Stochastic MethodsThe Power of Numbers

Presented byRoger M. Hayne, PhD, FCAS, MAAA

CAS Spring Meeting16-18 June 2008Quebec City, Quebec

2 April 22, 2023

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 April 22, 2023

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 April 22, 2023

Why is This Important?

Consider the following very simple loss development triangle:

AY 12 24 36 482004 1.20 1.10 1.02 ??

2005 1.25 1.08 ?? ??

2006 1.15 ?? ?? ??

2007 ?? ?? ?? ??

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 April 22, 2023

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 April 22, 2023

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 April 22, 2023

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 April 22, 2023

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 April 22, 2023

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 April 22, 2023

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 April 22, 2023

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