Chpt 5, Retrospective accumulation.

[Ark raw]

1.) Sec 5.1, retrospective accumulation, pure endowment.

I couldn't understand how the results were found for these limits:

lim F(N)/N^1 = E[F(1)]
N->∞

lim N^1/N = n_p_x (for this limit why can't we simply write : N^1/N = n_p_x)
N->∞
2.)sec 5.2 , term assurance

why is the distribution of F(1) defined that way on page 16?

 

[Robert Chadburn]

Let's take this first:

lim N^1/N = n_p_x N->∞

N1 is the random number of survivors out of the N starting lives aged x. So N1/N is the observed proportion of survivors, and this will differ from the probability of surviving, due to random sampling error. But, as the number of lives (that you start with) increases towards infinity, the random sampling error gets proportionally smaller and smaller until, in the limit, the ratio will exactly equal the probability.

You have mis-quoted your first example. I presume you mean:

lim F(N)/N = E[F(1)]
N->∞

The logic is similar. F(N) is the total fund accumulated by time N, from N investors alive at the start, and by dividing this by N we are expressing this as the Fund at time N accumulated per initial investor. F(N) is random. As N increases towards infinity the ratio becomes closer and closer to the expected ( = long-term average) amount of fund that would be accumulated per initial investor, and will ultimately equal the expected value in the limit.

On page 16, F(1) is the accumulated fund. The term assurance pays 1 at the end of the year of death - that is at time k+1. So the first line of F(1) is the accumulated value of this payment, from time k+1 to time n. This is the value if he dies during the term. The value is zero if he survives the n years (as no payment is made) which is the second line.

BUT please put this proof in context: I have not known it ever come up in the exam. All you need for the exam is the ability to perform accumulations. Also, have a look at other threads with the "retrospective accumulations" title (eg over the last 2-3 pages of posts.)

PROOFS in CT5 are not important, as they will make up around 5% of the exam. You need to be able to apply the methods to solve problems - looking at past exam questions is always the best way to prepare. When studying the notes don't get bogged down in theory - you don't need to follow every step of every derivation - try and keep a high level view. Then, get trying some questions as soon as you possibly can, and that way you will get to know what's important.

Robert

 

[Ark raw]

lim F(N)/N = E[F(1)]
N->∞

The logic is similar. F(N) is the total fund accumulated by time N, from N investors alive at the start, and by dividing this by N we are expressing this as the Fund at time N accumulated per initial investor. F(N) is random.

1.)The notes state that F(N) is the accumulated fund at age x+n, so was it suppose to be time n instead of time N

As N increases towards infinity the ratio becomes closer and closer to the expected ( = long-term average) amount of fund that would be accumulated per initial investor, and will ultimately equal the expected value in the limit.

2.) I didn't understand how " as N ->∞ the ratio becomes closer and closer to the expected ( = long-term average) amount of fund that would be accumulated per initial investor"?

F(1) is the accumulated fund. The term assurance pays 1 at the end of the year of death - that is at time k+1. So the first line of F(1) is the accumulated value of this payment, from time k+1 to time n. This is the value if he dies during the term. The value is zero if he survives the n years (as no payment is made) which is the second line.

3.) If F(1) is the accumulated fund, then why F(1) doesn't contain an expression for the premium paid by the policy holder in both the cases i.e. k<n and k>=n? And why does it accumulate benefit amount?

Thank you

 

[Robert Chadburn]

1.)The notes state that F(N) is the accumulated fund at age x+n, so was it suppose to be time n instead of time N

No, N is the number of people alive at age x. F[N] is the fund at time n which has accumulated from these N lives. The notation that the Core Reading uses is very confusing!

2.) I didn't understand how " as N ->∞ the ratio becomes closer and closer to the expected ( = long-term average) amount of fund that would be accumulated per initial investor"?

It's the law of large numbers. If you throw a coin N times and calculate the proportion of times you throw a head. This proportion will be approximately 0.5, but will differ due to random error. But as N tends to infinity then the random error disappears and it becomes equal to 0.5.

3.) If F(1) is the accumulated fund, then why F(1) doesn't contain an expression for the premium paid by the policy holder in both the cases i.e. k<n and k>=n? And why does it accumulate benefit amount?

Ok, if you look again at the notes, it is just the benefit amount it is referring to. F(1) is the fund that is accumulated from the payment of the benefit amount only. It has not included any other payment.
Robert