# Discount rate

For the fees charged to merchants for accepting credit cards, see Discount Rate under Merchant Account.
For discount rate as a term in investment financing, see Discounted cash flow
German central bank discount interest rates

The discount rate can mean

• the annual effective discount rate, which is the annual interest divided by the capital including that interest; 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

## 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. 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, suppose there is a government bond that sells for $95 and pays$100 in a year's time. The discount rate according to the given definition is

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

The interest rate is calculated using 95 as its base:

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

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

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

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.