09-12-2006, 10:11 AM
I am having hard time to understnad how the solution of R/S is obtained. Any thought?
WP is constant over time
Effective at beg of year.
Loss paid at end of year for 4 years
LR = 0.8
I = 8%
P /S = 2
What is the expense ratio to get 20% ROE?
R / s = R /P * P /S = R/P *2
I am struck here for the R / P.
09-12-2006, 02:20 PM
This problem is a little tricky. I get a different answer, but I think I can shed some light anyway.
Take the basic equation: T/S = I/A(1+R/S) + U/S.
It may be easier to approach this problem by using T/P = T/S*S/P = 0.2*0.5 = 0.1, then multiplying by P/S = 2, so we should look at the equation:
T/P = I/A(S/P+R/P) + U/P = I/A(0.5 + R/P) + U/P.
Now, what are the units for each term? The units of T/P is total return % (over the policy period). The units on I/A are % return/year, so the units on R/P must be in years, in other words, how many years, as a percent of premium you have to invest. Think in terms of simple interest: if you invest 50% of your funds for 4 years at 5% return, you're looking at a return of 10% = 5% * 4 * 50%; in this case you can say you've invested "2" premium years.
Since the loss ratio is 80%, paid evenly at the end of each of the four years, you'll have
0.8P in reserves the first year,
0.6P in reserves the second year,
0.4P in reserves the third year,
0.2P in reservese the fourth year
for a total of 2P years invested, so R/P = 2 years. Now we solve for U/P:
0.1 = 8%(0.5 + 2) + U/P
U/P = -0.1, for a combined ratio of 110%.
So, the expense ratio = 110% - 80% = 30%.
Hope this helps.
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