What’s the Point (Estimate)? Casualty Loss Reserve Seminar September 12-13, 2005 Roger M. Hayne, FCAS, MAAA

I Don’t Get the Point What is the reserve point estimate? 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…”

What’s the Point? 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 Is it an estimate of the midpoint of the range of amounts that might be paid? No It is an estimate of the amount to be paid

The Reality Given current knowledge there is a distribution of possible future outcomes – the reserve properly is a DISTRIBUTION The “estimate of future payments” definition says the reserve is a point on this distribution It does not say which one We are measured by how close the “estimate” is to what actually happens Even if we completely know the process, we can be “wrong”

Getting the Point Across The reserve is actually a distribution Need a vocabulary to talk about distributions The accounting entry is a single number Need to be clear what number the accounting entry represents Is it – The mean (expected) value? – The point on the distribution that will actually happen? – Something else?

Point, Counterpoint Actuaries like to think about the average – It all evens out in the end (pluses and minuses cancel over the long term) – It may even be possible to predict the mean (or at least get “close”) – May make life “easier” – But usually not “verifiable” by actual events Our publics, however, often think in terms of “what will happen” We are measured by how close our estimates are to what “actually happens”

The Point is Not Clear “Best Estimate” is no longer good enough We need a vocabulary to talk about uncertainty Concepts are easy to understand but sometimes difficult to communicate Sources of uncertainty – Process – Parameter – Model/Specification

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)

Parameter Uncertainty about the parameters of models 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 Many parameter estimation methods also give estimates of parameter uncertainty

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

Simple Example Write policy 1/1/2005, roll fair die and hide result Reserves as of 12/31/2005 Claim to be settled 1/1/2006 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

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

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 models whose expected values duplicate some traditional methods (chain ladder, Bornhuetter-Ferguson) Traditional methods do not give distributions

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” You need distributions to get the mean, mode, etc. Distributions are normally possible only after added assumptions

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

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 Really does not deal with distributions

Models and Methods 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

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

Careful – You Might Get Burnt Some claim all information about variability in future payments is summed up in the paid loss triangle Thus the single-triangle focus of most stochastic methods is not really a problem Not so quickly, consider the following example

Looks Can Be Deceiving Incurred Losses Acc.Months of Devel. Yr Paid Losses Acc.Months of Devel. Yr

Maybe Bayes Gives Insight Following thinking is influenced by conversations with Glenn Meyers Once you have a (stochastic) model you have some sense of process uncertainty Under the assumption of no parameter or model uncertainty the issue is simply using that (process) model to pick your “favorite” reserve number – Some statistic? It can be calculated – “What will happen?” the process can guide you as to the estimate and ranges

Simple Inclusion of Parameter Uncertainty Adding parameter uncertainty is not that difficult Very simple example – Losses have lognormal distribution, parameters m (unknown) and σ 2 (known), respectively the mean and variance of the related normal – The parameter m itself has a normal distribution with mean μ and variance τ 2

Simple Example Continued Expected is lognormal – Parameters μ+ σ 2 /2 and τ 2 – c.v. 2 of expected is exp(τ 2 )-1 “What will happen” is lognormal – Parameters μ and σ 2 +τ 2 – c.v. 2 is exp(σ 2 +τ 2 )-1 c.v. = standard deviation/mean, measure of relative dispersion Note expected is much more certain (smaller c.v.) than “what will happen”

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

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: /6 1/212/213/214/215/216/21 5/214/213/212/211/21

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?

How Likely Is It? Likelihood of observing a 2 : – Distribution 11/6 – Distribution 22/21 – Distribution 35/21 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)

Evolutionary Approach Revised prior is now: – Distribution – Distribution – Distribution 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”)

Next Iteration Second observation of 1 Revised prior is now (based on observing a 2 and a 1): – Distribution – Distribution – Distribution 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

Not-So-Conclusive Example Observations 3, 4, 3, 4 Revised prior is now : – Distribution – Distribution – Distribution Revised posterior distribution is unchanged from the start: /6 As are the overall mean and chances for various states

Some “Take-Aways” Always be clear on what you are describing – The entire distribution of future outcomes (“what will happen”)? – Some statistic representing the distribution? Our publics probably think in terms of the former – we in terms of the latter The reality is that the distribution of future outcomes (“what will happen”) is quite disperse – maybe too wide for “comfort” Disclosure, disclosure, disclosure