### 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*.

## Contents

- Annual discount rate convertible \,pthly 1
- Business calculations 2
- See also 3
- References 4

## Annual discount rate convertible \,pthly

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 \,pthly.

- 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 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.

## Business calculations

Businesses consider this discount rate when deciding whether to invest profits to buy equipment or whether to deliver the profit to shareholders. In an ideal world, they would buy a piece of equipment if shareholders would get a bigger profit later. The amount of extra profit a shareholder requires to prefer that the company buy the equipment rather than giving them the profit now is based on the shareholder's discount rate. A common way of estimating shareholders' discount rates uses share price data is known as the capital asset pricing model. Businesses normally apply this discount rate by calculating the net present value of the decision.

## See also

## References

http://www.mcu.edu.tw/department/management/stat/ch_web/etea/Theory%20of%20Interest/interest2.pdf