# Perpetuity

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1. Valuation.

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 = 5%

The present value of the fixed perpetuity is:

= \$10m x (1 / 0.05)

= \$10m x 20

= \$200m

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 = 5%.

Periodic growth rate = 2%

The present value of the growing perpetuity is:

= A1 x 1 / (r - g)

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

= \$10m x (1 / 0.03)

= \$10m x 33.3

= \$333m

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:

= A1 x 1 / (r - g)

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

= \$10m x (1 / 0.07)

= \$10m x 14.3

= \$143m

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.

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