Question

Juan has an annuity that pays him $9400 at the beginning of each year. Assume the economy will grow at a rate of 3.4% annually. What is the value of the annuity if he received it now instead of over a period of 10 years?

Answer 1

4. Let P be the current balance of the annuity and let Pn denote the amount remaining in the annuity AT THE BEGINNING of year n (i.e. before the $9,400 is paid out).

P0 = P

P0 was the balance at the start of year 0, and then $9,400 was deducted and paid to Juan. So the interest begins accumulating after the deduction, and at the beginning of year 1 the annuity now has:
P1 = 1.034(P0 - 9,400)
P1 = 1.034P0 - 1.034(9400)
P1 = 1.034P - 1.034(9400)

P2 = 1.034(P1 - 9400)
P2 = 1.034P1 - 1.034(9400)
P2 = 1.034[1.034P - 1.034(9400)] - 1.034(9400)
P2 = (1.034^2)P - (1.034^2)(9400) - 1.034(9400)
P2 = (1.034^2)P - (1.034^2 + 1.034)(9400)

-- etc --

Pn = (1.034^n)P - (1.34^n + 1.34^(n-1) + ... + 1.34)(9400)
Pn = (1.034^n)P - [(1.034^n - 1)/(1.034 - 1) - 1](9400)

The annuity pays out for 10 years (years 0 thru 9), so at the beginning of year 10 (n=10) the remaining balance is zero.

0 = (1.034^10)P - [(1.034^10 - 1)/(1.034 - 1) - 1](9400)
[(1.034^10 - 1)/(1.034 - 1) - 1](9400) = (1.034^10)P
[(1.034^10 - 1)/(1.034 - 1) - 1](9400)/(1.034^10) = P

And when you work out all the math, you end up with P = $81,243.05.

from yahoo answers

Answer 2

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