# Annuity payable n years after death

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Hi

Could you please explain the reasoning behind the solution given in the memo? Why is the guaranteed annuity of duration n being paid first and then the annuity with duration t being paid afterwards (since it is being discounted back n years after the initial annuity)? I did it differently and so why would my expression be incorrect in this case?

Both answers follow mathematically when I try and simplify the integral of the annuity with term T+n depending on whether I split the integrals at t or n, but intuitively the other expression makes more sense to me as surely the guaranteed annuity with n payments occurs after the death of the policyholder and the annuity dependant on t occurs whilst the policyholder is still alive?

Thanks

by (2.9k points)
The solution in the memo is a mathematical solution that you need to think about from a less-intuitive angle. Remember that we are dealing with a timeline of cashflows and all we are trying to estimate is a level amount of cashflows that end n years after death.

We know that the annuity has a minimum of n years so we start there. We then are just, separately, adding the annuity till the policyholder dies and we are ordering the guaranteed annuity to happen first to simplify the calculation.

The key point you need to remember is that whilst the cashflows of the complete life annuity are being discounted by more than we expect, the ages remain the same which is why this works (in a less-intuitive way).

Where your answer wouldn't be correct is that you still need to take the expectation of the expression you have. V^t and a^(bar)_t are random variables, not an answer so you would need to take the expectation of your expression which would result in your answer which would be accepted.

The problem with your answer is that now how will you get the variance of your expression? It'll be much more challenging and time-consuming which is why the memo went down the route that it did (making the next steps much easier).
by (1.1k points)
edited by
Thank you. Regarding the last bit on how the variance will be calculated, can't one just do it the same way as the memo and treat the guaranteed annuity as a constant (since it's not dependant on t?), thereby factoring it out? I don't get the same answer as the memo though, the variance will be higher with the guaranteed annuity occurring after death.
by (2.9k points)
You treat the guaranteed annuity as constant but you need to work out the variance of V^t and appropriate add that to the variance of the other annuity w.r.t. so that it should end up being equal (otherwise you didn't simplify the expression before taking the variance).

I get the exact same answer as the memo when I take the variance (see photo below).
by (1.1k points)
Thanks, I can't seem to see the photo. Could you please re-upload it?
by (2.9k points)

Here is the answer I worked out:

by (1.1k points)
Thanks for the assistance!