# Periodic to Continuous Interest Rate Formula

## 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 ern
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.

Periodic to Continuous Interest Rate Formula November 6th, 2016