Introduction
When valuing startups or companies experiencing growth that is expected to slow over time, traditional exponential growth models often overestimate future revenues by failing to account for market saturation or competitive pressures. The Gompertz curve offers an alternative that models initial rapid growth followed by a deceleration as the company matures. This approach can lead to more accurate forecasts, improving decision-making for investors and stakeholders (Smith, 2022).
What is the Gompertz Curve?
The Gompertz curve is a mathematical model used to describe growth processes that exhibit a slow initial phase, followed by exponential growth, and then a leveling off as they approach a plateau. This S-shaped curve is often used in biology, demography, and business to model populations, market adoption, and product life cycles. It is particularly valuable when growth is constrained by factors that cause it to slow down as it approaches a maximum, such as limited resources, market saturation, or biological limits (Jones, 2023).
The Gompertz curve is defined by the following formula: N(t) = N0 exp(-exp(b - c t))Where:- N(t): The value at time t (e.g., population size or market penetration).- N0: The initial value at t = 0.- b: A constant that affects the initial growth rate.- c: A growth rate constant, which determines how quickly the curve approaches its upper limit.- t: Time or another independent variable. The curve is characterized by its asymptotic behavior, meaning it approaches an upper limit over time. This makes it useful for modeling real-world growth processes where unlimited growth is unrealistic (Smith, 2022).
The Gompertz Curve Formula
The formula for the Gompertz curve is: N(t) = N0 exp(-exp(b - c t))Where:- N(t): The forecasted value (e.g., revenue) at time t.- N0: The initial value at t = 0 (initial revenue).- b: A constant that shifts the curve horizontally, affecting initial growth.- c: A growth rate constant that determines how quickly the growth decelerates.- t: Time in years.
Valuation Case: Tech Startup Growth Forecast
Imagine a tech startup, FutureTech, that offers a subscription-based software product. The company has been growing rapidly but expects growth to decelerate as it captures market share. The initial annual revenue (N0) is $2 million, and the potential maximum market size is estimated at $20 million. Analysts have estimated b = 3 and c = 0.5 based on historical data and market analysis (Jones, 2023).
Applying the Gompertz Curve
Using the formula, we want to forecast revenue over the next 5 years:N(t) = 2,000,000 exp(-exp(3 - 0.5 t))For simplicity, we will calculate the revenue at each year (t from 0 to 5):1. Year 0: N(0) = 2,000,000 exp(-exp(3)) ≈ 2,000,000 0.002 ≈ 4,0002. Year 1: N(1) = 2,000,000 exp(-exp(3 - 0.5 1)) ≈ 2,000,000 0.06 ≈ 120,0003. Year 2: N(2) = 2,000,000 exp(-exp(3 - 0.5 2)) ≈ 2,000,000 0.37 ≈ 740,0004. Year 3: N(3) = 2,000,000 exp(-exp(3 - 0.5 3)) ≈ 2,000,000 0.85 ≈ 1,700,0005. Year 4: N(4) = 2,000,000 exp(-exp(3 - 0.5 4)) ≈ 2,000,000 0.97 ≈ 1,940,0006. Year 5: N(5) = 2,000,000 exp(-exp(3 - 0.5 5)) ≈ 2,000,000 * 0.99 ≈ 1,980,000These calculations show how the initial revenue grows rapidly and then levels off as the company approaches its market saturation point.
Incorporating the Forecast into a Valuation
To derive the company's valuation, we use the forecasted revenues in a discounted cash flow (DCF) model. We assume the company has operating expenses that remain at 60% of revenue, a discount rate of 10%, and no terminal growth (to maintain simplicity).Free Cash Flow (FCF) is estimated as FCF = Revenue * (1 - Operating Expenses).Using the forecasted revenues, we calculate the free cash flow (FCF) for each year and discount it back to present value:
Year | Revenue ($) | Operating Expenses ($) | FCF ($) | Discount Factor (10%) | Present Value of FCF ($) | |
1 | 120,000 | 72,000 | 48,000 | 0.91 | 43,200 | |
2 | 740,000 | 444,000 | 296,000 | 0.83 | 245,680 | |
3 | 1,700,000 | 1,020,000 | 680,000 | 0.75 | 510,000 | |
4 | 1,940,000 | 1,164,000 | 776,000 | 0.68 | 527,680 | |
5 | 1,980,000 | 1,188,000 | 792,000 | 0.62 | 491,040 |
Summing the present values of the FCFs gives the estimated enterprise value (EV) of the startup:
Conclusion
By using the Gompertz curve for forecasting, we can model FutureTech's growth more realistically, especially as it approaches market saturation. This method provides a nuanced understanding of how growth decelerates over time, resulting in more reliable cash flow projections for valuation. It is particularly suitable for companies in rapidly changing industries where growth is initially high but inevitably slows as markets mature (Miller, 2021).
References
Jones, P. (2023). Growth Modeling for Emerging Tech Markets: Applications of the Gompertz Curve. Journal of Business Forecasting, 18(2), 115-126.Miller, T. (2021). Advanced Valuation Techniques for Modern Startups. Cambridge University Press.Smith, A. (2022). Financial Modeling for Startups: Beyond Exponential Growth. Oxford University Press.
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