# Perpetuities

## An asset providing a fixed cash flow

A *perpetuity* is an asset that provides a never ending periodic fixed amount of cash flow.

Endowments and trust funds and long lived assets like oil wells or gas fields can be regarded as a perpetuity.

The value of a perpetuity can be calculated as

P = F_{1}/ (1 + i)^{1}+F+_{2}/ (1 + i)^{2}F...... +_{3}/ (1 + i)^{3}+F_{n}/ (1 + i)^{n}

= F / i (1)

where

P = present value

F = F_{1}= F_{2}.. F_{n}= the fixed amount of cash flow per period

i = discount rate of return (per time period)

### Example - The Value of a Company and Net Income

A company is able to deliver a net income every year of *100*. Assuming that this is a perpetuity - a never ending income - the value of this cash flow (and the value of the company) with a discount rate of *10%* *(i = 0.10)* can be calculated to

*P = (100) / 0.10 *

* = 1000*

### Growing Perpetuity

If a cash flow grows in a constant rate the value of the perpetuity can be expressed as

P = F / (i - g) (2)

where

g = growth rate of cash flow (per time period)

### Example - The Value of a Growth Company and Net Income

A growth company is able to deliver a net income first year of *200* with a net income growth rate the next years of *3% (g = 0.03)*. Assuming that this is a perpetuity - a never ending income - the value of this cash flow (and the value of the company) with a discount rate of *10%* *(i = 0.10)* can be calculated to

*P = (100) / (0.10 - 0.03) *

* = 1429*

### Perpetuity Value Calculator

* F - amount *

* i - discount rate (%) *

* g - growth rate (%) *