If interest is compounded more than once per year, with i as the nominal annual interest rate, n as the number of years, and m as the number of compounding periods per year, what is the compound amount factor (CAF) that multiplies the present amount P to yield future amount F?

Introduction / Context:
Many financial instruments compound interest more frequently than annually (e.g., monthly, quarterly). To compute the future amount F from a present amount P under nominal annual rate i with m compounding periods per year, we use the compound amount factor (CAF). Understanding this factor is essential for accurate time-value-of-money calculations in project cash flows and equipment financing.

Given Data / Assumptions:

• Nominal annual rate i.
• m compounding periods per year.
• n years of investment.

Concept / Approach:

Each compounding period accrues interest at i/m. Over total periods N = n * m, the accumulation is geometric: F = P * (1 + i/m)^(nm). Therefore, the CAF equals (1 + i/m)^(nm). This reduces to (1 + i)^n when m = 1, confirming consistency with annual compounding.

Step-by-Step Solution:

Verification / Alternative check:

Example: i = 12% nominal, m = 12, n = 1 → CAF = (1 + 0.12/12)^(12) ≈ 1.1268, matching monthly compounding tables.

Why Other Options Are Wrong:

• (1 + im)^n incorrectly multiplies i by m instead of dividing.
• (1 + i)^(n/m) reduces compounding count instead of increasing it.
• (1 - i/m)^(nm) would discount, not compound.
• (1 + ni/m) is linear (simple interest), not compound.

Common Pitfalls:

• Mixing nominal and effective annual rates.
• Using m incorrectly (e.g., treating it as years).

Final Answer:

(1 + i/m)^(n*m)