# Exam FM problem. What does this problem mean?

Danny borrows 4,000 from Genevive at an annual effective rate of interest i. He agrees to pay back 4,000 after 14 years and 5,440.32 after another 14 years. Danny repays the outstanding balance after 7 years after his first payment. What is the amount of Danny's second payment?

I am not sure what the problem wants me to find.

Danny clearly only pays twice, so the second payment should be 5,440.32, right?

I also don't understand why Danny's loan is paid off between his first and second payment...what is going on?

• Do you know about Present Value (PV)? It helps a lot. If his second payment is after $7$ years, then less interest has accumulated than if he waits $14$ years after the first payment, so he will owe less than $5440.32$. If you still have trouble after giving it a good try, and you have not been given a solution, and if the question has not been put on hold, please leave a message for me, and I can write out a solution. Probably not before tomorrow, it is getting latish here. Jan 23, 2015 at 5:26
• Make an equation you can solve for the interest rate - all the information to do that is in the question. Then, plug that into the chain of events, where instead of 14 years for the second period, you'll use 7. Jan 23, 2015 at 5:43

The PV (present value) of the two payments is $4000(1+i)^{-14}+5440.32(1+i)^{-28}$. This is $4000$. Thus we obtain the equation $$4000(1+i)^{-14}+5440.32(1+i)^{-28}=4000.$$ Let $x=(1+i)^{-14}$. Then the above equation can be written as $$5440.32x^2+4000x-4000=0.$$ This is a quadratic equation in $x$. Solving the equation using the Quadratic Formula, and discard the negative root. Now we know $x$, so we know $(1+i)^{-14}$. We could use the result to calculate $i$, though it turns out we don't need to.

Now let's look at the second scenario. Let $P$ be the payment $7$ years after the first payment. Then a present value calculation shows that $$4000(1+i)^{-14}+P(1+i)^{-21}=4000.\tag{1}$$ We know $(1+i)^{-14}$, so we can calculate $(1+i)^{-7}$ by taking the square root, and then we can calculate $(1+i)^{-21}$. Now we know everything in Equation (1) except $P$, so we can solve for $P$.

Remark: As was mentioned in a comment, in the second scenario the second payment comes after $21$ years, not $28$ years. So less interest has accumulated, and therefore in the second scenario the second payment $P$ is less than $5440.32$. Exactly how much less is not obvious, so requires the above calculation.

• You know what, I think I understand the problem a bit better. The borrower "agreed" to paying back with two payments that was suggested to be 4000 in 14 years and 544.32 in 28 years, but he has decided not to do that. I think the problem was not well stated. Either way I really appreciate your input. Jan 24, 2015 at 4:32
• You are welcome. The wording is not optimal, though the author does fairly well given that (s)he wanted to make the problem statement short. Jan 24, 2015 at 5:00

My answer is equivalent with the validated answer but my approach is more intuitive- I hope.

First we need to agree that the structure of payments extinguishes the debt. Otherwise said neither part takes advantage of the other in the second scenario of payments.

Now let’s analyze the first structure:

1. The residual debt after first payment is $$4000(1+i)^{14}-4000$$
2. The residual debt after second payment is zero: $$4000(1+i)^{28}-4000(1+i)^{14}-5440.32=0$$

Since the residual debt must be zero, then we could solve for $$x=(1+i)^{14}$$:

$$4000x^2-4000x-5440.32=0, x=\frac{4000+\sqrt{4000^2+4\cdot 4000\cdot 5440.32}}{2\cdot 4000}$$

Therefore, $$(1+i)^{14}=1.7689\Rightarrow i=4\%$$

The debt that in the last 14 years amounts to 5440.32 started at $$\frac{5440.32}{(1+i)^{14}}$$

Now to extinguish in 7 years a debt which in 14 years would amount to 5440.32 at an interest rate i needs to be paid $$\frac{5440.32}{(1+i)^{14}}\cdot (1+i)^7=4090.48$$.

Final note: both answers produce the same numerical results. However, there is a mistake: while the borrower might have a reason to extinguish the debt sooner, the lender’s interest is only in revenue from interest on lending. In this light then the lender should charge full 14-years interest even when the borrower obliges to pay sooner. The context of the problem is well presented as borrowing between friends but it would not apply to an institutional lender. The only reason an institutional lender would agree on the terms presented in this problem was if the lender already had another borrower lined and willing to accept terms worse than the actual borrower, otherwise said it would advantage the lender to end the current contract sooner.