233k views
0 votes
A loan of R750 000 is taken out at an interest rate of 12,5% p.a. compounded biannually. It is repaid by equal biannual payments of R55 000 and a final payment less than R55 000. a) How many payments are required to settle the loan? b) What is the outstanding balance on the loan after the final payment of R55 000? c) What is the value of the final payment?​

User Artuc
by
7.6k points

1 Answer

1 vote

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.

User Prakhar Agarwal
by
7.8k points
Welcome to QAmmunity.org, where you can ask questions and receive answers from other members of our community.