Formula and 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 ern 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.