Reserve Ranges

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

2. Changing Scene • 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: • A reserve is really a probability statement consisting of an amount x plus the probability that the final settlement will not exceed x

3. A Range – Gas or Electric? • Start simple – a range around what? • 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 and appropriate methods…”

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

5. Simple Example • Reserves as of 12/31/2005 • 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) • Mean and median are $3.5 million

6. An Almost-Simple Example • Reserves as of 12/31/2005 • 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? • Mean and median still $3.5 million • “Most likely” is either $1 million or $6 million

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

8. Traditional 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 are normally possible only after added assumptions

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

10. Reasonable? • Range of reasonable results an attempt to quantify an actuary’s “gut feel” or “judgment” • 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 – estimate is quite subjective

11. Model and Method • A method is a general approach • Chain ladder • Bornhuetter-Ferguson • A model usually specifies an underlying process or distribution and the focus is on identifying the parameters of the model • Most traditional actuarial forecasting approaches are methods and not models

12. Stochastic Methods • Stochastic methods have assumptions about underlying models • 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) • Mack/Quarg method not yet stochastic

13. Some Vocabulary • Components of uncertainty: • Process • Parameter • Model/Specification • Any true estimate of the distribution of outcomes ordinarily would recognize all three

14. Process • Uncertainty that cannot be avoided • Inherent in the process • 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 distributions (law of large numbers)

15. Parameter • Uncertainty about the parameters of models (Note: Some models are not parametric) • The underlying process is known • Just the position of some “knobs” is not • Example – flip of a weighted coin • Uncertainty regarding the expected proportion of heads

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

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

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

19. A Range Idea • Take largest and smallest forecast by accident year • Add these together • Is this a “reasonable range” • 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).

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

21. Why Can’t the Actuaries 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 • 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

22. 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 • Note expectation taken over all possible reserve outcomes (along with their probabilities) • Economically rational – focuses on the impact of the final settlement on a company’s net worth

23. Least Pain • Since any single number will be “wrong” let me submit a reasonable estimate of reserves (compliments of Rodney Kreps) • Suppose • (a really BIG suppose) we know the probability density function of future claim payments and expenses is f(x) • For simplicity assume a 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.

24. Least Pain (Cont.) • 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

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

26. Example Distribution I

27. Example Distribution II