# Perpetuity

1.

A series of cash flows modelled to carry on for an infinite amount of time in the future.

2. **Fixed perpetuity**

A fixed perpetuity is a periodic cash flow starting one period in the future, then carrying on for ever thereafter.

Each cash flow is an equal fixed amount.

The present value of a fixed perpetuity is calculated - assuming a constant periodic cost of capital (r) for all periods from now to infinity - as:

Present Value = A_{1} x 1/r

where:

A_{1} = Time 1 cash flow

r = periodic cost of capital

**Example 1: Fixed perpetuity valuation**

Time 1 cash flow = $10m, continuing at the same amount each period thereafter in perpetuity.

Periodic cost of capital = 5%

The present value of the fixed perpetuity is:

= $10m x (1 / 0.05)

= $10m x 20

= $**200**m

3. **Growing perpetuity**

A growing perpetuity is an infinite series of cash flows, modelled to grow by a constant proportionate amount every period.

For a growing perpetuity, the present value formula is modified to take account of the constant periodic growth rate, as follows:

Present Value = A_{1} x 1 / (r - g)

where g = the periodic rate of growth of the cash flow.

**Example 2: Growing perpetuity valuation**

Time 1 cash flow = $10m, growing by a constant percentage amount each period thereafter in perpetuity.

Periodic cost of capital = 5%.

Periodic growth rate = 2%

The present value of the growing perpetuity is:

= A_{1} x 1 / (r - g)

= $10m x (1 / (0.05 - 0.02) )

= $10m x (1 / 0.03)

= $10m x 33.3

= $**333**m

The modest rate of growth in the cash flow has added substantially to the total present value.

4. **Declining perpetuity**

Growth can be negative, in other words, decline.

For a declining perpetuity, the present value formula is the same as the growing perpetuity, but the growth rate (g) is entered as a negative number as follows:

**Example 3: Declining perpetuity valuation**

Time 1 cash flow = $10m, declining by a constant percentage amount each period thereafter in perpetuity.

Periodic cost of capital = 5%.

Periodic growth rate = -(2)% negative = -0.02

The present value of the declining perpetuity is:

= A_{1} x 1 / (r - g)

= $10m x (1 / (0.05 - -0.02) )

= $10m x (1 / 0.07)

= $10m x 14.3

= $**143**m

The small negative rate of growth in the cash flow has reduced the total present value very substantially.

The growing / declining perpetuity concept is applied in many contexts.

For example, the Dividend growth model for share valuation.

## See also

- Annuity
- Dividend growth model
- Growing annuity
- Growing perpetuity
- Irredeemable
- Perpetuity due
- Perpetuity factor
- Simple annuity

## Student article

*Real rates of corporate decline often lead to miscalculation, overpaying for acquisitions and disastrous losses.*

*Read this article to discover how to avoid the most common errors and add value for your organisation.*