Stochastic Methods The Power of Numbers

1. Stochastic MethodsThe Power of Numbers • Presented by • Roger M. Hayne, PhD, FCAS, MAAA CAS Spring Meeting 16-18 June 2008 Quebec City, Quebec

2. Why Bother with Stochastic Methods? • We all know stochastic methods are: • Complicated • Black boxes • Leave no room for actuarial judgment • Are impossible to describe • Take way too long to implement • Take far more data than we could ever imagine obtaining • Don’t answer the interesting questions • Bottom line, inconvenient and too complex for their own good • Well, some of you might know that, others (yours truly included) need to be convinced

3. What is A Stochastic Actuarial Model? • Many definitions – lets use: A simplified statement about of one or more aspects of a loss process that explicitly includes the potential for random effects • Two main features • Simplification • Explicit incorporation of random effects • Both important and both informative • In effect it is a statement about all possible outcomes along with their relative likelihood of occurring, that is a statement of the distribution of outcomes and not just a single “selection”

4. Why is This Important? • Consider the following very simple loss development triangle: • Simple chain ladder method: • First pick a “typical” number for each column • Square the triangle with those numbers • Not a stochastic model, though a simplified statement of loss process

5. Traditional “Deterministic” Approaches • Chain ladder – pick factors thought to be representative of column • What happens “next year” when new information available? • Often entire exercise is repeated “afresh” • Sometimes we ask “what did we pick last year?” • If “actual” varies “too much” from “expected” then we might reevaluate the “expected” • How much is “too much” is often dictated by experience, with line of business or particular book being reviewed • That indefinable quality – “actuarial judgment”

6. Let’s Parse The Traditional • Start out with the chain ladder recipe, i.e. a “model” • We pick “selections” that are somehow representative of a particular age • Experience and “actuarial judgment” often inform us as to what we expect to see (e.g. auto physical damage = stable, umbrella = volatile) • Wait a minute – we have a simplified statement about the loss process and an implicit statement about random fluctuation • The traditional is almost stochastic already! • Why not write down the recipe and expectation of randomness explicitly

7. More Info in a Stochastic Context • Stochastic approaches gain significant advantages over deterministic ones from many sources • Practitioner is forced to explicitly state his/her assumptions • Not only will a good model give projections, but also estimates of certain data points along the way – we can measure next year’s actual vs. expected • Parametric models have some advantages too • They allow for extrapolation beyond the observed data under the assumptions of the model • Good methods for estimating the model parameters also provide estimates of how volatile those parameters themselves are • Maximum likelihood • Bayesian

8. We May Never Pass This Way Again • Two schools of statistical thought • Frequentist • Bayesian • Two distinct approaches in dealing with uncertainty • Frequentist makes the most sense with repeatable experiments • Bayesian attempts to incorporate prior experience in a rational, rigorous fashion • Actuarial problems usually do not relate to repeatable experiments, unless you use the dice example… • Actuarial judgment is essentially a Bayesian “prior distribution” • Bayesian prior is also a way to handle model uncertainty

9. All Models are Wrong … • The banking sector has “sophisticated” risk models setting capital to be adequate at very high (well above 99) percentiles • All is fine … until something like the “subprime crisis” comes along • But the models were well founded and based on “considerable” data • Think about it – using 10 years of data to estimate a 1-in-1,000 year, or even a 1-in-100 year event really does not make a whole lot of sense • The only way to extrapolate from such data is to assume an underlying parametric model and assume that you can extrapolate with it

10. Model Uncertainty • Mentioned before a good parameter estimation method also gives an estimate of uncertainty in the parameter estimates within that model • The subprime issue was not one of parameter estimation but one of model mis-estimation • Traditional methods long recognized this problem and solved it by using several forecast techniques • At end of day an actuary “selected” his/her “estimate” based on the projections of the various models – stochastically he/she calculated an expected value “forecast” using weights (probabilities) that were determined by “actuarial judgment” • Thus there was a Bayesian prior dealing with model uncertainty

11. More is Better • Stochastic methods can be thought of as extensions of traditional approaches can • Be based on same recipes as traditional methods • Give rigor in “making selections” avoiding the ever-present temptation to “throw out that point – it is an obvious outlier” • Provide more information as to the distribution of outcomes within the scope of the particular model • Provide more information as to how well model fits with reality • Be evolutionary and evolve as data indicate • Be adapted to recognize “actuarial judgment” as well as a multiplicity of potential models • All in all stochastic reserving models can give you everything that traditional methods do and much, much more