CT5- ch.2

[Bharti Singla]

Hi all
Could anyone please explain why it is written here that it is not the definition of ax ?

If we wish to see the expected benefit we have to pay under whole life annuity contract to a life aged x now, we need to take all possible cases of how many years he/she will survive and do its expectation.
That is Summation of (benefit × prob. of surviving for that period).
And this is what done here.
Then why it is not the definition of ax?

Thanks

 

[KaustavSen]

Hello Bharti,

I believe what the core reading is trying to imply is that the actual "definition" of \(a_x\) is that it is the Expected Present Value of an annuity payable annually in arrears until the life currently aged \(x\) is alive, ie,
\[
a_x = \mathbb{E}[a_{\overline{K_x}|}]
\]

However, if we simplify the expression we arrive at the formula that you have highlighted in the attached image. So, technically it is not a definition of \(a_x\) but what a formula to find the underlying value.

So, if in the exam we are asked to define \(a_x\), I believe we would be required to given its "definition" in terms of Expected Present Value and NOT simply quote the formula.

Hope this makes sense to you. Please let me know if I missed out on anything. Again, these are just my interpretation and they may be wrong .

Regards,
Kaustav.

 

[Bharti Singla]

Hello Bharti,

I believe what the core reading is trying to imply is that the actual "definition" of \(a_x\) is that it is the Expected Present Value of an annuity payable annually in arrears until the life currently aged \(x\) is alive, ie,
\[
a_x = \mathbb{E}[a_{\overline{K_x}|}]
\]

However, if we simplify the expression we arrive at the formula that you have highlighted in the attached image. So, technically it is not a definition of \(a_x\) but what a formula to find the underlying value.

So, if in the exam we are asked to define \(a_x\), I believe we would be required to given its "definition" in terms of Expected Present Value and NOT simply quote the formula.

Hope this makes sense to you. Please let me know if I missed out on anything. Again, these are just my interpretation and they may be wrong .

Regards,
Kaustav.

Hi Kaustav, thanks for the reply.
Yes, I am getting your point. That's okay, it is not the definition but the formula for ax.

But I think there is some other reason why they wrote so. See in the pic below. They have written the definition of any assurance or annuity must be given in terms of future lifetime random variable.
But I'm not getting it completely. Future lifetime random variable means all possible values of Kx (future lifetime) with their probabilities. And we are considering it in the formula of Ax and ax by taking k|qx.
Then why

 

[Mark Mitchell]

Strictly speaking, assurances and annuities are defined in terms of the future lifetime random variables Tx and Kx.

In chapters 1 and 2 of the course, the present value random variables (PVRVs) for different kinds of assurances and annuities are given. The definition of symbols that relate to the expected present value of an assurance or an annuity (Ax or ax for example), are strictly E(PVRV) (for whatever the relevant PVRV is).

When the benefits are paid at discrete points in time, we can also write down summation expressions for these expectations, and when the benefits are paid continuously/immediately on death, we can also write down integral expressions for these expectations.

For the assurance, Ax, there is only really one basic form that the summation can take (given on page 11 of Ch 1).
For the annuity, ax, there are two forms: the one given at the top of page 6 of Ch 2 (the formal definition), and the simplified formula arrived at on page 7 (after some algebra), which is a useful formula, but not the formal definition.

 

[Bharti Singla]

Thankyou for the reply.
What I'm getting now is:
It is written that ax should be in the form of future lifetime random variable. That means if we would have been a random variable of Kx,
Like- For a life aged 50 now, we have his/her future lifetime random variable (some values of Kx i.e. 10, 20, 30 and their respective probabilities). Then we could actually calculate EPV of benefit payable by Summing PV of benefits into probabilities. But it is not that easy!

What we are doing here is, taking all possible values of Kx till infinity and then taking out its expectation. Which is not actually the definition of ax.

I know it may seems not realistic or logical to have a random variable of Kx like that. But I just took example for understanding purpose. Please clarify if I'm getting it wrong.
Many thanks.

 

[Mark Mitchell]

I think you're about right.

To calculate any expectation for a discrete RV (recall CT3), we take the possible values the random variable can take, multiply by the corresponding probability and then sum over all possible values. This is what the formal definition is doing. (Note that you need to sum over ALL values that Kx can take, rather than just 10, 20 30 etc.)

This approach can be a bit long-winded though for calculation purposes. So we tend instead to use the alternative formula (summing k_p_x over all values of k) for that. This formula is directly equivalent to the formal definition for calculation purposes.

To be honest, I strongly doubt that such niceties will have any bearing on how you perform on the exam!

 

[Bharti Singla]

Okay. Thankyou sir.