Question
Derek decides that he needs $130,817.00 per year in retirement to cover his living expenses. Therefore, he wants to withdraw $130817.0 on each birthday from his 66th to his 90.00th. How much will he need in his retirement account on his 65th birthday? Assume a interest rate of 5.00%.
Answers
GPT-4o mini
To find out how much Derek needs in his retirement account on his 65th birthday, we will calculate the present value of a series of withdrawals he plans to make from age 66 to age 90, which is a total of 25 withdrawals (from age 66 to 90 inclusive).
The formula to calculate the present value of an annuity (where payments are made at regular intervals) is:
\[
PV = P \times \left(1 - (1 + r)^{-n}\right) / r
\]
Where:
- \( PV \) = Present Value (the amount Derek needs at age 65)
- \( P \) = Annual payment (withdrawal amount)
- \( r \) = Interest rate per period
- \( n \) = Total number of payments
Given:
- \( P = 130817.00 \)
- \( r = 0.05 \)
- \( n = 25 \)
Now, we can plug in the values:
1. Calculate \( (1 + r)^{-n} \):
\[
(1 + 0.05)^{-25} \approx (1.05)^{-25} \approx 0.295302
\]
2. Now calculate \( 1 - (1 + r)^{-n} \):
\[
1 - 0.295302 \approx 0.704698
\]
3. Substitute back into the formula:
\[
PV = 130817 \times \left(0.704698 / 0.05\right)
\]
\[
PV \approx 130817 \times 14.09396 \approx 1847102.05
\]
Therefore, Derek will need approximately **$1,847,102.05** in his retirement account on his 65th birthday to withdraw $130,817.00 per year from age 66 to 90 at an interest rate of 5.00%.
The formula to calculate the present value of an annuity (where payments are made at regular intervals) is:
\[
PV = P \times \left(1 - (1 + r)^{-n}\right) / r
\]
Where:
- \( PV \) = Present Value (the amount Derek needs at age 65)
- \( P \) = Annual payment (withdrawal amount)
- \( r \) = Interest rate per period
- \( n \) = Total number of payments
Given:
- \( P = 130817.00 \)
- \( r = 0.05 \)
- \( n = 25 \)
Now, we can plug in the values:
1. Calculate \( (1 + r)^{-n} \):
\[
(1 + 0.05)^{-25} \approx (1.05)^{-25} \approx 0.295302
\]
2. Now calculate \( 1 - (1 + r)^{-n} \):
\[
1 - 0.295302 \approx 0.704698
\]
3. Substitute back into the formula:
\[
PV = 130817 \times \left(0.704698 / 0.05\right)
\]
\[
PV \approx 130817 \times 14.09396 \approx 1847102.05
\]
Therefore, Derek will need approximately **$1,847,102.05** in his retirement account on his 65th birthday to withdraw $130,817.00 per year from age 66 to 90 at an interest rate of 5.00%.