Nominal Discount Rate Convertible m-thly

Study guide: Exam FM-2

The nominal discount rate convertible $m$ -thly, denoted $d^{(m)}$ , is an annual discount rate under which interest is paid at the beginning of each of $m$ sub-periods per year at rate $d^{(m)}/m$ .

The relationship to the effective annual rate $i$ (and effective discount rate $d$ ) is:

$\left(1 - \frac{d^{(m)}}{m}\right)^m = 1 - d$

$= v$

$= \frac{1}{1+i}$

$d^{(m)} = m\!\left[1 - v^{1/m}\right]$

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

As $m \to \infty$ , $d^{(m)} \to \delta$ (the Force of Interest).
For a positive interest rate, $d^{(m)} < i^{(m)}$ for all $m$ ; more fully, $d < d^{(m)} < \delta < i^{(m)} < i$ for $m > 1$ .

Finding Nominal Discount Rate The effective annual interest rate is $6\%$ . Find $d^{(12)}$ .
Answer With $v = 1/1.06$ :
 $\begin{align*} d^{(12)} &= 12\left[1 - (1.06)^{-1/12}\right] \\ &= 12(1 - 0.9951560) \\ &= 12(0.0048440) \\ &= 0.058128 = 5.813\% \end{align*}$ 
As expected, $d^{(12)} = 5.813\%$ is below $\delta = \ln 1.06 = 5.827\%$ and $i^{(12)} = 12\left[(1.06)^{1/12} - 1\right] = 5.841\%$ .