Hi Duc,
The definition of 'profits' in such a context would normally include change in reserves.
Prof(t) = P(t) + I(t) - C(t) - E(t) - dR(t), where dR(t) = R(t) - R(t-1) is the change in reserves over period (t-1,t]
Let's have a look at a single contract assuming gross premium reserves are used and assumptions in the valuation basis are borne out in practice.
Year 1:
Change in reserves is just the reserve at t=1, and this can be calculated (ignoring interest for simplicity) as:
R(1) = [C(2)+...+C(n)] + [E(2)+...+E(n)] - [P(2)+...+P(n)]
If we now put it into the profit definition above we'll get:
Prof(1) = Sum of premiums - Sum of claims - Sum of expenses, where we're adding up cash flows over the whole duration of the contract (years 1 to n).
Year 2:
Change in reserves is now equal to dR(2) = R(2) - R(1) = P(2) - C(2) - E(2)
Hence Prof(2) = P(2) - C(2) - E(2) - [P(2) - C(2) - E(2)] = 0
So in year 2 and beyond we recognise no additional profits; this is all assuming that experience turns out exactly as in the valuation basis (meaning actual premiums, claims, etc are exactly the same as assumed in the reserve calculation).
There are various reasons why in years 2+ we can have profits higher or lower than 0 (actual vs expected variance, change in assumptions, etc.), but this example is simplistic on purpose and does not cover all important aspects.
Last edited: Oct 24, 2020