# Perpetuity

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Perpetuity can be well defined as an annuity without any end, or it can be said that perpetuity features a stream of cash payments continuing forever. To describe in detail, perpetuity is an annuity wherein the periodic payments commence on a specific date and continue to an indefinite time. Sometimes, it is even referred to as perpetual annuity. Some of the prime examples perpetuities include fixed coupon payments on permanently invested money, or consol (the British issued bond). The concept of perpetuity is, very often, used in financial theory, like the Dividend Discount Model (DDM).

Calculation (formula) of perpetuity

The value of perpetuity or a perpetual annuity is calculated by a simple formula: where,

• PV represents the present value of the perpetuity,
• A represents the amount of periodic payments, and
• r represents the discount rate, yield, or interest rate.

Besides, the present value of perpetuity can also be determined by the following steps:

• Find out the annual payment, rate of interest, and growth rate of the perpetuity.
• Put the apt numbers into the formula:

Present Value of a growing perpetuity = P / (i – g),

Where ‘P’ represents the annual payment, ‘i’ represents the interest or discount rate, and “g” is the growth rate.

You can apply the following example as a point of reference so as to crosscheck the calculations. Let us assume that a XYZ company pays a dividend of \$100 per annum forever at an interest rate of 5% and a growth rate of 1%. To estimate the present value of this perpetuity, you can apply the following formula:

PV of preferred stock in XYZ = \$100 / (\$0.05-0.01)

PV of preferred stock in XYZ = \$100 / 0.04

PV of preferred stock in XYZ = \$2,500

Therefore, the present value of a share of XYZ’s preferred stock is expected to be \$2,500.

Although the payments made on perpetuity are assumed to be indefinite, there is still a finite value for the perpetuity’s present value which is accrued to the expected returns in future featuring a low present value.

The value of perpetuity can, however, change over the time period, in spite of same amount of payments. This might be due to changes in discount rate. The value of perpetuity increases with a decrease in the discount rate and vice versa.