1 Analyzing Loss Sensitive Treaty Terms 2001 CARe Seminar g ERC ® Jeff Dollinger, FCAS, Employers Reinsurance Co Kari Mrazek, FCAS, GE Reinsurance Co.

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Analyzing Loss Sensitive Treaty Terms

2001 CARe Seminar

gERC®

Jeff Dollinger, FCAS, Employers Reinsurance Co

Kari Mrazek, FCAS, GE Reinsurance Co

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Introduction to Loss Sharing Provision

• Definition: A reinsurance contract provision that varies the ceded premium, loss, or commission based upon the loss experience of the contract

• Purpose: Client shares in ceded experience & could be incented to care more about the reinsurer’s results

• Typical Loss Sharing Provisions– Profit Commission– Sliding Scale Commission– Loss Ratio Corridors– Annual Aggregate Deductibles– Swing Rated Premiums– Reinstatements

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Simple Profit Commission Example

• A property prorata contract has the following profit commission terms– 50% Profit Commission after a reinsurer’s margin of 10%.

– Key Point: Reinsurer returns 50% of the contractually defined “profit” to the cedant

– Profit Commission Paid to Cedant = 50% x (Premium - Loss - Commission - Reinsurers Margin)

– If profit is negative, reinsurers do not get any additional money from the cedant.

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Simple Profit Commission Example

• Ceding Commission = 30%

• Loss ratio must be less than 60% for us to pay a profit commission

• Contract Expected Loss Ratio = 70%

• $1 Prem - $0.7 Loss - $0.3 Comm - $0.10 Reins Margin = minus $0.10

• Is the expected cost of profit commission zero?

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Simple Profit Commission Example

• Answer: The expected cost of profit commission is not zero

• Why: Because 70% is the expected loss ratio. – There is a probability distribution of potential outcomes

around that 70% expected loss ratio.

– It is possible (and may even be likely) that the loss ratio in any year could be less than 60%.

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Cost of Profit Commission: Simple Quantification

• Earthquake exposed California property prorata treaty

• LR = 40% in all years with no EQ

• Profit Comm when there is no EQ = 50% x ($1 of Premium - $0.4 Loss - $0.30 Commission - $0.1 Reinsurers Margin)

= 10% of premium

• Cat Loss Ratio = 30%.– 10% chance of an EQ costing 300% of premium, 90% chance no

EQ loss

Cost of Profit Comm =

Profit Comm Costs 10% of Prem x 90% Probability of No EQ

+ 0% Cost of PC x 10% Probability of EQ Occurring = 9% of Premg

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Basic Mechanics of Analyzing Loss Sharing Provisions

• Build aggregate loss distribution

• Apply loss sharing terms to each point on the loss distribution or to each simulated year

• Calculate a probability weighted average cost (or saving) of the loss sharing arrangement

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Example of Basic Mechanics: PC: 50% after 10%, 30% Commission, 65% Expected LR

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Cost of PC CRat avg LR at avg LR

Low High Avg in Band Probability in Band in Band20% 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%

Loss Ratio Band

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

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Determining an Aggregate Distribution - 2 Methods

• Fit statistical distribution to on level loss ratios– Reasonable for prorata treaties.

• Determine an aggregate distribution by modeling frequency and severity– Typically used for excess of loss treaties.

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Fitting a Distribution to On Level Loss Ratios

• Most actuaries use the lognormal distribution

– Reflects skewed distribution of loss ratios

– Easy to use

• Lognormal distribution assumes that the natural logs of the loss ratios are distributed normally.

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Skewness of Lognormal Distribution

0.00%

5.00%

10.00%

15.00%

20.00%

25.00%

0-10%

10-20%

20-30%

30-40%

40-50%

50-60%

60-70%

70-80%

80-90%

90-100%

100-110%

110-120%

Loss Ratios

Incr

emen

tal P

rob

abili

ty

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Fitting a Lognormal Distribution to Projected Loss Ratios

• Fitting the lognormal

^2 = LN(CV^2 + 1) = LN(mean) - ^2/2

Mean = Selected Expected Loss Ratio

CV = Standard Deviation over the Mean of the loss ratio (LR) distribution.

• Prob (LR X) = Normal Dist(( LN(x) - )/ ) i.e.. look up (LN(x) - )/ ) on a standard normal distribution table

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Fitting a Lognormal Loss Ratio Distribution

• Producing a distribution of loss ratios– For a given point i on the CDF, the following Excel

command will produce a loss ratio at that CDFi:

Exp ( + Normsinv(CDFi) x )

• Key Question: Is the resulting LR distribution reasonable– Analyze that issue by reviewing historical data

– Discuss this issue with your underwriter

– If the distribution is not reasonable, adjust the CV selection.

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Sample Lognormal Loss Ratio Distribution

On Level ModeledYear LR CDF LR1993 65.5% 10.0% 48.8%1994 70.0% 20.0% 52.6%1995 55.0% 30.0% 55.5%1996 48.0% 40.0% 58.1%1997 72.0% 50.0% 60.6%1998 65.0% 60.0% 63.3%1999 55.0% 70.0% 66.2%

Mean LR: 61.5% 80.0% 69.9%standard deviation: 8.92% 90.0% 75.3%Calculated CV: 0.15 95.0% 80.0%Selected CV: 0.17 98.0% 85.8%Lognormal Mu: (0.500) 99.0% 89.8%Lognormal Sigma: 0.169

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Additional Considerations for Fitting a Lognormal Loss Ratio Distribution• Selected CV should usually be above indicated

– 5 to 10 years of data does not reflect full range of possibilities

• Parameter Uncertainty: Do you really know the true mean of the loss ratio distribution for the upcoming year?

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Modeling Parameter Uncertainty: One Possible Method

• Select 3 equally likely expected loss ratios

• Assign weight to each loss ratio so that the weighted average ties to your selected expected loss ratio– Example: Expected LR is 65%, assume 1/3 probability

that true mean LR is 60%, 1/3 probability that it is 65%, and 1/3 probability that it is 70%.

– Simulate the “true” expected loss ratio

• Simulate the loss ratio for the year modeled using the lognormal based on simulated expected loss ratio & your selected CV

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Example of Modeling Parameter Uncertainty

1) Simulate Expected Loss RatioSimulated 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: 0.8Simulated Expected Loss Ratio: 70.0%

2) Calculate New Lognormal ParametersSigma (same as original selection): 0.17Simulated Lognormal Mu: (0.37) Mu = LN(Expected LR) - Sigma^2/2

3) Simulate Loss Ratio for Year Based on New Lognormal MuSimulated 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)g

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Common Loss Sharing Provisions for Prorata Treaties

• Profit Commissions– Already covered

• Sliding Scale Commission

• Loss Ratio Corridor

• Loss Ratio Cap

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Sliding Scale Comm

• Commission initially set at Provisional amount

• Ceding commission increases if loss ratios are lower than expected

• Ceding commission decreases if losses are higher than expected

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Sliding Scale Commission Example

• Provisional 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 ratio

• If 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 ratio

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Sliding Scale Commission - Solution

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Low HighAvg LR in Band Probability

Ceding Comm @ avg LR in

BandCR @ avg LR in Band Lognormal Parameters

0.0% 52.5% 45.0% 11.91% 35.0% 80.0% Mean LR: 65.0%

52.5% 57.5% 55.0% 14.18% 35.0% 90.0% Selected CV: 17.0%

57.5% 62.5% 60.0% 18.08% 35.0% 95.0% Lognormal Mu: (0.45)

62.5% 67.5% 65.0% 17.98% 30.0% 95.0% Lognormal Sigma: 0.17

67.5% 72.5% 70.0% 14.67% 27.5% 97.5%

72.5% 77.5% 75.0% 10.22% 25.0% 100.0% LR Comm

77.5% 87.5% 82.5% 9.73% 25.0% 107.5% Max Comm 60% 35%

87.5% 100.0% 93.8% 2.82% 25.0% 118.8% Prov Comm 65% 30%

100.0% 200.0% 135.0% 0.42% 25.0% 160.0% Min Comm 75% 25%

200.0% 300.0% 228.0% 0.00% 25.0% 253.0%

Prob Wtd Avg 64.9% 30.7% 95.5%

Conclusion: Expected cost of commission is not 30%.

Loss Ratio Band

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Loss Ratio Corridors

• A loss ratio corridor is a provision that forces the ceding company to retain losses that would be otherwise ceded to the reinsurance treaty

• Loss ratio corridor of 100% of the losses between a 75% and 85% LR– If 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 ???

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Loss Ratio Cap

• This is the maximum loss ratio that could be ceded to the treaty.

• Example: 200% Loss Ratio Cap– If LR before cap is 150%, then ceded LR is

150%– If LR before cap is 250%, then ceded LR is

200%

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Loss Ratio Corridor Example

• Reinsurance 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 ceded LR net of the corridor

Low HighAvg LR in Band Probability

0.0% 50.0% 45.0% 14.23%

50.0% 60.0% 55.0% 33.82%

60.0% 65.0% 62.5% 17.47%

65.0% 70.0% 67.5% 13.71%

70.0% 75.0% 72.5% 9.28%

75.0% 80.0% 77.5% 5.58%

80.0% 85.0% 82.5% 3.05%

85.0% 100.0% 92.5% 2.61%

100.0% 200.0% 135.0% 0.25%

200.0% 300.0% 228.0% 0.00%

Loss Ratio Band

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Loss Ratio Corridor Example - Solution

Loss Ratio Corridor50.0% between 70.0% & 80.0%

Low HighAvg LR in Band Probability

Savings from Corridor

LR Net of Corridor

0.0% 52.5% 48.0% 11.91% 0.0% 48.0%52.5% 57.5% 55.0% 14.18% 0.0% 55.0%57.5% 62.5% 60.0% 18.08% 0.0% 60.0%62.5% 67.5% 65.0% 17.98% 0.0% 65.0%67.5% 72.5% 70.0% 14.67% 0.0% 70.0%72.5% 77.5% 75.0% 10.22% 2.5% 72.5%77.5% 82.5% 80.0% 9.73% 5.0% 75.0%82.5% 100.0% 92.5% 2.82% 5.0% 87.5%

100.0% 200.0% 135.0% 0.42% 5.0% 130.0%200.0% 300.0% 228.0% 0.00% 5.0% 223.0%

Prob Wtd Avg: 64.9% 0.9% 64.0%

Loss Ratio Band

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Modeling Property Treaties with Significant Cat Exposure

• Model noncat & cat LR’s separately – Non Cat LR’s fit to a lognormal curve

– Cat LR distribution produced by commercial catastrophe model

• Combine (convolute) the noncat & cat loss ratio distributions

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Convoluting Noncat & Cat LR’s - Example

0% 30% 60% 100%LR Prob 60% 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%

These probabilities 40% 70% 100% 140%correspond to 55% 85% 115% 155%these total LR's 65% 95% 125% 165%

77% 107% 137% 177%100% 130% 160% 200%

Total Loss Ratios

Disretized Cat LR'sNon cat

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Truncated Loss Ratio Distributions

• Problem: To reasonably model the possibility of high LR requires a high lognormal CV

• High lognormal CV often leads to unrealistically high probabilities of low LR’s, which overstates cost of PC

• Solution: Don’t allow LR to go below selected minimum, e.g.. 0% probability of LR<30%– Adjust lognormal mean so that aggregate distribution

will probability weight back to initial expected LR

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Summary of Loss Ratio Distribution Method

• Advantage: – Easier and quicker than separately modeling frequency and

severity

– Reasonable for most prorata treaties

• Usually inappropriate for excess of loss contracts– Continuous distribution around the mean does not reflect the

hit or miss nature of many excess of loss contracts

– Understates probability of zero loss

– Understates potential of losses much greater than the expected loss

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Excess of Loss Contracts: Separate Modeling of Frequency and Severity

• Used mainly for modeling excess of loss contracts

• A detailed mathematical explanation is beyond the scope of this session

• Software that can be Used to do the above modeling– Crystal Ball

– Crimcalc

– @Risk

– Excel

• Most aggregate distribution approaches assume that frequency and severity are independent

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Common Frequency Distributions

• Poisson

f(x|) = exp(-) ^x / x!

where = mean of the claim count distribution and

x = claim count = 0,1,2,...

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Fitting a Poisson Claim Count Distribution

• Estimate ultimate claim counts by year

• Multiply ultimate claim counts by frequency trend factor to bring them to the frequency level of the upcoming treaty year

• Adjust for change in exposure levels, ie.Adjusted Claim Count year i =

Trended Ultimate Claim Count i x

(SPI for upcoming treaty year / On Level SPI year i)

• Poisson parameter equals the mean of the ultimate, trended, adjusted claim counts from above

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Example of Simulated Claim Count

SPI at Reported Count Est Ult Annual Freq Trended Exposure Level2001 Rate Claim Devel Claim Freq Trend to Claim Adj Claim

Year Level Count Factor Count Trend 2001 Count Factor Count1991 10,000 2.0 1.0 2.0 0.0% 1.104 2.21 1.60 3.53 1992 10,500 1.0 1.0 1.0 0.0% 1.104 1.10 1.52 1.68 1993 11,025 1.0 1.0 1.0 0.0% 1.104 1.10 1.45 1.60 1994 11,576 1.0 1.1 1.1 0.0% 1.104 1.16 1.38 1.60 1995 12,155 3.0 1.1 3.3 0.0% 1.104 3.64 1.32 4.80 1996 12,763 - 1.2 - 0.0% 1.104 - 1.25 - 1997 13,401 - 1.3 - 2.0% 1.082 - 1.19 - 1998 14,071 - 1.5 - 2.0% 1.061 - 1.14 - 1999 14,775 1.0 2.0 2.0 2.0% 1.040 2.08 1.08 2.25 2000 15,513 1.0 3.5 3.5 2.0% 1.020 3.57 1.03 3.68 2001 16,000 2.0%

Average: 1.92 Variance: 2.82

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

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Modeling Frequency- Negative Binomial

• Negative Binomial: Same form as the poisson distribution, except that it assumes that is not fixed, but rather has a gamma distribution around the selected – Claim count distribution is negative binomial if the variance

of the count distribution is greater than the mean

– The gamma distribution around has a mean of 1

• Negative Binomial is the preferred distribution– Reflects parameter uncertainty regarding the true mean claim

count

– The extra variability of the Negative Binomial is more in line with historical experience

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Algorithm for Simulating Claim Counts

• Poisson

– Manually create a poisson cumulative distribution table

– Simulate the CDF (a number between 0 and 1) and lookup the number of claims corresponding to that CDF. This is your simulated claim count for year 1

– Repeat the above two steps for however many years that you want to simulate

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Additional Steps for Simulating Claim Counts using Negative Binomial

• Determine contagion parameter, c, of claim count distribution:

(^2 / ) = 1 + c If the claim count distribution is poison, then c=0

If it is negative binomial, then c>0

• Solve for the contagion parameter:c = [(^2 / ) - 1] /

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Additional Steps for Simulating Claim Counts using Negative Binomial

• Simulate gamma random variable with a mean of 1– Gamma distribution has two parameters: and

= 1/c; = c

– Using Excel, simulate gamma random variable as follows

Gammainv(Simulated CDF, , )

– Simulated Poisson parameter =

= x Simulated Gamma Random Variable Above

– Use the poisson distribution algorithm using the above simulated poisson parameter, , to simulate the claim count for the year

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Year 1 Simulated Negative Binomial Claim Count

(A) Selected Mean Claim Count/Poisson Gamma 1.92 (B) Selected Variance of Claim Count Distribution 3.11 (C) Contagion Parameter [(Variance / Mean -1) / Mean] 0.32 (D) Gamma Distribution Alpha 3.08 (E) Gamma Distribution Beta 0.32 (F) Simulated Gamma CDF 0.412 (G) Simulated Gamma Random Variable 0.78 (H) Simulated Poisson Parameter (A) X (G) 1.50

Simulated Poisson Parameter 1.50 Simulated Poisson CDF: 0.808 Year 1 Simulated Claim Count: 2

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Year 1 Simulated Negative Binomial Claim Count

Simulated Poisson Gamma 1.50 Simulated Poisson CDF: 0.808 Year 1 Simulated Claim Count: 2

Prob ProbClaim Poisson Count ClaimPoisson CountCount Probability <= X CountProbability <= X

0 22.39% 22.39% 5 1.40% 99.56%1 33.51% 55.90% 6 0.35% 99.91%2 25.07% 80.97% 7 0.07% 99.98%3 12.51% 93.48% 8 0.01% 100.00%4 4.68% 98.16% 9 0.00% 100.00%

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Year 2 Simulated Negative Binomial Claim Count

Selected Mean Claim Count/Poisson Gamma 1.92 Simulated Gamma CDF 0.668 Simulated Gamma Random Variable 1.15 Simulated Poisson Gamma (A) X (G) 2.20

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Year 2 Simulated Negative Binomial Claim Count

Simulated Poisson Gamma 2.20 Simulated Poisson CDF: 0.645 Year 2 Simulated Claim Count: 3

Prob ProbClaim Poisson Count Claim Poisson CountCount Probability <= X Count Probability <= X

0 11.13% 11.13% 5 4.73% 97.53%1 24.44% 35.57% 6 1.73% 99.26%2 26.83% 62.40% 7 0.54% 99.80%3 19.63% 82.03% 8 0.15% 99.95%4 10.77% 92.80% 9 0.04% 99.99%

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Modeling Severity

• Common Severity Distributions– Lognormal

– Pareto

– Mixed Exponential (currently used by ISO)

– Truncated Pareto. This curve was used by ISO before moving to the Mixed Exponential and will be the focus of this presentation.

Key Point: The ISO Truncated Pareto focused on modeling the larger claims. Typically those over $50,000

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Truncated Pareto

• Truncated Pareto Parameters

t = truncation point.

s = average claim size of losses below truncation point

p = probability claims are smaller than truncation point

b = pareto scale parameter

q = pareto shape parameter

• Cumulative Distribution Function

F(x) = 1 - (1-p) ((t+ b)/(x+ b))^q

Where x>t

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Algorithm for Simulating Severity to the Layer

• For each loss to be simulated, choose a random number between 0 and 1. This is the simulated CDF

• Transformed CDF for losses hitting layer (TCDF) =

Prob(Loss<Reins Att Pt) +

Simulated CDF x (1 - Prob(Loss<Reins Att Pt))

• Find simulated ground up loss, x, that corresponds to simulated TCDF

Doing some algebra, find x using the following formula:x = Exp{ln(t+b) - [ln(1-TCDF) - ln(1-p)]/Q} - b

• From simulated ground up loss calculate loss to the layerg

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Year 1 Loss # 1 Simulated Severity to the Layer

B Q P S TPareto Parameters 79,206 1.39 0.858 6,090 50,000

Reinsurance Layer: 750,000 xs 250,000 Pareto Probability of Loss < Reins Att Point: 96.13%Simulated CDF: 0.4029Transformed CDF for Losses Simulated to the Excess Layer: 0.9769Simulated Loss: 397,876 Simulated Loss to Layer: 147,876

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Year 1 Loss # 2 Simulated Severity to the Layer

B Q P S TPareto Parameters 79,206 1.39 0.858 6,090 50,000

Reinsurance Layer: 750,000 xs 250,000 Pareto Probability of Loss < Reins Att Point: 96.13%Simulated CDF: 0.8400Transformed CDF for Losses Simulated to the Excess Layer: 0.9938Simulated Loss: 1,151,131 Simulated Loss to Layer: 750,000

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Simulation SummaryClaim LossesCount to Layer

Year 1 Simulation 2 147,876 750,000

Total: 897,876

Year 2 Simulation 3 576,745 281,323

54,726 Total: 912,794

Run about 1,000 more years and we have our aggregate distribution to the excess of loss layer

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Common Loss Sharing Provisions for Excess of Loss

Treaties• Profit Commissions

– Already covered

• Swing Rated Premium

• Annual Aggregate Deductibles

• Limited Reinstatements

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Swing Rated Premium• Ceded premium is dependent on loss experience• Typical Swing Rating Terms

– Provisional Rate: 10% of subject premium– Ceded premium is adjusted to equal ceded loss

times 100/80 loading factor, subject to a minimum rate of 5% and a maximum rate of 15%

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Swing Rated Premium - Example

• Burn (ceded loss / SPI) = 10%. Rate = 10% x 100/80 = 12.5%

• Burn = 2%. Calculated Rate = 2% x 100/80 = 2.5%. Rate = 5% minimum rate

• Burn = 14%. Calculated Rate = 14% x 100/80 = 17.5%. Rate = 15% maximum rate

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Swing Rated Premium Example

• Swing Rating Terms: Ceded premium is adjusted to equal ceded loss times 100/80 loading factor, subject to a minimum rate of 5% and a maximum rate of 15%

• Use the aggregate distribution to your right to calculate the ceded loss ratio under the treaty

Low High Average Probability0.0% 0.0% 0.0% 9.0%0.0% 2.5% 1.3% 6.0%2.5% 5.0% 3.8% 9.0%5.0% 7.5% 6.3% 10.2%7.5% 10.0% 8.8% 11.4%

10.0% 12.5% 11.3% 15.0%12.5% 15.0% 13.8% 12.0%15.0% 17.5% 16.3% 9.0%17.5% 20.0% 18.8% 7.8%20.0% 25.0% 21.9% 6.0%25.0% 50.0% 30.3% 4.8%

Band of Burns

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Swing Rated Premium Example - Solution

Min Rate Prov Rate Max Rate

Loss Load

FactorSwing Rated Terms 5.0% 10.0% 15.0% 125.0%

FinalLow High Average Probability Rate

0.0% 0.0% 0.0% 9.0% 5.0%0.0% 2.5% 1.3% 6.0% 5.0%2.5% 5.0% 3.8% 9.0% 5.0%5.0% 7.5% 6.3% 10.2% 7.8%7.5% 10.0% 8.8% 11.4% 10.9%

10.0% 12.5% 11.3% 15.0% 14.1%12.5% 15.0% 13.8% 12.0% 15.0%15.0% 17.5% 16.3% 9.0% 15.0%17.5% 20.0% 18.8% 7.8% 15.0%20.0% 25.0% 21.9% 6.0% 15.0%25.0% 50.0% 30.3% 4.8% 15.0%

Prob Wtd Avg: 11.1% 11.3%

Proj LR (Expected Burn/Expected Final Rate): 98.1%

Band of Burns

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Annual Aggregate Deductible

• The annual aggregate deductible (AAD) refers to a retention by the cedant of losses that would be otherwise ceded to the treaty

• Example: Reinsurer provides a $500,000 xs $500,000 excess of loss contract. Cedant retains an AAD of $750,000– Total Loss to Layer = $500,000. Cedant retains all $500,000. No

loss ceded to reinsurers

– Total Loss to Layer = $1 mil. Cedant retains $750,000. Reinsurer pays $250,000.

– Total Loss to Layer =$1.5 mil. Cedant retains? Reinsurer pays?

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Annual Aggregate Deductible

• Discussion Question: Reinsurer writes a $500,000 xs $500,000 excess of loss treaty.– Expected Loss to the Layer is $1 million (before AAD)

– Cedant retains a $500,000 annual aggregate deductible.

– Cedant says, “I assume that you will decrease your expected loss by $500,000.”

– How do you respond?

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Annual Aggregate Deductible Example

• Your expected burn to a $500K xs $500K reinsurance layer is 11.1%. Cedant adds an AAD of 5% of subject premium

• Using the aggregate distribution of burns to your right, calculate the burn net of the AAD.

Low High Average Probability0.0% 0.0% 0.0% 9.0%0.0% 2.5% 1.3% 6.0%2.5% 5.0% 3.8% 9.0%5.0% 7.5% 6.3% 10.2%7.5% 10.0% 8.8% 11.4%

10.0% 12.5% 11.3% 15.0%12.5% 15.0% 13.8% 12.0%15.0% 17.5% 16.3% 9.0%17.5% 20.0% 18.8% 7.8%20.0% 25.0% 21.9% 6.0%25.0% 50.0% 30.3% 4.8%

Prob Wtd Avg: 11.1%

Band of Burns

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Annual Aggregate Deductible Example - SolutionAnnual Aggregate Deductible as % of SPI: 5.0%

Low High Average Probability

Savings from AAD

Burn Net of AAD

0.0% 0.0% 0.0% 9.0% 0.0% 0.0%0.0% 2.5% 1.3% 6.0% 1.3% 0.0%2.5% 5.0% 3.8% 9.0% 3.8% 0.0%5.0% 7.5% 6.3% 10.2% 5.0% 1.3%7.5% 10.0% 8.8% 11.4% 5.0% 3.8%

10.0% 12.5% 11.3% 15.0% 5.0% 6.3%12.5% 15.0% 13.8% 12.0% 5.0% 8.8%15.0% 17.5% 16.3% 9.0% 5.0% 11.3%17.5% 20.0% 18.8% 7.8% 5.0% 13.8%20.0% 25.0% 21.9% 6.0% 5.0% 16.9%25.0% 50.0% 30.3% 4.8% 5.0% 25.3%Prob Wtd Avg: 11.1% 4.2% 6.8%

Band of Burns

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Limited Reinstatements

• Limited reinstatements refers to the number of times that the occurrence or risk limit of an excess can be reused.

• Example: $1 million xs $1 million layer– 1 reinstatement: It means that after the cedant uses up the first

limit, they also get a second occurrence limit

• Treaty Aggregate Limit =

= Occurrence Limit x (1 + number of Reinstatements)

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Limited Reinstatements Example

$1 million xs $1 million layer1 reinstatement

Individual Ceded Individual Ceded Individual CededLosses Loss Losses Loss Losses Loss$000's $000's $000's $000's $000's $000's

2,000 1000 3,000 1000 3,000 ?2,000 1000 1,500 500 1,500 ?2,000 0 1,500 500 1,500 ?

2,000 ?

Simulated Year 1 Simulated Year 2 Simulated Year 3

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Reinstatement Premium

• In many cases to “reinstate” the limit, the cedant is required to pay an additional premium

• Choosing to reinstate the limit is almost always mandatory

– Reinstatement premium can simply be viewed as additional premium that reinsurers receive depending on loss experience

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Reinstatement Premium Example 1

$1 million xs $1 million layer1 reinstatement at 100%Upfront Ceded Premium = $250,000

Individual Ceded Reinst Individual Ceded Reinst Individual Ceded ReinstLosses Loss Prem Losses Loss Prem Losses Loss Prem$000's $000's $000's $000's $000's $000's $000's $000's $000's

2,000 1,000 250 1,500 500 125 1,250 ? ?2,000 1,000 - 1,500 500 125 2,000 ? ?2,000 - - 1,500 500 - 2,000 ? ?

Simulated Year 1 Simulated Year 2 Simulated Year 3

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Reinstatement Premium Example 2

$1 million xs $1 million layer2 reinstatements: 1st at 50%, 2nd at 100%.Upfront Ceded Premium = $250,000

Individual Ceded Reinst Individual Ceded Reinst Individual Ceded ReinstLosses Loss Prem Losses Loss Prem Losses Loss Prem$000's $000's $000's $000's $000's $000's $000's $000's $000's

3,000 1,000 125 1,500 500 62.5 1,250 ? ?2,000 1,000 250 1,500 500 62.5 2,000 ? ?2,000 1,000 - 1,500 500 125.0 2,000 ? ?2,000 - -

Simulated Year 1 Simulated Year 2 Simulated Year 3

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Reinstatement Example 3

TotalLoss$000's Probability

- 70.47%1,000 24.66%2,000 4.32%3,000 0.50%4,000 0.04%

• Reinsurance Treaty:

$1 mil xs $1 mil

Upfront Prem = 400K

2 Reinstatements: 1st at 50%, 2nd at 100%

• Using the aggregate distribution to the right, calculate our expected ultimate loss, premium, and loss ratio

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Reinstatement Example 3 - Solution

Upfront Premium = 400K2 Reinstatements: 1st at 50%, 2nd at 100%

Total Loss NetLoss of Reinst Reinst Total$000's ProbabilityLimitation Premium Premium

- 70.47% - - 400 1,000 24.66% 1,000 200 600 2,000 4.32% 2,000 600 1,000 3,000 0.50% 3,000 600 1,000 4,000 0.04% 3,000 600 1,000

Prob Wtd Avg: 349 79 479Projected Loss Ratio: 73.0%

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Reinstatement Example 4

Upfront Premium = 1.2M1 Reinstatement at 50%

# of Expected Loss Net of Reinst TotalClms Prob Loss (000's) Reinst Limit Premium Premium

0 90.48% 0 0 0 1,2001 9.05% 10,000 10,000 600 1,8002 0.45% 20,000 20,000 600 1,8003 0.02% 30,000 20,000 600 1,8004 0.00% 40,000 20,000 600 1,8005 0.00% 50,000 20,000 600 1,800

Prob Wtd Avg 100.0% 1,000 998 57 1,257Projected Loss Ratio: 79.5%

• Note: Reinstatement provisions are typically found on high excess layers, where loss tends to be either 0 or a full limit loss.

• Assume: Layer = 10M xs 10M, Expected Loss = 1M, Poisson Frequency with mean = .1

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Deficit Carryforward

• Treaty terms may include Deficit Carryforward Provisions, in which some losses are carried forward to next year’s contract in determining the commission paid.

Comm LR SlideMin 25.0% 75.0% 50.0%Prov 30.0% 65.0% 100.0%Max 35.0% 60.0%

• Example:- Provisional Commission: 30%- Min Comm 25% at 75% LR - Sliding .5-to-1 to a 30% comm at a 65% LR- Sliding 1-to-1 to a max comm of 35% at 60% LR

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Deficit Carryforward Example

w/o CF w/ CF Prob w/o CF w/ CF40.0% 45.0% 3.49% 35.0% 35.0% Last Year's Treaty LR: 80.0%57.5% 62.5% 8.23% 35.0% 32.5% Deficit Carryforward: 5.0%62.5% 67.5% 15.22% 32.5% 28.8% (80.0% - 75.0% = 5.0%)67.5% 72.5% 19.77% 28.8% 26.3% Current Expected LR: 71.5%72.5% 77.5% 19.30% 26.3% 25.0% Expected LR w/ CF: 76.5%77.5% 82.5% 14.94% 25.0% 25.0%85.0% 90.0% 14.79% 25.0% 25.0% Comm LR Slide95.0% 100.0% 3.60% 25.0% 25.0% Min 25.0% 75.0% 50.0%

150.0% 155.0% 0.66% 25.0% 25.0% Prov 30.0% 65.0% 100.0%225.0% 230.0% 0.00% 25.0% 25.0% Max 35.0% 60.0%71.5% 76.5% 28.3% 26.8%

Exp Loss Ratios Ceding Comm

• Option 1 - Add Deficit Carryforward % to each simulated LR and recalculate average commission.

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Deficit Carryforward Example

Last Year's Treaty LR: 80.0%Ceding Deficit Carryforward: 5.0%

Exp LR Prob Comm (80.0% - 75.0% = 5.0%)40.0% 3.49% 35.0% Current Expected LR: 71.5%57.5% 8.23% 32.5% Expected LR w/ CF: 76.5%62.5% 15.22% 28.8%67.5% 19.77% 26.3% Original Comm LR Slide72.5% 19.30% 25.0% Min 25.0% 75.0% 50.0%77.5% 14.94% 25.0% Prov 30.0% 65.0% 100.0%85.0% 14.79% 25.0% Max 35.0% 60.0%95.0% 3.60% 25.0%

150.0% 0.66% 25.0% Shifted Comm LR Slide225.0% 0.00% 25.0% Min 25.0% 70.0% 50.0%71.5% 26.8% Prov 30.0% 60.0% 100.0%

Max 35.0% 55.0%

• Option 2 - Shift Sliding Scale Commission terms.

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DCF/Multi-Year Block

• Question: How much credit do you give an account for Deficit Carryforwards, other than using the CF from the previous year (e.g. unlimited CFs)?

• Can estimate using an average of simulated “years”, but this method should be used with caution:

– Assumes independence (probably unrealistic)

– Accounts for both Deficit and Credit carryforwards

– Deficits are often forgiven, treaty terms may change, or treaty may be terminated before the benefit of the deficit carryforward is felt by the reinsurer.

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DCF/Multi-Year Block - Example

3-YearYear 1 Year 2 Year 3 Block

Average LR 71.52% 71.39% 71.69% 71.54%Std Dev 9.98% 9.95% 10.08% 5.84%

Avg Comm 28.25% 28.28% 28.20% 27.39%

Simulation1 69.62% 69.42% 52.09% 63.71%2 67.96% 63.91% 68.91% 66.93%3 77.54% 71.13% 74.77% 74.48%4 73.85% 58.66% 46.96% 59.82%5 88.54% 91.61% 72.24% 84.13%6 55.43% 79.21% 65.86% 66.83%7 67.49% 78.55% 80.54% 75.53%8 71.83% 78.42% 73.05% 74.43%9 63.93% 59.58% 47.51% 57.01%

10 75.92% 70.11% 72.82% 72.95%g

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Loss Sharing Summary

• Modeling loss sharing provisions is easy.

• Selecting your expected loss and aggregate distribution is hard

• Steps to analyzing loss sharing provisions– Build aggregate loss distribution

– Apply loss sharing terms to each point on the loss distribution or to each simulated year

– Calculate probability weighted average of treaty results

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Additional Issues & Uses of Aggregate Distributions

• Correlation between lines of business• Aggregate distributions are just a guess• Reserving for loss sensitive treaty terms• Using aggregate distributions to measure risk & allocate

capitalCapital = 99th percentile Discounted Loss x Correlation Factor

• Fitting Severity Curves: Don’t Ignore Loss Development– Increases average severity

– Increases variance.

– See “Survey of Methods Used to Reflect Development in Excess Ratemaking” by Stephen Philbrick, CAS 1996 Winter Forum

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