Jason takes a personal loan for 4 years with equal quarterly repayments. The lender quotes 12% per annum flat rate (simple interest) on the original principal. What is the approximate effective annual interest rate (in %) implied by this flat-rate loan structure?

Introduction / Context:
This question tests the difference between a quoted flat (simple) rate and the effective rate when repayments happen throughout the term. A flat rate applies interest on the original principal for the entire duration, but because the borrower repays principal gradually, the lender’s yield (effective rate) is much higher than the flat rate. We can model the cashflows: the borrower receives principal P at time 0, then repays equal quarterly instalments over 4 years (16 quarters). Total flat interest over 4 years at 12% is P * 0.12 * 4 = 0.48P, so total repaid is 1.48P, split equally over 16 payments. The effective rate is the internal rate that discounts these payments back to P.

Given Data / Assumptions:

• Flat rate = 12% per annum on original principal
• Term = 4 years
• Quarterly repayments: 4 per year, total n = 16 payments
• Total flat interest = P * 0.12 * 4 = 0.48P
• Total repaid = P + 0.48P = 1.48P
• Quarterly payment = (1.48P) / 16 = 0.0925P

Concept / Approach:
Let i be the effective quarterly interest rate. Present value of an annuity must equal principal:
P = Payment * (1 - (1 + i)^(-16)) / i Divide both sides by P to solve i. Then convert to effective annual rate:
Effective annual rate = (1 + i)^4 - 1

Step-by-Step Solution:
Payment/P = 0.0925 So, 1 = 0.0925 * (1 - (1 + i)^(-16)) / i Solving this gives i ≈ 0.05035 per quarter (about 5.035%) Effective annual rate = (1 + 0.05035)^4 - 1 ≈ 1.2171 - 1 = 0.2171 ≈ 21.71% per annum (effective)

Verification / Alternative check:
The result must be much higher than 12% because principal is repaid throughout the term while interest was computed as if the full principal stayed outstanding. An effective rate around 20%+ is consistent with flat-rate loan behavior.

Why Other Options Are Wrong:
12% is the quoted flat rate, not the effective rate. 15.50% and 18% are still too low for a 4-year flat-rate loan with frequent repayments. 24% is too high for this specific repayment frequency and flat rate.

Common Pitfalls:
Assuming the flat rate equals the effective rate, forgetting there are 16 quarterly payments, or converting quarterly rate to annual incorrectly (must compound using (1+i)^4 - 1).

Final Answer:
The approximate effective annual interest rate is 21.71%.