Session II: Reserve Ranges Who Does What Presented by Roger M. Hayne, FCAS, MAAA May 8, 2006 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 7/26/2016 2 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 7/26/2016 3 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 7/26/2016 4 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” 7/26/2016 5 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 7/26/2016 6 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 7/26/2016 7 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” 7/26/2016 8 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 7/26/2016 9 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 7/26/2016 10 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 7/26/2016 11 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 7/26/2016 12 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 7/26/2016 13 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 7/26/2016 14 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 7/26/2016 15 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” 7/26/2016 16 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 7/26/2016 17 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: 1 1/6 1/21 6/21 7/26/2016 2 1/6 2/21 5/21 3 1/6 3/21 4/21 4 1/6 4/21 3/21 5 1/6 5/21 2/21 6 1/6 6/21 1/21 18 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? 7/26/2016 19 How Likely Is It? Likelihood of observing a 2: – Distribution 1 – Distribution 2 – Distribution 3 1/6 2/21 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) 7/26/2016 20 Evolutionary Approach Revised prior is now: – Distribution 1 – Distribution 2 – Distribution 3 0.33 0.19 0.48 Revised posterior distribution is now: 1 2 3 4 5 6 0.20 0.19 0.17 0.16 0.15 0.13 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”) 7/26/2016 21 Next Iteration Second observation of 1 Revised prior is now (based on observing a 2 and a 1): – Distribution 1 – Distribution 2 – Distribution 3 0.28 0.05 0.67 Revised posterior distribution is now: 1 2 3 4 5 6 0.25 0.21 0.18 0.15 0.12 0.09 Now the mean is 3.0 The expected can be 2.67, 3.50, or 4.33 with 7/26/2016 probability 0.67, 0.28, and 0.05 respectively 22 Not-So-Conclusive Example Observations 3, 4, 3, 4 Revised prior is now : – Distribution 1 – Distribution 2 – Distribution 3 0.34 0.33 0.33 Revised posterior distribution is unchanged from the start: 1 2 3 4 5 6 1/6 1/6 1/6 1/6 1/6 1/6 As are the overall mean and chances for various 7/26/2016 states 23 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” 7/26/2016 24