Effective Rate

Study guide: Exam FM-2

The effective interest rate i is the rate of interest credited once per measurement period, at the end of the period. It is the rate that, applied once per period, produces the same accumulation as an equivalent Nominal Interest Rate compounded more frequently.

• For a nominal rate i^{(m)} Convertible m-thly:

i = \left(1 + \frac{i^{(m)}}{m}\right)^m - 1

• The effective rate is the standard benchmark for comparing interest rates across different compounding frequencies.
• The effective Discount Rate d satisfies d = iv = i/(1+i).

Comparing Two Rates Which is better: 7% compounded semi-annually, or 6.9% compounded monthly?

Answer Convert each to an effective annual rate:

\begin{align*}
i_{\text{semi}} &= \left(1 + \frac{0.07}{2}\right)^2 - 1 \\
  &= (1.035)^2 - 1 \\
  &= 0.0712250 = 7.12250\%
\end{align*}

\begin{align*}
i_{\text{monthly}} &= \left(1 + \frac{0.069}{12}\right)^{12} - 1 \\
  &= (1.00575)^{12} - 1 \\
  &= 0.0712245 = 7.12245\%
\end{align*}

The two are practically equivalent: 7% semi-annually is higher by only 0.00005 percentage points. (The nominal monthly rate equivalent to 7% semi-annually is 12\left[(1.071225)^{1/12} - 1\right] = 6.90005\%.) Neither rate is materially better; strictly, the semi-annual rate is marginally higher.