# Understanding Perpetuity in Finance with Formulas and Examples

AD

Summary:

Perpetuity is a unique type of annuity that offers investors a source of identical cash flows forever. Although the concept may seem old-fashioned, it remains essential in finance, helping investors understand the worth of an investment in perpetuity. The perpetuity formula divides cash flows by a discount rate, while the present value formula estimates the value of future cash flows in today’s money. The terminal value of a company is calculated using a more complex formula than the basic perpetuity formula. Growing perpetuities offer an income stream that adjusts with inflation. Investors should bear in mind that perpetuities may not always be the most practical investment option due to their infinite duration.

## What are perpetuities in finance?

If you’re looking for a never-ending source of income, you might be interested in perpetuities. These securities provide a constant stream of cash flows that never cease, making them a unique and intriguing investment opportunity.

But perpetuities aren’t just interesting because of their infinite lifespan; they also play a critical role in financial theories such as the dividend discount model (DDM), which values a company’s stock based on its future dividend payments. So whether you’re an investor or just curious about finance, it’s worth exploring the fascinating concept of perpetuity.

## A quick overview of perpetuities

Annuities are streams of cash flows. A perpetuity is a special type of annuity that lasts indefinitely, with an unending flow of identical cash payments. Understanding how to calculate the present value of a perpetuity is crucial in finance, as it helps investors determine the worth of an investment in perpetuity.

While perpetuities may seem like a thing of the past, they remain a critical concept in financial theory. For instance, the Bank of England issued consols, a type of bond with perpetual cash flows that lasted until their phase-out in 2015. When you invested in a consol, you were guaranteed annual interest payments for life.

Although it may sound strange, even an infinite series of cash flows can have a finite present value. This is due to the time value of money, where each payment is only worth a fraction of the previous one. So, the next time you hear the term perpetuity in finance, you know it’s not just an imaginary concept, but one with real-world applications.

The perpetuity formula calculates the amount of cash flows in the terminal year of operation. During the valuation, companies are considered to be a going concern, representing that it goes on forever. As such, the terminal year is considered a perpetuity, and analysts will utilize the perpetuity formula to calculate its value.

## The perpetuity present value formula

Let’s dive into the formula for calculating the present value of a perpetuity or security with perpetual cash flows:

PV = C / (1+r)^1 + C / (1+r)^2 + C / (1+r)^3 ⋯ = C / r

where:

where:

- PV = present value
- C = cash flow
- r = discount rate

The method used to calculate the perpetuity divides cash flows by a given discount rate. But how do you calculate the terminal value – the estimated cash flows beyond a specific period?

The terminal value is calculated using a slightly more complex formula than the basic perpetuity formula. To estimate the cash flows in year 10 of the company, multiply it by one plus the long-term growth rate, and then divide it by the difference between the cost of capital and the growth rate.

Essentially, the terminal value is the future cash flows of a company beyond a specified time period divided by a discount rate.

## Looking at a perpetuity example

Let’s look at an example of perpetuity. Say a company is projected to earn $100,000 in year 10, and the company’s cost of capital is at 8%, with a long-term growth rate of 3%, in this case, the calculation of the perpetuity would be like so:

(Cash FlowYear 10×(1 + g)) / (r – g)

($100,000 x 1.03) / (0.08 – 0.03)

($100,000 x 1.03) / (0.08 – 0.03)

($103,000) / (0.05) = $2.06 million

This shows that with $100,000 paid into a perpetuity, assuming a 3% growth rate and an 8% capital cost, would be worth $2.06 million in 10 years. But what if you want to know the value of that money today? That’s where the present value of a perpetuity formula comes in.

This shows that with $100,000 paid into a perpetuity, assuming a 3% growth rate and an 8% capital cost, would be worth $2.06 million in 10 years. But what if you want to know the value of that money today? That’s where the present value of a perpetuity formula comes in.

## Adjusting for inflation and growing perpetuities

A perpetuity’s net present value is not as large as it might seem because the time value of money lowers the value of dollars far into the future due to various factors like inflation. As such, the cash flows that are received by a fixed perpetuity many years from now may become insignificant in future buying power.

Growing perpetuities offer an income stream that adjusts with inflation to ensure that your buying power doesn’t decrease over time. The present value of a growing perpetuity will be higher than that of a fixed perpetuity because it factors in inflation. The higher the rate of growth, the more valuable the payments will be in today’s terms.

The formula for calculating the present value of a growing perpetuity is similar to the formula for a fixed perpetuity, but with an adjustment for inflation. To calculate it, subtract the inflation rate (also known as the growth rate) from the discount rate in the denominator:

PV = C / (r-g)

It’s important to note that the growth rate of a growing perpetuity remains constant throughout its infinite duration, which means it can only provide a general approximation of the average inflation over the long term.

## How do perpetuities work in securities?

Perpetuities are financial products that promise a never-ending stream of cash flows, has been gaining attention among investors. One notable example of a perpetuity was the U.K. government-issued bond called “consol,” which used to provide an infinite series of interest payments until it was phased out in 2015. Unlike traditional bonds that have a fixed maturity date, perpetuities do not have a specified end date, making them a unique investment option.

## How valuable are perpetuities

Although a financial instrument that promises an endless stream of cash flows might appear to be valuable, it is actually not quite the case. Mathematically, the value of a perpetuity is finite and can be calculated by discounting its future cash flows to present value using a specific discount rate. This process is called discounted cash flow (DCF) analysis, which is a widely used method for valuing various types of securities like bonds, stocks, and real estate investments.

## How does a perpetuity compare to an annuity?

While both perpetuities and annuities provide a fixed stream of cash flows, there is a significant difference between them. An annuity has a specific end date, known as the maturity date, after which it no longer provides cash flows. Perpetuities offer cash flows without end. Despite this difference, both annuities and perpetuities can be evaluated using discounted cash flow (DCF) analysis.

## How long can a perpetuity last for?

Perpetuities are true to their name and last in perpetuity, or in other words, forever.

Perpetuities are the unicorns of the investment world, offering payments that continue indefinitely without any set end date. They provide a never-ending stream of cash flow. While perpetuities as financial products are quite rare these days, the concept of a perpetuity and the calculation of its present value remain an essential part of finance. To calculate the present value of a perpetuity, simply divide the cash flow amount by the discount rate. This fundamental concept is a cornerstone of financial analysis and is used in many investment valuation models.

## What is “in perpetuity” in the context of real estate?

“In perpetuity” is a legal term used in real estate to indicate that an agreement, restriction, or condition applies indefinitely, without any end date or limit. This term is often used in the context of easements, covenants, and conservation agreements, ensuring that specific rights or restrictions are maintained for the entire life of a property, regardless of changes in ownership. Here are some examples.

### Easements

An easement is a legal right to use someone else’s land for a specific purpose, such as access to a shared driveway or a public walkway. If an easement is granted “in perpetuity,” it means that the easement continues to exist indefinitely, regardless of any changes in property ownership. For example, if you grant your neighbor an easement to use a pathway on your property to reach a nearby lake, the easement could be specified “in perpetuity,” ensuring that future owners of both properties will still have the same rights and obligations related to the easement.

### Covenants

Covenants are agreements between property owners that restrict or control the use of their land, often to maintain the character of a neighborhood or protect property values. A covenant “in perpetuity” means that the restrictions or conditions outlined in the covenant will apply indefinitely. For example, a homeowner’s association (HOA) might create a covenant that restricts the color of houses in a neighborhood to maintain a consistent appearance. If the covenant is in perpetuity, it will continue to apply to all current and future property owners in the neighborhood, ensuring that the color restrictions remain in place indefinitely.

### Conservation Easements

A conservation easement is a voluntary agreement between a landowner and a land trust or government agency that restricts the development or use of a property to protect its natural resources, habitat, or historic value. When a conservation easement is established “in perpetuity,” it means that the land will be protected from development indefinitely, regardless of changes in ownership. For example, a landowner might grant a conservation easement to a land trust to preserve a wetland habitat on their property. The easement would restrict any future development or land use that could harm the wetland and would apply “in perpetuity” to ensure the wetland remains protected for generations to come.

## Key takeaways

- A perpetuity is essentially an annuity that never ends.
- The present value of a perpetuity can be calculated by dividing the regular cash flows by the discount rate.
- If the cash flows increase each period, it’s called a growing perpetuity.
- Although perpetuities are rare in modern finance, understanding the concept is still important in various financial theories, such as the dividend discount model.

AD

Allan Du is a personal finance writer passionate about helping people take control of their finances. Allan strives to present readers with the right knowledge and tools, so they can make informed decisions about their money and build wealth. When he is not writing about finance, Allan enjoys pursuing his other interests, including powerlifting, kickboxing, and investing. He is an active follower of economic and political trends, always keeping watch on the latest developments that could impact the financial world.