Question

A loan of $100,000 is made today. This loan will be repaid by 10 level repayments, followed by a final smaller repayment, i.e., there are 11 repayments in total.

The first of the level repayments will occur exactly 2 years from today, and each subsequent repayment (including the final smaller repayment) will occur exactly 1 year after the previous repayment. Explicitly, the final repayment will occur exactly 12 years from today.

If the interest being charged on this loan is 3.0% per annum compounded half-yearly, and the final smaller repayment is $310,

(a) Calculate the loan outstanding exactly 1 year from today.

(b) Calculate the loan outstanding exactly 11 years from today.

(c) Calculate the amount of the level repayments.

Answer 1

1. As the loan will be paid starting from 2 year from now, the loan outstanding after 1 year will be equal to the loan today,

i.e. the loan outstanding after 1 year will be $100,000

2. The smaller payment is the last payment of the loan, that implies that after that payment the loan will be repaid, so that means, the loan outstanding before that period will be $310. So the loan outstanding at the end of 11^th year is $310.
3. To calculate the level payment,

Firstly, here the interest is compounded semiannually while the payment is made annually. So we need to calculate the effective annual interest rate for annual repayment

Effective annual interest rate= (1+ r/n)^n -1

So, EAR= (1+3%/2)^2 -1

= 0.030225 or 3.0225%

Now,

Payment = r*(PV)/ 1-(1+r)^(-n)

PV= 100000

r= 3.0225%

N= 10

= (100000*3.0225%)/(1-(1+0.030225)^(-10))

= 11,736.52

So the level payment= $11726.17