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: