# Exam FM problem with loans. $(1.0075)^2$ or $(1.0075)^3$?

I am a bit confused about the following problem and I would like to have clarification.

A loan of $12,000$ was made with annual rate of $12\%$ convertible quarterly. Smith plans to make a payments of $750$ at the end of every 6 months until he completely pays off the loan. However, 3 months before the 9th payment Smith refinances the loan to a new rate of $9\%$ convertible monthly and pays $R$ each month. In exactly 30 payments after refinancing, Smith pays off the loan. Calculate $R$.

I understand thus far.

Until the 8th payment, the loan accrues $3\%$ interest every 3 months.

Smith is planning to pay the 9th payment 6 months after he pays the 8th payment, so 3 months before the 9th payment means 3 months after the 8th.

So, the oustanding balance right after the 8th payment, $OB_8$ accrues $3\%$ interest the moment Smith refinances.

The following is what my argument is.

The moment Smith makes his 9th payment, which is the moment he starts his monthly payments is 3 months after the refinancing, so the effective monthly interest $0.75\%$ is acrrued three times. i.e. The outstanding balance right before the 9th payment must be $OB_9=(1.03)(1.0075)^3OB_8$.

The following is my question.

The book that I am working on tells me that the outstanding balance is $(1.03)(1.0075)^2OB_8$.

This makes a rather significant difference in the answer I get and I cannot afford such mistakes in future practice. Can someone tell me what's going on?

• The key question here seems to be when the interest accrues. So the question appears to be one of financial semantics, which would be off-topic for this site. – epimorphic Jan 16 '15 at 7:27
• Thanks for the comment. Where would it be more appropriate to ask this question so that I can get help and it would not bother others? – hyg17 Jan 16 '15 at 8:58
• I don't know about this question in particular. To be sure, most of your existing and future finance/actuarial questions might still be fine; as long as you can use your knowledge to translate the problem into a mathematical one, it would be on topic here. I just don't think math has anything to say here. – epimorphic Jan 16 '15 at 9:19
• Understood! I will keep that in mind. – hyg17 Jan 16 '15 at 22:22

## 1 Answer

Am I missing something here ? A loan of 12,000 attracting 3% quarterly with end-of-six-month payments will stand at 11,809.35 just after the 8th payment. In another three months this balance will be 12,163.63 and will be paid off and refinanced with another loan attracting 0.75% monthly. Assuming the new loan is paid off with end-of-month timings then surely the first payment will be one month after the refinancing. So, if we assume a 1 Jan initial draw-down date, the refinancing date will be 1 April and the first payment on the new loan would be 1 May. If this is correct, then in order to clear a loan of 12,163.63 at 0.75% monthly in 30 payments, this will require a monthly payment of 454.29. However ... if you are saying that the payments do not switch to monthly payments until the 9th payment date on the original loan - meaning that what would usually be the first two months' payment dates were missed - then by the time that the first (end-of-)month payment would be made, the balance of the loan for that month (at the beginning) would indeed be larger by two months' interest, at 12,346.77. The 30 payments required to clear this loan would be 461.13. Either way, you must consider the outstanding balance at the beginning of the period for which the payment is being made. Please advise.