To figure out how much Derek needs to deposit annually in order to reach his retirement goal of $3,220,614 by the time he turns 65, we will use the future value of an annuity formula. We'll first need to determine the number of years he will be making deposits and the interest rate.
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Determine the number of deposits:
- Derek starts depositing on his 28th birthday and stops on his 65th birthday.
- The number of deposits is from age 28 to age 65, which is \(65 - 28 + 1 = 38\) deposits.
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Annual interest rate:
- The interest rate is given as 7.00%, which we can express as a decimal: \(r = 0.07\).
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Future Value of an Annuity Formula: The future value of an annuity can be calculated using the formula: \[ FV = P \times \frac{(1 + r)^n - 1}{r} \] where:
- \(FV\) is the future value ($3,220,614),
- \(P\) is the annual deposit,
- \(r\) is the annual interest rate (0.07),
- \(n\) is the number of deposits (38).
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Rearranging the Formula to Solve for P: We can rearrange the formula to solve for \(P\): \[ P = \frac{FV \times r}{(1 + r)^n - 1} \]
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Plugging in the values: \[ P = \frac{3,220,614 \times 0.07}{(1 + 0.07)^{38} - 1} \]
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Calculating \((1 + r)^n\): \[ (1 + 0.07)^{38} \approx 7.684 \]
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Calculating the denominator: \[ (1 + 0.07)^{38} - 1 \approx 7.684 - 1 \approx 6.684 \]
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Calculating P: \[ P = \frac{3,220,614 \times 0.07}{6.684} \approx \frac{225,444.98}{6.684} \approx 33,746.88 \]
Thus, the amount Derek must deposit annually into his retirement account from his 28th birthday to his 65th birthday is approximately $33,746.88.