Flat-Rate Loan and Effective Interest: 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 %) that this flat-rate loan structure implies?

Introduction / Context:
This question explores the difference between a quoted flat rate of interest and the true effective annual interest rate on a loan repaid in equal instalments. Jason's loan carries a 12% per annum flat rate on the full principal, but payments are made quarterly over 4 years. You must compute the approximate effective annual rate implied by this structure using the concept of internal rate of return (IRR).

Given Data / Assumptions:

• Flat simple interest rate = 12% per annum on the original principal.
• Loan term = 4 years.
• Repayments are equal and made quarterly (every 3 months).
• Total number of payments = 4 years × 4 quarters per year = 16 payments.
• We assume a principal of P = 1 (for rate calculation, any positive principal works proportionally).

Concept / Approach:
Under a flat rate, total interest over the entire term is computed on the original principal as I = (P * r_flat * t) / 100. This total interest is then spread evenly over all instalments. To find the effective rate, treat the loan as an investment by the lender: an initial outflow of P at time 0 and an inflow of equal instalments each quarter. Solve for the quarterly rate r_q that equates the present value of instalments to P, then convert r_q to an effective annual rate using (1 + r_q)^4 − 1.

Step-by-Step Solution:
Step 1: Assume principal P = 1 for convenience. Step 2: Flat rate r_flat = 12% per annum, term t = 4 years. Step 3: Total interest over the full term: I_total = (1 * 12 * 4) / 100 = 0.48. Step 4: Total amount repaid = principal + interest = 1 + 0.48 = 1.48. Step 5: Number of quarterly payments = 4 years × 4 = 16. Each instalment = 1.48 / 16 = 0.0925. Step 6: Let r_q be the effective quarterly rate. The present value of all instalments must equal the original principal: 0.0925 / (1 + r_q)^1 + 0.0925 / (1 + r_q)^2 + ... + 0.0925 / (1 + r_q)^16 = 1. Step 7: Solving this equation (using numerical methods) gives r_q ≈ 0.05035, i.e., about 5.04% per quarter. Step 8: Convert this to an effective annual rate: r_eff = (1 + r_q)^4 − 1 ≈ (1.05035)^4 − 1 ≈ 0.2171 or 21.71%.

Verification / Alternative check:
The flat rate of 12% per annum seems low, but because the borrower is repaying the principal gradually while paying interest as if the full principal were outstanding for the whole 4 years, the economic cost is much higher. An effective annual rate above 20% is typical for such flat rate loans, so 21.71% is a reasonable and consistent result.

Why Other Options Are Wrong:

• 18.00% and 15.50%: These are below the computed effective rate and underestimate the true cost of the loan.
• 12.00%: This is just the flat rate and ignores the impact of declining principal due to instalments.
• 24.00%: Slightly higher than the computed IRR and not supported by the detailed present value calculation.

Common Pitfalls:
Many students mistakenly equate the flat rate with the effective rate or linearly scale the total interest over time without considering the timing of repayments. Others simply double the flat rate when payments are frequent, which is incorrect. The correct approach uses present value and internal rate of return concepts to reflect the actual cost of borrowing.

Final Answer:
The approximate effective annual interest rate implied by the flat-rate loan is 21.71% per annum.