# 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

${\frac {100-95}{100}}=5.00\%$ The interest rate is calculated using 95 as the base

${\frac {100-95}{95}}=5.26\%$ For every effective interest rate, there is a corresponding effective discount rate, given by

$d={\frac {i}{1+i}}$ or inversely,

$i={\frac {d}{1-d}}$ Given the above equation relating $\,d$ to $\,i$ it follows that

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

or equivalently,

$v=1-d$ Since $\,d=iv$ ,it can readily be shown that

$id=i-d$ 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 $\,p$ 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 $\,p$ 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.