You wish to retire in 12 years and currently have $50,000 in a savings account yielding 5 percent annually and $100,000 in quality "blue chip" stocks yielding 10 percent. If you expect to add $30,000 at the end of each year to your stock portfolios, how much will you have in your retirement fund when you retire? What rate of return must you earn on your retirement funds if you want to withdraw $102,000 per year for the next 15 years after retiring?) .

Question

Reiny · Answer

first let the 50,000 and the 100,000 ride for the 12 years.

Amount = 50000(1.05)^12 + 100000(1.10)^12
= 403,635.65
PLus our annuity of 30,000 for 12 years at 10%
= 30000(1.1^12 - 1)/.1
=641,528.51
for a total of $1,045,164.17

Last part:
let the rate be i

1,045,164.17 = 102,000 (1 - (1+i)^-15)/i
10.24670752 i = 1 - (1+i)^-15

that is going to be hard to solve, and will need something like Newton's Method,
I will "cheat" and use Wolfram
http://www.wolframalpha.com/input/?i=10.24670752x+%3D+1+-+%281%2Bx%29%5E-15

(I had to switch to x, since Wolfram reads i as √-1)

x = .0519134
So you will need a return of 5.19% per annum

check:
102000(1 - 1.0519134^-15)/.0519134
= 1,045,164.20

(how about that? Out by 3 cents in over 1 Million dollars !!!!)