# Continuous Compounding

Normally when computing compound interest the compounding period is a discrete interval, annually, quarterly, monthly, weekly etc. There is nothing however to stop the compounding period getting smaller and smaller until eventually interest is calculated on the balance of the principal amount plus accumulated interest on a continuous basis.

Each discrete time value of money formula has a corresponding continuous compounding formula which has been derived from it, the most important of these are summarized below:

## Present Value Continuous Compounding (PV)

The present value continuous compounding formula is shown below:

`PV = FV / ein`
Variables used in the formula
PV = Present Value
FV = Future Value
i = Discount rate
n = Number of periods
e = Base of the natural logarithm (LN) ≈ 2.71828

### Present Value Continuously Compounded Example

If an amount of 3,000 is received at the end of year 4 and the discount rate is 6% compounded continuously, then the present value is calculated as follows:

```Present value continuous compounding = FV / ein
Present value continuous compounding = 3,000 / e(6% x 4)
Present value continuous compounding = 2,359.88
```

## Future Value Continuous Compounding (FV)

The future value continuous compounding formula is shown below:

`FV = PV x ein`
Variables used in the formula
PV = Present Value
FV = Future Value
i = Discount rate
n = Number of periods
e = Base of the natural logarithm (LN) ≈ 2.71828

### Future Value Continuously Compounded Example

If an amount of 2,000 is compounded continuously at a discount rate of 4%, then the future value at the end of year 7 is calculated as follows:

```Future value continuous compounding = PV x ein
Future value continuous compounding = 2,000 x e(4% x 7)
Future value continuous compounding =  2,646.26
```

## Effective Annual Rate (EAR) Continuous Compounding

The effective annual rate continuous compounding formula is shown below:

`EAR = ein - 1`
Variables used in the formula
i = Discount rate
n = Number of periods
e = Base of the natural logarithm (LN) ≈ 2.71828

### EAR Continuously Compounded Example

If an amount is compounded continuously at a rate of 8%, then the effective annual rate (EAR) is calculated as follows:

```Effective annual rate = ein - 1
Effective annual rate = e(8% x 1) - 1
Effective annual rate = 8.329%
```

## Length of Time (n) Continuous Compounding

The formula for calculating the time period for an amount PV to compound to an amount FV at a discount rate of i compounded continuously, is calculated as follows:

`n = LN(FV / PV) / i`
Variables used in the formula
PV = Present Value
FV = Future Value
i = Discount rate
n = Number of periods
e = Base of the natural logarithm (LN) ≈ 2.71828

### Length of Time Continuously Compounded Example

The length of time it takes an amount of 4,000 to grow to an amount of 9,000 when compounded continuously at an annual discount rate of 3% is calculated as follows:

```n = LN(FV / PV) / i
n = LN(9000 / 4000) / 3%
n = 27.03 years
```

## Discount Rate (i) Continuous Compounding

The discount rate required to grow an amount PV to an amount FV in a time period n is given by the following formula.

`i = LN(FV / PV) / n`
Variables used in the formula
PV = Present Value
FV = Future Value
i = Discount rate
n = Number of periods
e = Base of the natural logarithm (LN) ≈ 2.71828

### Discount Rate Continuously Compounded Example

The discount rate needed to grow an amount of 7,000 to an amount of 12,000 when compounded continuously for a period of 10 years is calculated as follows:

```i = LN(FV / PV) / n
i = LN(12000 / 9000) / 10
i = 2.877%
```

## Continuous Compounding Formulas Cheat Sheet 