## Formula

i = m x (e^{(r/m)}- 1)

**Variables used in the annuity formula**

i = Periodic interest rate

r = Continuous interest rate

m = Number of times compounding takes place in a period

e = Base of the natural logarithm (LN) ≈ 2.71828

## Use

The continuous to periodic interest rate formula is used to convert a continuous interest rate (r) to a periodic interest rate (i) with compounding taking place (m) times in a period.

## Example 1: Using the Continuous to Periodic Interest Rate Formula

If an amount is invested at a continuous interest rate of 5%, then the equivalent periodic interest rate with monthly compounding is given a follows:

Periodic interest rate = i = m x (e^{(r/m)}- 1) r = 5% m = 12 (monthly compounding) Periodic interest rate = i = 12 x (e^{(5%/12)}- 1) Periodic interest rate = i = 5.0104%

This means that continuous compounding at a rate of 5% is the same as periodic compounding at a rate of 5.0104% when compounded monthly.

To show that the two rates do in fact give the same result, suppose for example an amount of 5,000 is invested for 3 years and compounded continuously at a rate of 5%, then its future value is given as follows:

FV = PV x e^{rn}PV = 5,000 r = 5% n = 3 FV = 5000 x e^{(5% x 3)}FV = 5,809.17

If the same amount is invested with periodic compounding, then using our calculated rate above, the future value is given as follows:

FV = PV x (1 + i)^{n}PV = 5,000 i = 5.0104%/12 (monthly rate) n = 3 x 12 = 36 months FV = 5000 x (1 + 5.0104%/12)^{36}FV = 5,809.17

The future value is each case is the same.

## Example 2: Using the Continuous to Periodic Interest Rate Formula

If an amount is invested at an annual rate of 6% compounded continuously, then the equivalent periodic interest rate with quarterly compounding is given as follows:

Periodic interest rate = i = m x (e^{(r/m)}- 1) r = 6% m = 4 (quarterly compounding) Periodic interest rate = i = 4 x (e^{(6%/4)}- 1) Periodic interest rate = i = 6.0452%

This means that continuously compounding at a rate of 6% is the same as quarterly compounding at a periodic interest rate of 6.0452%.

## Example 3: Using the Continuous to Periodic Interest Rate Formula

If an amount is invested at an annual rate of 8% compounded continuously, then the equivalent periodic interest rate with yearly compounding is given as follows:

Periodic interest rate = i = m x (e^{(r/m)}- 1) r = 8% m = 1 (annual compounding) Periodic interest rate = i = 1 x (e^{(8%/1)}- 1) Periodic interest rate = i = 8.3287%

This means that continuously compounding at a rate of 8% is the same as annual compounding at a periodic interest rate of 8.3287%.

The continuous to periodic interest rate formula is one of many used in time value of money calculations, discover another at the links below.

## About the Author

Chartered accountant Michael Brown is the founder and CEO of Double Entry Bookkeeping. He has worked as an accountant and consultant for more than 25 years in all types of industries. He has been the CFO or controller of both small and medium sized companies and has run small businesses of his own. He has been a manager and an auditor with Deloitte, a big 4 accountancy firm, and holds a BSc from Loughborough University.