The Credibility of the Overall Rate Indication Paper by Joseph Boor, FCAS Florida Office of Insurance Regulation Presented by Glenn Meyers, FCAS, MAAA, ISO
Background-Why is this needed? Actuaries in North America and elsewhere base general insurance rates on claims data, not tables. Claims data often has too few claims to fully rely on it. Actuaries combine claims data with some other data in a credibility formula cost estimate = Z*(claims data) + (1-Z)*(other data) Z is called the credibility
Background-Why is this needed? For making broad (i.e., not individual insured) rates, two general approaches are regularly used Square root rule [(proxy for expected claims or losses)/(some F)] 1/2 F is full credibility standard Bühlmann credibility (expected loss proxy P )/(P+K) K is based on variance structure of two data elements combined
Background-Why is this needed? Problems with square root rule Square root rule is not optimum credibility in terms of predicting claims costs Merely guarantees, up to a certain probability, that the effect of random fluctuations in the claims data is contained at a certain level. Does not address fluctuations or uncertainty in the other data (called complement of credibility)
Background-Why is this needed? Problems with Bühlman credibility Not designed for the overall rate indication, suitable for making rates for a class that is part of a much larger line of business. Most actuaries do not know a good method to compute the constant K
What is needed Credibility formula for credibility producing greatest accuracy estimate of future claims costs Suitable for use for a line of business as a whole Complement of credibility is claims costs expected in current rate, updated for inflation, etc. Reasonably simple formula Advice to practitioners on how to compute key constants
Overview of Results Formula is Z = d 2 Ł 1+ 2e 2 e 4 2 d 2-1 ł
Constants in formula Constants are: δ² is the coefficient of variation(squared) of the change in claims cost levels factor each year Each year s cost change is (1+T)*(1+Δ) where δ² is the variance of Δ and everything else is constant. ε² is the coefficient of variation(squared) of the observed results around the real underlying loss costs each year Each year s observed claims costs are (L)*(1+E) where ε² is the variance of E, X is the true expected costs and everything else is constant.
Underlying assumptions Almost all credibility models involve assumptions To understand δ² and ε² you must understand the assumptions of this model The true underlying expected losses follow a geometric Brownian motion The data does not contain the true expected losses from each prior year, the expected losses are imperfectly observed
Geometric Brownian motion assumption Expected change in losses for each year is increase of T E.g., next years losses L(y+1) have expected value L(y)*(1+T) Actual changes vary from expected by multiplicative factor of (1+ Δ) so L(y+1) =L(y)(1+T)*(1+ Δ) Δ s for each consecutive year are independent, but identically distributed with mean 0 and variance δ².
Geometric Brownian motion assumption GBM requires slightly more restrictive assumptions than those of prior page GBM not required for the formula GBM is however, a very popular financial model that meets criteria above. Model should also accommodate the Levy processes that are also popular with financial professionals
Observation Error Typically, actuary s data never quite represents the true expected claims costs Law of Large Numbers never guarantees exact calculation of costs, just approximation More importantly much US data is loss development estimates, not true costs
Observation Error Model assumes each years observed data is multiplicatively distributed around true expected costs Instead of seeing true costs L(y) we see Lˆ (y + 1) = L(y) [ 1+ E(y)] E(y) s are independent and identically distributed, with mean zero and variance ε².
Estimating δ² and ε² Reference item- Bühlmann credibility significantly underused in US due to lack of general knowledge on how to compute constant K.
Estimating δ² and ε² Paper presents methods for estimation Methods based on subtracting squared differences of L ˆ( y)' s Squared differences between terms different #years apart are different linear combinations of δ² and ε². Credibility that would have worked in the past Fitting parameters across a wider group of data Estimating δ² or ε² structurally (e.g estimating ε² from loss development variance and collective risk) Suggest use two methods and understand the error in each.
Other relevant items Formula is for steady-state credibility and updating the rate with one year of data. Over a series of papers, the author intends to expand the analysis to embrace non-steady state credibility, conversion from non-optimal credibility, multiple years of data, rates made less often that annually, etc.
Other relevant items For geometric Brownian motion, the formula is actually slightly different The formula given is essentially the formula for linear (standard) Brownian motion with additive error It is a high quality approximation to the slightly more complex formula for geometric Brownian motion
Other relevant items Other papers have dealt with this issue, but not to this level E.g. Gerber and Jones (Transactions of the Society of Actuaries, 1975) dealt essentially with linear type problems and had more elegant formulas This paper uses nonlinear geometric Brownian motion, more realistic for North American general insurance Paper sapproach is more calculation-oriented Technique works in a wider variety of situations and is in itself an important aspect of the paper.
Other relevant items This paper employs a more calculationoriented approach The technique can be applied to a wider variety of situations and is in itself an important aspect of the paper. Involves computing the variance of the observed data from the detrended true future cost level Lengthy basic statistics and summing of mathematical series does lead to a more robust approach
Enhancement to Bühlmann model Bonus item in paper- Several authors, including this presenter, have proven that, when losses change from year to year as with this model, the Z= P/(P+K) model should be Z = P/[P(1+J)+K] An appendix in the paper shows that this is also true when the data has loss development uncertainty.
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Contact Details Joseph Boor, FCAS Actuary, Office of Insurance Regulation 200 East Gaines St, Tallahassee, FL 32399, USA Phone: 00-1-850-413-5330 Email: joe.boor@floir.com Email2: joeboor@embarqmail.com