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
$$x(1+2(1.07)^{-3})=500(1.05)^{-3}+500(1.03)^{-8}$$
$$x= \frac{500(1.05)^{-3}+500(1.03)^{-8}}{1+2(1.07)^{-3}}$$
$$x=313.99\approx 314.00 $$
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*}