Retrospective Accumulations

[tatos]

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

 

[tatos]

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)]

 

[jensen]

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?

 

[DevonMatthews]

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

 

[jm_kinuthia]

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.

 

[John Potter]

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