How do you evaluate a joint life temporary annuity?

[Danny]

Hi,

Suppose I want to evaluate the EPV of a joint life temporary annuity paying annually in advance, for two lives, one age 60, one age 50, with a term of 20.

I am aware of the formula for this payable continuously i.e the integral of the survival probability multiplied by the discount factor. By this logic, I know from first principles I could write these 20 terms out, but this would take too long. I also considered a geometric series, but this doesn't seem to be possible from what I can see, either.

So, what would be a formula that I can actually use to evaluate a temporary joint life annuity practically?

 

[Richie Holway]

Hi,

Suppose I want to evaluate the EPV of a joint life temporary annuity paying annually in advance, for two lives, one age 60, one age 50, with a term of 20.

I am aware of the formula for this payable continuously i.e the integral of the survival probability multiplied by the discount factor. By this logic, I know from first principles I could write these 20 terms out, but this would take too long. I also considered a geometric series, but this doesn't seem to be possible from what I can see, either.

So, what would be a formula that I can actually use to evaluate a temporary joint life annuity practically?

Hi Danny,

We could use the approach: whole life joint life annuity minus discounted whole life joint life annuity, ie adue:xy:<n> = adue:xy – adue:x+n:y+n * npxy * v^n.

The first whole life joint life annuity (adue:xy ) would value all possible payments, including both those within the term and beyond. So the 2nd (aduex+n:y+n), which we deduct, needs to value all of those payments beyond the term (n), as these are not possible payments under the temporary annuity, hence why we add n to both ages within that function. In addition, as that 2nd annuity is valuing payments that begin n years into the future, we need to discount those back to today using both financial discounting (v^n) and the probability of survival of both lives (npxy, or npx * npy).

I hope this helps,
Richie

 

[Danny]

Hi Danny,

We could use the approach: whole life joint life annuity minus discounted whole life joint life annuity, ie adue:xy:<n> = adue:xy – adue:x+n:y+n * npxy * v^n.

The first whole life joint life annuity (adue:xy ) would value all possible payments, including both those within the term and beyond. So the 2nd (aduex+n:y+n), which we deduct, needs to value all of those payments beyond the term (n), as these are not possible payments under the temporary annuity, hence why we add n to both ages within that function. In addition, as that 2nd annuity is valuing payments that begin n years into the future, we need to discount those back to today using both financial discounting (v^n) and the probability of survival of both lives (npxy, or npx * npy).

I hope this helps,
Richie

Thanks!