Reserve Ranges Roger M. Hayne, FCAS, MAAA C.K. “Stan” Khury, FCAS, MAAA Robert F. Wolf, FCAS, MAAA 2005 CAS Spring Meeting

Changing Scene Changes: Changes: –Changes in the 2005 NAIC reporting requirements (best estimate, ranges, etc.) –SEC pending rule changes about disclosures with respect to items involving uncertainty –Pending changes in the reserving principles –Pending changes in the ASOP Unifying theme driving all of these changes: Unifying theme driving all of these changes: –A reserve is really a probability statement consisting of an amount x plus the probability that the final settlement will not exceed x

A Range – Gas or Electric? Start simple – a range around what? Start simple – a range around what? Accountants say it is to be a “reasonable estimate” of the unpaid claim costs Accountants say it is to be a “reasonable estimate” of the unpaid claim costs CAS says that “an actuarially sound loss reserve … is a provision, based on estimates derived from reasonable assumptions …” CAS says that “an actuarially sound loss reserve … is a provision, based on estimates derived from reasonable assumptions …”

First Question – An Estimate of What? An “estimate” of amount unpaid. An “estimate” of amount unpaid. Is it an estimate of the average amount to be paid? No Is it an estimate of the average amount to be paid? No It is an estimate of the most likely amount to be paid? No It is an estimate of the most likely amount to be paid? No It is an estimate of the amount to be paid It is an estimate of the amount to be paid

Simple Example Reserves as of 12/31/2005 Reserves as of 12/31/2005 Claim to be settled 1/1/2006 with immediate payment of $1 million times roll of fair die Claim to be settled 1/1/2006 with immediate payment of $1 million times roll of fair die All results equally likely so some accounting guidance says book low end ($1 million), others midpoint ($3.5 million) All results equally likely so some accounting guidance says book low end ($1 million), others midpoint ($3.5 million) Mean and median are $3.5 million Mean and median are $3.5 million

An Almost-Simple Example Reserves as of 12/31/2005 Reserves as of 12/31/2005 Claim to be settled 1/1/2006 as $1 million times toss of loaded die: Claim to be settled 1/1/2006 as $1 million times toss of loaded die: –Prob(x=1)=Prob(x=6)=1/4 –Prob(x=2)=Prob(x=5)=1/6 –Prob(x=3)=Prob(x=4)=1/12 What do you book now? What do you book now? Mean and median still $3.5 million Mean and median still $3.5 million “Most likely” is either $1 million or $6 million “Most likely” is either $1 million or $6 million

Traditional Approach Traditional actuarial methods: Traditional actuarial methods: –“Deestribution? We don’ need no steenkin’ deestribution.” Traditional methods give “an estimate” Traditional methods give “an estimate” No assumptions, thus no conclusions on distributions No assumptions, thus no conclusions on distributions There are stochastic versions of some methods (chain ladder, Bornhuetter- Ferguson) There are stochastic versions of some methods (chain ladder, Bornhuetter- Ferguson)

Traditional Estimates Traditional methods give “estimates” Traditional methods give “estimates” –Not estimates of the mean –Not estimates of the median –Not estimates of the mode –Not estimates of a percentile –Not estimates of any statistic of the distribution –Just “estimates” Distributions come only after added assumptions Distributions come only after added assumptions

Range of Reasonable Results Designed for traditional analysis Designed for traditional analysis Does not address or even talk about distributions Does not address or even talk about distributions Definition is “soft” and talks about results of “reasonable” methods under “reasonable” assumptions Definition is “soft” and talks about results of “reasonable” methods under “reasonable” assumptions Does not refer to the distribution of potential outcomes Does not refer to the distribution of potential outcomes

Reasonable? Range of reasonable results an attempt to quantify an actuary’s “gut feel” Range of reasonable results an attempt to quantify an actuary’s “gut feel” Typically you do a lot of methods Typically you do a lot of methods –If they “bunch up” you feel “good” –If they are “spread out” you feel “uncomfortable” In the end – quite subjective In the end – quite subjective

Model and Method A method is a general approach A method is a general approach –Chain ladder –Bornhuetter-Ferguson A model specifies an underlying process or distribution and the focus is in parameterizing the model A model specifies an underlying process or distribution and the focus is in parameterizing the model Many traditional actuarial forecasting approaches are methods and not models Many traditional actuarial forecasting approaches are methods and not models

Stochastic Methods Stochastic methods have assumptions about underlying models Stochastic methods have assumptions about underlying models Nearly all focus on a single data set (paid loss triangle, incurred loss triangle, etc.) Nearly all focus on a single data set (paid loss triangle, incurred loss triangle, etc.) Do not directly model multiple sources of information (e.g. counts, paid, and incurred at the same time) Do not directly model multiple sources of information (e.g. counts, paid, and incurred at the same time) Mack/Quarg method not yet stochastic Mack/Quarg method not yet stochastic

Some Vocabulary Components of uncertainty: Components of uncertainty: –Process –Parameter –Model/Specification Any true estimate of the distribution of outcomes should recognize all three Any true estimate of the distribution of outcomes should recognize all three

Process Uncertainty that cannot be avoided Uncertainty that cannot be avoided Inherent in the process Inherent in the process Example – the throw of a fair die Example – the throw of a fair die –You completely know the process –You cannot predict the result with certainty Usually the smallest component of insurance distribuitons (law of large numbers) Usually the smallest component of insurance distribuitons (law of large numbers)

Parameter Uncertainty about the parameters of models Uncertainty about the parameters of models The underlying process is know The underlying process is know Just the position of some “knobs” are not Just the position of some “knobs” are not Example – flip of a weighted coin Example – flip of a weighted coin –Uncertainty regarding the expected proportion of heads

Model/Specification The uncertainty that you have the right model to begin with The uncertainty that you have the right model to begin with Not just what distributions, but what form the model should take Not just what distributions, but what form the model should take Most difficult to estimate Most difficult to estimate Arguably unestimable for P&C insurance situations Arguably unestimable for P&C insurance situations

Distribution of Outcomes Combines all sources of uncertainty Combines all sources of uncertainty Gives potential future payments at point in time along with associated likelihood Gives potential future payments at point in time along with associated likelihood Must be estimated Must be estimated Estimation is itself subject to uncertainty, so we are not away from “reasonable” issues Estimation is itself subject to uncertainty, so we are not away from “reasonable” issues

Example Distribution I

Example Distribution II

What is Reasonable? I use a series of methods I use a series of methods My “range of reasonable estimates” is the range of forecasts from the various methods My “range of reasonable estimates” is the range of forecasts from the various methods Is this reasonable? Is this reasonable? What if one or more of the methods is really “unreasonable”? What if one or more of the methods is really “unreasonable”? Is something outside this range “reasonable”? Is something outside this range “reasonable”?

A Range Idea Take largest and smallest forecast by accident year Take largest and smallest forecast by accident year Add these together Add these together Is this a “reasonable range” Is this a “reasonable range” Example: Example: –Roll of single fair die, 2/3 confidence interval is between 2 an 5 inclusive –Roll of a pair of fair dice, 2/3 confidence interval is between 5 and 9 inclusive, not 4 to 10 (5/6).

You Missed Again!! Your best estimate is $x Your best estimate is $x Actual future payments is $y (>$x) Actual future payments is $y (>$x) Conclusion – you were “wrong” Conclusion – you were “wrong” Why? The myth that the estimate actually will happen Why? The myth that the estimate actually will happen Problem – a reserve is a distribution not a single point, any treatment otherwise is doomed to failure Problem – a reserve is a distribution not a single point, any treatment otherwise is doomed to failure

Why Can’t the Actuaries Get it Right? Actually, why can’t the accountants get it right? Actually, why can’t the accountants get it right? The accountants need to deal with the fact rather than the myth that the actual payments will equal the reserve estimate The accountants need to deal with the fact rather than the myth that the actual payments will equal the reserve estimate Need to Need to –Be able to book a distribution –Recognize the entire distribution –Recognize context (company environment) –Realize that future payments = reserves is an accident with a nearly 0% chance of happening

An Economically Rational Reserve Why not set reserves so that the loss in company value when actual payments turn out different is the least expected Why not set reserves so that the loss in company value when actual payments turn out different is the least expected Note expectation taken over all possible reserve outcomes (along with their probabilities) Note expectation taken over all possible reserve outcomes (along with their probabilities) Economically rational – focuses on company worth Economically rational – focuses on company worth

Least Pain Since any number will be “wrong” let me submit a reasonable estimate of reserves (complements of Rodney Kreps) Since any number will be “wrong” let me submit a reasonable estimate of reserves (complements of Rodney Kreps) Suppose Suppose –(a really BIG suppose) we know the probability density function of future claim payments and expenses is f(x) –For simplicity one year time horizon –g(x,μ) denotes the decrease in shareholder (policyholder) value of the company if reserves are booked at μ but payments are actually x.

Least Pain (Cont.) A rational reserve (i.e. “estimate of future payments”) is that value of μ that minimizes A rational reserve (i.e. “estimate of future payments”) is that value of μ that minimizes i.e. the expected penalty for setting reserves at μ over all reserve outcomes i.e. the expected penalty for setting reserves at μ over all reserve outcomes

A Reasonable g Not likely symmetric Not likely symmetric Likely flat in a region “near” μ Likely flat in a region “near” μ Increases faster when x is above μ than when x is below Increases faster when x is above μ than when x is below Likely increases at an increasing rate when x is above μ Likely increases at an increasing rate when x is above μ Such a function generally gives an estimate above the mean Such a function generally gives an estimate above the mean