What is Perpetuity and how to calculate its Present Value?

Perpetuity is an annuity that lasts forever. It consists of several cash flows in a series where the period between two payments is equal. Moreover, in the case of perpetuity, the amount paid after each period remains the same. The periodic cash flows occurring in perpetuity are of utmost importance because they provide the structure of the perpetuity.

Perpetuities are very common in finance. For example, if a government creates a fund for scholarships to girls paying INR 1 crore every quarter, it is perpetuity. Here the payments are fixed and will be made forever after a period of three months.

Irredeemable preference shares are also perpetuity. In this case, a constant sum of money is paid to the shareholders at equal intervals. Many types of investments and liabilities are also covered by the term perpetuity. House rentals that are paid forever could also be termed as perpetuities.

Note − Perpetuities are continuing annuities where the investments after regular intervals remain unchanged. Perpetuities are quite common in government initiatives where the perpetuity is formed for a defined requirement.

How to Calculate the Present Value of a Perpetuity?

For the scholarship example above, let the endowment value be PV, the annual scholarship withdrawals be PMT and i being the periodic interest rate. If the perpetuity continues, then we can write,

PMT = PV × i

Rearranging the above equation,

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

Now, if the rate of return is 8%, we can find the scholarship value each year.

$$PV of Perpetuity=\frac{10000000}{8%}=1250000$$

Growing Perpetuity

In some circumstances, the perpetuity may grow at a certain rate. The present value of perpetuity that grows is given by,

$$PV of Perpetuity=\frac{PMT}{(i-g)}$$

where g is the growth rate.

Now, if perpetuity grows at a rate of 4%,

$$PV of Perpetuity=\frac{PMT}{(i-g)}=\frac{10000000}{(8-4)}=2500000$$

Note − The PV of perpetuity changes when the rate of interest changes. The growth rate is usually lower than the original rate of return.