Interest as % of final balance
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
100 − 95 100 = 5.00 % {\displaystyle {\frac {100-95}{100}}=5.00\%}The effective interest rate is calculated using 95 as the base
100 − 95 95 = 5.26 % {\displaystyle {\frac {100-95}{95}}=5.26\%}which says that
95
%
{\displaystyle 95\%}
For every effective interest rate
i
{\displaystyle i}
Using this, we can derive the following expression of d {\displaystyle d} and i {\displaystyle i} .
d = i 1 + i {\displaystyle d={\frac {i}{1+i}}}We usually define
v
{\displaystyle v}
using the above relationships between i {\displaystyle i} and d {\displaystyle d} .
Annual discount rate convertible pthly
A discount rate applied
p
{\displaystyle \,p}
where
d
(
p
)
{\displaystyle \,d^{(p)}}
d
(
∞
)
=
δ
{\displaystyle \,d^{(\infty )}=\delta }
The rate d ( p ) {\displaystyle \,d^{(p)}} is always bigger than d because the rate of discount convertible p {\displaystyle \,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.
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
- Notation of interest rates
References
//web.archive.org/web/20131230232348///www.mcu.edu.tw/department/management/stat/ch_web/etea/Theory%20of%20Interest/interest2.pdf
- ^ Kellison, S. G.: The Theory of Interest (Irwin: Illinois, 2008, 3rd edition), p.17.
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