Continuously payable annuity approximation - does it hold for temporary annuities?

[Danny]

Hi,

Does the approximation for a continuously payable whole of life annuity also apply to a temporary continuously payable annuity?

I.e, we know that:

abar_x is approx equal to adue_x-1/2

But is it also the case that:

abar_x:n is approx equal to adue_x:n-1/2

Assuming the above doesn't hold, I also have a second proposition for what might work.

We know that:

adue_x:n=adue_x+v^nnpxadue_(x+n)

and:

aarrears_x:n=aarrears_x+v^nnpxaarrears_(x+n)

So is it the case that:

acontinuous_x:n=acontinuous_x+v^nnpxacontinuous_(x+n)

 

[Joe Hook]

Hi Danny,

For the formula to apply please see chapter 14 page 34:

abarx:<n> = aduex:<n> - 1/2 *(1 - v^n * npx)

This formula comes from the general principles that we apply for term assurances and other temporary annuities "whole of life minus discounted whole of life". So if I calculate as:

abar:x - abar:x+n * v^n * npx

and then replace abar:x and abar:x+n with adue:x - 1/2 and adue:x+n - 1/2 I can rearrange to the result above.

This I think is the result you have at the bottom of your post but you've used a plus rather than a minus ie you are adding two annuities together rather than taking one away from the other.

Joe