Question

A 30-year $115000 loan has a level amortization payments at the end of each year. The effective annual interest rate is 3.75%. Let P be the ratio of total interest paid and total payments made during the life of the loan.

(a) Find P.

(b) At the end of year n, the outstanding balance is less than half of the loan amount for the first time. Find n.

Please do the problem without Excel.

Answer 1

Loan L=115000

Number of Years N=30

Interest rate r=3.75%

Annual Payment A=L*r/(1-(1+r)^-N)

A= 115000*3.75%/(1-(1+3.75%)^-30)

A=$6450.08

Total Interest paid = N*A-L= 30*6450.08-115000=78502.30

Total Payment =N*A=12*6450.08=193502.30

A)

P=Total Interest paid / Total Payment = 78502.30 / 193502.30 =0.41

B)

Outstanding balance after nth year F=A*(1-(1+r)^-(30-n))/r

F=6450.08*(1-(1+3.75%)^-(30-n))/3.75%

Given F<L/2

6450.08*(1-(1+3.75%)^-(30-n))/3.75% <57500

(1+3.75%)^-(30-n)>0.67

Taking Log on both sides

-(30-n) *ln(1+3.75%)>ln(0.67)

n=19 Years