected Loss 4.5 22.75 27.25 4.5 5.25 9.75 17.5
Margin Expected Loss 3.945 19.95 23.89 3.069 3.581 6.65 17.24
Scaled EPD 1.806 22.09 23.89 1.806 4.844 6.65 17.24
Scaled TVaR 1.806 22.09 23.89 1.806 4.844 6.65 17.24
Scaled VaR 1.806 22.09 23.89 1.806 4.844 6.65 17.24
Equal Risk EPD 1.806 22.09 23.89 1.806 4.844 6.65 17.24
Equal Risk TVaR 1.806 22.09 23.89 1.806 4.844 6.65 17.24
Equal Risk VaR 1.806 22.09 23.89 1.806 4.844 6.65 17.24
coTVaR inf
Covar 0.324 23.57 23.89 1.477 5.173 6.65 17.24
Premium Expected Loss 8.445 42.7 51.14 7.569 8.831 16.4 34.74
Scaled EPD 6.306 44.84 51.14 6.306 10.09 16.4 34.74
Scaled TVaR 6.306 44.84 51.14 6.306 10.09 16.4 34.74
Scaled VaR 6.306 44.84 51.14 6.306 10.09 16.4 34.74
Equal Risk EPD 6.306 44.84 51.14 6.306 10.09 16.4 34.74
Equal Risk TVaR 6.306 44.84 51.14 6.306 10.09 16.4 34.74
Equal Risk VaR 6.306 44.84 51.14 6.306 10.09 16.4 34.74
coTVaR inf
Covar 4.824 46.32 51.14 5.977 10.42 16.4 34.74
Loss Ratio Expected Loss 0.533 0.533 0.533 0.595 0.595 0.595 0.504
Scaled EPD 0.714 0.507 0.533 0.714 0.52 0.595 0.504
Scaled TVaR 0.714 0.507 0.533 0.714 0.52 0.595 0.504
Scaled VaR 0.714 0.507 0.533 0.714 0.52 0.595 0.504
Equal Risk EPD 0.714 0.507 0.533 0.714 0.52 0.595 0.504
Equal Risk TVaR 0.714 0.507 0.533 0.714 0.52 0.595 0.504
Equal Risk VaR 0.714 0.507 0.533 0.714 0.52 0.595 0.504
coTVaR 0
Covar 0.933 0.491 0.533 0.753 0.504 0.595 0.504
Capital Expected Loss 8.068 40.79 48.86 6.277 7.323 13.6 35.26
Scaled EPD 3.694 45.16 48.86 3.694 9.906 13.6 35.26
Scaled TVaR 3.694 45.16 48.86 3.694 9.906 13.6 35.26
Scaled VaR 3.694 45.16 48.86 3.694 9.906 13.6 35.26
Equal Risk EPD 3.694 45.16 48.86 3.694 9.906 13.6 35.26
Equal Risk TVaR 3.694 45.16 48.86 3.694 9.906 13.6 35.26
Equal Risk VaR 3.694 45.16 48.86 3.694 9.906 13.6 35.26
coTVaR
Covar 0.663 48.19 48.86 3.02 10.58 13.6 35.26
Rate of Return Expected Loss 0.489 0.489 0.489 0.489 0.489 0.489 0.489
Scaled EPD 0.489 0.489 0.489 0.489 0.489 0.489 0.489
Scaled TVaR 0.489 0.489 0.489 0.489 0.489 0.489 0.489
Scaled VaR 0.489 0.489 0.489 0.489 0.489 0.489 0.489
Equal Risk EPD 0.489 0.489 0.489 0.489 0.489 0.489 0.489
Equal Risk TVaR 0.489 0.489 0.489 0.489 0.489 0.489 0.489
Equal Risk VaR 0.489 0.489 0.489 0.489 0.489 0.489 0.489
coTVaR
Covar 0.489 0.489 0.489 0.489 0.489 0.489 0.489
Leverage Expected Loss 1.047 1.047 1.047 1.206 1.206 1.206 0.985
Scaled EPD 1.707 0.993 1.047 1.707 1.019 1.206 0.985
Scaled TVaR 1.707 0.993 1.047 1.707 1.019 1.206 0.985
Scaled VaR 1.707 0.993 1.047 1.707 1.019 1.206 0.985
Equal Risk EPD 1.707 0.993 1.047 1.707 1.019 1.206 0.985
Equal Risk TVaR 1.707 0.993 1.047 1.707 1.019 1.206 0.985
Equal Risk VaR 1.707 0.993 1.047 1.707 1.019 1.206 0.985
coTVaR
Covar 7.273 0.961 1.047 1.979 0.985 1.206 0.985
Assets Expected Loss 16.51 83.49 100 13.85 16.15 30 70
Scaled EPD 10 90 100 10 20 30 70
Scaled TVaR 10 90 100 10 20 30 70
Scaled VaR 10 90 100 10 20 30 70
Equal Risk EPD 10 90 100 10 20 30 70
Equal Risk TVaR 10 90 100 10 20 30 70
Equal Risk VaR 10 90 100 10 20 30 70
coTVaR
Covar 5.488 94.51 100 8.997 21 30 70

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: Discrete Example, gross twelve plot with roe distortion.
Figure Figure(1080x1200)
(TN) Figure 15.3: Discrete Example, net twelve plot with tvar distortion.
Figure Figure(900x325)
(U) Figure 15.8: Discrete Example, capital density for Discrete Example, with roe gross and Tail VaR, 0.487 net distortion.
(V) Table 13.35: Constant 0.49 ROE pricing for Discrete Example, distortion, SRM methods.
    Gross Net Ceded
  line X1 X2 Total X1 X2 Total Diff
stat Method              
Loss Expected Loss 4.50 22.75 27.25 4.50 5.25 9.75 17.50
Margin Expected Loss 3.95 19.95 23.89 3.07 3.58 6.65 17.24
Dist ROE 1.81 22.09 23.89 1.81 4.84 6.65 17.24
Dist PH 1.47 22.42 23.89 1.47 4.93 6.40 17.49
Dist Wang 1.53 22.36 23.89 1.53 4.93 6.46 17.43
Dist Dual 1.72 22.17 23.89 1.72 4.90 6.62 17.27
Dist Tvar 2.53 21.37 23.89 2.53 4.74 7.27 16.62
Dist Blend 0.46 22.09 22.54 0.46 4.84 5.30 17.24
Premium Expected Loss 8.45 42.70 51.14 7.57 8.83 16.40 34.74
Dist ROE 6.31 44.84 51.14 6.31 10.09 16.40 34.74
Dist PH 5.97 45.17 51.14 5.97 10.18 16.15 34.99
Dist Wang 6.03 45.11 51.14 6.03 10.18 16.21 34.93
Dist Dual 6.22 44.92 51.14 6.22 10.15 16.37 34.77
Dist Tvar 7.03 44.12 51.14 7.03 9.99 17.02 34.12
Dist Blend 4.96 44.84 49.79 4.96 10.09 15.05 34.74
Loss Ratio Expected Loss 0.53 0.53 0.53 0.59 0.59 0.59 0.50
Dist ROE 0.71 0.51 0.53 0.71 0.52 0.59 0.50
Dist PH 0.75 0.50 0.53 0.75 0.52 0.60 0.50
Dist Wang 0.75 0.50 0.53 0.75 0.52 0.60 0.50
Dist Dual 0.72 0.51 0.53 0.72 0.52 0.60 0.50
Dist Tvar 0.64 0.52 0.53 0.64 0.53 0.57 0.51
Dist Blend 0.91 0.51 0.55 0.91 0.52 0.65 0.50
Capital Expected Loss 8.07 40.79 48.86 6.28 7.32 13.60 35.26
Dist ROE 3.69 45.16 48.86 3.69 9.91 13.60 35.26
Dist PH 3.54 45.32 48.86 3.68 10.16 13.85 35.01
Dist Wang 3.54 45.32 48.86 3.79 9.99 13.79 35.07
Dist Dual 3.17 45.69 48.86 3.69 9.94 13.63 35.23
Dist Tvar 2.73 46.13 48.86 3.38 9.60 12.98 35.88
Dist Blend 1.32 48.89 50.21 2.24 12.71 14.95 35.26
Rate of Return Expected Loss 0.49 0.49 0.49 0.49 0.49 0.49 0.49
Dist ROE 0.49 0.49 0.49 0.49 0.49 0.49 0.49
Dist PH 0.42 0.49 0.49 0.40 0.48 0.46 0.50
Dist Wang 0.43 0.49 0.49 0.40 0.49 0.47 0.50
Dist Dual 0.54 0.49 0.49 0.47 0.49 0.49 0.49
Dist Tvar 0.93 0.46 0.49 0.75 0.49 0.56 0.46
Dist Blend 0.35 0.45 0.45 0.20 0.38 0.35 0.49
Leverage Expected Loss 1.05 1.05 1.05 1.21 1.21 1.21 0.99
Dist ROE 1.71 0.99 1.05 1.71 1.02 1.21 0.99
Dist PH 1.69 1.00 1.05 1.62 1.00 1.17 1.00
Dist Wang 1.70 1.00 1.05 1.59 1.02 1.18 1.00
Dist Dual 1.97 0.98 1.05 1.69 1.02 1.20 0.99
Dist Tvar 2.58 0.96 1.05 2.08 1.04 1.31 0.95
Dist Blend 3.76 0.92 0.99 2.21 0.79 1.01 0.99
Assets Expected Loss 16.51 83.49 100.00 13.85 16.15 30.00 70.00
Dist ROE 10.00 90.00 100.00 10.00 20.00 30.00 70.00
Dist PH 9.52 90.48 100.00 9.66 20.34 30.00 70.00
Dist Wang 9.58 90.42 100.00 9.83 20.17 30.00 70.00
Dist Dual 9.39 90.61 100.00 9.91 20.09 30.00 70.00
Dist Tvar 9.75 90.25 100.00 10.40 19.60 30.00 70.00
Dist Blend 6.27 93.73 100.00 7.20 22.80 30.00 70.00
Figure Figure(900x975)
(W) Figure 15.11: Discrete Example, 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 = 0.489∕1.4889999999999999 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: Discrete Example, percentile layer of capital allocations by asset level, showing 1.0 capital. (Same distortions.)
(Y) Table 15.38: Discrete Example percentile layer of capital allocations compared to distortion allocations.
  Gross Net Ceded
line X1 X2 Total X1 X2 Total Diff
Method              
Expected Loss 16.51 83.49 100 13.85 16.15 30 70
Dist ROE 10 90 100 10 20 30 70
Dist PH 9.518 90.48 100 9.659 20.34 30 70
Dist Wang 9.575 90.42 100 9.828 20.17 30 70
Dist Dual 9.391 90.61 100 9.912 20.09 30 70
Dist Tvar 9.751 90.25 100 10.4 19.6 30 70
Dist Blend 6.275 93.73 100 7.198 22.8 30 70
PLC 10.35 89.65 100 10.92 19.08 30 70

Created 2024-06-12 22:03:43.54673

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.