Ma 162 Hw 21: Problem 1

Question

Bob makes his first $850 deposit into an IRA earning 7% compounded annually on the day he turns 27 and his last $850 deposit on the day he turns 46 (20 equal deposits in all). With no additional deposits, the money in the IRA continues to earn 7% interest compounded annually until Bob retires on his 65th birthday. How much is the IRA worth when Bob retires?

Answer 1

It helps to separate a problem as such into two segments:

1) Let's calculate the Future Value of an $850 annuity occurring annually for 20 years. Because he makes his first payment on the very day he turns 27, this first payment would be considered to be at the beginning of the first year, thus making this an annuity due rather than an ordinary annuity. The future value of an annuity due formula is PMT * {[(1+i)ⁿ-1]*(1+i)}/i. Therefore, we have 850*[(1.07²⁰-1)1.07]/.07 = $37,285.40.

2) Now, the $37,285.40 represents what is in the IRA after he finishes depositing money. Now the deposit will continue to earn interest until he is 65 years old, despite no payments being contributed annually. This represents 19 total years. Thus, the calculation will be as follows: $37,285.40×(1+0.07)¹⁹ = $134,843.68