You plan to retire 33 years from now. You expect that you will live 27 years after retiring.
You want to have enough money upon reaching retirement age to withdraw $180,000 from
the account at the beginning of each year you expect to live, and yet still have $2,500,000 left
in the account at the time of your expected death (60 years from now). You plan to
accumulate the retirement fund by making equal annual deposits at the end of each year for
the next 33 years. You expect that you will be able to earn 12% per year on your deposits.
However, you only expect to earn 6% per year on your investment after you retire since you
will choose to place the money in less risky investments.
Required:
Calculate the equal annual deposits you must make each year to reach your retirement
goal.
To calculate the equal annual deposits required, we can use the future value of an annuity formula:
FV = PMT * [(1 + r)^n - 1] / r
Where:
FV = future value (desired retirement savings)
PMT = equal annual deposits
r = annual interest rate
n = number of years
Given:
FV = $2,500,000 (desired amount at retirement)
PMT = $180,000 (annual withdrawal during retirement)
r1 = 12% (annual interest rate before retirement)
r2 = 6% (annual interest rate after retirement)
n1 = 33 (number of years until retirement)
n2 = 27 (number of years in retirement)
First, let's calculate how much the annual deposits accumulate to at retirement using the formula:
FV = PMT * [(1 + r)^n - 1] / r
2,500,000 = PMT * [(1 + 0.12)^33 - 1] / 0.12
Simplifying the equation:
PMT * (1.12^33 - 1) / 0.12 = 2,500,000
PMT * 64.8015 = 2,500,000
PMT = 2,500,000 / 64.8015
PMT ≈ $38,574.09
Therefore, the equal annual deposits you must make each year to reach your retirement goal is approximately $38,574.09.