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Credibility for Excess (Re)insurance Casualty Actuaries in Reinsurance (CARe) 2007 David R. Clark, Vice President Munich Reinsurance America, Inc.

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2 Brief Review of Credibility Theory3 Mashitz and Patrik Model7 The Problem of Dependence14 A Solution Using Relativities instead of Rates17 Discussion on unresolved questions21 Agenda

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3 Credibility for Excess (Re)insurance The purpose of applying Credibility Theory: Experience Rate= E[Loss | Account Loss Experience ] Exposure Rate = E[Loss | External Information] Final Rate= E[Loss | Account Loss Experience & External Information] The question: How do we calculate the “best” expected loss E[Loss] given ALL of the information that is available to us?

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4 Credibility for Excess (Re)insurance Linear Approximation for Bayesian Credibility:

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5 Credibility for Excess (Re)insurance Two Key assumptions: The two estimates are UNBIASED The information in both estimates should be relevant for the contract being priced. This means we are “shooting at the right target” (see next slide). The two estimates are INDEPENDENT We can modify our formula if there is dependence… We will look at an alternative way of addressing the dependence between experience and exposure rates.

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6 Credibility for Excess (Re)insurance Emmons Loses Gold Medal After Aiming at Wrong Target Monday, August 23, 2004; Page D16 Matt Emmons was just focusing on staying calm. He wishes he had been more concerned with where he was shooting. Emmons fired at the wrong target on his final shot, a simple mistake that cost the American a commanding lead in the 50-meter three-position rifle final and ruined his chance for a second gold medal. Ahead after nine shots and needing only to get near the bull's-eye to win, Emmons fired at the target in Lane 3 while he was shooting in Lane 2. He had cross-fired -- an extremely rare mistake in elite competition -- and received a score of zero. That dropped Emmons to eighth place at 1,257.4 points and lifted Jia Zhanbo of China to the gold at 1,264.5. "On that shot, I was just worrying about calming myself down and just breaking a good shot, and so I didn't even look at the number," said Emmons, 23. "I probably should have. I will from now on.“ © 2004 The Washington Post Company. Reprinted with permission

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7 Credibility for Excess (Re)insurance The 1990 paper by Mashitz & Patrik applies Bayesian Credibility to the problem of excess reinsurance treaty pricing. Assumptions in the Mashitz & Patrik Model: 1)Restrict the credibility formula to frequency 2)Each risk (treaty) has claim counts distributed as Poisson, the Poisson means for all of the risks in the portfolio are distributed as Gamma 3) For a given risk, each historical year has the same volume of exposure (we will relax this assumption later)

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8 Credibility for Excess (Re)insurance Credibility for Ground-Up Claim Counts: n i Actual number of claims in year “i” mNumber of years in the historical period E(λ)A Priori expected number of annual claims α1/CV λ 2 “shape” parameter of the prior gamma distribution for the distribution of mean frequencies

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9 Credibility for Excess (Re)insurance Credibility for Excess Counts, when Severity is Known: d“deductible” or excess attachment point n i (d)Actual number or claims above “d” in year “i” mNumber of years in the historical period E(λ)A Priori expected number of annual claims α1/CV λ 2 “shape” parameter of the prior gamma distribution for the distribution of mean frequencies qProbability that a ground-up loss would exceed “d”

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10 Credibility for Excess (Re)insurance Credibility for Excess Counts, when Severity is Unknown: The credibility constant “k” changes from α to the value:

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11 Credibility for Excess (Re)insurance Observations: The credibility assigned to the experience is based on the expected counts, NOT based on the actual counts. When severity is known, the “k” in the Credibility = n/(n+k) is the same for ground- up and excess counts. When the severity is not known with certainty, the credibility constant “k” is lower. This means that we give more credibility weight to the experience when the severity distribution is uncertain.

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12 Credibility for Excess (Re)insurance Sample for Including Growth and Development (relaxing assumption of constant exposure by year) All numbers for illustration only

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13 Credibility for Excess (Re)insurance How should we set the credibility constant “k” in practice? Mashitz and Patrik recommend a survey of questions based on consistency of the historical business, data quality, etc. Adjust “process variance” based on variance of the historical counts. Practical Rule of thumb is that “k” represents the number of expected claims for which you would assign 50%/50% weights between the two methods.

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14 Credibility for Excess (Re)insurance The problem of Dependence between the two estimates: Mashitz and Patrik consider an analogy with the application of credibility in primary insurance: Primary Insurance: Final Price = Actual Experience ·Z + Manual Rate·(1-Z) Excess Reinsurance: Final Price = Experience Rating·Z + Exposure Rating·(1-Z) But does this analogy really hold?

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15 Credibility for Excess (Re)insurance Excess Layer Subject Premium * ELR (but what is the source of the ELR?) Treaty Limit Treaty Retention Exposure Rating “Layers” and overall loss:

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16 Credibility for Excess (Re)insurance If the Expected Loss Ratio (ELR) used in the exposure rating is based on account experience, then it is not truly an a priori ELR. The reason for this is that in reinsurance we have subject premium, but not exposures and rates to calculate a true loss cost by layer. Mark Cockroft (2004) describes this: “…in the real world there are many instances when exposure and experience methods do interact already, blurring the credibility weighting” This is where the original INDEPENDENCE assumption is violated.

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17 Credibility for Excess (Re)insurance An Alternative Approach using Relativities instead of Rates: The industry-based severity distributions provide us with a means for “layering” losses, but they do not provide an absolute frequency to produce a rate. Instead, we typically base the exposure-rating on an ELR that is a ground-up experience rating. The “exposure-rate” therefore is already dependent on the excess experience. An alternative is to select a base layer – considered to be 100% credible – and use layer relativities to estimate higher layers. The final relativity is a credibility weighted average of an experience relativity and a relativity from the industry severity distribution.

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18 Credibility for Excess (Re)insurance Cumulative Distribution Function (CDF) for Severity

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19 Credibility for Excess (Re)insurance A full Bayesian model for excess layer counts: LetN 1 = # claims in lower (base) layer p= probability of a loss in lower layer reaching higher layer “survival ratio” N 2 = binomial random variable E[N 2 |p] = N 1 ·p Var( N 2 |p) = N 1 ·p·(1-p) Let the survival ratio p have a prior beta distribution, with parameters ν and ω. f(p) = constant·p v ·(1-p) ω E[p] = ν/(ν+ω)

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20 Credibility for Excess (Re)insurance The credibility-weighted average of the predictive survival ratio p is a linear average of the actual experience and the a priori expectation from the prior Beta distribution. If we have an estimate of expected claims for the lower layer, we can then calculate an estimate of expected claims to the higher layer. The credibility is based on the number of claims in the lower layer, or equivalently, on the expected number of claims in the higher layer. This procedure can be repeated for higher layers.

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21 Credibility for Excess (Re)insurance Unresolved Questions: Under what circumstances should multiple lines of business be combined? How to modify the credibility when only certain years are included in the “selected” loss cost. Adjusting credibility when data quality is poor or possibly irrelevant How to simultaneously include the credibility in the development pattern with the credibility in the severity curve.

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22 Credibility for Excess (Re)insurance Select Bibliography: Cockroft, Mark; Bayesian Credibility for Excess of Loss Reinsurance Rating; GIRO Conference 2004. Dale, Andrew; Most Honourable Remembrance: The Life and Work of Thomas Bayes; Springer 2003. Mashitz, Isaac, and Gary Patrik; Credibility for Treaty Reinsurance Excess Pricing; CAS Discussion Paper Program on Pricing, 1990. Philbrick, Stephen; An Examination of Credibility Concepts; CAS Proceedings, 1981. Venter, Gary; Credibility Theory for Dummies; CAS Forum, Winter 2003.

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Thank you very much for your attention. David R. Clark, Vice President Munich Reinsurance America, Inc. © Copyright 2007 Munich Reinsurance America, Inc. All rights reserved. The Munich Re America name is a mark owned by Munich Reinsurance America, Inc. The material in this presentation is provided for your information only, and is not permitted to be further distributed without the express written permission of Munich Reinsurance America. This material is not intended to be legal, underwriting, financial, or any other type of professional advice. Examples given are for illustrative purposes only.

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