The total exposure area for the mature claims-made policy written 1/3 of the
way through year 1 is 1/2*1*1 = 1/2. (the shaded are in the graph)
In order to find the earned exposures for this policy, you have to take
4/9'ths of the exposure from report year 1, lag 0 (4/9)*(1/2)(1)(1) = 4/18,
Plus the 1/9th of the exposure from report year 2, lag 0 (1/9)*(1/2)(1)(1) =
1/18
Plus 2/9th the exposure from Report year 2, lag 1 (2/9)*(1) = 4/18
If you add these up, they add up to 1/2 which is the area in the graph which
represents the mature claims made policy written 1/3 of the way through year
j.
I don't know if this explanation is very clear. I hope it helps.