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

Discussion in 'CM1' started by Danny, Feb 26, 2024.

  1. Danny

    Danny Active Member

    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)
     
    Last edited: Feb 26, 2024
    Danny, Feb 26, 2024
    #1
  2. Joe Hook

    Joe Hook ActEd Tutor Staff Member

    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
     
    Joe Hook, Feb 27, 2024
    #2

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