Hi everyone I'm reading through the notes currently. I'm a bit stuck with something in Chapter 4 Retrospective Accumulations, and hoping someone can help: For a pure endowment Starting off with N people in the group F(N) is the accumulated fund What I don't understand is the following limit: Lim F(N) = E[F(1)] (n tends to ∞) N
oops, now I know why there's a 'preview post' basically, as n tends to infinity, the limit of F(N) / N is equal to E[F(1)]
Hi With reference to the retrospective reserves formula in chapter 5 section 5.4, I understand why (for example) the accumulated value of the benefits is calculated for up to time t, then indexed to time t using 1+i . How come the formula then is multiplied by l_x/l_x+t and not the inverse?
Because the pot is split between the people who are expected to be alive at the end i believe. If you only have one person you expect to have t_p_x alive at the end, therefore u divide by l_x+t / l_x which is the same as multiplying by the factor you stated. xx
Can someone please explain to me how the limit ends up being an expectation i.e the limit of F(N) / N is equal to E[F(1)]. Also, I still don't understand the rationale for multiplying by Lx and dividing by Lx+t. Please explain.
I'm not really sure there's much to add to DevonMatthews explanation, which I think is very good. F(1) is the accumulated value of the fund at age x+n for 1 survivor (ie fund per survivor) starting at age x. What do we expect this to be? ie What is E[F(1)]? Each survivor's share is the total fund F(N) divided by the survivors N1. So, by definition, E[F(1)] is the limit of this as the number of people gets large - there's no calculation that leads to it, it's the definition of an average. tpx = probability that somebody aged x survives t years = lx+t / lx So, 1/tpx = lx / lx+t, that's where that bit comes from, Good luck! John
