Actuarial Modeling

# Theoretical vs. Practical Solutions(6 min read)

What’s 6 + 2? 8. It’s obvious. But the next question is not: How do you know that is true?

There are a couple of different ways. One, you just remember. You learned some kind of fact long ago (similar to “The American Revolution began in 1776” or “Cats have five claws on front paws and four on back paws”). Now, you just know that it’s true.

Another way would be to build a model of that problem. Take a piece of paper, draw six dots, and then draw two additional dots. Count the total, and you get eight.

Similarly, for actuarial problems, there are two ways to go about getting an answer: either a theoretical solution (finding an exact formula) or creating a model to estimate it.

How do you know which is right for you? Let’s explore some of the reasons professionals need to turn to models, and then we’ll give you a couple of thinking points to help you determine whether you can solve the problem from first principles or you need a model.

#### Life Insurance: A Model Waiting to Happen?

Think about life insurance. The simplest products are term and whole life, in which the value of the insurance remains fixed each time period. You can actually write a formula for the “cost” of that insurance.

That is, given that we know the mortality rate and the face amount, we can figure out how much to charge each period. You don’t need a model for this kind of product. Heck, your intern may be able to create a reasonable facsimile of the cost of term insurance in an afternoon.

But as you increase the complexity of the products you’re evaluating, you start to add more and more need for a modeled solution. Why?

Because the theoretical becomes more complex. Sometimes, like many variable annuities, it becomes unsolvable without differential equations. (Differential Equations incorporate both the variable and its derivative into a function. It’s a feedback loop that makes solving the problem much harder.)

Since most actuaries forgot DiffEq about a day after they completed the final (since it wasn’t going to be on any more actuarial exams), solving these kinds of problems isn’t really in our wheelhouse.

#### Circular Reference Product Example – Deferred Annuities with GMxBs

Deferred annuities have lots of things to like about them: they pay out regular payments, they can allow an investor to harness a diversified investment portfolio, they don’t do their business on the carpet when guests come over, etc., etc.

And in order to reduce some risks and make these products a little more attractive, many insurance companies add additional guarantees to their variable annuities. They’re called a Guaranteed Minimum Death Benefit, or Guaranteed Minimum Annuitization Benefit, or Guaranteed Accumulation Benefit, etc. We’ll follow the modern convention and call them GMxBs.

Each of these GMxBs must be paid for somehow, either in lower profit for the insurer (not likely) or by the policyholder. Maybe it is a lower interest rate credited than what the insurance company earns. Maybe it’s a front-end load that gets taken out before any interest starts to accumulate.

Or, and this is pretty popular, it could be a fee that is withdrawn from the account balance each month to pay for the additional bit of insurance that the policyholder has bought for their account value.

The problem is that simply having the fees taken out reduces the amount of the account balance available, and if you’re looking at a guaranteed minimum something, then the insurance company actually has more at risk after the charge has been taken out than before. Because, you know, that guaranteed minimum didn’t actually go anywhere. [If it did, that’s quite the odd guarantee, no?]

This becomes a circular reference: paying out the GMxB will cost the insurance company money, which will mean that there must be some charge for it, which means that there will be less money in the account, which means the guarantee will be worth more, which means the charge needs to be higher, and so on.

You might ask, can we write the (equation) for something like this? Perhaps. But remember, each insurance policy is going to last for hundreds of these charges (and benefit) periods, as annuities often pay out over 20, 30, or 40 years. Month 1 charges affect the benefits available in month 2, month 2 benefits available impact the charges that should be deducted in month 1, and so on. That’s a differential equation.

Which means you may not be able to solve for a rate that provides a reasonable profit, risk margin, return on investment, or other metric you use. Thus, a model becomes likely. That model may be simple enough to run in a spreadsheet. Generally, though, you’re going to want advanced actuarial software to incorporate all of the features of that insurance product, like expenses, reserves, taxes, and risk capital, so you don’t have to build it from scratch.

#### A Second Reason for a Model: Broader Insight

Another reason you’ll need modeled solutions is if you’re not looking for a point answer, but some kind of distribution. For the simple life insurance policy above, you’ve got only one number that tells you the expected value of claims over the lifetime of the policy. But what’s the distribution? How likely is it you’re going to have death claims more than 10% higher than that expected value? Assuming your mortality assumption is correct (again, big assumption there), what’s the year-to-year variance around that middle value?

Theoretically, we can solve for the variance of mortality for a single policy as (q x * (1 – q x )) (binomial), and then create the sum of all those policies with the total expected value and total variance. But do you really want to do that across 50,000 policies in your portfolio? Remember, each of those has a different expected mortality and different face amount. That’s a lot of algebra.

A better way might be to model each policy with a stochastic all-or-nothing mortality rate, and see what that distribution looks like. Yes, it might be a little more modeling work, but it would certainly give you a chance to get a better idea, with a repetitive process, what the range of potential values might be.

These are just two reasons why you might want a modeled solution instead of a theoretical, or solved, one. We’re sure there are others. How about sending us a note and letting us know your thoughts on why actuaries create models?

I know what you’re thinking: so what’s the point of this article?

#### Two Tips For Deciding Whether To Solve An Equation Or Model Your Solution

Well, we wouldn’t be Slope without giving you some real, practical advice for your consideration of modeled versus theoretical solutions. We’ve got two criteria you’ll need to consider when deciding to solve the question exactly or create a model.

First, consider the complexity of your product. If it’s super simple (like no inherent feedback loops), it’s much more likely that you’re going to get by with the theoretical solution. If it’s complex, you’re going to need to expand and build a model, one which can incorporate multiple interactions between inputs and outputs.

Second, understand the breadth of the solution you’re looking for. Point estimates are fine for some tasks. But maybe you need to know how often the surplus in between now and 2030 dips below zero when the S&P 500 is expected to follow a regime-switching lognormal model without regression to the mean. Or you’re asked to measure the Conditional Tail Expectation of the 90th percentile of your mortality claims. It’s possible you could do that after some hardcore calculus review. But is it practical for you to go get your advanced degree in linear algebra and combinatorics before you come back to work?

Nope. That’s just not going to fly.

Those are best solved with modeled solutions, not theoretical ones.

So, when you’ve found yourself with a complex, circular-reference product, or a question that requires more than just a best-estimate value, you need a model.

At Slope, we’re modeling experts. We’d love to talk about how applying cloud technologies to actuarial models can improve your information and get you better results. Give us a call anytime at 855.756.7373.

You’ll find that we’re great at helping with your actuarial models. Getting flesh-and-blood models to notice us…

well, that’s a whole other story.