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.
The growing perpetuity concept is applied in many contexts.
For example, the Dividend growth model for share valuation.
