# Deferred Term-Limited Annuity

I'm looking for a formula to calculate the EPV of an annuity (in terms of $$a_x$$ , $$a_{xy}$$ etc) that is payable during life y, for a maximum of n years, but only beginning on the death of life x.

I can work out the EPV of a deferred unlimited annuity - it's simply $$a_y - a_{xy}$$. Intuitively that's life y receiving an annuity, but "paying it back" as long as he and life x are still living.

I can also solve it for a guaranteed annuity payable on death of life x. That's simply $$a_{10}|$$ * $$A_x$$.

Here, however, the present value of the annuity varies depending on the lifetime of x and relies on life y being alive, so it's more complicated.

Is there some kind of intuitive answer to this?

• If the annuity were payable during the life of $y$ but for no more than $10$ years, the present value would be an immediate life annuity to $y$ minus a ten-year deferred annuity to $y.$ I don't remember the notation. I think you can expand on this to get the joint annuity. – saulspatz Feb 2 at 22:54