Ted needs to take out a loan to buy a new bicycle.He is able to repay the loan by means of 15 semi-annual payments of R2500,starting in 6months from now,when interest is 16% p.a compounded quartely.Calculate the amount of the loan
To calculate the amount of the loan, we need to use the formula for the present value of an annuity. In this case, the formula becomes:
Loan Amount = Payment Amount * [(1 - (1 + r)^-n) / r]
Where:
Payment Amount = R2500 (the semi-annual payment)
r = interest rate per period (quarterly rate)
n = total number of periods
First, let's calculate the quarterly interest rate (r). Since the given annual interest rate is 16% compounded quarterly, the quarterly interest rate can be calculated as follows:
r = (1 + (annual interest rate / number of compounding periods)) ^ (number of compounding periods / total number of periods) - 1
= (1 + (16% / 4)) ^ (4 / 2) - 1
= (1 + 0.04) ^ 2 - 1
= (1.04) ^ 2 - 1
= 1.0816 - 1
= 0.0816
Next, we need to calculate the total number of periods (n). Since there are 15 semi-annual payments, there will be 30 quarters in total:
n = total number of payments * number of periods per payment
= 15 * 2
= 30
Now, we can substitute the values into the formula:
Loan Amount = R2500 * [(1 - (1 + 0.0816)^-30) / 0.0816]
= R2500 * [0.43182 / 0.0816]
= R2500 * 5.28799
≈ R13,219.975
Therefore, the amount of the loan is approximately R13,219.975.