To solve for the annual deposits Derek needs to make into his retirement account to reach his goal of $3,220,614.00 by his 65th birthday, we need to use the future value of an annuity formula.
We can break down the components as follows:
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FV (Future Value): This is the amount Derek wants to have at retirement. In this case, \( FV = 3,220,614.00 \).
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PMT (Payment): This is what we are trying to find — the amount that Derek needs to deposit annually.
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i/y (Interest Rate): This is the annual interest rate the retirement account will earn. Given that the interest rate is 7%, we can enter \( i/y = 7% \) or 0.07 in decimal form.
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n (Number of Periods): Derek will begin making deposits at age 28 and continue until he is 65. That gives him \( 65 - 28 = 37 \) deposits.
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PV (Present Value): Since he starts with nothing today (he makes no initial deposit), \( PV = 0 \).
The formula for the future value of an annuity can be expressed as:
\[ FV = PMT \times \frac{(1 + r)^n - 1}{r} \]
Where:
- \( FV \) is the future value of the annuity,
- \( PMT \) is the annual payment,
- \( r \) is the interest rate per period (7% or 0.07),
- \( n \) is the total number of deposits (37).
To solve for PMT, we can rearrange the formula:
\[ PMT = \frac{FV}{\frac{(1 + r)^n - 1}{r}} \]
Now, plug in the values:
- \( FV = 3,220,614.00 \)
- \( r = 0.07 \)
- \( n = 37 \)
Now, calculate PMT using these values:
\[ PMT = \frac{3,220,614.00}{\frac{(1 + 0.07)^{37} - 1}{0.07}} \]
Now, you'll want to perform that math.
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Calculate \((1 + r)^n\):
- \( (1 + 0.07)^{37} \)
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Subtract 1 from the result.
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Divide by the interest rate (0.07).
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Finally, divide the FV by that result to find PMT.
This will give you the annual deposit amount that Derek needs to make to ensure he has the desired amount in his retirement account at age 65.