Reserve Uncertainty 1999 CLRS by Roger M. Hayne, FCAS, MAAA Milliman & Robertson, Inc.

Reserves Are Uncertain? n Reserves are just numbers in a financial statement n What do we mean by “reserves are uncertain?” –Numbers are estimates of future payments »Not estimates of the average »Not estimates of the mode »Not estimates of the median –Not really much guidance in guidelines n Rodney’s presentation will deal with this more

Let’s Move Off the Philosophy n Should be more guidance in accounting/actuarial literature n Not clear what number should be booked n Less clear if we do not know the distribution of that number n There may be an argument that the more uncertain the estimate the greater the “margin” n Need to know distribution first

“Traditional” Methods n Many “traditional” reserve methods are somewhat ad-hoc n Oldest, probably development factor –Fairly easy to explain –Subject of much literature –Not originally grounded in theory, though some have tried recently –Known to be quite volatile for less mature exposure periods

“Traditional” Methods n Bornhuetter-Ferguson –Overcomes volatility of development factor method for immature periods –Needs both development and estimate of the final answer (expected losses) –No statistical foundation n Frequency/Severity (Berquist, Sherman) –Also ad-hoc –Volatility in selection of trends & averages

“Traditional” Methods n Not usually grounded in statistical theory n Fundamental assumptions not always clearly stated n Often not amenable to directly estimate variability n “Traditional” approach usually uses various methods, with different underlying assumptions, to give the actuary a “sense” of variability

Basic Assumption n When talking about reserve variability primary assumption is: Given current knowledge there is a distribution of possible future payments (possible reserve numbers) Given current knowledge there is a distribution of possible future payments (possible reserve numbers) n Keep this in mind whenever answering the question “How uncertain are reserves?”

Some Concepts n Baby steps first, estimate a distribution n Sources of uncertainty: –Process (purely random) –Parameter (distributions are correct but parameters unknown) –Specification/Model (distribution or model not exactly correct) n Keep in mind whenever looking at methods that purport to quantify reserve uncertainty

Why Is This Important? n Consider an example n “Usual” development factor projection method n Assume: –Reserves can be estimated by development factor method –Age-to-age factors lognormal –Age-to-age factors independent –You can estimate age-to-age parameters using observed factors

Conclusions n Use “customary” parameterization of lognormal (based on transformed normal) n Parameters for distribution of age-to-age factors can be estimated using: –  i = Average of logs of observed age-to-age factors –  i 2 = (Sample corrected) variance of logs of observed age-to-age factors

Conclusions n Given assumptions distributions of age-to-ultimate factors are lognormal with parameters: – i– i– i– i – i2– i2– i2– i2 n Given amounts to date one derives a distribution of possible future payments for one exposure year n Convolute years to get distribution of total reserves

Sounds Good -- Huh? n Relatively straightforward n Easy to implement n Gets distributions of future payments n Job done -- yes? n Not quite n Why not?

An Example n Apply method to paid and incurred development separately n Consider resulting distributions n What does this say about the distribution of reserves? n Which is correct?

A “Real Life” Example

What Happened? n Conclusions follow unavoidably from assumptions n Conclusions contradictory n Thus assumptions must be wrong n Independence of factors? Not really (there are ways to include that in the method) n What else?

What Happened? n Obviously the two data sets are telling different stories n What is the range of the reserves? –Paid method? –Incurred method? –Extreme from both? –Something else? n Main problem -- the method addresses only one method under specific assumptions

What Happened? n Not process (that is measured by the distributions themselves) n Is this because of parameter uncertainty? n No, can test this statistically (from normal distribution theory) n If not parameter, what? What else? n Model/specification uncertainty

Why Talk About This? n Almost every paper in reserve distributions considers –Only one method –Applied to one data set n Only conclusion: distribution of results from a single method n Not distribution of reserves

Discussion n Some proponents of some statistically- based methods argue analysis of residuals the answer n Still does not address fundamental issue; model and specification uncertainty n At this point there does not appear much (if anything) in the literature with methods addressing multiple data sets

Moral of Story n Before using a method, understand underlying assumptions n Make sure what it measures what you want it to n The definitive work may not have been written yet n Casualty liabilities very complex, not readily amenable to simple models

All May Not Be Lost n Not presenting the definitive answer n More an approach that may be fruitful n Approach does not necessarily have “single model” problems in others described so far n Keeps some flavor of “traditional” approaches n Some theory already developed by the CAS (Committee on Theory of Risk, Rodney Kreps, Chairman)

Collective Risk Model n Basic collective risk model: –Randomly select N, number of claims from claim count distribution (often Poisson, but not necessary) –Randomly select N individual claims, X 1, X 2, …, X N –Calculate total loss as T =  X i n Only necessary to estimate distributions for number and size of claims n Can get closed form expressions for moments (under suitable assumptions)

Adding Parameter Uncertainty n Heckman & Meyers added parameter uncertainty to both count and severity distributions n Modified algorithm for counts: –Select  from a Gamma distribution with mean 1 and variance c (“contagion” parameter) –Select claim counts N from a Poisson distribution with mean  –Select claim counts N from a Poisson distribution with mean  –If c 0, N is negative binomial

Adding Parameter Uncertainty n Heckman & Meyers also incorporated a “global” uncertainty parameter n Modified traditional collective risk model –Select  from a distribution with mean 1 and variance b –Select N and X 1, X 2, …, X N as before –Calculate total as T =   X i n Note  affects all claims uniformly

Why Does This Matter? n Under suitable assumptions the Heckman & Meyers algorithm gives the following: –E(T) = E(N)E(X) –Var(T)= (1+b)E(X 2 )+ 2 (b+c+bc)E 2 (X) n Notice if b=c=0 then –Var(T)= E(X 2 ) –Average, T/N will have a decreasing variance as E(N)= is large (law of large numbers)

Why Does This Matter? n If b  0 or c  0 the second term remains n Variance of average tends to (b+c+bc)E 2 (X) n Not zero n Otherwise said: No matter how much data you have you still have uncertainty about the mean n Key to alternative approach -- Use of b and c parameters to build in uncertainty

If It Were That Easy … n Still need to estimate the distributions n Even if we have distributions, still need to estimate parameters (like estimating reserves) n Typically estimate parameters for each exposure period n Problem with potential dependence among years when combining for final reserves

CAS To The Rescue n CAS Committee on Theory of Risk commissioned research into –Aggregate distributions without independence assumptions –Aging of distributions over life of an exposure year n Paper on the first finished, second nearly so n Will help in reserve variability n Sorry, do not have all the answers yet