To find out the contributions Derek must make, we will break the problem down into several phases and apply the future value (FV) and present value (PV) formulas related to retirement savings.
Phase 1: Contributions from Age 26 to Age 65
Let's denote:
- C = annual contribution
- t_1 = 65 - 26 = 39 years of contributions
- r = 8% = 0.08 (annual interest rate)
The future value of a series of contributions at the end of 39 years is given by the future value of an annuity formula: \[ FV = C \times \frac{(1+r)^{t_1} - 1}{r} \]
Phase 2: Growth of the retirement account from Age 65 to Age 73
At age 65, Derek makes no additional contributions until he turns 73. The retirement account will grow for 8 years (from age 65 to age 73). The future value of the account at age 73 based on the growth of the account is: \[ FV_{73} = FV \times (1 + r)^{t_2} \] Where:
- \( t_2 \) = 73 - 65 = 8 years
Phase 3: Withdrawals from Age 74 to Age 93
Derek will begin withdrawing $144,792 annually starting from age 74 for 20 years (from age 74 to age 93). The present value of these withdrawals at age 73 (just before the withdrawals start) must equal the total future value Derek has accumulated at that time.
The present value of annuity formula is given by: \[ PV_{withdrawals} = PMT \times \frac{1 - (1 + r)^{-n}}{r} \] Where:
- \( PMT = 144,792 \)
- \( n = 20 \) years (from age 74 to age 93)
Now putting everything together:
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Total future value at age 73 from contributions: \[ FV_{65} = C \times \frac{(1 + 0.08)^{39} - 1}{0.08} \]
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Future value at age 73 accounting for growth: \[ FV_{73} = FV_{65} \times (1 + 0.08)^{8} \]
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Present value of withdrawals at age 73: \[ PV_{withdrawals} = 144,792 \times \frac{1 - (1 + 0.08)^{-20}}{0.08} \]
Equating the future value at age 73 from contributions (FV_73) to the present value of withdrawals (PV_withdrawals) gives us the equation to solve for C: \[ C \times \frac{(1 + 0.08)^{39} - 1}{0.08} \times (1 + 0.08)^{8} = 144,792 \times \frac{1 - (1 + 0.08)^{-20}}{0.08} \]
Calculation Step-by-step:
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Calculate \((1 + 0.08)^{39}\): \[ (1 + 0.08)^{39} \approx 14.8347 \] So: \[ \frac{(1 + 0.08)^{39} - 1}{0.08} \approx \frac{14.8347 - 1}{0.08} \approx \frac{13.8347}{0.08} \approx 172.935 \]
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Calculate \((1 + 0.08)^{8}\): \[ (1 + 0.08)^{8} \approx 1.8509 \]
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Calculate \((1 + 0.08)^{-20}\): \[ (1 + 0.08)^{-20} \approx 0.2145 \]
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Calculate Present Value: \[ PV_{withdrawals} \approx 144,792 \times \frac{1 - 0.2145}{0.08} \approx 144,792 \times \frac{0.7855}{0.08} \approx 144,792 \times 9.8181 \approx 1,419,849.55 \]
Now we can set the two equal: \[ C \times 172.935 \times 1.8509 = 1,419,849.55 \]
Calculate \( 172.935 \times 1.8509 \): \[ 172.935 \times 1.8509 \approx 320.689 \]
Now rearranging: \[ C \approx \frac{1,419,849.55}{320.689} \approx 4,426.67 \]
Conclusion
Derek must contribute approximately $4,426.67 annually to his retirement account from his 26th birthday until his 65th birthday in order to meet his financial goal of making annual withdrawals of $144,792 starting at age 74 until age 93.
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