Session II: Reserve Ranges Who Does What

1. Session II: Reserve Ranges Who Does What Presented by Roger M. Hayne, FCAS, MAAA May 8, 2006

2. Reserve – A Loaded Term • “Reserve” is an accounting term • A “reserve” to be a “reasonable estimate” of the unpaid claim costs • Definition is vague at best • Only requires a “reasonable estimate” (whatever that is) • Seems to imply an amount that will happen • Even if we knew entire process this is not much help in what to record • One number is not enough – the “reserve” is really a set of outcomes with related probabilities, a distribution

3. Why A Distribution? • Future payments are uncertain • A single number cannot convey that uncertainty • If we had a distribution then concepts like mean, mode, median, probability level, value at risk, or any other statistic have specific meaning • If accountants insist on a single number embodying all that “reserve” means then we can talk about which of (infinitely) many numbers best convey that message

4. Our Solution • Our solution – punt • We will try not to talk about “reserves” • We will try to focus on the “distribution of outcomes” under a policy or group of policies (or for an insurer) • The “distribution of outcomes” will inform the reserve to be recorded • Better having the “what to book” discussion with this knowledge than without it

5. How Did We Get Here? • Accounting definition seems to be intentionally vague • Current CAS Statement of Principles on reserves also somewhat vague • Reasonable estimates • Ranges of reasonable estimates • No mathematical/statistical precision • To quote Tevye – “Tradition”

6. Tradition • Traditional actuarial forecast methods are deterministic • Do not have an underlying model (more on this later) • They produce “estimates” • Not estimates of the expected (mean) • Not estimates of the most likely (mode) • Not estimates of the middle-of-the-road (median) • Just “estimates” without quantifying variability or uncertainty

7. Measuring Uncertainty In Times Past • Traditional methods do not give underlying distributions • Our “Fore-parents” knew this and tradition included the application of a variety of methods • Bunching up of methods gave a sense of variability or uncertainty • Methods gave similar answers => little uncertainty • Methods gave disperse answers => much uncertainty

8. Reasonable Ranges and Ranges of Reasonable Results • Need for “Range of Reasonable Estimates” • Still important to discuss uncertainty • Still a need to quantify how uncertain an estimate is • Lack of statistical qualities in traditional forecasts • Solution “Range of Reasonable Estimates” • Definition still “soft” without any statistical meaning • Depends on nebulous term “reasonable”

9. A Hint of the Future? • Consider a more reasoned approach • Assume that a “reserve” still needs to be a single number but that (big assumption here) we all agree on which statistic to use • (Presenter’s digression – I like Rodney Kreps’ “Least Pain” statistic) • What statistical sense do terms like “reasonable estimate” and “range of reasonable estimates” convey? • To help let’s define a few terms

10. Talking About Uncertainty • Ultimate future payments on insurance claims are generally unknown • Theoretically, for a given amount there is a probability that future payments will not exceed that amount • Problem, we usually need to estimate those probabilities • The way we do this can (should?) involve several steps

11. Simple Example • Write policy 1/1/2006, roll fair die and hide result • Reserves as of 12/31/2006 • Claim to be settled 1/1/2007 with immediate payment of $1 million times the number already rolled • 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, there is no mode • What would you book as a reserve? • Note here there is no model or parameter uncertainty • If only one statistic is “reasonable” then “range of reasonable estimates” is a single point

12. Almost Simple Example • Claim process as before • This time die is not fair: • Prob(x=1)=Prob(x=6)=1/4 • Prob(x=2)=Prob(x=5)=1/6 • Prob(x=3)=Prob(x=4)=1/12 • Mean and median still $3.5 million • “Most likely” is either $1 million or $6 million • What do you book now? • The means are the same but is the reality? • Still no parameter or model uncertainty • Again, if only one statistic is “reasonable” then “range of reasonable estimates” is a single point

13. Steps In Estimating • Define one or more models of the future payment process • Estimate the parameters underlying the model(s) • Assess the volatility of the process under the assumption that the model(s) and parameters are all correct • Aggregate the uncertainty from each of these steps • Particular contributions called respectively model/specification, parameter and process uncertainty

14. Some Context • The aggregate distribution you get in the end is useful in talking about the “range of potential outcomes” • The “range of reasonable results” is not this range of potential outcomes • If one defines a particular statistic (mean, “least pain,” value at risk, etc.) as a “reasonable” reserve estimate then it makes sense to look at the distribution of that statistic under different selections of models and parameters

15. Simple Inclusion of Parameter Uncertainty • Adding parameter uncertainty is not that difficult • Very simple example • Losses have lognormal distribution, parameters m (unknown) and σ2 (known),respectively the mean and variance of the related normal • The parameter m itself has a normal distribution with mean μ and variance τ2

16. Simple Example Continued • Expected (“reasonable estimate”) is lognormal • Parameters μ+ σ2/2 and τ2 • c.v.2 of expected is exp(τ2)-1 • “What will happen” (“possible outcome”) is lognormal • Parameters μ and σ2+τ2 • c.v.2 is exp(σ2+τ2)-1 • c.v. = standard deviation/mean, measure of relative dispersion • Note expected is much more certain (smaller c.v.) than “what will happen”

17. Carry the Same Thought Further • Suppose that judgmentally or otherwise we can quantify the likelihood of various models • Think of each of them as different possible future states of the world • Why not use this information similar to the way the normal distribution was used in the example to quantify parameter/model uncertainty • Simplifies matters • Quantifies relative weights • Provides for a way to evolve those weights

18. An Evolutionary (Bayesian) Model • Again take a very simple example • Use the die example • For simplicity assume we book the mean • This time there are three different dice that can be thrown and we do not know which one it is • Currently no information favors one die over others • The dice have the following chances of outcomes:

19. Evolutionary Approach • “What will happen” is the same as the first die, equal chances of 1 through 6 • The expected has equally likely chances of being 2.67, 3.50, or 4.33 • If you set your reserve at the “average” both have the same average, 3.5, the true average is within 0.83 of this amount with 100% confidence • There is a 1/3 chance the outcome will be 2.5 away from this pick. • We now “observe” a 2 – what do we do?

20. How Likely Is It? • Likelihood of observing a 2: • Distribution 1 1/6 • Distribution 2 2/21 • Distribution 3 5/21 • Given our distributions it seems more likely that the true state of the world is 3 (having observed a 2) than the others • Use Bayes Theorem to estimate posterior likelihoods Posterior(model|data)likelihood(data|model)prior(model)

21. Evolutionary Approach • Revised prior is now: • Distribution 1 0.33 • Distribution 2 0.19 • Distribution 3 0.48 • Revised posterior distribution is now: • Overall mean is 3.3 • The expected still takes on the values 2.67, 3.50, and 4.33 but with probabilities 0.48, 0.33, and 0.19 respectively (our “range”)

22. Next Iteration • Second observation of 1 • Revised prior is now (based on observing a 2 and a 1): • Distribution 1 0.28 • Distribution 2 0.05 • Distribution 3 0.67 • Revised posterior distribution is now: • Now the mean is 3.0 • The expected can be 2.67, 3.50, or 4.33 with probability 0.67, 0.28, and 0.05 respectively

23. Not-So-Conclusive Example • Observations 3, 4, 3, 4 • Revised prior is now : • Distribution 1 0.34 • Distribution 2 0.33 • Distribution 3 0.33 • Revised posterior distribution is unchanged from the start: • As are the overall mean and chances for various states

24. Summary • Though our publics seem to want certainty future payments are uncertain • It is virtually certain actual future payments will differ from any estimate • Quantifying the distribution of future payments will inform discussion • Keep process, parameter and model/specification uncertainty in mind • Models contain more information than methods • “Ranges of reasonable estimates” are different than “ranges of possible outcomes”