Excess layer pricing

I had to solve an interesting problem yesterday relating to pricing an excess layer which was contained in another layer which we knew the price for – I didn’t price the initial layer, and I did not have a gross loss model. All I had to go on was the overall price and a severity curve which I thought was reasonably accurate. The specific layers in this case were a 9m xs 1m, and I was interested in what we would charge for a 6m xs 4m.

Just to put some concrete numbers to this, let’s say the 9m xs 1m cost \$10m

The xs 1m severity curve was as follows:

​Which we can convert into a survival function for xs 4m as follows:

​Pricing the layer

We can reason as follows, we are interested in:
$$ \text{Price}_{\text{6 xs 4}} = \text{(freq xs 4)(average loss to layer s.t. loss xs 4)/(loss ratio)} $$ 
And we know the following:
$$ \text{Price}_{\text{9 xs 1}} = \text{(freq xs 1)(average loss to layer s.t. loss xs 1)/(loss ratio)} = 10m$$
Solving for freq xs 1:
$$  \text{(freq xs 1)} = \frac{10m}{\text{(average loss to layer s.t. loss xs 1)/(loss ratio)}} $$
And then noting that $ \text{freq xs 4} = \text{freq xs 1} * S(4) = \text{freq xs 1} * 22\% $, which when combined with the above gives us:
$$\text{Price}_{\text{6 xs 4}} = \frac{10m}{\text{(average loss to layer s.t. loss xs 1)}/\text{(loss ratio)}} * 22\% * \text{(average loss to layer s.t. loss xs 4)} * \text{(loss ratio)} $$
And then, rearranging and cancelling the loss ratios:
$$\text{Price}_{\text{6 xs 4}} = 10m * 22\% \frac{\text{(average loss to layer s.t. loss xs 4)} }{\text{(average loss to layer s.t. loss xs 1)}}$$
i.e. the price for the 6 xs 4 is the price for the 9 xs 1 multiplied by the probability of a loss being excess 4, scaled by the quotient of the average loss into each layer. In this case, we have:
$$\text{Price}_{\text{6 xs 4}} = 10m* 22\% \frac{2,136,364}{1,490,000} = 3,154,362$$