# Annual effective discount rate

The annual effective discount rate expresses the amount of interest paid/earned as a percentage of the balance at the end of the (annual) period. This is in contrast to the effective rate of interest, which expresses the amount of interest as a percentage of the balance at the start of the period. The discount rate is commonly used for U.S. Treasury bills and similar financial instruments.

For example, consider a government bond that sells for $95 and pays$100 in a year's time. The discount rate is The interest rate is calculated using 95 as the base For every effective interest rate, there is a corresponding effective discount rate, given by or inversely, Given the above equation relating to it follows that where is the discount factor

or equivalently, Since ,it can readily be shown that This relationship has an interesting verbal interpretation. A person can either borrow 1 and repay 1 + i at the end of the period or borrow 1 - d and repay 1 at the end of the period. The expression i - d is the difference in the amount of interest paid. This difference arises because the principal borrowed differs by d. Interest on amount d for one period at rate i is id.

## Annual discount rate convertible thly

A discount rate applied times over equal subintervals of a year is found from the annual effective rate d as where is called the annual nominal rate of discount convertible thly.  is the force of interest.

The rate 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.