F2016 - Q13 Sample 2 Solution
The 2nd sample solution of this problem uses the Balance Equations to solve for the Tax Multiplier. However, I'm having a hard time following their algebra, particularly for the values of G and H. It seems like they pull out of nowhere that G = 1.25 (1/80% LR ??), yet then eliminate H to solve for T. If we had a value for G, wouldn't we have an analogous value for H? (1/LR does not make sense)
My first attempt at this alternate approach led me to trying to set the Max LR equal to L_G/SP, where L_G can be solved to equal (G/T -B)/c. Then 0.8 = (G/T - B)/c ---> G/T = 1.2304 and equivalently H/T = 0.5794. Both of these 2 numbers pop up in my algebra when solving the equations derived in the CAS solution.
Interestingly, the tax rate we solve for is exactly 1.25/1.2304 = 1.0159. So I feel like I'm on the right track, but I still don't quite see where the value of 1.25 comes from, and why if we can directly estimate G so easily, then why we don't also obtain H then solve for T using 0.651xT = G - H.
Thanks.
Comments
Great question - the sample answer is definitely scant on some pretty important points.
I'll assume you're fine with part a.
You are correct that there aren't handy equations for getting the maximum and minimum premiums (G and H). The only way to get them is to know the maximum and minimum ratable losses and use R = (B+cL)T.
We know r_G = Actual maximum loss/ Expected loss = actual maximum loss ratio / expected loss ratio = 80%/60% = 1.333.
Similarly, r_H = actual minimum loss / expected loss = 20%/60% = 0.333.
So we get 1 = r_G - r_H = (G - H)/(c*E[A]*T). The right hand side of this equation can be divided by 1 in the form standard premium/standard premium to express it in terms of things seen in this question if you prefer.
Using our answers from part a and psi(r) = phi(r) + r - 1 we can get phi(r_H) - phi(r_G) = 0.733 - 0.133 = 0.6.
We also have the other balance equation, phi(r_H) - phi(r_G) = ((e + E[A])*T - H) / (c*E[A]*T)
So 3 unknowns to solve for with 2 equations = we're missing some information.
In the question we're told the actual ultimate losses are 8.7m. This is 87% of the standard premium, i.e. above the loss ratio at maximum premium. Therefore, the final premium charged to the insured will be the maximum premium, G. If you prefer to work with the balance equations in terms of percentage of standard premium, then divide this by the 10m. Hence we get G = 12.5m or rather (G/SP = 1.25).
Now we can substitute this in and have 2 equations in 2 unknowns and solve.
Oh of course, crazy I used this information in my alternate solution yet skipped past drawing the connection on this approach. Thanks.
In part b, why don't we use E[L] = E[A] - c*I in the retrospective premium formula? Instead, the solution just uses 8M directly.
If we use E[L] then we're working with the expected ratable losses. However, we're told the value of the final retrospective premium which means we need to use the actual ratable losses the policy experienced.
Oh that makes perfect sense!