SOLUTION: Calculate the purchase price of an annuity paying $200 per month for 10 years with a lump payment of $2000 on the same day as the last payment of $200, at 6.5% compounded monthly.

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Question 1207349: Calculate the purchase price of an annuity paying $200 per month for 10 years with a lump payment of $2000 on the same day as the last payment of $200, at 6.5% compounded monthly.
Found 2 solutions by Theo, ikleyn:
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website!
PREVIOUS SOLUTION, IF YOU RECEIVED IT, MAY BE IN ERROR.

THE CORRECTED VERSION FOLLOWS:

the calculator at https://arachnoid.com/finance/index.html can help you solve this.

this calculator is used to determine the present value of the annuity.
the present value of the annuity is what you would have to pay up front for it.

the first pass finds the present value of the payments you receive at the end of each month for 120 months at 6.5% interest rate compounded monthly.

inputs are:
present value = 0
future value = 0
payment at the end of each month = 200
interest rate per month = 6.5/12 .541666666....%
number of time periods = 10 years * 12 = 120 months.

calculator says that the present value of the payments is equal to -17613.70.

the payments of 200 are shown as positive because that's what you will received.
the present value is shown as negative because that's what you will have to pay up front for the annuity.

the second pass finds the present value of the additional 2000 you receive at the same time as the last payment is made.

inputs are:
present value = 0
future value = 2000
payment at the end of each month = 0
interest rate per time period = 6.5/12 .541666666....% per month.
number of time periods = 10 * 12 = 120 months.

calculator says that the present value of the future value is equal to -1045.92.

the present value is negative because that's what you will have to invest today to receive 2000 dollars at the same time that you receive the last payment of 200 from the annuity.

the total present value of the payments is equal to -17613.70 + -1045.92 = -18659.62.

that's how much the annuity is worth today and how much you will have to pay up front to receive the benefits from the annuity.

you could have derived this with one pass of the calculator which i'll call the final pass of the calculator.

the inputs are:

present value = 0
future value = 2000
payment at the end of each time period = 200
interest rate per month = 6.5/12 .541666666....%
number of time periods = 10 * 12 = 120.

click on pv and the calculator tells you that the present value is equal to -18,659.62.

the first two passes of the calculator give you the same result as the final pass of the calculator.

your solution is that you would have to pay 18,659.62 up front to obtain an annuity that will provide you with 200 dollars at the end of each month plus 2000 at the same time as the last payment of 200 dollars.

i also used excel to model the problem.
excel finds the present value of 200 received at the end of each month plus an additional 2000 at the end of the last payment of 200.

here are the results of the first two passes of the calculator followed by the third pass of the calculator follow by the excel results excel for the first few time periods and the last few time periods.

Answer by ikleyn(52835)   (Show Source):
You can put this solution on YOUR website!
.
Calculate the purchase price of an annuity paying $200 per month for 10 years
with a lump payment of $2000 on the same day as the last payment of $200,
at 6.5% compounded monthly.
~~~~~~~~~~~~~~~~~~~~~~~~~

        The problem does not say if the regular payments (withdrawals) are made at the beginning or at the end of each month.
        Also, from the problem, it is unclear if the lump payment of $2000 does include the last regular payment of $200 or is an addition to it.
        Due to this reason, the problem formulation is mathematically lame.
        In my solution below, I assume that the regular payments (withdrawals) are made at the end of each month.
        I also assume that the lump payment of $2000 is an addition to the last regular payment of $200.

The sough purchase price is the sum of two amounts. First amount is the starting amount of an annuity that provides paying (from annuity to you) $200 per month for 10 years. Second amount is the starting amount which provides a lump payment (from the annuity to you) of $2000 in 10 years from now. To find first amount, use the formula X = . In this case, W = $200 is the monthly withdrawal; r is the effective monthly compounding rate r = 0.065/12; p = 1 + 0.065/12; n is the number of withdrawals (the same as the number of months, n = 10*12 = 120). So, X = = 17613.70 dollars for the first amount. To find the value of the second amount, Y, use this equation 2000 = . From this equation, Y = = 1045.924586, or 1045.92 dollars (rounded). Thus the ANSWER to the problem's question is this sum X + Y = 17613.70 + 1045.92 = 18659.62 dollars.

Solved.