Question

You have taken out a loan of $300 000 at an interest rate of 3% per annum compounded quarterly. You plan to repay the loan in 240 reducing-principal payments made at the end of each quarter and starting at the end of the first quarter.

(a) How much is the principal reduced by after each payment?

(b) Write down an expression for the loan balance before repayment Am at the end of period m. This means that A1 should give the initial loan balance.

(c) Write down an expression for the interest Im owed at the end of period m. Do not round values.

Answer 1

Here, loan amount : P = $300,000

interest rate : r = (3/4)% = 0.75% = 0.0075

time : n = 240 quarters

Now, quarterly payment is :

Pr Q 1 (1r-

i.e.,

300000 0.0075 $ 1 (10.0075)-240

i.e.,

2250 $ 1 (10.0075)-240

Then, the expression for the loan balance before repayment at the end of period m is :

Am P(1r)- - (1r)m-1 1]

i.e.,

Am P(1r)- - (1+r)m-1

i.e.,

P- Am (1r)m

i.e.,

300000 300000 m-1 Am (1.0075) 300000 1 1.0075-240 1 1.0075-240

i.e.,

1 - 1.0075-240 1 1,0075-240 1 (1.0075)-1 300000 Am 300000 1 1.0075-240

i.e.,

Am 300000 -1.0075-241 1 1.0075-240

And, the principle reduced after each payment is :

1 1.0075m-241 1 1.0075-240 Am -Am+1300000 1 1.0075m-240 1 1,0075-240 300000

i.e.,

300000 0.0075 1.0075m-241 Am Am+1 1-1.0075-240

i.e.,

2250 1.0075m-241 1 1.0075-240 Am Am+1