A toy model of Pareto claims vs Lognormal claims in Insurance.

A few weeks ago Nassim Taleb recommended a book that introduces some of the work of Harald Cramér:

To my surprise (and great joy!) the first few chapters of the book tied together some ideas that have been kicking around in the back of my mind since I read these two books in the late 1990s:

The book by Peters actually led to my first programming assignment in finance when a friend from college asked me to help code in some of the integral equations in the book. I didn’t know what I was doing finance wise, but I could crunch through the equations. That was fine – it was just sort of a hobby in graduate school.

The beginning of “Modelling Extremal Events” lays out some fairly simple ideas in insurance but then – akin to Peters – starts asking what happens when you move way from easy-to-use probability distributions. These charts, for example, compares lognormal claims versus claims from a Pareto distribution:

The point of those charts is that you have a very high chance of being ruined if you think claims are coming from a lognormal instead of Pareto distribution (the ruin happens when the value of your company hits $0).

My suspicion, though, is that using the lognormal distribution instead of a Pareto distribution happens all the time -> especially in insurance (or insurance-like products) relating to the financial markets.

With some unexpected free time today I decided to make a little toy model. It isn’t remotely robust or anything (and this blog post is mostly just meant as a note to myself, so the blog post isn’t really that robust either . . . ), just a simple look at the difference in solvency when insurance claims come from the two distributions.

Here’s the simple idea of my model:

I used the Mathematica function RandomVariate[ParetoDistribution[1,alpha]] to generate 100 draws from a Pareto distribution with minimum value 1 and tail parameter alpha ranging from 1.0 to 2.0 in increments of 0.1.

I then used EstimatedDistribution[] to find the lognormal distribution that best fit that data, and picked 100 numbers from that distribution.

These draws will represent the annual claims that the insurance company faces. I set the annual insurance premium level to be 20% higher than the mean of the claims and started the company with total capital equal to 10 times the mean claims. To be clear, if the mean from the 100 draws was 7.5 of claims per year, the annual premium would be 9 and the starting capital of the company would be 75.

Here’s an example showing draws from the Pareto distribution with alpha = 1.2 (top chart) and the corresponding lognormal distribution (bottom chart). The dots plot the annual total capital of the hypothetical insurance company as premium and claims come in over time. When the numbers go negative around year 15 in the top chart the game is over . . . .


For each alpha I repeated the process 100,000 times to see how often the hypothetical insurance company went insolvent. The results are in the chart below. “count 1” corresponds to the Pareto claims with a given alpha, and “count2” corresponds to lognormal claims:


So, roughly speaking, under the lognormal assumption the company almost never fails. Under Pareto claims, however, it falls quite a bit in a 100 year time frame. Around 1/4 of the time when alpha = 1.1, 1/5 of the time when alpha = 1.2, and 1/6 of the time when alpha = 1.3.

When I see these results and think about the way things like pensions or life insurance product linked to the equity markets are typically modeled, I’m pretty worried.

So I don’t lose it later, here’s the complete code:


Exploring the Monty Hall problem with kids

I learned of the death of the game show host Monty Hall this morning:

“Monty Hall, Host of Game Show ‘Let’s Make a Deal,’ Dies at 96” on Bloomberg

Although we’ve talked about the problem before, seeing Monty Hall’s name made me think of talking through the famous “Monty Hall” problem with the boys for our project today.

I started by reminding them of the problem:

As a way to explore the problem, we started by playing the game and not switching. Through this exploration the boys were able to see that the “not switching” strategy wins the good prize 1/3 of the time.

Next we played the game using the switching strategy. Through this exploration the boys were able to see that the “switching” strategy wins the good prize 2/3 of the time. However, at the end of this video they were still struggling to explain why – I think they were pretty surprised.

Two wrap up the project we looked at three cases in detail for the switching and non-switching strategy. At the end I asked them again why they thought the switching strategy was so much better. Now they seemed to have a better grasp of the situation.

The Monty Hall problem is definitely fun, though it has always amazed me how controversial the problem is. I struggle to think of another problem that causes people to hold on to their answers so strongly. Understanding how this problem works is a great lesson in problem solving and in probability.