# 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