# 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 = A1 x 1/r

where:

A1 = 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 = 10%

The present value of the fixed perpetuity is:

= \$10m x (1 / 0.10)

= \$10m x 10

= \$100m

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 = A1 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 = 10%.

Periodic growth rate = 2%

The present value of the growing perpetuity is:

= A1 x 1 / (r - g)

= \$10m x (1 / (0.10 - 0.02) )

= \$10m x (1 / 0.08)

= \$10m x 12.5

= \$125m

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 = 10%.

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

The present value of the declining perpetuity is:

= A1 x 1 / (r - g)

= \$10m x (1 / (0.10 - -0.02) )

= \$10m x (1 / 0.12)

= \$10m x 8.33

= \$83.3m

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

The growing perpetuity concept is applied in many contexts.

For example, the Dividend growth model for share valuation.