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Question 1105815: Scheduled payments of $2000 due in three months with interest at 3.1% compounded quarterly and $1800 due in 21 months with interest at 3.1% compounded quarterly are to be replaced by two equal payments. The first replacement payment is due today and the second payment is due in four years. Determine the size of the two replacement payments if interest is 2% compounded monthly and the focal date is today.
Answer by Theo(13342)   (Show Source):
You can put this solution on YOUR website! i have to make some assumptions that may or may not be what you're looking for, depending on what assumptions you have to go by.
the first part of the problem says the payments are due in 3 months and in 21 months.
my assumption is that the payment is due at the end of the third month and the end of the twenty first month.
that's equivalent to the end of the first quarter and the end of the seventh quarter.
since the compounding rate is quarterly, i worked with quarters.
the interest rate per quarter was .031/4.
to get the present value of the payments, the payment at the end of the first quarter was divided by (1 + .031/4)^1, and the payment at the end of the seventh quarter was divided by (1 + .031/4)^7
that got me a present value of 3659.519657.
that present value now needed to be split into two equal payments.
the first payment was due immediately and the second payment was due at the end of the 4th year.
since the interest rate was compounded monthly, i worked in months.
payment due at the end of the 4th year is equivalent to payment due at the end of the 48th month.
interest rate per month was .02/12.
to get the present value of the payments, the payment due at the beginning of the first month was divided by (1 + .02/12)^0, and the payment due at the end of the forty eighth month was divided by (1 + .02/12)^48.
note that the beginning of month 1 is the end of month 0.
since the payments were equal, i let x equal the value of each payment.
present value of payments became x / (1 + .02/12)^0 + x / (1 + .02/12)^48.
since (1 + .02/12)^0 is equal to 1, the formula became:
present value of payments = x + x / (1 + .02/12)^48.
since present value of payments was equal to 3659.519657, the formula became:
3659.519657 = x + x / (1 + .02/12)^48
i factored out the x to get 3659.519657 = x * (1 + 1/(1 +.02/12)^48)
i divided both sides of the equation by (1 + 1/(1 +.02/12)^48) to get:
3659.519657 / (1 + 1/(1 +.02/12)^48) = x
i solved for x to get:
x = 1902.850385.
that's the payment that has to be made up front right away and at the end of the fourth year.
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