Class Ratemaking and Return on Surplus (Spr08 Q44)
(The attached PDF file has better formatting.)
CAS Spring 2008 Exam 5 Question 44 (3 points)
You are given the following information.
Class A
Class B
Premium First Year
$633.80
X
Loss Cost First Year
$500
$1,000
Fixed Expense
First Year
$90
$120
Subsequent Years
$80
$110
Variable Expense
10%
5%
Persistency Rate
80%
60%
Loss costs for renewal business are 10% lower than for new business.
There is no premium trend and there are no rate changes.
There is no expense trend.
Interest rate for discount is 10%.
Premium to surplus ratio is 2:1.
Target pre-tax return on equity is 6%.
Calculate the indicated rate relativity for Class B as compared to the base class (Class A) using the asset share pricing model and a two year time horizon.
Solution 1.1: This exercise uses simple factors: no premium or expense trends, the same fixed expenses by class, and a two year time horizon. We solve the exercise and explain how to solve a more general problem.
Step 1: Verify pre-tax return on equity for Class A
We verify the pre-tax return on equity for Class A to make sure we use the right formula.
Income in the first year is $633.80 – $500 – $90 – 10% × $633.80 = ($19.58).
Income in the second year is $633.80 – $500 × 90% – $80 – 5% × $633.80 = $72.11.
Only 80% of policies renew for the second year, so the income per dollar of original premium = $72.11 × 80% = $57.69.
The present value at the underwriting date of the second year income using a 10% discount rate is $57.69 / 1.1 = $52.44.
The present value of underwriting income for all years is $52.44 – $19.58 = $32.86.
The present value of the second year premium using a 10% discount rate is $633.80 × 80% / 1.1 = $460.95.
The present value of premium for all years is $633.80 + $460.95 = $1,094.75.
The return on premium is $32.68 / $1,094.75 = 2.99%
At a 2:1 premium to surplus ratio, this is a 2.99% × 2 = 5.98%
We verified the 6% return on surplus for Class A. We use the same procedure to back into the premium needed for a 6% return on surplus for Class B.
Step 2: Determine the lifetime return on premium for Class B. Denote the premium as Z.
Income in the first year is Z – $1,000 – $120 – 10% × Z = 90% × Z – $1,120.00.
Income in the second year is Z – $1,000 × 90% – $110 – 5% × Z = 95% × Z – $1,010.00.
Only 60% of policies renew for the second year, so the income per dollar of original premium = (95% × Z – $1,010.00) × 60% = 57.00% × Z – $606.00.
The present value at the underwriting date of the second year income using a 10% discount rate is (57.00% × Z – $606.00) / 1.1 = 51.82% × Z – $550.91.
The present value of underwriting income for all years is 90% × Z – $1,120.00 + 51.82% × Z – $550.91 = 141.82% × Z – $1,670.91.
The present value of the second year premium using a 10% discount rate is Z × 60% / 1.1 = 54.55% × Z.
The present value of premium for all years is 154.55% × Z.
The return on premium is (141.82% × Z – $1,670.91) / (1.5455 × Z).
At a 2 to 1 premium to surplus ratio, the pre-tax return on surplus is 2 × (141.82% × Z – $1,670.91) / (1.5455 × Z).
We compare the return on surplus with 6%:
2 × (141.82% × Z – $1,670.91) / 1.5455Z = 6%
A
Take heed: For optimal exam preparation, know the likely variations on this problem.
(1) Most problems include trend rates for premium, fixed expenses, and losses.
The ratio of renewal losses to new business losses assumes no trend.
Both trend and loss improvement must be applied to the renewal losses.
To verify the return on surplus in Class A and to compute the return on surplus in Class B, adjust the second year dollars by the trend factors.
(2) Variable expenses, premium to surplus ratios, discount rates, and the return on surplus do not generally differ by class.
(3) A more complete problem will not give the return on surplus. The problem will say
The premium to surplus ratio is the same for all classes.
The return on surplus is the same for all classes.
To solve the problem, compute the return on surplus for Class A and then back into the premium for Class B that gives the same return on surplus.
We showed the procedure in Step 1: "Verifying the 6% Return on Surplus." Don’t neglect this step. In future problems, you won’t be given the return on surplus and you must derive it from the premium rates.
Take heed: You generally don’t need to compute the return on surplus. If the premium to surplus ratio and the return on surplus is the same for both classes, the return on premium is the same for both classes.
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