# [solved]Compound interest problem

I must admit this is my homework, but I have tried to solve it from many different angles but I can't solve it.
The problem is as follows:
A person has a 500 dollar debt to be paid in 3 years at an interest rate of 5% compounded annually and a debt of 500 dollars to be paid in 4 years at an interest rate of 6% compounded every 6 months. The debtor wants to pay off his debt with 2 payments: one payment now and a second one, which will be the double of the first one, at the end of the third year. If the money has a value of 7% compounded annually. What is the amount of the first payment?

Tried several things:
Tried to get the present value of both of the debts and equate that to 3x assuming x was the first payment and 2x was the second payment. Tried to get the future value of both debts and equate that to x + 2x(1.07)^3. x is the amount of the first payment and 2x(1.07)^3 is the amount of the second payment, 2 times the first one(2x) at an interest rate of 7% compounded annually for 3 years(1.07^3).

I also tried playing with the numbers in general, specially with the present/future value, since I know the answer is 314 but I can't get the answer. Help is greatly appreciated

Got the answer finally. let $x$ be the amount of the first payment. We get the present value of the second payment the payment 3 years from now
$$2x(1.07)^{-3}$$
Then we get the present value of the debts.
$$500(1.05)^{-3}+500(1.03)^{-8}$$ As an equation
$$x+2x(1.07)^{-3}=500(1.05)^{-3}+500(1.03)^{-8}$$
Now some algebra
$$x(1+2(1.07)^{-3})=500(1.05)^{-3}+500(1.03)^{-8}$$ $$x= \frac{500(1.05)^{-3}+500(1.03)^{-8}}{1+2(1.07)^{-3}}$$ $$x=313.99\approx 314.00$$

• Please clarify how "money has a value of 7% compounded annually" should be interpreted mathematically. It sounds like a statement about inflation, but that doesn't seem to relate to the rest of the problem at all. Is amount of the debt increasing that much? Is the 3x supposed to be interpreted as "3x of the 'value' rather than direct amount" of money? Jan 3, 2014 at 10:16
• Does your question want the present value of the second payment to be twice the value of the first payment? I can't replicate $314$, but I get close on this basis provided I also choose to pay off the most expensive debt first. I am assuming that the interest gets rolled up into capital. Jan 3, 2014 at 10:37
• @DanielV I think it implies that the money is earning some interest. The amount of money is worth more in 3 years than now. Of course I may be wrong and maybe my interpretation is wrong, and the money is worth less at the end of the 3 years. Jan 3, 2014 at 19:31
• @MarkBennet that is actually a good question because, it must be the present value. But I may give it a go assuming the second payment is twice the future value of the first. And yes I get pretty close to 314 the closes I got was 329 even tried to reverse engineer with the answer and I couldn't. Jan 3, 2014 at 19:38
• I am an accountant (in my spare time) so I did lots of these calculations. I can't see how you get the advertised answer. It is rational to keep debt running if the time value of money (to you) is greater than the cost of debt [true in your case], and to pay off the most expensive debt first. If you have the money, invest it, rather than paying off the debt. Maybe that makes a difference to the calculation?? Jan 3, 2014 at 22:15

I get \$363 as the first payment$P\$, solving \begin{align*} P + 2P v^3 & = 500 \times 1.05^3 v^3 + 500 \times (1+\frac{6\%}{2})^{(2\times4)} v^4, \\ \mbox{where }v &= 1/1.07. \\ \mbox{I.e.} P & = 500 \frac{ 1.05^3 \times 1.07^{-3} + 1.03^8 \times 1.07^{-4} }{1 + 2 \times 1.07^{-3}} \\ & = 363. \end{align*}