View source for Annual effective discount rate

{{Short description|Interest as % of final balance}}

The '''annual effective discount rate''' expresses the amount of [[interest]] paid or earned as a ''percentage'' of the [[Balance (accounting)|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 [[United States Treasury security#Treasury bill|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

: <math>\frac{100-95}{100} = 5.00\%</math>

The effective interest rate is calculated using 95 as the base

: <math>\frac{100-95}{95} = 5.26\%</math>

which says that <math>95\%</math> of $105.26 is $100. 

For every [[effective interest rate]] <math>i</math>, there is a corresponding effective discount rate <math>d</math> that can produce the same 
[[future value]] as <math>i</math> if a given amount of [[Debt#Principal|principal]] is invested for the same amount of time at each of the rates <math>i</math> and <math>d</math>, and they are said to be ''equivalent''.<ref>Kellison, S. G.: The Theory of Interest (Irwin: Illinois, 2008, 3rd edition), p.17.</ref> Therefore, we have the following relationship between two equivalent rates <math>i</math> and <math>d</math>.

:<math>1+i=\frac{1}{1-d}.</math>

Using this, we can derive the following expression of <math>d</math> and <math>i</math>.

: <math> d = \frac{i}{1+i} </math>, and

:<math> i = \frac{d}{1-d}.</math>

We usually define <math>  v  </math> as the discount factor which is given by

:<math> v = \frac{1}{1+i} </math>, then we can derive that

:<math> v = 1-d </math>, and

:<math> d = iv </math>

using the above relationships between <math>i</math> and <math>d</math>.

==Annual discount rate convertible {{mvar|p}}thly==
{{Unreferenced section|date=June 2023}}

A discount rate applied <math>\,p</math> times over equal subintervals of a year is found from the annual effective rate d as

:<math>1-d = \left(1-\frac{d^{(p)}}{p}\right)^p</math>

where <math>\,d^{(p)}</math> is called the annual nominal rate of discount convertible <math>\,p</math>thly.

:<math>1-d = \exp (-d^{(\infty)})</math>

<math>\,d^{(\infty)}=\delta</math> is the [[force of interest]].

The rate <math>\,d^{(p)}</math> is always bigger than d because the rate of discount convertible <math>\,p</math>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.

==Business calculations==
{{Unreferenced section|date=June 2023}}
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==
* [[Actuarial notation#Interest_rates|Notation of interest rates]]

==References==
https://web.archive.org/web/20131230232348/http://www.mcu.edu.tw/department/management/stat/ch_web/etea/Theory%20of%20Interest/interest2.pdf
{{Reflist}}

[[Category:Interest rates]]