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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?

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  • $\begingroup$ 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. $\endgroup$ – saulspatz Feb 2 at 22:54

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