They want to have $10,000 in 3 years time.
They currently put $1500 into the account.
The future value of $1500 at 6% a year compounded yearly means that the $1500 is worth 1786.524 at the end of 3 years time.
This means that they need to make equal payments that would have a value of $10,000 - $1786.524 in 3 years time.
The future value they require to make equal payments for is therefore equal to $8213.476.
The future value of those payments plus the $1786.524 would make a total future value of $10,000.
They will make equal year end payments for 3 years at 6% per year compounded yearly.
The payments at the end of each year would have to be equal to $2579.9333409.
The Future value of these payments would be equal to $8213.476.
Add that to the $1786.524 that they invested up front to get a total of $8213.476 + $1786.524 = $10,000 in 3 years time.
The yearly cash flow breakdown would be as follows:
Time Point 0 equals when you start saving.
Time Point 1 is exactly 1 year after time point 0
Time Point 2 is exactly 2 years after time point 0
Time Point 3 is exactly 3 years after time point 0.
Your Present Value is at time point 0.
Your Future Value is at time point 3.
Here's how the yearly breakdown works.
Each succeeding time point is the balance in the previous time point * 1.06 + the investment in that time point.
Example:
Balance in time point 0 = 1590 because that's how much you put in the savings account at time point 0.
balance in time point 1 = 1590 * 1.06 + 2579.933409 = 4169.933409
Here's the yearly breakdown.
Time Point Investment Balance
0 $1500 $1500
1 $2579.933409 $4169.933409
2 $2579.933409 $7000.062822
3 $2579.933409 $10000
The assumptions made are that interest is compounded yearly and payments are made yearly.
The first payment is made at the beginning of the study period.
That would be time point 0.
The other 3 payments are made at the end of each time period.
That would be time point 1, 2, and 3.
Time point 0 is the beginning of time period 1
Time point 1 is the end of time period 1 and the beginning of time period 2.
Time point 2 is the end of time period 2 and the beginning of time period 3.
Time point 3 is the end of time period 3.
Each time period represents 1 year in this problem.
The formulas used to solve this problem were:
Future Value of a Prsent Amount.
Payment for a Future Value.
Because the Payment Formula required equal payments, the $1500 up front had to be removed from the equation for the Payment for a Future Value.
Since that was an up front payment, the Future Value of a Present Amount Formula made it into an equivalent Future Value.
The formulas used are shown below:
FUTURE VALUE OF A PRESENT AMOUNT
FV = future value
PA = present amount
i = interest rate per time period
n = number of time periods
PAYMENT FOR A FUTURE VALUE
PMT = Payment per Time Period
FV = Future Value
i = Interest Rate per Time Period
n = Number of Time Periods
The 2 step process took the present value of $1500 and got a future value of $1786.24
We used the Future Value of a Present Amount formula to get that value.
That formula is:
PA = $1500
i = .06
n = 3
Formula becomes:
= 1786.524
1786.524 was then subtracted from the $10,000 to get a future value of $8213.476 that we needed to make equal payments at the end of each time period to get.
We then used the Payment for a Future Value formula to find out what the payments had to be.
That formula is:
FV = $8213.476
i = .06
n = 3
Formula becomes:
which becomes:
which becomes:
which becomes:
which becomes:
That becomes your answer:
Equal payments of $2579.933409 at the end of each time period are required plus the initial investment of $1500 in order to get a future value of $10,000 at the end of 3 years.
To check your answer, you can do the yearly analysis that I did above, or you would use the Future Value of a Present Amount formula to get the value of the $1500 3 years from now, and you would use the Future Value of a payment formula to get the future value of the payments 3 years from now.
That formula would be:
FUTURE VALUE OF A PAYMENT
FV = future value
PMT = payment per time period
i = interest rate per time period
n = number of time periods
You would then add the Future Value of the Present amount of $1500 and the future value of the payments of 2579.933409 and you would wind up with a future value of $10,000.