Lesson CONTINUOUS COMPOUNDING FORMULAS USED IN FINANCIAL ANALYSIS.

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This lesson provides an overview of CONTINUOUS COMPOUNDING FORMULAS USED IN FINANCIAL ANALYSIS.

REFERENCES

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<a href = "http://cs.selu.edu/~rbyrd/math/continuous/" target = "_blank">Continuous Compound Interest Formula</a>
<a href = "http://finance.wharton.upenn.edu/~pbond/teaching/Review%20of%20continuous%20compounding.pdf" target = "_blank">Basics of Continuous Compounding in PDF Format</a>
<a href = "http://www.frickcpa.com/tvom/TVOM_Continuous.asp" target = "_blank">Continuous Compounding and other Financial Topics of Interest</a>
<a href = "http://www.mathsisfun.com/money/compound-interest-periodic.html" target = "_blank">Compound Interest</a>
<a href = "http://pages.stern.nyu.edu/~wsilber/Continuous%20Compounding.pdf" target = "_blank">Continuous Compounding Formula in PDF Format</a>
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NOTE:

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CONTINUOUS COMPOUNDING FORMULA

The continuous compounding formula is:

{{{f = p*e^(r*t)}}}

f = future value
p = present amount
r = annual interest rate expressed as a proportion (percent divided by 100% = proportion)
t = time in years
e = the scientific constant that is equivalent to 2.718281828.........

Example of use:

p = $1,000
r = 12% / 100% = .12
t = 20

{{{f = p*e^(r*t)}}} becomes:

{{{f = 1000*e^(.12*20)}}} which becomes:

f = 11023.17638

It's a very simple formula to use.

This is one of the reasons for its popularity, even in situations that don't require continuous compounding.

The references expand on that statement in much more detail.

The following is a summary of the continuous compounding formula and its relationship to the compounding formulas.

DISCRETE COMPOUNDING

The Standard Financial Analysis Formulas divide time into discrete intervals.

Interest is usually compounded either annually, semi-annually, quarterly, monthly, or daily.

With Discrete Compounding, the Effective Interest Rate per year is defined by the formula:

{{{R[e] = (1+(r/c))^c)) - 1}}}

where:

R[e] is the effective interest rate per year.
r is the nominal interest rate per year.
c is the number of compounding periods per year

Example 1:

Nominal Interest Rate of 10% per year is compounded monthly.

r = 10% / 100% = .1
c = 12

{{{R[e] = (1+(.1/12))^12 - 1}}} = {{{(1.008333333)^12 = 1.104713067 - 1}}} = .104713067 * 100% = 10.4713067% per year.

The nominal rate of 10% per year, when compounded monthly, yields an effective rate of 10.4713067% per year.

Example 2:

Nominal Interest Rate of 10% per year is compounded daily.

r = 10% / 100% = .1
c = 365

{{{R[e] = (1+(.1/365))^365 - 1}}} = {{{(1.000273973)^365 = 1.105155781 - 1}}} = .105155781 * 100% = 10.5155781% per year.

The nominal rate of 10% per year, when compounded daily, yields an effective rate of 10.5155781% per year.

The higher the number of compounding periods, the greater the effective rate will be.

This happens up to a limit and no further.

Continuous Compounding defines that limit.

CONTINUOUS COMPOUNDING

With Continuous Compounding, the Effective Interest Rate per year is defined by the formula:

{{{R[e] = e^r - 1}}}

where:

R[e] is the effective interest rate per year.
e is the scientific constant of 2.718281828...
r is the nominal interest rate per year.

The continuous compounding formula is derived from the discrete compounding formula as follows:

{{{R[e] = e^r - 1}}} = the limit of {{{(1+(r/c))^c)) - 1}}} as c approaches infinity.

This means that the continuous compounding formula for R[e] equals the limit of the discrete compounding formula for R[e] as the number of compounding intervals approaches infinity.

Example 3:

Nominal Interest Rate of 10% per year is compounded continuously.

r = 10% / 100% = .1

{{{R[e] = e^r - 1 = e^.1 - 1 = 1.105170918 - 1}}} = .105170918 * 100% = 10.5170918% per year.

The nominal rate of 10% per year, when compounded continuously, yields an effective rate of 10.5170918% per year.

COMPARISON OF DISCRETE COMPOUNDING RESULTS WITH CONTINUOUS COMPOUNDING RESULTS

We have 3 results.

10% per year compounded monthly yields effective rate of 10.4713067% per year.
10% per year compounded daily yields effective rate of 10.5155781% per year.
10% per year compounded continuously yields effective rate of 10.5170918% per year.

The limit is 10.5170918% per year which is the maximum effective rate you could possibly achieve in 1 year given that the nominal interest rate is 10% per year.

REASONS FOR USING CONTINUOUS COMPOUNDING FORMULAS RATHER THAN DISCRETE COMPOUNDING FORMULAS IN ANALYSIS

The reasons are largely mathematical by making calculations easier for financial analysis as far as I have been able to determine from the referenced material.

The formula {{{R[e] = e^(r*y)}}} is certainly simpler and easier to work with than {{{R[e] = (1+(r/c))^(c*y))) - 1}}}

where:

r = nominal interest rate per year
y = number of years.
c = compounding periods per year

In normal, everyday, real world situations, Interest Rates on Investments and Loans are usually defined in discrete intervals.

Where it makes sense, mathematicians involved in financial analysis make use of continuous compounding formulas rather than discrete compounding formulas because it is less cumbersome for them to do so.

The study of derivatives is one such use.

CONVERTING FROM DISCRETE COMPOUNDING FORMULAS TO CONTINUOUS COMPOUNDING FORMULAS

Establishing equivalencies by converting between discrete compounding rates and continuous compounding rates facilitates using continuous compound formulas to model discrete compound rate situations.

Example:

Your Nominal Interest Rate is 10% per year.
This will be compounded monthly.

Your Effective Interest Rate becomes:

{{{R[e] = (1+(r/12))^c - 1}}} where:

r = .1
c = 12

This equation becomes:

{{{R[e] = (1.0083333333)^12 - 1 = .1047130674}}}

r = 10% a year
R[e] = .1047130674 * 100% = 10.47130674% per year.

To find the equivalent effective interest rate using continuous compounding, you would do the following:

Take the effective interest rate you calculated using monthly compounding and use it to find the nominal interest rate using continuous compounding.

{{{R[e] + 1 = e^r}}} becomes:

{{{1.10471340674 = e^r}}}

Take the natural log of both sides of this equation to get:

{{{ln (1.10471340674) = r*ln(e) = r*1 = r}}}

Solve for r to get:

r = .09958594 * 100% = 9.958594%

This means that a nominal rate of 10% a year with monthly compounding is equivalent to a nominal rate of 9.958594% a year with continuous compounding.

You can now use continuous compounding formulas to solve the financial analysis that began with discrete compounding formulas.

As an example:

If you have a nominal interest rate of 10% a year with monthly compounding, and an equivalent nominal interest rate of 9.958594% a year with continuous compounding, then:

The future value of a $1,000 investment at nominal rate of 10% with monthly compounding for a period of 20 years becomes:

{{{1000 * (1+(.1/12))^(12*20) = 7328.073628}}} = $7328.073628

The future value of a $1,000 investment at nominal rate of 9.958594% with continuous compounding for a period of 20 years becomes:

{{{1000 * e^(20*.09958594) = 7328.073628}}} = $7328.073628

The continuous compounding formula with a nominal interest rate of 9.958594% is equivalent to the monthly compounding formula with a nominal interest rate of 10%.

Either method can be used to perform the analysis.

Both will provide the same result.

CONVERTING FROM CONTINUOUS COMPOUNDING FORMULAS TO DISCRETE COMPOUNDING FORMULAS

If you are given a nominal interest rate using continuous compounding formulas, then you can convert to the equivalent discrete compounding formulas using the same method in reverse.

Example:

You are given that the nominal interest rate equals 10% using continuous compounding.

You want to find the equivalent monthly compounding rate using discrete compounding.

You first find the effective interest rate using continuous compounding.

{{{R[e] = e^(r) - 1 = e^(.1) = 1.105170918 - 1 = .105170918}}}

You then use the effective interest rate using continuous compounding to find the nominal interest rate using monthly compounding.

{{{R[e] = (1+(r/12))^12 - 1}}} becomes:

{{{1.105170918 = (1+(r/12))^12}}}

Take the 12th root of both sides of this equation to get:

{{{1.008368152 = 1 + r/12}}}

Subtract 1 from both sides of this equation and then multiply both sides of this equation by 12 to get:

r = .008368152 * 12 = .100417827 * 100% = 10.0417827%

You started with a continuous compounding nominal rate of 10%.

This became equivalent to a monthly compounding nominal rate of 10.0417827%

THE PROCESS OF CONVERTING FROM DISCRETE COMPOUNDING TO CONTINUOUS COMPOUNDING IN A NUTSHELL

You are given the nominal rate using discrete compounding.
You solve for the effective rate using discrete compounding.
You use the effective rate using discrete compounding to solve for the nominal rate using continuous compounding.

THE PROCESS OF CONVERTING FROM CONTINUOUS COMPOUNDING TO DISCRETE COMPOUNDING IN A NUTSHELL.

You are given the nominal rate using continuous compounding.
You solve for the effective rate using continuous compounding.
You use the effective rate using continuous compounding to solve for the nominal rate using discrete compounding.