A perpetuity is a constant and equal cash flow continuing infinitely. Deferred perpetuity is the same cash flow but starts at a deferred time.

Theoretically, the value of perpetuity continues infinitely. However, the present value of the future cash flows arising from perpetuity or deferred perpetuity can be discounted.

Let us discuss what is a perpetuity and deferred perpetuity.

## What is a Perpetuity?

In finance, perpetuity refers to a system of cash flows for an infinite time. It is a stream of cash flows with no end.

Often the term is used for annual payments associated with financial securities such as bonds, stocks, and real estate income.

The concept of perpetuity is used in business valuation models as well as discounted cash flow analysis commonly. Theoretically, the cash flows continue infinitely, however, in practice, it is rarely the case.

## Formula

The formula to calculate perpetuity can be written in different ways.

The present value of the perpetuity can be calculated by:

PV= C/(1+r)^{1}+ C/(1+r)^{2}+ C/(1+r)^{3}…… = C/r

Where C = Cash flow and r = discount rate

In the basic form, the present value of a perpetuity is the sum of all future cash flows. It can be discounted using a reasonable discount rate used by an entity.

Similarly, an entity can calculate the present value of a perpetuity that grows at a certain rate and starts after a certain period.

## Perpetuity with Growth

The formula for the PV of perpetuity with a growth rate is:

Value of Perpetuity = C_{n}× (1+g)/(r-g)

Where Cn is the cash flow in year n, r is the discount rate, and g is the growth rate of perpetuity. The value of perpetuity will be then discounted for the PV using the PV factor for year n.

## How Does a Perpetuity Work?

The total theoretical value of a perpetuity is infinite. However, the present value of a perpetuity is finite. The PV of future cash flows arising annually (annuity) is calculated by adding all discounted annuities and decreasing the discounted annuity period until it reaches close to zero.

In finance, the perpetuity formula can be used to calculate the PV of future cash flows arising infinitely. Similarly, it can be used in business valuations with expected future cash flows for an infinite period.

Perpetuity is any cash flow that arises constantly for an infinite period. Common examples of such cash flows include businesses, real estate income, stocks, bonds, and so on.

A real-life example of a perpetuity is income from UK bonds called consoles. These bonds are issued by the UK government and investors receive coupon payments infinitely.

In a nutshell, perpetuity is a constant stream of equal cash flows with no end. Simply, it is an annuity with infinite life.

## What is a Deferred Perpetuity?

A deferred or delayed perpetuity is a constant stream of cash flows starting at a predetermined future date with infinite life.

Deferred perpetuities work similarly to normal perpetuities. The only difference arises from the future cash flows originating at a specific date rather than starting immediately.

In finance, deferred perpetuity represents a more realistic cash flow stream. For instance, dividend stocks often start paying dividends at a certain future date or a specified period when the business accumulates sufficient profits.

The concept of deferred perpetuity is similar to a deferred annuity as well. It is a constant cash stream arising annually with a finite life. The only difference for delayed perpetuity would be the infinite life of the cash flow stream.

## Formula

The formula for perpetuity calculates the present value of cash flows starting in one year. The formula for deferred perpetuity starting after a specific period “n” can be calculated as:

PV = (Annual cash flow/discount rate) x discount factor for the year before the perpetuity starts

**Important:**

The calculation of deferred perpetuity will be done in two steps. The PV starting in year “n” will then be discounted again for time zero.

## How Does a Deferred Perpetuity Work?

Deferred perpetuity is perpetuity starting with a delayed stream of cash flows rather than immediately. A normal perpetuity cash flow is discounted for one year.

A common example of deferred perpetuity is the cash flow arising from a real estate project when it is rented out after completion. As long as the project functions successfully, the cash flows arising can be accounted for infinitely as perpetuities. However, since these cash flows start after a specified interval, they should ideally be accounted for as deferred perpetuities.

Similarly, growth stocks or dividend stocks represent deferred perpetuities. Most companies announce dividends for future years under certain conditions. Often dividend stocks start paying dividends after a specified period and continue infinitely.

The present value concept of discounting the infinite cash flows to an infinite present value works the same way as with the normal perpetuity or an annuity calculation.

## Working Example of a Perpetuity

Suppose a company ABC pays a $ 5 dividend to its shareholders infinitely. We assume that the current rate of return used by ABC company is 6%.

We can calculate the present value of this perpetuity using the formula:

PV of Perpetuity = C/r

PV of Perpetuity = $ 5/6% = $ 83.33

It means if the ABC company’s stock is worth $ 83.33 today, the investors would invest in it.

Now suppose, ABC company further estimates a 2% growth rate in its dividend amount. The present value of growth perpetuity can be calculated using the formula:

PV of Perpetuity = C/r-g

PV of Perpetuity = $ 5/ (6-2) % = $ 125

It means the investors would be willing to invest in ABC company’s stock at a price of $ 125 or less if it has a growth rate of 2% for its dividends.

## Perpetuity Vs. Deferred Perpetuity – Key Differences

Both types represent an infinite stream of cash flows. The basic difference is when the cash flow starts at a constant rate.

Perpetuity starts immediately. It means the first cash flow can be an advance yearly payment or a payment at the end of each year infinitely.

On the other hand, deferred perpetuity starts a stream of cash flows after a specified interval. For instance, a dividend starts after five years of inception for a new business.

Both types of cash flows can be discounted to their present values. Both can include a growth factor starting after one year or at a delayed date in the future.

## Perpetuity Vs. Annuity – Key Differences

An annuity is a constant cash flow arising annually. A perpetuity is the same as well. The only difference between the two types of cash flow streams is that an annuity comes with a known or finite life.

An annuity is a constant stream of cash flow that is often in equal amounts. However, an annuity may come with a growth rate as well. It means an annuity may grow at a specific rate yearly.

Inflation-adjusted financial securities are good examples of annuities with growth factors.

Similarly, perpetuity comes with equal and constant cash flows but with infinite life. Perpetuity can also have a growth rate.

Both types of cash flows can be discounted for the present value factors.