Question

Allysha just borrowed 38,900 dollars. She plans to repay this loan by making a special payment of 5,100 dollars in 3 years and by making regular annual payments of 6,700 dollars per year until the loan is paid off. If the interest rate on the loan is 4.51 percent per year and she makes her first regular annual payment of 6,700 dollars in one year, then how many regular annual payments of 6,700 dollars must Allysha make? Round your answer to 2 decimal places (for example, 2.89, 14.70, or 6.00).

Answer 1

The future value of an amount today is given by

Future value = Present value *(1+interest rate )^years

As special payment is done in year 3, we will take every payment to year 3 and tale year 3 as focal point.

Value of loan in year 3 if no payment is done = loan amount*(1+interest)^3 = 38900*(1.0451)^3 = 44404.1074

Value of 1st 6700 payment at year 3 = 6700*(1.0451^2) = 7317.9679

Value of 2nd payment of 6700 in year 3 = 6700*(1.0451) = 7002.17

Value of third payment at year 3 = 6700

Also a special payment is made at year 3 of 5100.

Thus, loan balance at year 3 = value of loan at year 3 - payments

= 44404.1074 - 5100 - 6700 - 7002.17 - 7317.9679 = 18283.9695

Thus , loan balance at the end of 3 years is $ 18283.9695

To calculate how much more time is required to payoff, se Excel function = nper

=Nper(rate, pmt, pv,fv)

Rate is 4.51%, pmt is regular payments periodically of 6700, pv is present value of loan balance which is -18283.9695. It will be written negative as cash flow is opposite in sign for inflows and outflows.

=Nper(.0451,6700,-18283.9695) = 2.98

Thus, 2.98 years more is required for paying off the loan completely after 3 years.

Thus, total number of payments is 3+2.98 = 5.98.

Comment in case of any query.