You plan to apply for a $50,000 loan. The APR is 5.5% compounded quarterly. If you plan to make payments of $2,000 per quarter, how many years do you need to pay off the loan?

Question

You plan to apply for a $50,000 loan. The APR is 5.5% compounded quarterly. If you plan to make payments of $2,000 per quarter, how many years do you need to pay off the loan?

BlackTom AI · Accepted Answer

To determine how many years are needed to pay off the loan, we must evaluate the structure of the cash flows and the interest compounding.
Option 1: =NPER(5.5%*12, 2000, 50000,0)*12
This uses an annualized rate by multiplying 5.5% by 12, which is incorrect for quarterly compounding. It also uses a positive payment value, which conflicts with the typical cash-flow convention in NPER (payments are usually negative when you are paying the loan). Finally, multiplying by 12 would yield a number of years without the proper per-period calculation, making this option inconsistent with the actual quarterly setup.
Option 2: =NPER(5.5%/4, -2000, 50000,0)/4
Here the rate is divided by 4 to reflect quarterly compounding (5.5% per year → 5.5%/4 per quarter). The payment is entered as -2000, which correctly represents an outflow each quarter, and the present value is 50000 as a positive amount. The NPER function returns the number of quarters needed to amortize the loan to 0; dividing by 4 converts that quarter count into years. This aligns with the problem statement asking for years.
Option 3: =NPER(5.5%/4, 2000, -50000,0)/12
This option flips the sign of the payment (positive 2000) and the present value (negative 50000), which breaks the standard cash-flow convention used by NPER for loans. Additionally, dividing by 12 would attempt to convert to years via months, which is not appropriate here since the payments are quarterly. Overall, this misrepresents both cash flows and the timing unit.
Option 4: =NPER(5.5%/12, -2000, 50000,0)/12
This one uses an interest rate of 5.5%/12, implying monthly compounding, while the problem specifies quarterly compounding. It also divides by 12, which would convert quarters to years through a monthly basis, not aligning with the given quarterly payments. Thus the rate basis and unit conversion are both inconsistent with the scenario.
In summary, Option 2 is the only choice that correctly matches the quarterly compounding (rate per quarter), uses the correct cash-flow signs for a loan payoff, and converts the resulting number of quarters into years by dividing by 4. The other options mishandle the compounding period, sign conventions, or unit conversions, leading to incorrect results.

CSE 2111 AU2025 (10550) Post-Test Quiz

You plan to apply for a $50,000 loan. The APR is 5.5% compounded quarterly. If you plan to make payments of $2,000 per quarter, how many years do you need to pay off the loan?

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