Bob makes his first $800 deposit into an IRA earning 7% compounded annually on the day he turns 27 and his last $800 deposit on the day he turns 35. (9 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?
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To calculate the worth of Bob's IRA when he retires, we need to consider the annual deposits and the interest earned over the years.
First, let's break down the problem into different periods:
1. Period 1: From age 27 to age 35 (9 years)
- Bob makes 9 equal deposits of $800 each, earning 7% interest compounded annually.
- To calculate the future value of these deposits, we can use the formula for the future value of an ordinary annuity:
Future Value = Payment [(1 + r)^n - 1] / r,
where Payment is the annual deposit, r is the interest rate (7%), and n is the number of years.
Calculation:
Payment = $800,
r = 7% = 0.07,
n = 9.
Future Value = $800 [(1 + 0.07)^9 - 1] / 0.07 = $8,383.90.
Therefore, at age 35, Bob's IRA is worth $8,383.90.
2. Period 2: From age 35 to age 65 (30 years)
- Bob does not make any additional deposits, but the money in the IRA continues to earn 7% interest compounded annually.
- To calculate the future value of this amount, we can use the formula for compound interest:
Future Value = Present Value * (1 + r)^n,
where Present Value is the current value of the IRA, r is the interest rate (7%), and n is the number of years.
Calculation:
Present Value = $8,383.90,
r = 7% = 0.07,
n = 30.
Future Value = $8,383.90 * (1 + 0.07)^30 = $47,316.71.
Therefore, at age 65, Bob's IRA is worth $47,316.71.
Hence, the IRA is worth $47,316.71 when Bob retires on his 65th birthday.