Nominal Discount Rate

A nominal discount rate $d^{(m)}$ is a quoted annual rate of discount payable (convertible) $m$ times a year: the year is split into $m$ periods of length $1/m$ , and each is charged discount at the effective rate $d^{(m)}/m$ , paid at the beginning of that period. A unit accumulates under $d^{(m)}$ as:

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

$d^{(m)}/m$ is the Effective Discount Rate for each $m$ th of a year; the quoted $d^{(m)}$ is $m$ times that per-period rate, not the rate of discount over the year.
Discounting runs the other way: the present value of $1$ due at time $t$ is $\left(1 - \frac{d^{(m)}}{m}\right)^{mt}$ .
$m$ need not be a whole number: a rate "compounded every two years" has $m = \tfrac{1}{2}$ , so discount of $d^{(1/2)}/0.5 = 2d^{(1/2)}$ is charged at the start of each two-year period.
The conversion to the effective rates $d$ and $i$ , $d^{(m)} = m\left[1 - v^{1/m}\right]$ , and how $d^{(m)}$ ranks against $\delta$ and $i^{(m)}$ are on Nominal Discount Rate Convertible m-thly; the interest-paid-at-the-end counterpart is the Nominal Interest Rate.

Same Quoted Rate, Discount vs. Interest A fund credits a nominal rate of discount of 8% convertible quarterly. Find the accumulated value after 3 years of a deposit of $1{,}000$ , and compare it with a fund crediting a nominal rate of interest of 8% convertible quarterly.
Answer Each quarter is charged discount at $0.08/4 = 0.02$ in advance, and there are $4 \times 3 = 12$ quarters:
 $\begin{align*} 1000\left(1 - \frac{0.08}{4}\right)^{-12} &= 1000(0.98)^{-12} \\ &= 1274.35 \end{align*}$ 
 $\begin{align*} 1000\left(1 + \frac{0.08}{4}\right)^{12} &= 1000(1.02)^{12} \\ &= 1268.24 \end{align*}$ 
At the same quoted rate the discount fund grows faster (8.417% effective a year against 8.243%): 2% charged in advance is $\frac{0.02}{0.98} = 2.041\%$ interest on the balance actually invested each quarter.

A Rate Compounded Every Two Years A payment of $5{,}000$ is due in 6 years. Find its present value at a nominal rate of discount of 10% compounded every two years, and the equivalent effective annual rate of discount.
Answer Here $m = \tfrac{1}{2}$ : each two-year period is charged $0.10/0.5 = 0.20$ , and 6 years is $0.5 \times 6 = 3$ periods:
 $\begin{align*} PV &= 5000\left(1 - \frac{0.10}{0.5}\right)^{0.5 \times 6} \\ &= 5000(0.8)^{3} \\ &= 2560 \end{align*}$ 
One year is half a period, so $1 - d = (0.8)^{1/2} = 0.894427$ and $d = 10.557\%$ .