Question

A loan of $10,000 is being repaid with 10 payments at the end of each year at an annual effective rate of 5%. The payments grow by 10% each year. Find the amount of interest and principal paid in the fifth payment. (Answer: $397.91, $837.97) Show all calculations.

Answer 1

Let the first payment be x

Present value of payments=x/(1+r)*(1-((1+g)/(1+r))^t)/(1-((1+g)/(1+r))

Loan=Present value of payments

=>10000=x/1.05*(1-(1.1/1.05)^10)/(1-(1.1/1.05))

=>x=10000*1.05*(1-(1.1/1.05))/(1-(1.1/1.05)^10)

=>x=844.1200

Loan oustanding after 4th payment=Loan*(1+r)^4-Payment/(1+r)*(1-((1+g)/(1+r))^4)/(1-((1+g)/(1+r)))*(1+r)^4
=10000*1.05^4-844.1200/1.05*(1-(1.1/1.05)^4)/(1-(1.1/1.05))*1.05^4=7958.2034

Interest in 5th payment=Loan after 4th payment*rate=7958.2034*5%=397.9102

Principal in 5th payment=Payment-Interest=First payment*(1+g)^4-Interest=844.1200*1.1^4-397.9102=837.9659