This is why you take the Q

... to keep from making a fool of yourself when you start publishing stuff. Sometimes the significance of lessons learned in the context of a class doesn't really become apparent until you review those lessons with an actual problem in mind.

Readers of this blog know that I've been grappling with why you can't make sense of random samples from financial data. The actuaries have been telling me this for years. My own research has confirmed it. But, why? Why has the Central Limit Theorem abandoned me?

Simply put, because I'm dividing.

Financial data is all about ratios. Return on Investment, Return on Economic Capital, Present Value, Future Value, Annual Percentage Yeild, etc.. Even values that look like they wouldn't be ratios actually are. Reserve Capital looks like just a regular asset, but it's not. It's based on Amount at Risk times a bunch of factors based on various risk parameters.

And it's that last little piece that gets ya.

Taking a sample of, say, policy face values, is pretty safe. The distribution is a little skewed, but it's certainly no problem to take a sample average or standard deviation. You don't have to sample too many to get a convergent estimate of the mean. Multiplying that by a fixed interest rate to get a future value is simply applying a linear transformation, which will also yield a nice, convergent mean.

But, what about those pesky risk factors? They could be all sorts of things and the rates applied to cover reserves will vary with each. Let's suppose both the amounts and the rates are coming from normal distributions (which is an unrealistically best-case scenario to be sure). What happens when you divide them to get your reserve requirements?

Well, in just about any decent text on distributions, you'll find an offhand mention of what happens when you divide two Normals. It's the sort of thing you just glide over in class figuring it won't likely come up much in real life. The result is a Cauchy distribution - a distribution so heavy-tailed it doesn't even have a mean, much less the higher moments needed to get the Central Limit Theorem to work.

None of this has solved my problem, but at least I can stop stressing over why I even have one.