Answer: To find the number of payments required to settle the loan, we can use the formula for the number of periods in a loan payment calculation.
Let:
P = Principal amount (loan amount) = R750,000
r = Interest rate per period (biannual) = 12.5% = 0.125 (as a decimal)
n = Number of payments (unknown)
A = Biannual payment amount = R55,000
a) Number of payments required to settle the loan (n):
Using the formula for the number of periods in a loan payment calculation:
A = P * r / (1 - (1 + r)^(-n))
Substitute the known values:
R55,000 = R750,000 * 0.125 / (1 - (1 + 0.125)^(-n))
Now, solve for n:
1 - (1 + 0.125)^(-n) = R750,000 * 0.125 / R55,000
(1 + 0.125)^(-n) = 1 - (R750,000 * 0.125 / R55,000)
Take the reciprocal of both sides:
(1 + 0.125)^n = 1 / (1 - (R750,000 * 0.125 / R55,000))
Now, solve for n:
n = log(1 / (1 - (R750,000 * 0.125 / R55,000))) / log(1.125)
Using a calculator, we find:
n ≈ 12.9
Since you cannot have a fraction of a payment, the number of payments required to settle the loan is 13.
b) Outstanding balance on the loan after the final payment of R55,000:
To find the outstanding balance after the 13th payment, we can use the formula for the future value of an ordinary annuity:
Future Value = A * ((1 + r)^n - 1) / r
Substitute the known values:
Future Value = R55,000 * ((1 + 0.125)^13 - 1) / 0.125
Using a calculator, we find:
Future Value ≈ R992,145.99
So, the outstanding balance on the loan after the final payment of R55,000 is approximately R992,145.99.
c) Value of the final payment:
The final payment is the difference between the outstanding balance after the 13th payment and the regular biannual payment (R55,000):
Final Payment = Future Value - A
Final Payment ≈ R992,145.99 - R55,000
Final Payment ≈ R937,145.99
So, the value of the final payment is approximately R937,145.99.