4

What annual installment will discharge a debt of $\$ 717.60$ due in $4$ years at $20\%$ p.a. simple interest, if the installments are paid at the each end of each year?

I tried the following:

$\$717.60$ corresponds to $\textrm{SI} = (717.60 * 4 * 20)/100$ simple interest. So, each installment should be $\textrm{SI}/4$. I do not understand what is wrong with my solution.

Answer to the question is $\$138$.

callculus42
  • 30,550
user3710832
  • 71
  • 3
  • After each payment, we owe less. – André Nicolas Jun 18 '14 at 17:26
  • The annual interest on $717.60 is more than $138, so you cannot possibly pay off the debt in 4 years with installments of $138. After 4 years you would owe more than at the beginning. – David K Jun 18 '14 at 18:29
  • 1
    As for your proposed solution, you're still only paying off interest, not principal. (Are you sure about the "simple interest" part, by the way? That's an unusual condition for a debt paid in installments.) – David K Jun 18 '14 at 18:37
  • 1
    I found the problem statement confusing. People seem to be interpreting it as, "What series of equal payments is equivalent to a balloon payment of $717.60 made 4 years in the future", rather than, "If you receive $717.60 today, what annual installments will pay the loan in 4 years?" I suppose "due in 4 years" does suggest a balloon payment (future value), but who uses future value to express the cost of a series of installment payments? – David K Jun 21 '14 at 12:42

2 Answers2

1

In my previous answer I've assumed compound interest, but it is simple interest. I reworked my answer. After $x$ years a capital of $C_0$ is $C_x=(1+i\cdot x)\cdot C_0$

We have sum the 4 payments, which are made on different years.

  1. The first payment bears interest for 3 years. The factor is $1+i\cdot 3$
  2. The second payment bears interest for 2 years. The factor is $1+i\cdot 2$
  3. The third payment bears interest for 1 years. The factor is $1+i\cdot 1$
  4. The fourth payment bears no interest. The factor is $1+i\cdot 0=1$

The sum is $4+6\cdot i=4+6\cdot 0.2=4+1.2=5.2$. Thus the equation is

$$C_0\cdot 5.2=717.6$$

$$C_0=\frac{717.6}{5.2}=138 \ \ \color{\limegreen}{\checkmark}$$

Now it´s fine.

callculus42
  • 30,550
1

See my answer to this question. Use the bracketed formula at the end (preferably, once you make sure you understand how it is derived) with $R=1\mathord. 2$, $T=4$, and $v_0=\$717\mathord.60/1.2^4$, where $v_0$ is the present value of the future debt. This works because owing $\$717\mathord.60$ in four years is essentially the same thing as owing $\$717\mathord.60/1.2^4$ today.

Community
  • 1
triple_sec
  • 23,377
  • The future value is $$717.6$-not the present value $v_0$. – callculus42 Jun 18 '14 at 20:17
  • @calculus You're right. I missed the part “due in 4 years.” Please disregard my earlier comment. I will also edit my answer to reflect this. Sorry about the confusion. – triple_sec Jun 18 '14 at 20:20
  • On first reading, I interpreted "due in 4 years" to mean, "You have 4 years to pay off the debt." I'm not familiar with any real-world installment plan where future value is used to describe the amount of the debt. Home mortgages and auto loans, for example, use present value. – David K Jun 21 '14 at 12:50
  • @DavidK If that is the case, you can reinterpret the question as follows: “You will owe $717.60 in four years. How much money do you need to deposit each year for you to be able to pay off this debt if your deposits yield 20% yearly interest?” – triple_sec Jun 21 '14 at 17:55
  • You certainly can interpret the question that way; the question is, why would you? The language used is somewhat imprecise; should it then be interpreted in light of the usual home-mortgage or consumer-loan experience, or in some more esoteric terms of finance? Or does it appear the question was merely transcribed incorrectly? – David K Jun 22 '14 at 05:38
  • I'm also wondering why we would use compound interest calculations when the question says "simple interest". (Then again, I found it odd that the question said simple interest, so perhaps this also might be an incorrect transcription of the intended problem.) – David K Jun 22 '14 at 05:40
  • On further thought I suppose it is plausible that you could have a balloon payment of $717.60 due four years from now, but you are allowed to pay down parts of the debt early (as you can do with many real-world loans), and you want to restructure the payments into four annual installments. This just didn't jump out at me from the posted question. – David K Jun 22 '14 at 05:47
  • @DavidK I agree with you in that I have most certainly overthought this problem and read more into it than it was actually about. It would be helpful if the OP clarified, and double-checked the transcription of, the problem. – triple_sec Jun 22 '14 at 07:18
  • @triple_sec See my updated answer – callculus42 Nov 13 '21 at 12:51