I'm struggling to follow the solution to this question. I understand how you get to R = 0.02 * Theta, and I also understand how Capital = 115.12 / Theta. However, I don't the next bit of the solution. Where does the bit Theta = 10% * Capital / 10m, and the following Theta^2 = 115.13m / 100m term come from? This makes absolutely no sense to me - Anyone understand this?
You're happy that Capital = 115.13m/theta. If we now think about the Return on capital (or profit) for Company A, we notice that: Return on capital * Capital = theta * Expected claims since theta is the security loading i.e. the extra amount loaded onto the premiums to represent the profit. We know that the Expected claims = 10m (the loss cost for A) We also know that the required Return on capital = 10% so: 10% * Capital = theta * 10m and so Capital = (10m * theta) / 10% Equating this with the fact that Capital = 115.13m/theta from above gives the result that theta^2 = 115.13/100m