I think that the explanation is as follows. Assume you are trying to calculate the 2006 UPR c/f (gross of reinsurance):
Annual policies:
(a) the amount of premium written in the first year is 24,000 x 100. Of this, only 24,000 x 100 x 0.75 is written in the first year (the first three months are missing);
(b) the policies in (b) will incept on average in the middle of the period going from 1-7 to 31-3, which is 16-11, i.e. 4.5 months (from 16-11 to 31-3) in the first year, and for 7.5 months (from 1-4 to 15-11) in the second year à it will earn 7.5/12 of its premium in the following year;
(c) the contribution of the annual premium policies is therefore UPR(1) = 24,000 x 100 x 0.75 x 7.5/12;
Two weeks policies:
(d) the amount of premium written in the first year related to two-weeks policies is 780,000 x 20.Of this, only those in the last two weeks of 2006 can contribute to the UPR, since all the others expire before the end of the year. This corresponds to 780,000 x 20 x 2/52 premium;
(e) only half of this amount will be earned the following year, according to the usual reasoning (pick a representative policy in the middle of the period – which in this case is exactly one week before the end of year – etc). Therefore, the contribution of two-weeks policies is UPR(2) = 780,000 x 20 x 1/52;
Overall:
(f) the total UPR is UPR(1) + UPR(2) = 0.75*(24,000 x 100 x 7.5/12 + 780,000 x 20 x 1/39) [notice that 0.75/39 = 52]
And yes, I think there are general ways to deal with these questions, using integration as Ian suggests. For example, in the case where the risk doesn’t depend on external factors but on the age of the policy, then the earned premium in year [0,1] is (please check):
EP([0,1]) = integral(0 --> 1) p(t) x R(1-t) x dt
Where p(t) = density of premium written and R(t) = cumulative risk between 0 and t (R(T)=1 when T = duration of the policy).
The likelihood of coming up with the correct answer under exam conditions using a formula like that is however null (at least for me). And a different formula applies if the risk is distributed seasonally, regardless of policy inception...
