Boot Camp on Reinsurance Pricing Techniques – Loss Sensitive Treaty Provisions July 2005

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  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

3 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.

4 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?

5 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%.

6 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 loss Expected 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

7 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

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

9 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.

10 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.

11 Skewness of Lognormal Distribution

12 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  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  )

13 Sample Lognormal Loss Ratio Distribution

14 Is the resulting LR distribution reasonable?  Compare resulting distribution to historical results Focus on level LR’s, 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.

15 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

16 Modeling Parameter Uncertainty: One Suggestion  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 (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)

17 Example of Modeling Parameter Uncertainty

18 Common Loss Sharing Provisions for Pro-rata Treaties  Profit Commissions Already covered  Sliding Scale Commission  Loss Ratio Corridor  Loss Ratio Cap

19 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

20 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%?

21 Sliding Scale Commission - Solution

22 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 ???

23 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%

24 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

25 Loss Ratio Corridor Example – Solution

26 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

27 Convoluting Non-cat & Cat LR’s - Example

28 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

29 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

30 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 Excel

31 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 = 6 Poisson distribution assumes the mean and variance of the claim count distribution are equal.

32 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

33 Example of Simulated Claim Count

34 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

35 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

36 Negative Binomial Contagion Parameter  Determine contagion parameter, c, of claim count distribution: (  ^2 /  ) = 1 + c  If the claim count distribution is Poisson, then c=0 If it is negative binomial, then c>0, i.e. variance is greater than the mean  Solve for the contagion parameter: c = [(  ^2 /  ) - 1] / 

37 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

38 Year 1 Simulated Negative Binomial Claim Count

39 Year 1 Simulated Negative Binomial Claim Count

40 Year 2 Simulated Negative Binomial Claim Count

41 Year 2 Simulated Negative Binomial Claim Count

42 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

43 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 - larger b results in larger unlimited average loss q = pareto shape parameter - lower q results in thicker tailed distribution  Cumulative Distribution Function F(x) = 1 - (1-p) ((t+ b)/(x+ b))^q Where x>t

44 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  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 layer

45 Year 1 Loss # 1 Simulated Severity to the Layer

46 Year 1 Loss # 2 Simulated Severity to the Layer

47 Simulation Summary

48 Common Loss Sharing Provisions for Excess of Loss Treaties  Profit Commissions Already covered  Swing Rated Premium  Annual Aggregate Deductibles  Limited Reinstatements

49 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?

50 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

51 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

52 Swing Rated Premium Example - Solution

53 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?

54 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?

55 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.

56 Annual Aggregate Deductible Example - Solution

57 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)

58 Limited Reinstatements Example

59 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

60 Reinstatement Premium Example 1

61 Reinstatement Premium Example 2

62 Reinstatement Example 3  Reinsurance Treaty: $1 mil xs $1 mil Upfront Premium = 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

63 Reinstatement Example 3 – Solution

64 Reinstatement Example 4  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

65 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:

66 Deficit Carry forward Example Solution - Shift Sliding Scale Commission terms.

67 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 independence (probably unrealistic) – Accounts for both Deficit and Credit carry forwards – Deficits are often forgiven, treaty terms may change, or treaty may be terminated before the benefit of the deficit carry forward is felt by the reinsurer.

68 DCF/Multi-Year Block - Example

69 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

70 Additional Issues & Uses of Aggregate Distributions  Correlation between 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

71 Risk transfer  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.  SFAS 62 governs statutory accounting. Requirements are similar to FASB 113.  Recent regulator concerns have centered on pro-rata reinsurance.

72 Risk Transfer Recent Developments  New York State Draft Bifurcation Proposal: Bifurcation applies to any pro-rata treaty that contains one of the following features: profit commissions, sliding scale commissions, loss ratio corridors or caps, occurrence limits below an unspecificed % of premium, etc. Excess of loss and facultative contracts are excluded If above conditions are met, premium must be split as follows: Premium covering exposure in excess of the 90 th percentile of the loss distribution counts as reinsurance. The remaining premium should be booked as a deposit. Rule would be applied retroactively to business written 1/1/94 and later. Appears unlikely that NAIC will approve this proposal, but proposal emphasizes regulators concerns.

73 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.