## 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.