A man borrows $4000 and agrees to repay $200 at the end of each quarter. If the interest charged is 4%, how Long does it take him to clear his debt
To determine how long it takes to clear the debt, we need to know whether the interest rate of 4% is per quarter or per year. Most commonly, interest rates are given on an annual basis unless otherwise specified. Assuming the interest rate is 4% per year, we need to calculate the effective quarterly interest rate and then use an amortization formula or method to determine when the debt will be paid off.
Since the interest is compounded quarterly, we can divide the annual interest rate by 4 to find the quarterly interest rate:
Quarterly interest rate = Annual interest rate / 4
Quarterly interest rate = 4% / 4 = 1% per quarter
Now, using the formula for the present value of an annuity (because the loan repayments are being made at regular intervals), we can figure out the number of quarters it will take to repay the loan. The following annuity formula calculates the present value (PV) of a series of future annuity payments:
PV = P * [(1 - (1 + r)^-n) / r]
In this formula:
- PV is the present value of the annuity, which is the initial loan amount of $4000.
- P is the periodic payment amount, which is $200.
- r is the periodic interest rate, which is 0.01 (1% in decimal form).
- n is the number of periods (quarters, in this case) it takes to repay the loan.
Rearranging the formula to solve for n gives us:
n = -log(1 - (PV * r / P)) / log(1 + r)
Substituting the given values into the formula:
n = -log(1 - (4000 * 0.01 / 200)) / log(1 + 0.01)
n = -log(1 - (40 / 200)) / log(1.01)
n = -log(1 - 0.2) / log(1.01)
n = -log(0.8) / log(1.01)
Now calculate the value of n:
n ≈ -log(0.8) / log(1.01)
n ≈ -(-0.09691) / 0.00432
n ≈ 22.435
Since n represents the number of quarters, and we can't have a fraction of a quarter payment, we round up to the nearest whole number. Therefore, it will take the borrower approximately 23 quarters (rounded up from 22.435) to repay the entire loan.
To verify and have a precise understanding of the total time and interest, it might be necessary to set up an amortization schedule for the 23 quarters and ensure the final payment lines up with completely paying off the loan. If a slight balance remains after 23 quarters, the borrower may make a smaller final payment in the 24th quarter to clear the debt.