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 9.6% per annum compounded half-yearly, and the final smaller repayment is $720, Calculate the loan outstanding exactly 11 years from today.

Answer 1

Data Given in Question:

Loan Given in Year 0 = $100,000

Loan Repayment Period = 10 years level payment amount (stating from Year 2 to Year 11) + $720 (in year 12)

Annual Rate = 9.6% compounded quarterly.

Required:

So the answer required is to find the total Loan Outstanding after the installment of Year 10 is paid. In other words the value of last level installment + present value of $720 in Year 11 discounted at effective annual rate

Solution:

The easiest way to understand the question is through making a money timeline:

Year 0

Year 1

Year 2

Year 3

Year 4

Year 5

Year 6

Year 7

Year 8

Year 9

Year 10

Year 11

Year 12

100000

0

X

X

X

X

X

X

X

X

X

X

720

In the above table we have to calculate X which is the Level amount paid every year for 10 years. In Year 0, Total outstanding amount is 100000. In year 1 there is no payment made. And in Year 12, $720 is paid. All of this is discounted at Effective annual rate (EAR) for stated annual rate of 9.6% compounded quarterly.

We have to calculate the following to get the answer

1. Effective Annual Rate when stated rate is 9.6% compounded annually
2. Calculate Present Value of $720 (paid in Year 12) in Year 0
3. Calculate the Value of Loan Outstanding in Year 1
4. Value of Equal Monthly Instalment
5. Value of $720 (paid in Year 12) in Year 11.

Our final answer would be 4 + 5

1)

Effective annual rate (EAR) = (1 + Stated Annual Rate/n)^n - 1 , (where n is compounding frequency per year, for semi annual n = 2)

So, EAR = (1 +.096/2)^2 - 1 = .0983 or 9.83%

2) Present Value = Future Value /((1 + EAR/T)^T), here T is time period

Present Value of $720 in Year 0 = 720/((1 + .0983)^12) = $ 233.70

3) Loan Amount left for calculating level payments in Year 0 = $100000 - $233.70 = $99766.30 (PV0)

Value of Loan Amount left in Year 1 = $99766.30*(1+.0983) = $109573.72 (Using formula FV = PV0*(1+EAR)

4) Now we have to calculate level amount (X or PMT), for this we can use the financial calculator or PMT formula in Excel with inputs as follows

Present Value (PV) = $109573.72

Nper (No of Years) = 10

Rate = EAR

So, the PMT or X, comes out to be $ 17,702.90

5) Present Value of $720 in Year 11 = 720/((1 + .0983)^1) = $ 655.56

Final Answer of Total Loan Amount due after 11 years (n other words the value of last level installment + present value of $720 in Year 11 discounted at effective annual rate) = (4) + (5) = $17,702.90 + $655.56 = $18358.46