## Formula

r = m x LN(1 + i / m)

**Variables used in the annuity formula**

r = Continuous interest rate

i = Periodic interest rate

m = Number of times compounding takes place in a period

LN = Natural logarithm

## Use

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

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

If an amount is invested at an annual rate of 5% compounded monthly, then the equivalent continuous interest rate is given as follows:

Continuous interest rate = r = m x LN(1 + i / m) i = 5% annual m = 12 (monthly compounding) Continuous interest rate = r = 12 x LN(1 + 5% / 12) Continuous interest rate = r = 4.9896%

This means that monthly compounding at a rate of 5% is the same as continuous compounding at a rate of 4.9896%.

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

FV = PV x (1 + i)^{n}PV = 5,000 i = 5%/12 (monthly rate) n = 3 x 12 = 36 months FV = 5000 x (1 + 5%/12)^{36}FV = 5,807.36

If the same amount is invested with continuous compounding, then using our calculated rate above, the future value is given by

FV = PV x e^{rn}PV = 5,000 r = 4.9896% n = 3 FV = 5000 x e^{(4.9896% x 3)}FV = 5,807.36

The future value is each case is the same.

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

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

Continuous interest rate = r = m x LN(1 + i / m) i = 6% annual m = 4 (quarterly compounding) Continuous interest rate = r = 4 x LN(1 + 6% / 4) Continuous interest rate = r = 5.9554%

This means that quarterly compounding at a rate of 6% is the same as continuous compounding at a rate of 5.9554%.

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

If an amount is invested at an annual rate of 8% compounded annually, then the equivalent continuous interest rate is given as follows:

Continuous interest rate = r = m x LN(1 + i / m) i = 8% annual m = 1 (annual compounding) Continuous interest rate = r = 1 x LN(1 + 8% / 1) Continuous interest rate = r = 7.6961%

This means that annual compounding at a rate of 8% is the same as continuous compounding at a rate of 7.6961%.

The periodic to continuous interest rate formula is one example of an annuity formula used in time value of money calculations, discover another at the link below.