Rate conventions: The effective annual interest rate equals the nominal rate only under which compounding schedule?

Introduction / Context:
The nominal annual interest rate is a quoted rate that may be compounded multiple times per year. The effective annual rate (EAR) reflects the actual once-per-year growth including compounding effects. Understanding when EAR equals the nominal rate is important for comparing financial offers fairly.

Given Data / Assumptions:

• Nominal annual rate r_nom.
• Compounding m times per year (m ≥ 1).
• Effective annual rate defined as EAR = (1 + r_nom/m)^m − 1.

Concept / Approach:
If m = 1 (annual compounding), then EAR = (1 + r_nom)^1 − 1 = r_nom, so effective and nominal rates are equal. For any m > 1 (e.g., monthly, daily), EAR exceeds r_nom because of intra-year compounding. Therefore, the equality holds only when interest is compounded annually (once per year).

Step-by-Step Solution:
Write the EAR formula: EAR = (1 + r_nom/m)^m − 1.Set m = 1 → EAR = (1 + r_nom) − 1 = r_nom.For m > 1, EAR > r_nom due to compounding gains.

Verification / Alternative check:
Example: r_nom = 12%. Monthly compounding (m = 12) gives EAR = (1 + 0.12/12)^12 − 1 ≈ 12.68%, which is greater than 12%. With annual compounding (m = 1), EAR = 12% exactly.

Why Other Options Are Wrong:
Fortnightly / monthly / half-yearly / daily: All involve m > 1, producing EAR > nominal.

Common Pitfalls:
Comparing loans or investments with different compounding intervals using nominal rates alone; always convert to EAR for apples-to-apples comparisons.

Final Answer:
Compounded annually (once per year)