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I'm currently enrolled in a theory of interest class that covers the same material that's on exam 2/FM. At the instructors recommendation, I'm working through all of the problems in the book on my own as preparation for the exam. The following question has me stuck, it is not a homework question.

The two sets of grandparents for a newborn baby wish to invest enough money immediately to pay 10000 per year for four years toward college costs starting at age 18. Grandparents A agree to fund the first two payments, while Grandparents B agree to fund the last two payments. If the effective rate of interest is 6% per annum, find the difference between the contributions of Grandparents A and B.

Ok, so my understanding is as follows:

At time 0, set of Grandparents A contributes some payment W to the fund. At time 1, set of Grandparents A contributes some payment X to the fund. At time 2, set of Grandparents B contributes some payment Y to the fund. At time 3, set of Grandparents B contributes some payment Z to the fund.

At time 18, interest is paid and immediately afterwards, 10,000 is subtracted from the fund. This is repeated 4 times and we end up with 0 in the fund at time 21.

This means that at time 18, we must have 36,730.12 in the fund so that we can pay out 10000 per year for four years.

I discounted this 15 times to obtain the amount in the fund at time = 3 which is when all of the initial payments would have been made into the fund. This is 15,326.20. At this point, I need to figure out what W, X, Y, and Z are. It seems like there could be an infinite number of solutions. W could just be the entire initial payment and the rest could be 0 as the question does not stipulate that the payments must be equal. It's seemingly implied that they're not, as the answer in the back is 748.97.

Am I misunderstanding the question? It seems poorly posed in my opinion... Also, all quantities above are in dollars.

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1 Answer 1

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Do PV calculations. Let us assume that $10000$ will be withdrawn from the fund $18$, $19$, $20$, and $21$ years from now.

The PV of the first two withdrawals is $$10000(1.06)^{-18}+10000(1.06)^{-19}.$$ Call this $A$. The PV of the next two payments is $$10000(1.06)^{-20}+10000(1.06)^{-21}.$$ Call this $B$.

The required difference is $A-B$.

Remark: Present Value, and to a lesser extent Future Value, are powerful conceptual tools for comparing transactions at different times.

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