S17Q21 - Feldblum IRR

For part b, the crux of the question is about the impact of different surplus allocation choices, and how using a reserve based method will result in more allocation to longer tailed lines compared to a premium based method. Learning objectives D.1.e and D.1f. See Felblum pg 23.

In the footnote Felblum writes

Using the present value of future loss payments instead of undiscounted statutory reserves reduces the spread between slow paying and quick paying lines of business. An alternative adjustment for the illustration in the text is to substitute a discounted loss ratio for the 75% undiscounted WC loss ratio. The higher expected loss ratio in Workers’ Compensation than in Homeowners’ reflects the greater investment income in the former line.

S17Q21 Given:
Surplus is $25 million
The after-tax investment yield is 5%

Line A Written Premium = $30M
Line A Loss Ratio = 65%
Line A average time from loss to payment = 2.5 years

Line B Written Premium = $20M
Line B Loss Ratio = 60%
Line B average time from loss to payment = 0.5 years

Then based on the footnote:
Line A discounted LR = .65*(1/1.05)^(2.5) = .5958
Line B discounted LR = .60*(1/1.05)^.05 = .5855

PV of losses Line A = .5958 * 30 = 17.26
PV of losses Line B = .5855 * 20 = 11.71

allocation to line A = 25*(17.26/(17.26+11.71)) = 14.89
allocation to line B = 25*(11.71/(17.26+11.71)) = 10.11

Would this be a valid alternative solution to the problem? The examiner’s report only shows one sample solution, and it lists, “Attempting to discount the surplus allocation using the after-tax investment yield” as a common error.

Edit for additional info:
The problem also states,

The company’s pricing actuary is recommending to change the surplus allocation method to be:
o committed when losses are incurred
o released when losses are paid
o committed in proportion to reserves

But I don’t interpret that as being in conflict with the alternative method mentioned in the footnote. It seems to me like it should be a valid solution.

Thought about it some more, and what the proposed solution above overlooks is the “steady state” effect of the reserves. That is the reasoning behind multiplying the undiscounted nominal reserves by line by 2.5 and 0.5 in the sample solution.

But I still think that per the footnote, discounting the loss reserves is valid.

Perhaps, but the question says “committed in proportion to reserves”, not “committed in proportion to the present value of reserves”. I wouldn’t use discounting unless the question specifically asks for it - it just makes the calculations more complicated.

Also, using the 2.5 and 0.5 factors for scaling no longer works if you are using discounted reserves. You would have to actually know the true distribution of payments.