Question 1166517: You want to be able to withdraw $35,000 from your account each year for 30 years after you retire.
You expect to retire in 25 years.
If your account earns 8% interest, how much will you need to deposit each year until retirement to achieve your retirement goals?
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Found 2 solutions by CPhill, ikleyn:
Answer by CPhill(2138)   (Show Source):
You can put this solution on YOUR website! This is a two-step problem involving time value of money: first, determining the total amount needed at retirement (the **Present Value of an Annuity**), and second, calculating the annual deposit required to reach that goal (the **Payment for a Future Value Annuity**).
The required annual deposit is **\$5,384.97**.
***
## 1. Phase 1: Calculate the Required Retirement Nest Egg
The amount you need to have saved by the time you retire (at the end of 25 years) is the **Present Value** of the stream of 30 years of withdrawals.
* **Annual Withdrawal ($W$):** \$35,000
* **Interest Rate ($i$):** 8% (0.08)
* **Withdrawal Period ($n_W$):** 30 years
The formula for the Present Value of an Ordinary Annuity ($PV_R$) is:
$$PV_R = W \times \left[ \frac{1 - (1 + i)^{-n_W}}{i} \right]$$
$$PV_R = 35,000 \times \left[ \frac{1 - (1.08)^{-30}}{0.08} \right]$$
$$PV_R = 35,000 \times 11.25778$$
$$PV_R \approx \$393,669.94$$
You will need **\$393,669.94** in your account when you retire.
***
## 2. Phase 2: Calculate the Required Annual Deposit
Now we need to find the equal annual deposits ($D$) required to accumulate the target amount of \$393,669.94 over 25 years.
* **Target Future Value ($FV_D$):** \$393,669.94
* **Interest Rate ($i$):** 8% (0.08)
* **Accumulation Period ($n_A$):** 25 years
The formula to find the required payment ($D$) is derived from the Future Value of an Ordinary Annuity:
$$D = \frac{FV_D}{\left[ \frac{(1 + i)^{n_A} - 1}{i} \right]}$$
$$D = \frac{393,669.94}{\left[ \frac{(1.08)^{25} - 1}{0.08} \right]}$$
$$D = \frac{393,669.94}{73.10594}$$
$$D \approx \mathbf{\$5,384.97}$$
Answer by ikleyn(53339)  (Show Source):
You can put this solution on YOUR website! .
You want to be able to withdraw $35,000 from your account each year for 30 years after you retire.
You expect to retire in 25 years.
If your account earns 8% interest, how much will you need to deposit each year
until retirement to achieve your retirement goals?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
The numbers, the calculations and the answer in the post by @CPhill are incorrect.
I came to bring correct numbers, correct calculations and correct answer.
First calculate the amount at the bank account, sufficient to withdraw $35,000 every year
during 30 years after retiring.
Unfortunately, the problem does not specify when the money is withdraw - at the beginning of each year
or at the end of the year. Obviously, the person who created this problem, does not know the subject
and do not understand the difference between these options. For simplicity of my calculations,
I will assume that $35,000 is withdraw at the end of each year.
Use the formula relating the starting amount A of the sinking fund with the withdraw value
A =  = 394022.42 (rounded to the closest cent).
(compare this correct value with incorrect value of 393669.94 in the post by @CPhill).
Now, having this value A = 394022.42 as the starting value for withdrawing, we can determine the
annual deposit D during 25 years.
Use the formula for ordinary annuity
D =  = 5389.75 (rounded to the closest greater cent).
ANSWER. The required annual deposit is $5,389.75 at the end of each year during 25 years.
Solved.
The errors that @CPhill admits in his calculations, are not allowed in banking, in Finance and in solving Math problems in Finance.
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