# Annual effective discount rate

The annual effective discount rate is the annual interest divided by the capital including that interest, which is the interest rate divided by 100% plus the interest rate. This rate is lower than the interest rate; it corresponds to using the value after a year as the nominal value, and seeing the initial value as the nominal value minus a discount. It is used for U.S. Treasury bills and similar financial instruments. It is the annual discount factor to be applied to the future cash flow, to find the discount, subtracted from a future value to find the value one year earlier.

For example, consider a government bond that sells for $95 and pays$100 in a year's time. The discount rate is

$\frac{100-95}{100} = 5.00\%$

The interest rate is calculated using 95 as the base

$\frac{100-95}{95} = 5.26\%$

For every annual effective interest rate, there is a corresponding annual effective discount rate, given by

$d = \frac{i}{1+i}\approx i-i^2$

or inversely,

$i = \frac{d}{1-d}\approx d+d^2$

where the approximations apply for small i and d; in fact i - d = id.

## Annual discount rate convertible $\,p$thly

A discount rate applied $\,p$ times over equal subintervals of a year is found from the annual effective rate d as

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

where $\,d^{(p)}$ is called the annual nominal rate of discount convertible $\,p$thly.

$1-d = \exp (-d^{(\infty)})$

$\,d^{(\infty)}=\delta$ is the force of interest.

The rate $\,d^{(p)}$ is always bigger than d because the rate of discount convertible pthly is applied in each subinterval to a smaller (already discounted) sum of money. As such, in order to achieve the same total amount of discounting the rate has to be slightly more than 1/pth of the annual rate of discount.