Question

8. Your company has received a $50,000 loan from an industrial finance company. The annual payments are $6,202.70. If the company is paying 9 percent interest per year, how many loan payments must the company make? Round to the nearest number of periods.

Answer 1

Present value of  annual payments= annual payments*Present value of discounting factor(9%,time period)

=6202.7/1.09+6202.7/1.09^2+................+6202.7/1.09^n

=6202.7[1/1.09+1/1.09^2+........+1/1.09^n]

For [1/1.09+1/1.09^2+........+1/1.09^n];this is a geometric series with the:

First term (a)=1/1.09

Common ratio(r)(2nd term/1st term)=1/1.09

Hence sum of n terms=a(1-r^n)/(1-r)

=1/1.09[1-(1/1.09)^n]/(1-(1/1.09)

=0.917431192[1-0.917431192^n]/(1-0.917431192)

=11.111111(1-0.917431192^n)

Hence total present value=6202.7*[11.111111(1-0.917431192^n)]

50000=6202.7*[11.111111(1-0.917431192^n)]

(50000/6202.7)=11.111111(1-0.917431192^n)

1-0.917431192^n=(50000/6202.7)/11.111111

1-0.725490512=0.917431192^n

Taking log on both sides;

log 0.274509487=n*log0.917431192

Hence n=log 0.274509487/log0.917431192

which is equal to

=15 years(Approx).