Everybody involved in risk measurement knows what a 1 in 200 event is, don’t they? It’s such a fundamental concept, anybody reading this is probably insulted that I’m even asking the question. But bear with me.
Let’s make things really simple and imagine an insurer with only two uncorrelated risks. The profits/losses arising from risks are both normally distributed with zero mean and equal standard deviations. Each risk considered in isolation has a 0.5% probability of resulting in a loss of over £1. This is as simple as it gets.
Now let’s get a pencil and paper. We’ll draw x and y axes corresponding to profits/losses from the two risks. We’ll draw a unit circle centred on the origin. It goes through the (0,1) and (1,0) “univariate 1 in 200 points” and the (0.5sqrt(2), 0.5sqrt(2)) “biting scenario” point. Finally, we’ll add a straight line, sloped downwards at 45 degrees and touching the circle at the biting scenario point.
You’ll have seen diagrams like this countless times. But now I have a question for you. Where on the diagram is/are the 1 in 200 scenarios? Have a think about this before reading further.
Well, it was a badly worded question. To be able to identify 1 in 200 events, we need some sort of context. 1 in 200 after ordering the scenarios with respect to what? If this isn’t specified, then any line on the diagram that divides the plane into two areas with 99.5% and 0.5% probabilities could be described as a set of 1 in 200 scenarios. So let’s try introducing some context and looking at a couple of examples.
First let’s rank the scenarios by the loss that the insurer makes. In this example, the 1 in 200 scenarios lie along the 45 degree straight line. Above the line are the 0.5% of scenarios where the loss exceeds the SCR, below the line are the 99.5% of scenarios where the loss is less than the SCR. It’s a reminder that there’s not really a single biting SCR scenario, but a whole set of them all up and down that line. What people refer to as the biting scenario is only the most likely of the full set of 1 in 200 scenarios.
Time for another example. Let’s rank scenarios by their extremeness – their distance from the origin on our diagram. It’s easy to see that the 1 in 200 scenarios will form a circle around the origin. Some people may have answered my original question by describing such a circle. Some may have answered with the unit circle from the diagram. This latter set of people would be wrong! On the original diagram, there’s 99.5% probability below the sloping line. Unless there’s 0% probability between the unit circle and the line, there must be less than 99.5% probability within the unit circle. So the circle of 1 in 200 events (ranked by extremeness) must be bigger than the unit circle.
But (I hear the anonymous doubters say) the (1,0), (0,1) and biting scenario points are all 1 in 200 events! The problem is that they’re individual points in three different sets of 1 in 200 events. (1,0) is just one point on the x=1 line that separates off the 1 in 200 scenarios with the most extreme x. The biting scenario is just one point on a different straight line that separates off the 1 in 200 scenarios with the biggest losses. The points on the unit circle don’t divide up the plane into 0.5% and 99.5% probabilities so can’t be described as a set of 1 in 200 scenarios.
Finally, back to the title of this post. Is the SCR a 1 in 200 event? First, be honest with yourself. When I asked you for a set of 1 in 200 scenarios, were you thinking of a circle? I’m with you if you were. Without setting the context, distance from the origin feels like the obvious default. Next, I hope I’ve convinced you that the SCR lies inside this circle of 1 in 200 events, rather than on the circumference. In other words, the SCR scenario is not as extreme as a 1 in 200 event! And that means that the frequency with which we see events as extreme as those in the SCR “biting scenario” will be higher than 1 in 200. It’s just that most of them will result in profits or smaller losses.
Any comments welcome. I’m especially interested to hear whether people would have answered the question with circles or straight lines (or anything else).