Growing Perpetuities: Perpetual Income Streams That Increase Over Time

Imagine you are promised a payment, not just once, but forever. That’s the basic idea of a perpetuity: a stream of payments that goes on indefinitely. Think of it like winning a lottery where you receive a fixed amount of money every year, year after year, for as long as you live, and even beyond that – theoretically forever.

Now, what if this annual payment wasn’t fixed? What if it increased over time? That’s where the concept of a growing perpetuity comes in. A growing perpetuity is still a stream of payments that continues forever, but with a twist: each payment grows at a constant rate.

Let’s use a relatable example. Imagine you inherit a special type of rental property. Instead of the rent staying the same each year, the rental agreement is structured so that the rent automatically increases by, say, 2% every year to keep pace with inflation and potentially the increasing value of the property. This growing rental income, if it’s designed to continue indefinitely, acts like a growing perpetuity.

Think about another scenario. Consider a company that pays dividends to its shareholders. A regular perpetuity would be like receiving the same dividend amount from that company year after year. However, in reality, successful companies often aim to increase their dividends over time as their profits grow. If a company commits to increasing its dividend payments by a certain percentage each year, and is expected to continue doing so forever, this stream of growing dividend payments resembles a growing perpetuity for the shareholder.

The key difference between a regular perpetuity and a growing perpetuity is this growth factor. In a regular perpetuity, the payments are constant. In a growing perpetuity, the payments increase at a consistent rate. This growth rate is crucial. It’s usually expressed as a percentage, like 1%, 2%, or 3% per year. This growth rate is applied to each subsequent payment. So, if your first payment is one hundred dollars and the growth rate is 3%, the next payment would be one hundred and three dollars, the payment after that would be one hundred and three dollars plus 3% of one hundred and three dollars, and so on, forever.

Why is understanding growing perpetuities important? It’s particularly useful in finance and investment analysis. When we want to determine the present value of an asset that is expected to generate income that grows consistently forever, we use the concept of a growing perpetuity. For example, when evaluating a stock, some analysts might assume that the company’s dividends will grow at a constant rate indefinitely. In such cases, the growing perpetuity model can be used to estimate the intrinsic value of that stock.

It’s important to remember that the idea of payments growing forever at a constant rate is a theoretical concept. In the real world, nothing grows forever at a steady pace. Companies and economies experience cycles of growth and contraction. However, the growing perpetuity model is a valuable simplification that allows us to understand and analyze long-term income streams that are expected to increase over time. It provides a framework for valuation and decision-making, especially when dealing with assets or projects with a very long or indefinite lifespan and an anticipated pattern of consistent growth.

So, in essence, a growing perpetuity is a financial concept that describes a stream of payments that never ends and increases by the same percentage each period. It’s a powerful tool for understanding and valuing assets that are expected to provide ever-increasing income for the foreseeable future, even though in practice, perfect and constant growth forever is more of an ideal than a certainty.