Here's the relevant information:
* 10% rate decrease effective 7/1/87.
* 6 month policies.
* Exposures increase 25% annually beginning 1/1/87
* The level of exposures as of 1/1/87 is A.
The problem is to determine the earned exposures at the higher
rate level for the fiscal year ending 7/1/88.
Solution:
If you index the rates so that the rate level as of 1/1/87 is
1.0, then the situation described looks like this:
(Switch to a font like Courier if the diagram below is garbled)
_________________________
| | / |
| | / |
| 1.0 | 1.0 / 0.9 |
| | / |
| | / |
| | / |
| | / |
| |/ |
-------------------------
1/87 7/87 1/88 7/88
'The fiscal year ending 7/1/88' refers to the period from 7/1/87
to 7/1/88, so the 'earned exposures at the higher rate level'
correspond to triangular region in the diagram.
The exposure level is increasing 25% per year beginning 1/1/87
from a exposure level of A. This can be expressed as:
f(x) = A(1 + .25x) where x is the time (in years) from 1/1/87.
(It's not clear from the problem whether you should use
the formula above or f(x) = A(1 + .25)**x where the ** indicates
an exponent. By looking at the answer choices, however, you can
rule out this second possibility.)
To find the earned exposures in question one needs to integrate
f(x - ky) over the triangular region where k is the policy duration,
0.5 year. f(x - ky) = A(1 + .25(x - .5y))
Because x is the time since 1/1/88, let x vary from 0.5 to 1,
since the triangle goes from 7/1/87 to 1/1/88. Then y must vary
from the diagonal line, y = 2x - 1 to the top horizontal line y=1.
So the answer is E.
If you set up the integral in the opposite way, i.e., integrating
from y = 0 to y = 1 and from x = 0.5 to x = (y - 1)/2, then it would
be quite difficult to recognize E as the correct choice. It is
probably wise therefore to scan the answers before setting up the
integral.
-David