# Variance decomposition and price segmentation in Insurance

Today, I was giving a talk at the Economics department, and I got a very interesting question about some tables I keep showing to explain why insurance companies like segmentation. The tables illustrate three different case. Here, $S$ stands for the individual (random) loss.

• the first one is the case where the premium asked is the same for all the insured – i.e. the pure premium $\mathbb{E}[S]$

As explain, the loss is here on an individual basis, so, per policy, the insurer faces the (random) loss $S-\mathbb{E}[S]$, which is, on average, null. That’s the second line. For the last line, I keep saying that we look at the overall loss of the insurer, but that’s not correct. Here, with a factor $n$, we would have the variance of the total loss for the insurance company. We just removed the $n$ factor in the table

• then, we have perfectly observable heterogeneity : insured have a risk factor $\Omega$, obervable, and in that case, the ‘pure’ premium is $\mathbb{E}[S|\Omega]$

That’s what we have below. Here again, on average, the insured should have a null profit. And the total variance (which was $\text{Var}[S]$ in our previous example) is now splitted in two parts (that’s basically Pythagoras theorem).

The interpreration is the following

And then, I usually mention the third and last case, more realistic

• the risk factor $\Omega$ is not obervable, but segmentation is still possible using some proxy of the risk factor, obtained using some covariates, and the ‘pure’ premium is $\mathbb{E}[S|\boldsymbol{X}]$

And here also, there is a nice interpretation, because of the variance decomposition : there is one part that we observed previously, with some ‘perfect pricing’ and an additional part (that is positive) that is related to the fact that the covariates are just a proxy of the risk factor….

The term on the left is then a lower bound, obtained if actually, using our covariate, available for the pricing, we can get the risk factor.

That was my story, but the fact that $n$ (the portfolio size) was not mentioned in the tables was a bit confusing… So I decided to create some graphs to illustrate those three cases

Consider some simple simulations. On the graph on the left, we have on the x-axis the risk factor, and on the y-axis, the loss (going roughly from 0 to 20). The pure premium is the average of those losses. Here, it’s 10. That’s the plain red line (on the left). In the middle, the y-axis is the insured profit/loss per policy. Someone with a loss close to 0 means a gain of 10, someone with a loss close to 20 means a loss of 10. On average, there is no profit (that’s the plain line). And then, on the right, we have the distribution of the profit/loss (per contract). Again, on average it’s 0, with some variance,

Consider here is simple covariate $x$ : assume here that’s we’ve been able to create a binary variable, that can distinguish the low risks and the high risks. Here, there are two levels for the premium. The low premium is close to 6, and the high one is close to 14. That’s again the graph on the left
Assume now that $\Omega$ is observable. And that we use it for our pricing. The premium is now continuous, and it is the red line, on the left. The profit/loss (in the middle) is the difference between the loss, and its expected value (conditional on the risk factor). And on the right, we have the distribution.