@Old_Joe @Junkster - Aw, shucks.
@Dex - regarding the IRR (rate of return). From one perspective (especially on the insurer's side), the calculation is a lot more complicated, because the payout is not for a fixed term of
years, but a life expectancy. This involves actuarial tables, probabilities, analysis of customer base (purchasers will self-select for longer lifetimes), etc.
From your perspective, perhaps the calculation is simpler - you know your health, and are much more able to treat the annuity as a fixed term of
years, even if this is just an approximation.
In that case, the formula is relatively simple (but there's no closed form to compute the solution, i.e. IRR; a computer can calculate it by iterative approximation).
Let M be the number of
years until payments start, and N the number of
years of payments. Here, M is 15 (buy at age 70, start payments at age 85). Pick your own number for N.
By definition, the present value is the purchase price PP ($125K), and what you're interested in is the rate of return. You've got the right idea ... the value at year M (when payments start) is
PP * (1+r)^M = $125K * (1+r) ^ 15.
There's a standard formula for the value (price) of an annuity with N payments of $C ($55K). You can find it in a pretty nice paper
here. It is:
PV (present value at start of payments) = C/r * [1 - 1/(1+r)^N] = $55K/r * [1 - 1/(1+r)^N
So we set these two expressions, representing the value of the annuity at the time payments start, equal to each other, and solve.
$125K * (1+r) ^15 = $55K/r * [1 - 1/(1+r)^N] or
$125K * (1+r) ^15 - $55K/r * [1 - 1/(1+r)^N] = 0
(In case it matters, you can see this is a polynomial equation by multiplying both sides by (1+r)^N and by r to clear the fractions.)
So now you're left with an algebra problem in the form: f(r) = 0.
You want to find the real root of this equation with r somewhere between 0% and 20%.
There are various mathematical packages that will do this for you, e.g. Matlab's fzero function. If one is into programming, there are simple iterative methods to find roots, e.g. bisection and Newton's method. See, e.g.
http://www.math.niu.edu/~dattab/MATH435.2013/ROOT_FINDING.pdfOr you could look for online solvers. A quick search for online bisection method calculator turned up
http://keisan.casio.com/exec/system/1222999061(Bisection is slower, but you don't need to provide the derivative of your function as you would for Newton's method.)
I tried this calculator for N=10 (payments to age 95) and came up with 7.61% rate.
With N = 5 (payments to age 90), the return is 4.49%.(Use ^ for exponent and * for multiplication, as I did above. Also use a range between 0.01 and 0.2 - to avoid dividing by zero - see the $55K/r in the expression above. Finally, replace r in my expression with x for this calculator.)