To find the annual contributions Derek must make to his retirement account, we need to break the problem down into a series of steps.
Step 1: Calculate the amount needed at retirement.
Derek wants to have $2.00 million remaining in his retirement account after making annual withdrawals of $155,119 from ages 75 to 87 (12 years).
- Future Value of the Withdrawals
We need to compute the total balance required at the end of age 74 to accommodate all planned withdrawals and leave Derek with $2.00 million. The withdrawals form an annuity, and we can calculate the present value (PV) of these withdrawals at the time Derek turns 74.
The formula for the present value of an annuity is: \[ PV = PMT \times \left(1 - (1 + r)^{-n}\right) / r \] Where:
- \( PMT = 155,119 \) (the annual withdrawal),
- \( r = 0.05 \) (interest rate),
- \( n = 12 \) (number of withdrawals).
Substituting the values: \[ PV = 155,119 \times \left(1 - (1 + 0.05)^{-12}\right) / 0.05 \]
Calculating this: \[ 1 - (1 + 0.05)^{-12} \approx 0.556839 \] Thus, \[ PV \approx 155,119 \times \frac{0.556839}{0.05} \approx 155,119 \times 11.13678 \approx 1,727,982.45 \]
- Total Amount Needed at 74
To have $2.00 million remaining after the withdrawals, the total amount needed at 74 can be calculated as follows: \[ Total_Needed = PV + Ending\ Balance = 1,727,982.45 + 2,000,000 = 3,727,982.45 \]
Step 2: Calculate the amount needed at age 65 to achieve this total.
Derek will retire at 65, and his account will grow until he turns 74. We will calculate how much he needs at age 65 to reach $3,727,982.45 in 9 years.
The future value (FV) is given by: \[ FV = PV \times (1 + r)^n \] We need to solve for \( PV \): \[ PV = \frac{FV}{(1 + r)^n} = \frac{3,727,982.45}{(1.05)^9} \] Calculating \( (1.05)^9 \approx 1.551328 \): \[ PV \approx \frac{3,727,982.45}{1.551328} \approx 2,403,620.86 \]
Step 3: Calculate the annual contributions Derek must make from 26 to 65 to reach this amount.
Derek will make the contributions from age 26 to age 65 (39 years).
Using the future value formula for a series of equal contributions (ordinary annuity): \[ FV = C \times \left( \frac{(1 + r)^n - 1}{r} \right) \] Where:
- \( C \) is the annual contribution,
- \( r = 0.05 \),
- \( n = 39 \),
- \( FV = 2,403,620.86 \).
Rearranging for \( C \): \[ C = \frac{FV \times r}{(1 + r)^n - 1} \] Substituting the values: \[ C = \frac{2,403,620.86 \times 0.05}{(1.05)^{39} - 1} \]
Calculating \( (1.05)^{39} \approx 5.091270 \): \[ (1.05)^{39} - 1 \approx 5.091270 - 1 = 4.091270 \] Thus: \[ C \approx \frac{2,403,620.86 \times 0.05}{4.091270} \approx \frac{120,181.043}{4.091270} \approx 29,357.81 \]
Conclusion
Derek must contribute approximately $29,357.81 annually to his retirement account from his 26th birthday until his 65th birthday.