Nominal Interest Rate

Study guide: Exam FM-2

The nominal interest rate $i^{(m)}$ is a quoted annual rate convertible (compounded) $m$ times per year. Rather than crediting interest once at year-end, the year is divided into $m$ sub-periods each earning a periodic rate of $i^{(m)}/m$ .

$i^{(m)} = m\left[(1+i)^{1/m} - 1\right]$

The two-way relationship with the effective annual rate $i$ is:

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

$i^{(m)} = m\left[(1+i)^{1/m} - 1\right]$

As $m \to \infty$ the nominal rate converges to the Force of Interest: $\displaystyle\lim_{m\to\infty} i^{(m)} = \delta = \ln(1+i)$ .
For a fixed effective rate, $i^{(m)}$ is a decreasing function of $m$ — more frequent compounding requires a smaller stated rate to achieve the same year-end accumulation.

Converting a Nominal Rate to an Effective Rate A savings account advertises a nominal interest rate of $i^{(12)} = 6\%$ convertible monthly. Find the equivalent effective annual interest rate.

Answer The monthly periodic rate is $6\%/12 = 0.5\%$ , so: $i = \left(1 + \frac{0.06}{12}\right)^{12} - 1 = (1.005)^{12} - 1 \approx 6.168\%$