Milliman USA Reserve Uncertainty by Roger M. Hayne, FCAS, MAAA Milliman USA Casualty Loss Reserve Seminar September 23-24, 2002

Milliman USA Reserves Are Uncertain? Reserves are just numbers in a financial statement Reserves are just numbers in a financial statement What do we mean by “reserves are uncertain?” What do we mean by “reserves are uncertain?” – Numbers are estimates of future payments Not estimates of the average Not estimates of the average Not estimates of the mode Not estimates of the mode Not estimates of the median Not estimates of the median – Not really much guidance in guidelines Rodney Kreps has more to say on this subject Rodney Kreps has more to say on this subject

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

Milliman USA “Traditional” Methods Many “traditional” reserve methods are somewhat ad-hoc Many “traditional” reserve methods are somewhat ad-hoc Oldest, probably development factor 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

Milliman USA “Traditional” Methods Bornhuetter-Ferguson 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 Frequency/Severity (Berquist, Sherman) Frequency/Severity (Berquist, Sherman) – Also ad-hoc – Volatility in selection of trends & averages

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

Milliman USA Basic Assumption When talking about reserve variability primary assumption is: 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) Keep this in mind whenever answering the question “How uncertain are reserves?” Keep this in mind whenever answering the question “How uncertain are reserves?”

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

Milliman USA Why Is This Important? Consider “usual” development factor projection method, C ik accident year i, paid by age k Consider “usual” development factor projection method, C ik accident year i, paid by age k Assume: Assume: – There are development factors f i such that E(C i,k+1 |C i1, C i2,…, C ik )= f k C ik – {C i1, C i2,…, C iI }, {C j1, C j2,…, C jI } independent for i  j – There are constants  k such that Var(C i,k+1 |C i1, C i2,…, C ik )= C ik  k 2

Milliman USA Conclusions Following Mack (ASTIN Bulletin, v. 23, No. 2, pp ) Following Mack (ASTIN Bulletin, v. 23, No. 2, pp ) are unbiased estimates for the development factors f i are unbiased estimates for the development factors f i Can also estimate standard error of reserve Can also estimate standard error of reserve

Milliman USA Conclusions Estimate of mean squared error (mse) of reserve forecast for one accident year: Estimate of mean squared error (mse) of reserve forecast for one accident year:

Milliman USA Conclusions Estimate of mean squared error (mse) of the total reserve forecast: Estimate of mean squared error (mse) of the total reserve forecast:

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

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

Milliman USA “Real Life” Example Paid and Incurred as in handouts (too large for slide) Paid and Incurred as in handouts (too large for slide) Results Results PaidIncurred Case Reserve $96,917 Reserve Est. $358,45390,580 s.e.(Est.)41,63913,524

Milliman USA A “Real Life” Example

Milliman USA A “Real Life” Example

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

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

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

Milliman USA Why Talk About This? Most papers in reserve distributions consider Most papers in reserve distributions consider – Only one method – Applied to one data set Only conclusion: distribution of results from a single method Only conclusion: distribution of results from a single method Not distribution of reserves Not distribution of reserves

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

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

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

Milliman USA Collective Risk Model Basic collective risk model: 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 Only necessary to estimate distributions for number and size of claims Only necessary to estimate distributions for number and size of claims Can get closed form expressions for moments (under suitable assumptions) Can get closed form expressions for moments (under suitable assumptions)

Milliman USA Adding Parameter Uncertainty Heckman & Meyers added parameter uncertainty to both count and severity distributions Heckman & Meyers added parameter uncertainty to both count and severity distributions Modified algorithm for counts: 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

Milliman USA Adding Parameter Uncertainty Heckman & Meyers also incorporated a “global” uncertainty parameter Heckman & Meyers also incorporated a “global” uncertainty parameter Modified traditional collective risk model 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 Note  affects all claims uniformly Note  affects all claims uniformly

Milliman USA Why Does This Matter? Under suitable assumptions the Heckman & Meyers algorithm gives the following: 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) Notice if b=c=0 then 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)

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

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

Milliman USA An Example Consider the data set included in the handouts Consider the data set included in the handouts This is hypothetical data but based on a real situation This is hypothetical data but based on a real situation It is residual bodily injury liability under no-fault It is residual bodily injury liability under no-fault Rather homogeneous insured population Rather homogeneous insured population

Milliman USA An Example (Continued) Applied several “traditional” actuarial methods Applied several “traditional” actuarial methods – Usual development factor – Berquist/Sherman – Hindsight reserve method – Adjustments for Relative case reserve adequacy Relative case reserve adequacy Changes in closing patterns Changes in closing patterns

Milliman USA An Example (Continued) Reserve Estimates by Method AccidentPaidAdjustedCD Adjusted Paid YearIncurredDevel.Sev.Pure Prem.HindsightIncurredDevel.Sev.Pure Prem.Hindsight ,1431,7601,9091, ,9361,8421, ,3356,8475,5835,128 2,3486,0005,7905,2202, ,37119,76816,24613,45114,42810,39117,35216,43313,3998, ,78744,63136,88729,23232,19926,04839,24136,43128,51219, ,21183,76073,98761,84662,97455,73479,66770,24657,19243, ,093130,90795,28395,18578,61679,573154,26887,62584,68872,157

Milliman USA An Example (Continued) Now review underlying claim information Now review underlying claim information Make selections regarding the distribution of size of open claims for each accident year Make selections regarding the distribution of size of open claims for each accident year – Based on actual claim size distributions – Ratemaking – Other Use this to estimate contagion (c) value Use this to estimate contagion (c) value

Milliman USA An Example (Continued) AccidentReserveUnpaidSingle ClaimImplied YearSelectedStd. Dev.CountsAverageStd. Dev.c Value 19861, ,80218, ,2601, ,90919, ,8663, ,89420, ,2126,4281,89415,95123, ,51610,1983,34718,67827, ,01419,1664,07122,11132,

Milliman USA An Example (Continued) Thus variation among various forecasts helps identify parameter uncertainty for a year Thus variation among various forecasts helps identify parameter uncertainty for a year Still “global” uncertainty that affects all years Still “global” uncertainty that affects all years Measure this by “noise” in underlying severity Measure this by “noise” in underlying severity

Milliman USA An Example (Continued) AccidentSeverityEstimate YearSelectedFitted of 1/  19867,7237, ,5018, ,5778, ,9199, ,7399, ,19410, Variance0.019

Milliman USA An Example (Continued)

Milliman USA CAS To The Rescue Still assumed independence Still assumed independence CAS Committee on Theory of Risk commissioned research into CAS Committee on Theory of Risk commissioned research into – Aggregate distributions without independence assumptions – Aging of distributions over life of an exposure year Will help in reserve variability Will help in reserve variability Sorry, do not have all the answers yet Sorry, do not have all the answers yet