Payable m-thly

Study guide: Exam FM-2

An annuity payable $m$ -thly makes $m$ payments per year of $1/m$ each, totaling $1$ per year. The present value of an $n$ -year Annuity Immediate payable $m$ -thly uses the modified annuity symbol $a^{(m)}_{\overline{n}|}$ :

$a^{(m)}_{\overline{n}|} = \frac{1 - v^n}{i^{(m)}}$

Here $i^{(m)}$ is the Nominal Interest Rate Convertible m-thly and $v = 1/(1+i)$ .
Payable $m$ -thly annuities arise when cash flows occur monthly, quarterly, or semi-annually rather than annually.

Monthly Annuity Present Value A 10-year annuity pays $100 at the end of each month. Find the present value at $i = 6\%$ effective annual.
Answer The payments total $1{,}200$ a year, so $\text{PV} = 1{,}200\,a^{(12)}_{\overline{10}|}$ .
 $\begin{align*} i^{(12)} &= 12\left[(1.06)^{1/12}-1\right] \\ &= 12(0.00486755) \\ &= 0.0584106 \\ a^{(12)}_{\overline{10}|} &= \frac{1-(1.06)^{-10}}{0.0584106} \\ &= \frac{0.4416052}{0.0584106} \\ &= 7.56036 \\ \text{PV} &= 1{,}200 \times 7.56036 \\ &= 9{,}072.43 \end{align*}$ 