Question

A loan is to be repaid in end of quarter payments of $1,000 each, with there being 20 end of quarter payments total. The interest rate for the first two years is 6% convertible quarterly, and the interest rate for the last three years is 8% convertible quarterly. Find the outstanding loan balance right after the 6^th payment.

Please show/explain your work, I'd like to learn how to do it without excel

Answer 1

In this case we will have to calculate present value of the remaining payments pending afer 6th payment
Quarterly interest rate for first two years	1.50%	6%/4
Quarterly interest rate for last three years	2.00%	8%/4
Present value	Annual payment*(1-((1+r)^-n))/r
Interest rate is r and number of payment is n
No of payments for first two years	8	(2*4)
Remaining payments after 6th payment	2	(8-6)
No of payments of last three years	12	3*4
The interest rate are different and thus present value calculation is spilt
Loan outstanding after 6th payment	1000*(1-(1.015^-2))/0.015)+1000*(1-(1.02^-12)/0.02)
Loan outstanding after 6th payment	1000*1.9559+1000*10.57534
Loan outstanding after 6th payment	$12,531.22
Thus, remaining loan balance after 6th payment is $12,531.22