### What Is Perpetuity?

A Perpetuity is an endless consistent series of payments made at equal intervals of time. Said differently, a Perpetuity, or a perpetual annuity, is an infinite stream of cash flow payments.

### How to Calculate the Present Value of Perpetuity?

There are two basic methods used to calculate the Present Value of Perpetuity.

Method 1. Present Value (PV) of Perpetuity is calculated by dividing the Amount of the consistent payment by discount or interest rate.

$$PV = \frac{A}{r}$$

Where PV = Present Value of the Perpetuity, A = the Amount of the consistent payment, and r = the yield, discount or interest rate.

The total value of the Perpetuity is infinite since the payments are made endlessly, however the Present Value of the Perpetuity is finite because the farther in the future the payment is, the lower its present value.

This happens due to Time Value of Money, according to which, the same amount of money available at the present time is worth more than the same amount in the future.

Method 2. Since a Perpetuity is a form of Annuity, it could be also calculated as a Present Value of the future periodic payments. The Present Value of the Perpetuity can be written as:

$$PV = \frac{ A }{ 1 + r } + \frac{ A }{ (1 + r)^{2 } } + \frac{ A }{( 1 + r)^{3} } + ... + \frac{ A }{( 1 + r)^{n } }$$

As before, PV = Present Value of the Perpetuity, A = the Amount of the consistent payment, and r = the yield, discount or interest rate. In this formula, n = the number of periods.

This Present Value of Perpetuity formula can be simplified to the following:

$$PV = \displaystyle\sum_{n=1}^{∞} \frac{A}{(1+r)^{n}}$$

### Examples of a Present Value of Perpetuity Calculation

Example 1. An investor plans to invest in shares of a company. The dividends are $5 annually and will be paid indefinitely. The required rate of return is 7%. $$PV = \frac{\5}{7\%} = \71.43$$ That means the investment will be profitable if the investor pays$71.43 or less.

Example 2. A bond pays coupon payments of $15 annually and continues for an infinite amount of time. The discount rate is 5%. Present Value of Perpetuity is calculated as: $$PV = \frac{\15}{5\%} = \300$$ #### What is Growing Perpetuity? A Growing Perpetuity is similar to an ordinary Perpetuity. It’s also an endless consistent series of payments made at equal intervals of time, but the payments grow at proportional rates. #### How to Calculate the Present Value of Growing Perpetuity? For one period of time the formula of Present Value of Growing Perpetuity is calculated by dividing the Amount of the consistent payment by the difference between the discount (or interest) rate and the growth rate. $$PV\space of\space Growing\space Perpetuity = \frac{A_{1}}{r - G }$$ Where A1 = Amount of the consistent payment, r = yield, discount rate or interest rate, and G = growth rate. For this formula it’s important to notice that Discount/Interest rate must be always greater than the Growth rate. Otherwise, the Growing Perpetuity would have an infinite value. ### Example of a Present Value of Growing Perpetuity Calculation An investor plans an investment where the cash flow payments will be$5,000 per year. The required rate of return is 10%. The cash flow payments are expected to grow by 3% every year. Cash flow payments will be paid indefinitely.

$$PV\space of\space Growing\space Perpetuity = \frac{\5,000}{10\% - 3\% } = \71,428.57$$

Which means the investment will be profitable if the investor pays \$71,428.57 or less.

### How Perpetuity Calculation Is Used in Real Life

Despite the fact that it might seem that Perpetuity is not used in real life, there are some real-life forms of investment, that are based on Perpetuity.

• Real estate. Let’s assume an investor buys a flat and lets it. The flat is now their property and they also receive an infinite stream of rental payments.
• Common stocks. In theory, a company has an infinite life. Therefore, when investor buys a company’s stocks, they’re entitled to receive dividends endlessly.
• Consols (Consolidated Stocks). Consols were certain UK and USA types of perpetual redeemable at the option of the government bonds with fixed interest payments. First Consols were issued in 1750s by the Bank of England and were redeemed only in 2015.