Boot Camp on Reinsurance Pricing Techniques – Loss Sensitive Treaty

Provisions

August 2007Jeffrey L, Dollinger, FCAS

2

Introduction to Loss Sensitive Provision Definition: A reinsurance contract provision that varies the ceded

premium, loss, or commission based upon the loss experience of the contract

Purposes• Client shares in ceded experience & could be incented to care

more about the reinsurer’s results• Can compensate for differences between reinsurer and client view

of reinsurance program expected loss 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 pro-rata 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 Profit Commission: 50% after 10% Reinsurer’s Margin Ceding Commission = 30% Loss ratio must be less than 60% for us to pay a profit

commission Contract Expected Loss Ratio = 70% $1 Premium - $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%.

• Giving back some profits below a 60% loss ratio has a cost

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Cost of Profit Commission: Simple Quantification Earthquake exposed California property pro-rata 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 lossExpected Cost of Profit Comm = Profit Comm Costs 10% of Premium x 90% Probability of

No EQ + 0% Cost of PC x 10% Probability of EQ Occurring = 9% of

Premium

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Basic Mechanics of Analyzing Loss Sensitive Provisions Build aggregate loss distribution Apply loss sensitive terms to each point

on the loss distribution or to each simulated year

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

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

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 - Two Methods Fit statistical distribution to on level loss ratios

• Reasonable for pro-rata 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

roba

bilit

y

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Fitting a Lognormal Distribution to Projected Loss Ratios Fitting the lognormal

^2 = LN(CV^2 + 1) = LN(mean) - ^2/2Mean = Selected Expected Loss RatioCV = 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

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 )

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Sample Lognormal Loss Ratio Distribution

On Level ModeledYear LR CDF LR1998 65.5% 10.0% 48.8%1999 70.0% 20.0% 52.6%2000 55.0% 30.0% 55.5%2001 48.0% 40.0% 58.1%2002 72.0% 50.0% 60.6%2003 65.0% 60.0% 63.3%2004 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

Modeled LR = Exp(MU+Normsinv(CDFi)*Sigma)

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Is the resulting LR distribution reasonable?

Compare resulting distribution to historical results• On level LR’s should be the focus, but don’t

completely ignore untrended ultimate LR’s. Potential for cat or shock losses not captured

within historical experience Degree to which trended past experience is

predictive of future results for a book Actuary and underwriter should discuss the above

issues If the distribution is not reasonable, adjust the CV

selection.

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Process and Parameter Uncertainty Process Uncertainty: Random fluctuation of results

around the expected value. Parameter Uncertainty: Do you really know the true

mean of the loss ratio distribution for the upcoming year?• Are your trend, loss development & rate change

assumptions correct?• For this book, are past results a good indication of

future results?• Changes in mix and type of business• Changes in management or philosophy• Is the book growing, shrinking or stable

Selected CV should usually be above indicated• 5 to 10 years of data does not reflect full range of

possibilities

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Modeling Parameter Uncertainty: One Suggestion Select 3 (or more) possible true 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 (reflects Parameter Uncertainty)

Simulate the loss ratio for the year modeled using the lognormal based on simulated expected loss ratio above & your selected CV (reflects Process Variance)

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Example of Modeling Parameter Uncertainty

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 2̂/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)

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Common Loss Sharing Provisions for Pro-rata 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

If the expected loss ratio is 65% is the expected commission 30%?

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Sliding Scale Commission - Solution

Low HighAvg LR in Band Probability

Ceding Comm @ avg LR in

Band

CR @ 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 expected 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% 50.0% 45.0% 14.23% 0.0% 45.0%50.0% 60.0% 55.0% 33.82% 0.0% 55.0%60.0% 65.0% 62.5% 17.47% 0.0% 62.5%65.0% 70.0% 67.5% 13.71% 0.0% 67.5%70.0% 75.0% 72.5% 9.28% 1.3% 71.3%75.0% 80.0% 77.5% 5.58% 3.8% 73.8%80.0% 85.0% 82.5% 3.05% 5.0% 77.5%85.0% 100.0% 92.5% 2.61% 5.0% 87.5%

100.0% 200.0% 135.0% 0.25% 5.0% 130.0%200.0% 300.0% 228.0% 0.00% 5.0% 223.0%

Prob Wtd Avg: 61.5% 0.6% 60.9%

Loss Ratio Band

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Modeling Property Treaties with Significant Cat Exposure Model non-cat & cat LR’s separately

• Non Cat LR’s fit to a lognormal curve• Cat LR distribution produced by commercial

catastrophe model Combine (convolute) the non-cat & cat loss ratio

distributions Alternate easier method: Simulate non-cat loss

ratio, then simulate cat loss ratio

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Convoluting Non-cat & 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 the mean loss ratio used to calculate

the lognormal parameters to cause the aggregate distribution to 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 pro-rata treaties Usually inappropriate for excess of loss contracts

• Does not reflect the hit or miss nature of many excess of loss contracts

• Understates probability of zero loss• May understate the 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 Most aggregate distribution approaches assume

that frequency and severity are independent Different Approaches

• Simulation (Focus of this presentation)• Numerical Methods

• Heckman Meyers – Fast calculating approximation to aggregate distribution

• Panjer Method – • Select discrete number of possible severities (i.e. create

5 possible severities with a probability assigned to each)• Convolutes discrete frequency and severity distributions.

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

Software that can be used for simulations• @Risk• Excel

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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,...

f(x|) is the probability of x losses, given a mean claim count of

x! = x factorial, i.e. 3! = 3 x 2 x 1 = 6Poisson distribution assumes the mean

and variance of the claim count distribution are equal.

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Fitting a Poisson Claim Count Distribution Trend claims from ground up, then slot to

reinsurance layer. Estimate ultimate claim counts by year by

developing trended claims to layer. Multiply trended claim counts by frequency trend

factor to bring them to the frequency level of the upcoming treaty year.

Adjust for change in exposure levels, i.e..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

(Note) 2006SPI at Trended Count Est Ult Annual Freq Trended Exposure Level

2006 Rate Counts Devel Trended Freq Trend to Ult Claim Adj ClaimYear Level to Layer Factor Count Trend 2006 Count Factor Count1996 10,000 2.0 1.0 2.0 0.0% 1.104 2.21 1.60 3.53 1997 10,500 1.0 1.0 1.0 0.0% 1.104 1.10 1.52 1.68 1998 11,025 1.0 1.0 1.0 0.0% 1.104 1.10 1.45 1.60 1999 11,576 1.0 1.1 1.1 0.0% 1.104 1.16 1.38 1.60 2000 12,155 3.0 1.1 3.3 0.0% 1.104 3.64 1.32 4.80 2001 12,763 - 1.2 - 0.0% 1.104 - 1.25 - 2002 13,401 - 1.3 - 2.0% 1.082 - 1.19 - 2003 14,071 - 1.5 - 2.0% 1.061 - 1.14 - 2004 14,775 1.0 2.0 2.0 2.0% 1.040 2.08 1.08 2.25 2005 15,513 1.0 3.5 3.5 2.0% 1.020 3.57 1.03 3.68 2006 16,000 2.0%

Average: 1.92 Variance: 2.82

Note: Exposure Adj Factor Yr i = 2006 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 simulation

• Simulate (Poisson expected count)• Using simulated expected claim count, simulate claim

count for the year. Negative Binomial is the preferred distribution

• Reflects some 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 Using a Poisson Distribution 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 (pick the claim count with the CDF just below the simulated 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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Negative Binomial Contagion Parameter

Determine contagion parameter, c, of claim count distribution:

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

then c=0If it is negative binomial, then c>0, i.e.

variance is greater than the mean 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; c = contagion parameter

• 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

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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 Mixed Exponential (currently used by ISO) Pareto Truncated Pareto. This curve was used by ISO before moving to the Mixed

Exponential and will be the focus of this presentation.• The ISO Truncated Pareto focused on modeling the

larger claims. Typically those over $50,000

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Truncated Pareto

Truncated Pareto Parameterst = truncation point. s = average claim size of losses below truncation pointp = probability claims are smaller than truncation pointb = pareto scale parameter - larger b results in larger unlimited average lossq = pareto shape parameter - lower q results in thicker tailed distribution

Cumulative Distribution FunctionF(x) = 1 - (1-p) ((t+ b)/(x+ b))^qWhere 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 Prob (Loss > Reins Att. Pt)• If there is a 95% chance that loss is below attachment point,

then the transformed CDF (TCDF) is between 0.95 and 1.00.

Find simulated ground up loss, x, that corresponds to simulated TCDFDoing 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 layer

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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 Summary

Claim 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%• Minimum/Margin: 3%• Maximum: 15%• Ceded Rate = Minimum/Margin +

Ceded Loss as % of SPI x 1.1; subject to a maximum rate of 15%.

Why did 100/80 x burn subject to min and max rate become extinct?

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Swing Rated Premium - Example Burn (ceded loss / SPI) = 10%. Rate = 3% + 10%

x 1.1 = 14% Burn = 2%. Rate = 3% + 2% x 1.1 = 5.2%. Burn = 14%. Calculated Rate = 3% + 14% x 1.1 =

18.4%. Rate = 15% maximum rate

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Swing Rated Premium Example

Swing Rating Terms: Ceded premium is adjusted to equal to a 3% minimum rate + ceded loss times 1.1 loading factor, subject to 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/Margin Prov Rate Max Rate

Loss Load

FactorSwing Rated Terms 3.0% 10.0% 15.0% 110.0%

FinalLow High Average Probability Rate

0.0% 0.0% 0.0% 9.0% 3.0%0.0% 2.5% 1.3% 6.0% 4.4%2.5% 5.0% 3.8% 9.0% 7.1%5.0% 7.5% 6.3% 10.2% 9.9%7.5% 10.0% 8.8% 11.4% 12.6%

10.0% 12.5% 11.3% 15.0% 15.0%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% 11.8%

Proj LR = Expected Burn/Expected Final Rate 93.8%

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 - Solution

Annual 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 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 limit Treaty Aggregate Limit =

= Risk Limit x (1 + number of Reinstatements)

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Limited Reinstatements Example

$1 million xs $1 million layer1 reinstatement

Ground up Ceded Ground up Ceded Ground up 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 should 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 = $250,000Prorata as to amount 100% as to time

Ground Up Ceded Reinst Ground Up Ceded Reinst Ground Up 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

Ground Up Ceded Reinst Ground Up Ceded Reinst Ground Up 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

Loss$000's Probability

- 68.00%1,000 25.00%2,000 4.00%3,000 2.00%4,000 1.00%

Reinsurance Treaty:$1 mil xs $1 milUpfront Premium = 400K2 Reinstatements: 1st at 50%, 2nd at 100%

Using the aggregate distribution of losses to the layer 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 Loss Netto Layer of Reinst Reinst Total$000's ProbabilityLimitation Premium Premium

- 68.00% - - 400 1,000 25.00% 1,000 200 600 2,000 4.00% 2,000 600 1,000 3,000 2.00% 3,000 600 1,000 4,000 1.00% 3,000 600 1,000

Prob Wtd Avg: 420 92 492Projected Loss Ratio: 85.4%

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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 Carry forward

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

Example:

Comm LR SlideMin 25.0% 75.0%

0.5 to 1Prov 30.0% 65.0%

1 to 1Max 35.0% 60.0%

Defecit Carry Forward: 5% of Premium

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Deficit Carry forward 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%57.5% 8.23% 32.5% Original Comm LR Slide62.5% 15.22% 28.8% Min 25.0% 75.0% 50.0%67.5% 19.77% 26.3% Prov 30.0% 65.0% 100.0%72.5% 19.30% 25.0% Max 35.0% 60.0%77.5% 14.94% 25.0%85.0% 14.79% 25.0% Shifted Comm LR Slide95.0% 3.60% 25.0% Min 25.0% 70.0% 50.0%

150.0% 0.66% 25.0% Prov 30.0% 60.0% 100.0%225.0% 0.00% 25.0% Max 35.0% 55.0%71.5% 26.8%

• Solution - Shift Sliding Scale Commission terms.

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DCF/Multi-Year Block

• Question: How much credit do you give an account for Deficit Carry forwards, 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 years are independent (probably unrealistic)

–Treaty terms may change, or treaty may be terminated before the benefit of the deficit carry forward is felt by the reinsurer. Also, reinsurer with deficit could be replaced by new 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%

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Technical Summary

Modeling loss sensitive provisions is easy. Selecting your expected loss and aggregate

distribution is hard Steps to analyzing loss sensitive provisions

• Build aggregate loss distribution• Apply loss sensitive 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 – often higher than you

think due to directives from upper management influencing multiple lines of business

Reserving for loss sensitive treaty terms Some companies Use aggregate distributions to measure risk

& allocate capital. One hypothetical example:Capital = 99th percentile Discounted Loss x Correlation

Factor Fitting Severity Curves: Don’t Ignore Loss Development

• Increases average severity• Increases variance – claims spread as they settle.• See “Survey of Methods Used to Reflect Development in

Excess Ratemaking” by Stephen Philbrick, CAS 1996 Winter Forum

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Risk Transfer – Governing Regulations

FASB 113: A reinsurance contract should be booked using deposit accounting unless:• “The reinsurer assumes significant insurance risk”

• Insurance risk not significant if “the probability of a significant variation in either the amount or timing of payments by the reinsurer is remote”

• “It is reasonably possible that the reinsurer may realize a significant loss from the transaction.

• 10/10 Rule of Thumb: Is there a 10% chance that the reinsurer will have a loss of at least 10% of premium on a discounted basis

• Calculation excludes brokerage and reinsurer internal expense. Statutory Statements

• SSAP 62 is governing document: requirements are similar to FASB 113.• Also requires CEO’s and CFO’s attestation under penalty of perjury that

• No side agreements exist that alter reinsurance terms• For contracts where risk transfer is not self-evident, documentation

concerning economic intent and risk transfer analysis is available• Reporting entity in compliance with SSAP 62 & proper controls in

place Recent Developments: NY State and FASB proposed bifurcation proposals.

Very troubling, but seem to be going nowhere

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Report of 2005 CAS Working Party on Risk Transfer – Key Findings Three step risk transfer testing process

• Does contract transfer substantially all risk of ceding company? If yes, no testing required• Is reinsurer’s risk position the same as the ceding

companies?• Is risk transfer reasonably self evident? If yes, stop

• Facultative, Cat XOL, XOL contracts without significant loss sensitive features, and contracts with immaterial premium (less than $1 mil of premium or 1% of GEP)

• Remaining contracts: Perform risk transfer testing.• Calculate recommended risk metric & compare to

critical thresh-hold• Aggregate distribution should contemplate process &

parameter uncertainty• Recommend that 10/10 rule be replaced with

Expected Reinsurer Deficit Calculation (ERD) Above are only CAS’s working party recommendations.

Actual procedures and methods are determined by company management and accounting firm

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Exp. Reinsurer Deficit (ERD) Example

ERD = p * T / Premiump = Probability of loss to

reinsurer = 7%T = Average Severity of Loss

given a loss occurred= (3.5% * 35% + 2% * 80 +

1.5% * 125) / 7% = 67.1ERD = 7% * 67.1 / 10 = 47% CAS Working Party

implied a standard that ERD must be above 1%, which equates to 10/10 rule, although it is less conservative

Reinsurance Layer: 50 xs 50Ceded Premium: 10 (amounts in millions)

Loss to Layer Prob

Present Value of

Reinsurer Result

- 93.0% 10 50 3.5% (35)

100 2.0% (80) 150 1.5% (125)

Example from CAS article by David Ruhm and Paul Brehm, “Risk Transfer Testing of Reinsurance Contracts: A Summary of the Report by the CAS Research Working Party”

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Concluding Comment

Aggregate distributions are a critical element in evaluating the profitability of business.

They are frequently produced by (re)insurers as a risk management tool.

They are being used on a broader spectrum of contracts to review risk transfer.

Some accountants and regulators seem to treat these aggregate distributions as if they were gospel.

Critical to effectively communicate the difficulties in projecting aggregate distributions of future results.• Need to make regulators and accountants

understand the degree of parameter uncertainty