What Is a Growing Perpetuity? Formula and Uses

A growing perpetuity represents a financial concept where a series of payments is expected to increase at a consistent rate for an indefinite period. This theoretical model provides a framework for evaluating investments that generate cash flows anticipated to grow over an extended horizon. Calculating the present value of such a stream helps in understanding what these future, growing payments are worth in today’s terms.

Understanding the Core Concepts

A growing perpetuity is built upon two fundamental ideas: “perpetuity” and “growth.” A perpetuity signifies a stream of payments that continues without a defined end date, essentially forever. This assumption allows for simplified financial calculations for long-lived assets or income streams.

The “growing” aspect means that these payments do not remain fixed; instead, they increase at a constant, predictable rate over time. For example, each subsequent payment might be a specific percentage larger than the previous one, reflecting factors like inflation or business expansion. This distinguishes it from a simple perpetuity, where payments are constant, and from a growing annuity, which involves payments that grow but only for a finite period.

Calculating the Present Value

Calculating the present value of a growing perpetuity determines the current worth of future cash flows that are expected to increase consistently. This valuation helps investors and analysts make informed decisions by translating future income into a single, current dollar amount. The formula for the present value (PV) of a growing perpetuity is: PV = C1 / (r – g).

In this formula, C1 represents the cash flow expected in the first period. The variable ‘r’ signifies the discount rate, which is the required rate of return or the cost of capital, reflecting the opportunity cost of investing and the risk associated with the cash flows. The ‘g’ stands for the constant growth rate at which the payments are expected to increase indefinitely. For instance, if the first payment is $100, the growth rate is 5%, and the discount rate is 10%, the calculation would be $100 / (0.10 – 0.05) = $2,000.

A crucial condition for this formula to yield a finite and meaningful value is that the discount rate (r) must always be greater than the growth rate (g). If the growth rate were equal to or exceeded the discount rate, the mathematical series would not converge, resulting in an infinite or undefined present value. This condition ensures that the value of future cash flows, when discounted, eventually approaches zero due to the time value of money, even with continuous growth.

Common Applications

The concept of a growing perpetuity provides a useful framework for valuing various financial instruments and assets, particularly those with long-term, increasing cash flows. One prominent application is in stock valuation, specifically through the Dividend Discount Model (DDM). For companies that pay dividends expected to grow consistently over time, such as mature and stable businesses, the growing perpetuity formula can estimate the intrinsic value of their stock. This model assumes that a stock’s value is the present value of all its future dividend payments.

The growing perpetuity model is also relevant in real estate valuation, especially for properties that generate rental income expected to increase perpetually. When assessing the value of commercial properties, analysts might use this approach to capitalize future growing rental streams into a current valuation. Although it is a theoretical model, the growing perpetuity provides a practical tool for financial analysts to assess long-term investments with recurring and growing cash flows.