You borrow $400,000. The loan requires 24 quarterly repayments at an interest rate of 8% p.a. compounded quarterly. What amount of principal is repaid in the first three months? Group of answer choices $21,148.44 $8,000.00 $13,148.44 $386,851.56
Question
You borrow $400,000. The loan requires 24 quarterly repayments at an interest rate of 8% p.a. compounded quarterly. What amount of principal is repaid in the first three months? Group of answer choices
$21,148.44
$8,000.00
$13,148.44
$386,851.56
Solution
To calculate the amount of principal repaid in the first three months, we first need to calculate the quarterly payment using the formula for the payment of an ordinary annuity:
PMT = PV * r / [1 - (1 + r)^-n]
where:
- PMT is the regular payment per period (what we're trying to find)
- PV is the present value ($400,000)
- r is the interest rate per period (8% or 0.08 divided by 4)
- n is the number of periods (24 quarters)
Substituting the given values into the formula, we get:
PMT = $400,000 * (0.08/4) / [1 - (1 + 0.08/4)^-24]
After calculating the PMT, we can find the interest paid in the first quarter by multiplying the initial loan amount by the quarterly interest rate:
Interest = $400,000 * (0.08/4)
The principal repaid in the first quarter is then the difference between the total payment and the interest:
Principal = PMT - Interest
Calculating these expressions, we find that the amount of principal repaid in the first three months is approximately 13,148.44.
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