# Continuous to Periodic Interest Rate Formula

## 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.