Annual effective discount rate

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The annual effective discount rate is the annual interest divided by the capital including that interest, which is the interest rate divided by 100% plus the interest rate. This rate is lower than the interest rate; it corresponds to using the value after a year as the nominal value, and seeing the initial value as the nominal value minus a discount. It is used for U.S. Treasury bills and similar financial instruments. It is the annual discount factor to be applied to the future cash flow, to find the discount, subtracted from a future value to find the value one year earlier.

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 annual effective interest rate, there is a corresponding annual effective discount rate, given by

d = \frac{i}{1+i}\approx i-i^2

or inversely,

i = \frac{d}{1-d}\approx d+d^2

where the approximations apply for small i and d; in fact i - d = id.

Annual discount rate convertible \,pthly[edit]

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[edit]

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[edit]