Loss Aggregate 2extended

*Sliding Scale Commission ExampleProvisional Commission: 30%If the loss ratio is less than 65%, then the commission increases by 1 point for each point decrease in loss ratio up to a maximum commission of 35% at a 60% loss ratioIf the loss ratio is greater than 65%, the commission decreases by 0.5 for each 1 point increase in LR down to a minimum comm. of 25% at a 75% loss ratioIf the expected loss ratio is 65% is the expected commission 30%?

*Sliding Scale Commission - Solution

PC Example 1

Profit Commission Example

Profit Commission:50%After10%Reins Margin

Ceding Commission:30%

Cost of PCCR

Loss Ratio Bandat avg LRat avg LR

LowHighAvg in BandProbabilityin Bandin Band

20%30%25%2.8%17.5%72.5%

30%40%35%9.4%12.5%77.5%

40%50%45%15.2%7.5%82.5%

50%60%55%20.9%2.5%87.5%

60%70%65%17.4%0.0%95.0%

70%80%75%15.1%0.0%105.0%

80%90%85%10.1%0.0%115.0%

90%100%95%5.8%0.0%125.0%

100%150%125%1.4%0.0%155.0%

150%200%175%1.1%0.0%205.0%

200%300%250%0.5%0.0%280.0%

300%400%350%0.3%0.0%380.0%

Average:65.0%100.0%3.3%98.3%

Cost of Profit Comm & CR at expected LR doesn't equal expected Cost of Profit Comm and expected CR

LR Dist

Estimating a Lognormal Distribution of Loss Ratios

On LevelModeled

YearLRCDFLR

199365.5%10.0%48.8%

199470.0%20.0%52.6%

199555.0%30.0%55.5%30.0%40.0%37.9%

199648.0%40.0%58.1%

199772.0%50.0%60.6%

199865.0%60.0%63.3%

199955.0%70.0%66.2%

Mean LR:61.5%80.0%69.9%

standard deviation:8.92%90.0%75.3%

Calculated CV:0.1595.0%80.0%

Selected CV:0.1798.0%85.8%

Lognormal Mu:(0.500)99.0%89.8%

Lognormal Sigma:0.169

Convoluting Cat & Noncat LR's

Convolution of Cat and Noncat LR's

Disretized Cat LR's

Non cat0%30%60%100%

LRProb60%20%15%5%

40%10%6.0%2.0%1.5%0.5%

55%25%15.0%5.0%3.8%1.3%

65%35%21.0%7.0%5.3%1.8%

77%25%15.0%5.0%3.8%1.3%

100%5%3.0%1.0%0.8%0.3%

Total Loss Ratios

These probabilities40%70%100%140%

correspond to55%85%115%155%

these total LR's65%95%125%165%

77%107%137%177%

100%130%160%200%

Parameter Uncertainty

Parameter Uncertainty

1) Simulate Expected Loss Ratio

Simulated random variable from 0 to 0.33: Choose 60%

Simulated random variable from 0.33 to 0.67: Choose 65%

Simulated random variable from 0.67 to 1,00: Choose 70%

Simulated Random Variable:0.8

Simulated Expected Loss Ratio:70.0%

2) Calculate New Lognormal Parameters

Sigma (same as original selection):0.17

Simulated Lognormal Mu:(0.37)

Mu = LN(Expected LR) - Sigma^2/2

3) Simulate Loss Ratio for Year Based on New Lognormal Mu

Simulated Random Variable (CDFi):0.842

# of St. Deviations Away from Mean [Normsinv(CDFi)]:1.00

Simulated Loss Ratio:81.7%

Exp (mu + Normsinv(CDFi) x sigma)

Count Sim Data

Claim Count Simulation Data

(Note)2001

SPI atReportedCountEst UltAnnualFreqTrendedExposureLevel

2001 RateClaimDevelClaimFreqTrend toClaimAdjClaim

YearLevelCountFactorCountTrend2001CountFactorCount

199110,0002.01.02.00.0%1.1042.211.603.53

199210,5001.01.01.00.0%1.1041.101.521.68

199311,0251.01.01.00.0%1.1041.101.451.60

199411,5761.01.11.10.0%1.1041.161.381.60

199512,1553.01.13.30.0%1.1043.641.324.80

199612,7630.01.20.00.0%1.1040.01.250.0

199713,4010.01.30.02.0%1.0820.01.190.0

199814,0710.01.50.02.0%1.0610.01.140.0

199914,7751.02.02.02.0%1.0402.081.082.25

200015,5131.03.53.52.0%1.0203.571.033.68

200116,0002.0%

Average:1.92

Variance:2.82

Note: Exposure Adj Factor Yr i = 2001 SPI / SPI year iSelected Variance:3.11

Count Sim

Claim Count Simulation

(A)Selected Mean Claim Count/Poisson Gamma1.92

(B)Selected Variance of Claim Count Distribution3.11

(C)Contagion Parameter [(Variance / Mean -1) / Mean]0.32

(D)Gamma Distribution Alpha (1/C)3.08

(E)Gamma Distribution Beta (C)0.32

(F)Simulated Gamma CDF0.412

(G)Simulated Gamma Random Variable0.78

(H)Simulated Poisson Gamma (A) X (G)1.50

Simulated Poisson Parameter1.50

Simulated Poisson CDF:0.808

Year 1 Simulated Claim Count:2

ProbProb

ClaimPoissonCountClaimPoissonCount

CountProbability

*Loss Ratio CorridorsA loss ratio corridor is a provision that forces the ceding company to retain losses that would be otherwise ceded to the reinsurance treatyLoss ratio corridor of 100% of the losses between a 75% and 85% LRIf gross LR equals 75%, then ceded LR is 75%If gross LR equals 80%, then ceded LR is 75%If gross LR equals 85%, then ceded LR is 75%If gross LR equals 100%, then ceded LR is ???

*Loss Ratio CapThis is the maximum loss ratio that could be ceded to the treaty.Example: 200% Loss Ratio CapIf LR before cap is 150%, then ceded LR is 150%If LR before cap is 250%, then ceded LR is 200%

*Loss Ratio Corridor ExampleReinsurance treaty has a loss ratio corridor of 50% of the losses between a loss ratio of 70% and 80%.Use the aggregate distribution to your right to estimate the expected ceded LR net of the corridor

PC Example 1




Cost of PCCR



20%30%25%4.6%17.5%72.5%

30%40%35%8.1%12.5%77.5%

40%50%45%15.1%7.5%82.5%

50%60%55%20.9%2.5%87.5%

60%70%65%17.4%0.0%95.0%

70%80%75%14.4%0.0%105.0%

80%90%85%10.8%0.0%115.0%

90%100%95%5.4%0.0%125.0%

100%150%122%1.4%0.0%152.0%

150%200%270%1.1%0.0%300.0%

200%300%244%0.5%0.0%274.0%

300%400%235%0.3%0.0%265.0%

Average:65.26%100.0%3.48%98.7%


LR Dist


On LevelModeled

YearLRCDFLR

199365.5%10.0%48.1%

199470.0%20.0%52.1%

199555.0%30.0%55.1%

199648.0%40.0%57.9%

199772.0%50.0%60.5%

199865.0%60.0%63.3%

199955.0%70.0%66.5%

Mean LR:61.5%80.0%70.3%


Selected CV:0.1895.0%81.2%

Lognormal Mu:(0.502)98.0%87.3%

Lognormal Sigma:0.17999.0%91.7%

Count Sim Data


(Note)2001




199110,0002.01.02.00.0%1.1042.211.603.53

199210,5001.01.01.00.0%1.1041.101.521.68

199311,0251.01.01.00.0%1.1041.101.451.60

199411,5761.01.11.10.0%1.1041.161.381.60

199512,1553.01.13.30.0%1.1043.641.324.80

199612,7630.01.20.00.0%1.1040.01.250.0

199713,4010.01.30.02.0%1.0820.01.190.0

199814,0710.01.50.02.0%1.0610.01.140.0

199914,7751.02.02.02.0%1.0402.081.082.25

200015,5131.03.53.52.0%1.0203.571.033.68

200116,0002.0%

Average:1.92

Variance:2.82


Count Sim




(C)Contagion Parameter:0.32

(D)(Variance / Mean -1) / Mean

(E)Gamma Distribution Alpha3.08

(F)Gamma Distribution Beta0.32

(G)Simulated Gamma CDF0.412

(H)Simulated Gamma Random Variable0.78

(I)Simulated Poisson Gamma (A) X (I)1.50

Simulated Poisson Gamma1.50



ProbProb


CountProbability

*Loss Ratio Corridor Example Solution

PC Example 1




Cost of PCCR



20%30%25%4.6%17.5%72.5%

30%40%35%8.1%12.5%77.5%

40%50%45%15.1%7.5%82.5%

50%60%55%20.9%2.5%87.5%

60%70%65%17.4%0.0%95.0%

70%80%75%14.4%0.0%105.0%

80%90%85%10.8%0.0%115.0%

90%100%95%5.4%0.0%125.0%

100%150%122%1.4%0.0%152.0%

150%200%270%1.1%0.0%300.0%

200%300%244%0.5%0.0%274.0%

300%400%335%0.3%0.0%365.0%

Average:65.53%100.0%3.48%99.0%


LR Dist


On LevelModeled

YearLRCDFLR

199365.5%10.0%48.8%

199470.0%20.0%52.6%

199555.0%30.0%55.5%30.0%40.0%37.9%

199648.0%40.0%58.1%

199772.0%50.0%60.6%

199865.0%60.0%63.3%

199955.0%70.0%66.2%

Mean LR:61.5%80.0%69.9%

standard deviation:8.92%90.0%75.3%


Selected CV:0.1798.0%85.8%

Lognormal Mu:(0.500)99.0%89.8%

Lognormal Sigma:0.169

Convoluting Cat & Noncat LR's

Convolution of Cat and Noncat LR's

Disretized Cat LR's

Non cat0%30%60%100%

LRProb60%20%15%5%

40%10%6.0%2.0%1.5%0.5%

55%25%15.0%5.0%3.8%1.3%

65%35%21.0%7.0%5.3%1.8%

77%25%15.0%5.0%3.8%1.3%

100%5%3.0%1.0%0.8%0.3%

Total Loss Ratios

These probabilities40%70%100%140%

correspond to55%85%115%155%

these total LR's65%95%125%165%

77%107%137%177%

100%130%160%200%

Count Sim Data


(Note)2001




199110,0002.01.02.00.0%1.1042.211.603.53

199210,5001.01.01.00.0%1.1041.101.521.68

199311,0251.01.01.00.0%1.1041.101.451.60

199411,5761.01.11.10.0%1.1041.161.381.60

199512,1553.01.13.30.0%1.1043.641.324.80

199612,7630.01.20.00.0%1.1040.01.250.0

199713,4010.01.30.02.0%1.0820.01.190.0

199814,0710.01.50.02.0%1.0610.01.140.0

199914,7751.02.02.02.0%1.0402.081.082.25

200015,5131.03.53.52.0%1.0203.571.033.68

200116,0002.0%

Average:1.92

Variance:2.82


Count Sim




(C)Contagion Parameter [(Variance / Mean -1) / Mean]0.32

(D)Gamma Distribution Alpha (1/C)3.08

(E)Gamma Distribution Beta (C)0.32

(F)Simulated Gamma CDF0.412

(G)Simulated Gamma Random Variable0.78

(H)Simulated Poisson Gamma (A) X (G)1.50

Simulated Poisson Parameter1.50



ProbProb


CountProbability

Documents

Loss Aggregate 2extended