Perpetual growth model: Meaning, Criticisms & Real-World Uses

What Is Perpetual Growth Model?

The perpetual growth model is a financial modeling technique used to estimate the value of a business, asset, or cash flow stream that is expected to grow at a constant rate indefinitely. This model is a cornerstone in the broader field of financial modeling and plays a significant role in valuation methodologies, particularly within corporate finance. The core premise of the perpetual growth model is that, after an initial period of explicit forecasts, a company or its cash flows will continue to generate value into perpetuity, growing at a stable, long-term rate. This ongoing, predictable growth forms the basis for calculating a crucial component of many valuation models: the terminal value. By projecting a constant growth rate, the perpetual growth model simplifies the complex task of valuing assets with extremely long or infinite lives, such as businesses.

History and Origin

The conceptual underpinnings of the perpetual growth model, particularly its application in equity valuation, are closely associated with mid-20th-century financial theory. A notable figure in this area is Myron J. Gordon, whose work significantly contributed to the development of dividend-based valuation. Gordon's influential book, The Investment, Financing, and Valuation of the Corporation, published in 1962, laid out theoretical frameworks that included the concept of dividends growing at a constant rate into perpetuity.⁴ This became foundational for what is widely known as the dividend discount model, where a perpetual growth assumption is often employed to value a stock based on its future dividends. The idea of projecting a consistent, albeit low, growth rate for a company's earnings or dividends beyond a finite forecast period has since become a standard practice in various valuation contexts.

Key Takeaways

The perpetual growth model estimates the value of cash flows or dividends that are assumed to grow at a constant rate indefinitely.
It is a critical component in discounted cash flow (DCF) analysis for calculating terminal value, which represents the value of a business beyond the explicit forecast period.
The model relies heavily on two key assumptions: a stable, sustainable growth rate and an appropriate discount rate, both of which significantly influence the resulting valuation.
While simplifying long-term projections, the model's sensitivity to its inputs means small changes can lead to large variations in value.

Formula and Calculation

The perpetual growth model is most commonly applied within the terminal value calculation of a discounted cash flow (DCF) model. The formula calculates the present value of a perpetuity that grows at a constant rate.

The formula for the terminal value using the perpetual growth model is:

TV = \frac{FCFF_{t+1}}{WACC - g}

Or, for the Gordon Growth Model application (for dividends):

P_0 = \frac{D_1}{r - g}

Where:

(TV) = Terminal Value
(FCFF_{t+1}) = Free cash flow to firm in the first year beyond the explicit forecast period
(WACC) = Weighted average cost of capital (the discount rate for FCFF)
(g) = Perpetual growth rate of cash flows
(P_0) = Current stock price
(D_1) = Expected dividend per share in the next period
(r) = Required rate of return on equity (cost of equity)

This formula effectively discounts a growing stream of future cash flows or dividends back to their present value at a specified discount rate.

Interpreting the Perpetual Growth Model

Interpreting the perpetual growth model involves understanding its output as a significant portion of an asset's or company's total estimated value. In a typical discounted cash flow analysis, the terminal value calculated using the perpetual growth model can represent anywhere from 50% to 80% of the total valuation. This highlights the model's critical role and the substantial impact of its underlying assumptions. A higher assumed perpetual growth rate, for instance, implies greater future value, while a higher cost of capital or discount rate reduces the present value of those future cash flows. Analysts must ensure that the perpetual growth rate used is realistic and sustainable for the long term, typically no more than the expected long-term nominal gross domestic product (GDP) growth rate of the economy in which the company operates. This is because no single company can grow faster than its economy indefinitely.

Hypothetical Example

Consider a company, "Tech Innovations Inc.," which analysts are valuing using a discounted cash flow model. After explicitly forecasting its free cash flow for the next five years, they need to estimate its terminal value.

Year 5 Free Cash Flow (last explicit forecast year): $100 million
Perpetual Growth Rate (g): 3% (reflecting long-term economic growth)
Weighted Average Cost of Capital (WACC): 10%

To calculate the free cash flow for Year 6 ((FCFF_{t+1})), which is the first year of the perpetuity, we apply the growth rate:
(FCFF_{Year\ 6} = FCFF_{Year\ 5} \times (1 + g) = $100\ million \times (1 + 0.03) = $103\ million)

Now, using the perpetual growth formula:

TV = \frac{\$103\ million}{0.10 - 0.03} = \frac{\$103\ million}{0.07} \approx \$1,471.43\ million

This calculated terminal value of approximately $1,471.43 million would then be discounted back to the present day using the WACC for the explicit forecast period, and added to the present value of the explicit cash flows to arrive at the total estimated value of Tech Innovations Inc. This example demonstrates how the perpetual growth model provides a significant estimate of long-term value.

Practical Applications

The perpetual growth model is widely applied across various domains in finance. Its primary use is in mergers and acquisitions (M&A) and corporate finance, where it forms a critical component of valuation methodologies like the Discounted Cash Flow (DCF) model. When a company is acquired, the acquirer needs to determine the target's intrinsic value, which often extends beyond a typical five-to-ten-year forecast period. The perpetual growth component of the terminal value captures the ongoing value generation of the target company. As outlined by Reuters, discounted cash flow analysis is a commonly used income-based valuation technique in M&A, where estimated future cash flows are brought to their present value using a discount rate.³

Beyond M&A, the model is used in equity valuation for publicly traded companies, helping analysts determine a stock's intrinsic value based on projected dividends or free cash flows. It is also relevant for investment decisions involving long-lived assets or projects, and in the academic study of asset pricing.

Limitations and Criticisms

Despite its widespread use, the perpetual growth model faces significant limitations and criticisms, primarily due to its reliance on highly sensitive assumptions. The most notable challenge lies in accurately determining the perpetual growth rate and the appropriate discount rate. Even minor adjustments to these inputs can lead to substantial differences in the calculated terminal value, which often accounts for a large portion of the overall valuation. For instance, a small increase in the growth rate or a small decrease in the discount rate can inflate the terminal value dramatically.

Critics often point out that assuming a constant, indefinite growth rate for any business is inherently unrealistic, especially given competitive forces and market dynamics. The CFA Institute notes that the terminal value can account for up to 80% of a total valuation, resting on assumptions that a company will "survive and thrive for decades."² This highlights the "terminal value trap," where the bulk of the valuation is attributed to a period characterized by highly uncertain future assumptions. Furthermore, setting the perpetual growth rate higher than the long-term nominal GDP growth rate or the long-term inflation rate is generally considered unsustainable and a common error in DCF models. The Federal Reserve Bank of New York has also discussed the challenges in accurately estimating components like the equity risk premium, a key part of the discount rate, noting that historical averages may be misleading due to time-variation.¹ Analysts must exercise considerable judgment and conduct thorough sensitivity analysis when applying the perpetual growth model to mitigate these inherent subjective biases.

Perpetual Growth Model vs. Gordon Growth Model

While often used interchangeably or in similar contexts, the "perpetual growth model" is a broader concept, whereas the Gordon Growth Model (GGM) is a specific application. The perpetual growth model refers to the mathematical formula that calculates the present value of a stream of cash flows (or other financial metrics) that are assumed to grow at a constant rate into perpetuity. It is a fundamental component of various valuation methods, most notably in calculating the terminal value within a discounted cash flow (DCF) analysis, where the cash flows in question are typically free cash flows or earnings.

The Gordon Growth Model, however, specifically applies this perpetual growth formula to the valuation of a company's stock based on its future dividends. It assumes that dividends will grow at a constant rate indefinitely. Thus, while the GGM uses the principle of perpetual growth, it does so within the specific context of valuing equities via their dividend streams, making it a specialized form or direct application of the broader perpetual growth concept.

FAQs

What is the primary purpose of the perpetual growth model?

The primary purpose of the perpetual growth model is to estimate the terminal value of a business or asset in a valuation model, representing the value generated beyond an explicit forecast period. It simplifies the estimation of long-term cash flows that are assumed to grow at a constant, sustainable rate.

Can the perpetual growth rate be higher than the economy's growth rate?

No, in most realistic scenarios, the perpetual growth rate should not exceed the long-term nominal growth rate of the economy (e.g., nominal GDP growth). A company cannot realistically outgrow its entire market or economy indefinitely. Setting an unrealistically high growth rate is a common pitfall in valuation.

What inputs are required for the perpetual growth model?

The main inputs required are the cash flow (or dividend) in the first year of the perpetuity, the perpetual growth rate, and a discount rate (such as the weighted average cost of capital for free cash flows or the cost of equity for dividends).

Why is the perpetual growth model so sensitive to its inputs?

The model is highly sensitive to its inputs because the denominator involves the difference between the discount rate and the growth rate. A small change in either of these values, especially if they are close to each other, can drastically alter the resulting terminal value, impacting the overall present value calculation.