Question

Consider a $12,000 loan with 4 equal annual payments and 10% interest.

a. Calculate the annual payment, n = 4, r = 0.10.

b. Prepare a complete loan payment schedule table for this loan. You need the time period, the beginning principal, payment, interest paid, principal paid, and ending principal in your table.

c. Now assume that the loan is fully amortized over 4 years, however, the interest rate is variable. That is, the bank changes a different rate each year as money market conditions change. Assume the rates turn out to be:

Year 1 ‐ 10%

Year 2 ‐ 12%

Year 3 ‐ 14%

Year 4 ‐ 10%

Compute what each year's interest and principal would be under the above scenario. To do this you will need to amortize the principal left at the beginning of each year over the remaining years of the loan at the new interest rate. This will give you the payment. Then you can calculate the interest paid and the principal paid each year.

Answer 1

1. Loan Payment = Amount / Discount Factor or P = A / D

Number of Periodic Payments (n) = Payments per year times number of years

Periodic Interest Rate (i) = Annual rate divided by number of payment periods

Discount Factor (D) = {[(1 + i) ^n] - 1} / [i(1 + i)^n]

Here, Amount= 12000

N= 4

I= 10% or 0.10

D= {[(1 + 0.1) ^4] - 1} / [0.1(1 + 0.1)^4]

= 0.4641/0.14641

=3.169865

Loan payment= 12000/3.169865= $3785.65

2. 
   

Year	Opening balance	Interest	Principal= 3785.65- interest	Balance= Opening balance- principal paid
1	12,000.00	= 12000*10%= 1200	3785.65-1200= 2585.65	12000-2585.65= 9414.35
2	9,414.35	941.44	2844.21	6570.14
3	6,570.14	657.01	3128.64	3441.5
4	3,441.50	344.15	3441.5	0

3. 
   

The calculation of annual payment and the interest and principal paid goes one by one.

We need to calculate both for year and then pass to next year

Annual payments
Year	Loan amount(A) at the starting of the year	Period	interest	D= {[(1 + i) ^n] - 1} / [i(1 + i)^n]	Loan payment= A/D
1	12000	4.00	10%	{[(1 + 0.1) ^4] - 1} / [0.1(1 + 0.1)^4] = 3.17	3,785.65
2	9,414.35	3.00	12%	{[(1 + 0.12) ^3] - 1} / [0.12(1 + 0.12)^3] = 2.49	3,919.66
3	6,624.42	2.00	14%	{[(1 + 0.14) ^2] - 1} / [0.14(1 + 0.14)^2] = 1.74	4,022.94
4	3,528.90	1.00	10%	{[(1 + 01) ^1] - 1} / [0.1(1 + 0.1)^1] = 0.91	3,881.78

Repayment schedule:

Year	Opening balance	Interest rates	Interest	Principal= Annual payment - interest	Total amount paid	Balance= Opening balance- principal paid
1	12,000.00	10%	12000*10%= 1200	3785.65-1200= 2585.65	3,785.65	12000-2585.65= 9414.35
2	9,414.35	12%	9414.35*12% = 1129.72	3919.66- 1129.72= 2789.93	3,919.66	6,624.42
3	6,624.42	14%	6624.42*14%= 927.42	4022.94-927.42= 3095.52	4,022.94	3,528.90
4	3,528.90	10%	352.89	3881.78-352.89= 3528.90	3,881.78	0

First the first year loan payment us calculated, according to hich the interest and principal is calculated. So that the closing balance ( opening balance of the next yea can be calculated) . then the opening balance of the next year is used to calculate the repayment amount in the first table. And that amount is used in the second table to calculate the interest and principal paid. And so on.