To find out how much Derek needs to contribute to his retirement account, we can break down the problem into several steps:
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Determine the time frames:
- Derek will work and contribute from age 26 to 65, which is 39 years.
- He will then not make any contributions from age 65 to 73, a period of 8 years.
- After age 73, he will want to have $2,537,531 in his account one year later (age 74).
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Calculate the time until retirement at age 74:
- From age 65 to age 74 (the year he wants the total): 9 years.
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Future value calculation:
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To find out how much he needs to have at age 65 to grow to $2,537,531 in 9 years at 5% interest, we can use the Future Value formula:
\( FV = PV \times (1 + r)^n \)
Here:
- \( FV \) = future value ($2,537,531)
- \( PV \) = present value (the amount at age 65)
- \( r \) = interest rate (0.05)
- \( n \) = number of years (9)
Rearranging to solve for \( PV \):
\[ PV = \frac{FV}{(1 + r)^n} \] \[ PV = \frac{2,537,531}{(1 + 0.05)^9} \]
Calculate \( (1 + 0.05)^9 \):
\[ (1 + 0.05)^9 \approx 1.5513 \]
Now substitute back to find \( PV \):
\[ PV = \frac{2,537,531}{1.5513} \approx 1,632,126.47 \]
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Calculate the contributions made from age 26 to 65:
- Now we need to determine how much he needs to contribute annually from age 26 to 65 (39 years) to accumulate approximately $1,632,126.47 by age 65.
We use the Future Value of an Annuity formula:
\[ FV = P \times \frac{(1 + r)^n - 1}{r} \]
Where:
- \( FV \) = future value we found ($1,632,126.47)
- \( P \) = annual contribution (what we want to find)
- \( r \) = interest rate (0.05)
- \( n \) = number of contributions (39 years)
Rearranging to solve for \( P \):
\[ P = \frac{FV \times r}{(1 + r)^n - 1} \]
Substitute into the equation:
\[ P = \frac{1,632,126.47 \times 0.05}{(1 + 0.05)^{39} - 1} \]
Compute \( (1 + 0.05)^{39} \):
\[ (1 + 0.05)^{39} \approx 5.4013 \]
Now plug this back into the formula:
\[ P = \frac{1,632,126.47 \times 0.05}{5.4013 - 1} \] \[ P = \frac{81,606.32}{4.4013} \approx 18,527.54 \]
Thus, Derek must contribute approximately $18,527.54 annually from his 26th birthday to his 65th birthday in order to reach his retirement goal by the time he turns 74.