## Variations On Approximation – An Exploration in Calculation

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Before we get into the different approaches, why should you care about knowing multiple ways to calculate a distribution when we have a perfectly good symbolic formula that tells us the probability exactly?

As we shall soon see, having that formula gives us the illusion that we have the “exact” answer. We actually have to calculate the elements within. If you try calculating the binomial coefficients up front, you will notice they get very large, just as those powers of q get very small. In a system using floating point arithmetic, as Excel does, we may run into trouble with either underflow or overflow. Obviously, I picked a situation that would create just such troubles, by picking a somewhat large number of people and a somewhat low probability of death.

I am making no assumptions as to the specific use of the full distribution being made. It may be that one is attempting to calculate Value at Risk or Conditional Tail Expectation values. It may be that one is constructing stress scenarios. Most of the places where the following approximations fail are areas that are not necessarily of concern to actuaries, in general. In the following I will look at how each approximation behaves, and why one might choose that approach compared to others.

Author(s): Mary Pat Campbell

Publication Date: January 2014

Publication Site: CompAct, SOA

## Have Fun With Approximations!

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In the pre-computer days, people used these approximations due to having to do all calculations by hand or with the help of tables. Of course, many approximations are done by computers themselves — the way computers calculate functions such as sine() and exp() involves approaches like Taylor series expansions.

The specific approximation techniques I try (1 “exact” and 6 different approximation… including the final ones where I put approximations within approximations just because I can) are not important. But the concept that you should know how to try out and test approximation approaches in case you need them is important for those doing numerical computing.

Author(s): Mary Pat Campbell

Publication Date: 3 February 2016 (updated for links 2021)

Publication Site: LinkedIn, CompAct, Society of Actuaries