Aggregate loss limits

Comments

• admin

  August 2022

  Hi,

  Using the example on page 20 of the Fisher text, if the guaranteed cost premium is $1,000,000 and the multiple is 1.25 then the aggregate loss limit is whatever loss amount results in a maximum premium, G, of $1,250,000. How we calculate this depends on the exact provisions of the retrospective rating plan, such as whether there's also a minimum premium.

  Let's assume there's only a maximum premium constraint, in which case we can use the Table M balance equations to find expressions for phi(r_G) and r_G. Using the retrospective rating formula gives G = (B+c*L_G)T = (B + c*r_G*E[A])T so we're able to solve for the maximum ratable loss amount, L_G, implied by the maximum premium, G.

  The need to iterate comes from B contains a charge for losses in excess of the maximum ratable loss amount. So once we have an estimate for L_G we need to recalculate B. In turn this means we need to solve again for a new L_G. After iterating for a while, we'll reach stable values of B and L_G.
• tracyguo8

  August 2023

  Hi,

  To follow up on this, BattleWiki states "The implied maximum ratable loss increases the basic premium by adding a charge for expected losses above the maximum ratable loss amount. This reduces the implied maximum ratable loss and thus increases the insurance charge which reduces the maximum ratable loss further."

  I get that we need to recalculate B once we have estimate for L_G, but why increasing basic premium reduces implied max ratable loss? I understand that given G doesn't change, higher B means lower L_G in the equation, but I do not get the logic behind.

  Also, for the response above, why do we need  (B + c*r_G*E[A])T to calculate the max ratable loss?

  Thanks!
• admin

  August 2023

  The basic premium, B, consists of a fixed expense component and the converted net insurance charge. The assumption is changing the maximum ratable loss amount doesn't alter the fixed expense component and so if B increases then the converted net insurance charge must have increased.

  Again, all else being equal, i.e. assuming the loss conversion factor, c, and underlying unlimited and limited loss distributions, F(x) and F_D(x) respectively, do not change, this means the entry ratio(s) associated with the net insurance charge have changed. If we restrict ourselves to the case where there's only a maximum premium then this means the entry ratio, r_G, decreased so the insurance charge increased (it helps to draw a Lee diagram). A change in the associated entry ratio then means L_G = E[A] * r_G must have changed and so we need to iterate until we achieve stable values for B and L_G.

  We need (B + c*r_G*E[A])T to get from the new entry ratio at maximum premium to the new maximum loss amount L_G. Really all you're using though is L_G = r_G * E[A].