To determine the amount Derek must contribute annually from his 26th birthday to his 65th birthday in order to fund his withdrawals after retirement, we can break down the problem into several steps:
- Calculate the present value of the withdrawals that he will make from age 73 to 92.
- Calculate the future value needed at retirement age (age 65) to fund those withdrawals.
- Calculate the annual contributions needed from age 26 to age 65 to reach this future value.
Step 1: Present Value of Withdrawals
Derek will withdraw $192,814 each year from age 73 to 92 (which is 20 withdrawals). The present value \(PV\) of an annuity can be calculated using the formula:
\[ PV = PMT \times \frac{1 - (1 + r)^{-n}}{r} \]
where:
- \(PMT = 192,814\) (annual withdrawal),
- \(r = 0.04\) (interest rate),
- \(n = 20\) (number of withdrawals).
Substituting the values into the formula:
\[ PV = 192,814 \times \frac{1 - (1 + 0.04)^{-20}}{0.04} \]
Calculating the annuity factor:
\[ PV = 192,814 \times \frac{1 - (1.04)^{-20}}{0.04} \]
Calculating \((1.04)^{-20}\):
\[ (1.04)^{-20} \approx 0.2083 \]
Now calculate the full expression:
\[ PV = 192,814 \times \frac{1 - 0.2083}{0.04} \approx 192,814 \times \frac{0.7917}{0.04} \] \[ PV \approx 192,814 \times 19.7925 \approx 3,818,330.39 \]
Step 2: Future Value Needed at Retirement
Now, we need to find out how much Derek needs to have in his retirement account at age 65 to cover the present value we just calculated at the time of the withdrawals (which will actually occur at age 73). To find this future value, we need to account for the time value of money from age 65 to age 73 (8 years). The future value \(FV\) can be calculated with:
\[ FV = PV \times (1 + r)^n \]
where:
- \(PV \approx 3,818,330.39\),
- \(r = 0.04\),
- \(n = 8\).
Substituting in these values:
\[ FV = 3,818,330.39 \times (1 + 0.04)^{8} \]
Calculating \((1.04)^8\):
\[ (1.04)^{8} \approx 1.3685696 \]
Now substituting this value back in:
\[ FV \approx 3,818,330.39 \times 1.3685696 \approx 5,224,516.49 \]
Step 3: Annual Contributions to Age 65
Next, we need to calculate how much Derek must contribute annually to reach the future value of approximately $5,224,516.49 at age 65, starting from his 26th birthday and making contributions for 39 years (from age 26 to 65).
Using the future value of an annuity formula:
\[ FV = PMT \times \frac{(1 + r)^n - 1}{r} \]
Rearranging to solve for \(PMT\):
\[ PMT = \frac{FV \times r}{(1 + r)^n - 1} \]
Substituting in the values we have:
\[ FV = 5,224,516.49, \] \[ r = 0.04, \] \[ n = 39. \]
Calculating \((1.04)^{39}\):
\[ (1.04)^{39} \approx 5.297517 \]
Now plugging this value into our equation:
\[ PMT = \frac{5,224,516.49 \times 0.04}{(1.04)^{39} - 1} \] \[ PMT = \frac{209,020.6596}{5.297517 - 1} \] \[ PMT = \frac{209,020.6596}{4.297517} \approx 48,687.87 \]
Conclusion:
Derek must contribute approximately $48,687.87 annually from his 26th birthday until his 65th birthday to ensure he can withdraw $192,814 every year from ages 73 to 92, assuming a 4% interest rate.
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