Annual effective discount rate

The annual effective discount rate expresses the amount of interest paid or earned as a percentage of the balance at the end of the annual period. It is related to but slightly smaller than 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 ('balance' in the bond at the start of period) and pays $100 ('balance' in the bond at the end of period) in a year's time. The discount rate is

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

The effective interest rate is calculated using 95 as the base

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

which says that ${\displaystyle 95\%}$ of $105.26 is $100.

For every effective interest rate ${\displaystyle i}$, there is a corresponding effective discount rate ${\displaystyle d}$ that can produce the same future value as ${\displaystyle i}$ if a given amount of principal is invested for the same amount of time at each of the rates ${\displaystyle i}$ and ${\displaystyle d}$, and they are said to be equivalent.^[1] Therefore, we have the following relationship between two equivalent rates ${\displaystyle i}$ and ${\displaystyle d}$.

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

Using this, we can derive the following expression of ${\displaystyle d}$ and ${\displaystyle i}$.

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

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

We usually define ${\displaystyle v}$ as the discount factor which is given by

${\displaystyle v={\frac {1}{1+i}}}$, then we can derive that

${\displaystyle v=1-d}$, and

${\displaystyle d=iv}$

using the above relationships between ${\displaystyle i}$ and ${\displaystyle d}$.