Nominal vs. effective annual interest: When are the nominal and effective interest rates equal?

Introduction / Context:
The nominal annual rate states the rate without specifying the compounding effect. The effective annual rate incorporates compounding. Comparing them correctly is essential for consistent economic evaluations.

Given Data / Assumptions:

• Nominal rate r stated per year.
• Compounding m times per year.
• Effective annual rate reff computed including compounding.

Concept / Approach:
The relationship is reff = (1 + r/m)^(m) − 1. If m = 1 (annual compounding), then reff = (1 + r)^1 − 1 = r, so nominal equals effective.

Step-by-Step Solution:
Start with reff = (1 + r/m)^m − 1Set m = 1: reff = (1 + r)^1 − 1 = rTherefore nominal equals effective only under annual compounding.

Verification / Alternative check:
For quarterly compounding (m = 4), reff = (1 + r/4)^4 − 1 > r for r > 0, hence they differ.

Why Other Options Are Wrong:
Quarterly and semi-annual compounding increase effective rate above nominal. The claim that they are never equal is false because m = 1 makes them equal.

Common Pitfalls:
Comparing rates with different compounding bases without converting to effective annual terms.

Final Answer:
When compounding is annually (once per year)