Present Value of a Perpetuity

by Obaidullah Jan, ACA, CFA

Perpetuity is a perpetual annuity, it is a series of equal infinite cash flows that occur at the end of each period and there is equal interval of time between cash flows. Present value of a perpetuity equals the periodic cash flow divided by the interest rate.

Let’s say a government wants to set up an endowment that will off $1 million each year in scholarship for ever. This constitutes a perpetuity because the payment is fixed, there is equal duration between each payment, i.e. one year and there are infinite number of payments.

Some models treat different investments and liabilities are perpetuities. The dividend discount model values a share of common stock by treating it as a perpetuity of constant dividend payments. A real estate investment can be treated as a perpetuity of rentals.

Formula

Let’s follow the endowment example above. Let the endowment value be PV, the annual scholarship withdrawals be PMT and i being the periodic interest rate. If we want the endowment to finance scholarships each year perpetually, the interest earned on PV in one year must equal PMT. This can be expressed mathematically as follows:

$$ PMT=PV\times i $$

Rearranging the above equation, we get the formula to find present value of a perpetuity:

$$ PV=\frac{PMT}{i} $$

Growing Perpetuity

There might be a situation in which the payments comprising the perpetuity might grow at a rate g. The present value of a growing perpetuity can be calculated as follows:

$$ PV=\frac{PMT}{i-g} $$

The above equation is the basis of the Gordon Growth Model.

Example

Following the endowment example above, if the rate of return is 8%, we can find out the endowment value that can support $1 million payments each year:

$$ PV=\frac{$1,000,000}{8\%}=$12,500,000 $$

If the scholarship requirements grow at 4%, the endowment initial funding requirement increases:

$$ PV=\frac{$1,000,000}{8\%-4\%}=$25,000,000 $$