applet-magic.com
Thayer Watkins
Silicon Valley
USA

Instantaneous and
Compounded Annual Rates
for Interest

In finance there are two ways to express rates such as interest rates. The most common way is as the effective annual rates so that if the interest rate is r then \$1 deposited at the beginning of a year will grow to be (1+r) by the end of the year. The other way of expressing an interest rate is the rate of growth of the value of a deposit over an infinitesmal period of time. This rate is usually called the instantaneous rate. The value of a deposit after one year of accumulating interestwill depend not only upon the instantaneous interest rate but also upon how often the interest is compounded. To see the effect of compounding consider the following table that represents the growth of a deposit of \$1 over a one-year period when the instantaneous interest rate is 100 percent per year.

COMPOUNDING
PERIOD
VALUE AFTER
ONE YEAR
one year\$2.00
six months\$2.25
three months\$2.44
one month\$2.61
one week\$2.69
one day\$2.7146
one hour\$2.7181
one minute\$2.7179
one second\$2.7183

At the extreme the interest is compounded continuously. If R is the nominal instantaneous interest rate and n is the number times per year the interest is compounded then the value of \$1 deposit after one year is given by:

(1+R/n)n

The limit as n goes to infinity is 2.718281828...... This number is known as "e." Thus for an instantaneous interest rate of 100 percent the effective annual interest rate is about 171.28 percent. Generally the relationship between the annual interest rate r and the instantaneous rate R for continuous compounding is:

1 + r = eR
R = ln(1+r)

For small values of r and R there is not much of a difference between them.

The significance of the distinction between r and R is in the nature of formulas involving interest rates. For example, Fisher's theory of interest rates gives the relationship between the nominal interest rate i, the real interest rate r and the rate of inflation π as:

(1+i) = (1+r)(1+ π)

When we take the logarithm of both sides of this equation we get:

ln(1+i) = ln(1+r) + ln(1+ π)

Thus,

i = R + π

where i and π are the instanteous nominal interest rate and rate of inflation, respectively.

Given the simpler nature of this formula than Fisher's equation it might seem that the natural thing to do is express all rates in their continuously compounded form. The problem is that rates such as the rate of inflation are always given as annual rates and there would be confusion in interpreting their rates in the instantaneous form for continuous compounding.

In financial formulas if there are terms involving the exponential function then the rates are in their instantaneous form, but if there are terms involving (1+r) raised to a power then the rates are expressed in their effective annual rate form.

Below are calculators for converting rates from one form to another. The results are rounded to four decimal places.

THW's
Rate Converter
Instantaneous to Annual
Instantaneous Rate
(decimal)
=
Effective Annual Rate
(decimal)

THW's
Rate Converter
Annual to Instantaneous
Annual Rate
(decimal)
=
Instantaneous Rate
(decimal)

Below is a calculator which computes the instantaneous and annual rates for a quantity that increases over some period of time:

THW's
Rates Calculator
Beginning Value
Ending Value
Duration of Period
=
Instantaneous Rate
With Continuous
Compounding
Effective Annual Rate