A loan of $7,000,000 is being amortised over 48 months at an interest rate of 12% compounded monthly. Find:
(a) the monthly payment;
(b) the principal outstanding at the beginning of the 36th month;
(c) the interest in the 36th payment;
(d) the principal in the 36th payment and
(e) the total interest paid.
To solve this problem, we will use the formula for calculating the monthly payment on an amortizing loan:
M = P * (r * (1 + r)^n) / ((1 + r)^n - 1)
Where:
M: Monthly payment
P: Principal loan amount
r: Monthly interest rate
n: Total number of payments
First, let's find the values for the given variables:
P = $7,000,000 (Principal loan amount)
r = 12% / 12 = 1% = 0.01 (Monthly interest rate, calculated by dividing the annual interest rate by 12)
n = 48 (Total number of payments)
(a) Monthly Payment (M):
Using the formula mentioned above, plug in the values:
M = 7,000,000 * (0.01 * (1 + 0.01)^48) / ((1 + 0.01)^48 - 1)
Calculate this by breaking it down into steps:
Step 1: Calculate the value inside the parentheses:
(0.01 * (1 + 0.01)^48)
Step 2: Calculate the value at the denominator:
((1 + 0.01)^48 - 1)
Step 3: Divide the result from Step 1 by the result from Step 2 and multiply it by the loan amount:
M = 7,000,000 * (result from Step 1 / result from Step 2)
(b) Principal Outstanding at the Beginning of the 36th Month:
To find the principal outstanding at the given time, we need to calculate the remaining balance after 35 payments.
Since the loan is being amortized, we can use the formula:
B = P * ((1 + r)^n - (1 + r)^m) / ((1 + r)^n - 1)
Where:
B: Principal balance after m payments
P: Principal loan amount
r: Monthly interest rate
n: Total number of payments
m: Number of payments made
In this case, m = 35.
(c) Interest in the 36th Payment:
To find the interest payment for a particular month, we can use the formula:
Interest Payment = Principal Balance * Monthly Interest Rate
(d) Principal in the 36th Payment:
To find the principal payment for a particular month, we can use the formula:
Principal Payment = Monthly Payment - Interest Payment
(e) Total Interest Paid:
To find the total interest paid over the course of the loan, we can subtract the principal loan amount from the sum of all monthly payments.