Question

The repayment plan for a loan with an interest rate of 5.0% totaling 20,500 now is to fully pay off the loan in 11 years using end of year payments beginning at 1300 and increasing by the same amount beginning in the second so that the loan is completely paid off at the time of the last payment. What is the amount of the increase/decrease needed each year, the linear gradient, that results in the loan payoff as planned?

Answer 1

The payment series can be broken down into two series:

Series 1: Payment of 1300 for 11 years

Present value=Present value of ordinary annuity=payment/rate*(1-1/(1+rate)^n)=1300/5%*(1-1/1.05^11)=10798.33848

Series 2: Arithmetic gradient of G

Present value given as P is calculated using below formula

P=G/(5%*1.05^11)*((1.05^11-1)/0.05-11)=37.49884077G

Total present value=Present value of first series+Present value of second series=10798.33848+37.49884077G

Given, loan amount=20500

Hence,

10798.33848+37.49884077G=20500

=>G=(20500-10798.33848)/37.49884077

=>G=258.7189716

Increase of 258.7189716 each year