Growth perpetuity model

What Is the Growth Perpetuity Model?

The growth perpetuity model is a financial valuation tool used to estimate the present value of a series of future cash flows that are expected to grow at a constant rate indefinitely. It is a fundamental concept within the broader field of [Equity Valuation] and financial modeling, particularly in the context of [Discounted Cash Flow] (DCF) analysis. This model, often referred to as the Gordon Growth Model (GGM) when applied to dividends, assumes that an asset will generate a stream of payments that continues forever, with each payment increasing by a consistent [Constant Growth Rate] over time⁵⁵, ⁵⁶. Analysts utilize the growth perpetuity model to calculate the [Terminal Value] of a business or investment beyond a specific forecast period, representing the value of all cash flows into the perpetual future⁵⁴.

History and Origin

The conceptual underpinnings of the growth perpetuity model trace back to fundamental ideas of present value and the time value of money. While the idea of discounting future cash flows to determine a [Present Value] has long existed, the specific formulation involving a perpetual growth rate gained prominence with the work of economists like Myron J. Gordon. In 1962, Gordon published his definitive work, "The Investment, Financing, and Valuation of the Corporation," which solidified the application of a constant dividend growth rate in valuation. This approach built upon earlier concepts, such as those introduced by John Burr Williams in his 1938 book "The Theory of Investment Value," which laid the groundwork for the [Dividend Discount Model] (DDM). Today, the model remains a core component in [Valuation] methodologies, especially for mature companies with stable, predictable cash flows.

Key Takeaways

• The growth perpetuity model calculates the present value of an infinite series of cash flows that grow at a constant rate.
• It is a core component of [Discounted Cash Flow] valuation, primarily used to estimate the [Terminal Value] of an asset or company.
• Key inputs include the cash flow in the next period, the [Discount Rate] (or [Required Rate of Return]), and the perpetual growth rate.
• A critical assumption is that the discount rate must always exceed the growth rate for a meaningful and finite valuation.
• The model is highly sensitive to changes in both the growth rate and the discount rate assumptions.

Formula and Calculation

The formula for calculating the present value of a growth perpetuity model is:

Where:

• (PV) = Present Value of the growing perpetuity
• (CF_1) = The expected cash flow in the first period (e.g., year 1)⁵², ⁵³
• (r) = The [Discount Rate] or [Required Rate of Return] (often the [Cost of Equity] or Weighted Average Cost of Capital, WACC)⁵⁰, ⁵¹
• (g) = The [Constant Growth Rate] of the cash flow per period⁴⁸, ⁴⁹

For the formula to yield a sensible result, the discount rate ((r)) must be greater than the growth rate ((g))⁴⁷. If (g) equals or exceeds (r), the denominator becomes zero or negative, leading to an infinite or undefined present value, which is not practical in financial analysis⁴⁵, ⁴⁶.

Interpreting the Growth Perpetuity Model

Interpreting the result of the growth perpetuity model involves understanding its representation of an asset's value based on its long-term earnings potential. The calculated present value represents what an investor would theoretically pay today for a stream of cash flows expected to grow indefinitely at a constant rate. A higher calculated value suggests a more attractive investment, assuming the underlying assumptions are realistic. For example, in valuing a company, if the present value derived from the growth perpetuity model (often as part of [Terminal Value] in a larger DCF analysis) is significant, it indicates that a substantial portion of the company's total [Valuation] relies on its ability to generate growing [Cash Flow] well into the future. It is crucial to evaluate the assumptions used, especially the growth rate and the [Discount Rate], as small variations can lead to large differences in the final valuation⁴⁴.

Hypothetical Example

Consider a hypothetical scholarship fund that aims to provide annual scholarships forever, with the amount increasing each year to keep pace with rising education costs.

• Current scholarship awarded (CF₀) = $10,000
• Expected annual growth rate of the scholarship (g) = 3%
• Required rate of return on the fund's investments (r) = 7%

First, calculate the scholarship amount for the next period (CF₁):
(CF_1 = CF_0 \times (1 + g) = $10,000 \times (1 + 0.03) = $10,300)

Next, apply the growth perpetuity model formula:

This calculation suggests that an initial investment of $257,500 would be required to fund a scholarship that starts at $10,300 next year and grows at 3% indefinitely, given a 7% [Required Rate of Return]. This example illustrates how the model can quantify the [Present Value] of a perpetually growing income stream.

Practical Applications

The growth perpetuity model is widely applied in various areas of finance and [Equity Analysis], predominantly as a component of larger [Discounted Cash Flow] valuation models.

*⁴², ⁴³ Corporate Valuation: It is commonly used to calculate the [Terminal Value] of a company in a DCF analysis. After explicitly forecasting a company's [Cash Flow] for a short-to-medium period (e.g., 5-10 years), the growth perpetuity model estimates the value of all cash flows beyond that explicit forecast horizon. An⁴¹alysts often assume a stable, sustainable growth rate for this perpetual phase, usually aligned with long-term economic growth rates.

• Dividend Discount Models: The model forms the basis of the Gordon Growth Model (GGM), a popular [Dividend Discount Model] used to value dividend-paying stocks. In this context, (CF_1) represents the expected dividend per share in the next period. In³⁹, ⁴⁰vestors and analysts use this to assess the intrinsic value of a stock.
• ³⁷, ³⁸ Real Estate Valuation: For properties with stable, growing rental income that is expected to continue indefinitely (e.g., certain commercial leases or perpetual land leases), the model can provide a valuation estimate for the [Future Value] of those rental streams.
• Investment Planning: It can be applied in financial planning to estimate the capital needed to generate a perpetually growing income stream, such as for endowment funds or philanthropic endeavors.
• Regulatory Compliance: When companies prepare financial projections for regulatory filings or disclosures, especially in mergers and acquisitions, valuation models including components like the growth perpetuity model are often employed. The U.S. Securities and Exchange Commission (SEC) provides guidance on the use of projections in SEC filings, emphasizing that management must have a reasonable basis for its assessment of future performance, which would extend to the assumptions underpinning a growth perpetuity calculation.

T³⁴, ³⁵, ³⁶he model helps financial professionals determine whether a security is undervalued or overvalued by comparing its calculated intrinsic value to its current market price. Th³³e model can also be used to estimate how analysts value stocks in general.

#³¹, ³²# Limitations and Criticisms

While powerful, the growth perpetuity model is subject to several significant limitations and criticisms that can impact the reliability of its results.

• Assumption of Infinite Constant Growth: The most substantial critique is the assumption that cash flows will grow at a [Constant Growth Rate] forever. In²⁹, ³⁰ reality, very few, if any, companies or assets can sustain a perpetual, steady growth rate. Bu²⁸sinesses mature, industries change, and economic cycles fluctuate, making such a long-term forecast highly uncertain. An²⁷ unrealistically high growth rate can lead to an inflated [Terminal Value] and overvaluation of a business.
• ²⁶ Sensitivity to Inputs: The model is highly sensitive to small changes in its input variables, particularly the perpetual growth rate ((g)) and the [Discount Rate] ((r)). A ²⁴, ²⁵minor adjustment of even a fraction of a percentage point in either variable can result in a dramatically different present value. Th²³is sensitivity introduces considerable subjectivity into the valuation process.
• ²¹, ²² Required Rate of Return Exceeds Growth Rate: The mathematical requirement that the [Required Rate of Return] must be greater than the growth rate ((r > g)) limits the model's applicability. It cannot be used for high-growth companies where the growth rate might temporarily exceed the discount rate, nor for companies with zero or negative growth that fall outside its specific assumptions. Fo¹⁸, ¹⁹, ²⁰r such cases, multi-stage [Discounted Cash Flow] models, which incorporate varying growth phases, are generally more appropriate.
• ¹⁶, ¹⁷ Dependency on Predictable Cash Flows: The model performs best for mature, stable companies with predictable [Cash Flow] streams, such as utilities or well-established businesses. It¹⁴, ¹⁵ is less suitable for younger, rapidly evolving companies, or those with volatile earnings, where estimating a stable perpetual growth rate is particularly challenging. Ac¹³ademic research on valuation models often highlights these practical challenges in applying such models, especially in dynamic market conditions.

#¹²# Growth Perpetuity Model vs. Simple Perpetuity

The key distinction between the growth perpetuity model and a [Simple Perpetuity] lies in the behavior of the cash flows over time.

¹¹The simple [Perpetuity] is a special case of the growth perpetuity model where the growth rate ((g)) is assumed to be zero. The inclusion of a positive growth rate in the growth perpetuity model reflects the expectation that the stream of payments will increase over time, leading to a higher present value compared to an identical stream with no growth.

#⁹, ¹⁰# FAQs

What is the primary purpose of the growth perpetuity model?

The primary purpose of the growth perpetuity model is to estimate the [Present Value] of a stream of cash flows that are expected to grow at a [Constant Growth Rate] forever. It is frequently used in [Valuation] to determine the [Terminal Value] of a company or investment, representing its worth beyond a detailed forecast period.

#⁸## Can the growth rate be higher than the discount rate?
No, for the growth perpetuity model to yield a finite and sensible result, the [Discount Rate] ((r)) must be strictly greater than the growth rate ((g)). If the growth rate equals or exceeds the discount rate, the formula's denominator becomes zero or negative, leading to an infinite or undefined valuation, which is not financially logical.

#⁶, ⁷## Is the growth perpetuity model the same as the Gordon Growth Model?
The growth perpetuity model is the underlying mathematical framework that the Gordon Growth Model (GGM) utilizes. The GGM specifically applies this model to dividend payments to determine a stock's intrinsic value, where [Cash Flow] represents expected future dividends per share. Th⁴, ⁵erefore, while very similar in application and formula, the GGM is a specific application of the broader growth perpetuity model.

When is the growth perpetuity model most appropriate to use?

The growth perpetuity model is most appropriate for valuing mature companies or assets that are expected to generate stable and predictable cash flows that grow at a steady, sustainable rate for the foreseeable future. It³ is particularly useful for businesses in stable industries, such as utilities, or as the [Terminal Value] component in a [Discounted Cash Flow] analysis for companies that have reached a steady state of growth.¹, ²