To solve this problem using a financial calculator, you need to break it down into several steps. You'll determine how much Derek needs to deposit annually to ensure that his account will equal zero after year 12, considering the future withdrawals he wants to make.
Step 1: Calculate the Present Value (PV) of Future Withdrawals
Derek wants to withdraw $10,027 in 4 years and $12,798 in 12 years. First, you need to find the present value of those future withdrawals.
For the first withdrawal ($10,027 in 4 years):
- FV = $10,027
- i/y = 5.28%
- N = 4
Using the formula for present value: \[ PV_1 = \frac{FV}{(1 + i/y)^N} \] Plugging in: \[ PV_1 = \frac{10,027}{(1 + 0.0528)^4} \]
For the second withdrawal ($12,798 in 12 years):
- FV = $12,798
- i/y = 5.28%
- N = 12
Using the same formula for present value: \[ PV_2 = \frac{FV}{(1 + i/y)^N} \] Plugging in: \[ PV_2 = \frac{12,798}{(1 + 0.0528)^{12}} \]
Step 2: Sum the Present Values of the Withdrawals
Once you calculate \(PV_1\) and \(PV_2\), you'll sum them up to get the total present value of the future withdrawals:
\[ PV_{total} = PV_1 + PV_2 \]
Step 3: Determine the Amount that Needs to be Deposited Each Year
Since he currently has $2,364 in the account, we can find out how much needs to be deposited each year (PMT) for the next 12 years to account for the \(PV_{total}\) of the withdrawals:
Inputs for the financial calculator:
- \(PV\) = Current savings minus total present value of withdrawals = \(2,364 - PV_{total}\) (this will be negative if the total withdrawals exceed current savings)
- \(FV\) = 0 (account must equal zero at the end)
- \(i/y\) = 5.28%
- \(N\) = 12 (deposits made for 12 years)
Use the financial calculator to find PMT:
You will enter:
- N = 12
- PV = (Calculated using the above step)
- FV = 0
- i/y = 5.28%
Then compute PMT, and this will give you the amount Derek must deposit each year for the next 12 years.
Final steps:
Make sure to perform the above calculations correctly to ensure accuracy in deriving how much Derek should deposit annually. The inputs entering the financial calculator would look like this after finding the appropriate values for PV from the withdrawals:
- N = 12
- PV = (Current account - Present Value of Future Withdrawals)
- i/y = 5.28%
- FV = 0
- PMT = ?
Additional Notes:
If the calculated PV (the initial value minus the present value of the future withdrawals) is negative, then the amount will represent how much Derek needs to offset the shortfall over 12 years. If it's positive, he already has enough savings to cover the withdrawals and might not need to deposit anything additional.
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