# Math Help - Future Value of an Annuity Problem-HELP!!

1. ## Future Value of an Annuity Problem-HELP!!

Hi everyone,
Here is a problem that was given to me in one of my classes and I've been up nearly all night trying to figure this thing out! I would very much appreciate if anyone can help me out on this tough question:

Your father is 50 years old and will retire in 10 years. He expects to live for 25 years after he retires- until he is 85. He wants a fixed retirement income that has the same purchasing power at the time he retires as $40,000 has today. His retirement income will begin from the day he retires until 10 years from today, and he will then receive 24 additional annual payments. Annual inflation is expected to be 5 percent. He currently has$100,000 saved and he expects to earn 8% annually on his savings. How much must he save during the next 10 years to meet his retirement goal?

2. If we take the phrase “Your father wants a fixed retirement income that has the same purchasing power at the time he retires as $40,000 has today in 10 years time” to mean that at the end of 10 years it takes$65,155.79 {as in 40,000(1.05)^10 ≈ $65,155.78507} to buy what “Your father” could have bought for$40,000 at the beginning of 10 years, or at age 50, then it’s quite reasonable that “Your father” also expects his next 24 additional annual payments after age 60 to reflect the same purchasing power when he was 50. If this is indeed the case, then your problem is basically a growing (geometric) annuity problem.
(See Sample Problem #4 of page 5 at http://web.utk.edu/~jwachowi/growing_annuity.pdf and the reasoning used by Doctor Mitteldorf at Math Forum - Ask Dr. Math)

Accordingly, for “Your father” to meet his retirement goal, you must first solve what his accumulated savings must amount to in 9 years. You can easily do this by use of the following formula:

$
A_{{\text{ordinary geometric growing annuity}}} = R \cdot \frac{{1 - \left( {\tfrac{{1 + g}}
{{1 + i}}} \right)^n }}
{{i - g}}
$

where
R = 40,000(1.05)^10
g = 5%
i = 8%
n = 25 years

To determine how much “Your father" must save during the next 9 years (not 10 years as you stated), you simply solve for $
R_s$
in the following equation of value:

$
R \cdot \frac{{1 - \left( {\tfrac{{1 + g}}
{{1 + i}}} \right)^n }}
{{i - g}} = 100,000\left( {1 + i} \right)^9 + R_s \cdot \frac{{\left( {1 + i} \right)^9 - 1}}
{i}
$