# 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

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

The interest rate is calculated using 95 as the base

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

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

${\displaystyle d={\frac {i}{1+i}}}$

or inversely,

${\displaystyle i={\frac {d}{1-d}}}$

Given the above equation relating ${\displaystyle \,d}$ to ${\displaystyle \,i}$ it follows that

${\displaystyle d={\frac {1+i}{1+i}}-{\frac {1}{1+i}}\ =1-v}$ where ${\displaystyle v}$ is the discount factor

or equivalently,

${\displaystyle v=1-d}$

Since ${\displaystyle \,d=iv}$ ,it can readily be shown that

${\displaystyle id=i-d}$

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 ${\displaystyle \,p}$thly

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

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

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

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

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

The rate ${\displaystyle \,d^{(p)}}$  is always bigger than d because the rate of discount convertible ${\displaystyle \,p}$ thly 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.