When Growth Meets the Cost of Capital: Terminal Value under Measured Business Survival
Year: 2026
Type: preprint
Abstract: The constant-growth perpetuity fails when nominal growth reaches the discount rate: the factor 1/(r − g) becomes infinite at g = r and undefined beyond it. This paper shows that the failure is an artifact of the implicit assumption that the firm never ceases to exist. When each cash flow is weighted by the probability that the firm survives to generate it, the terminal-value factor under a constant exit hazard h becomes, to first order, 1/(r − g + h). The singularity moves from g = r to g ≈ r + h, a boundary that can be measured. Under an extrapolated Weibull curve the domain of convergence depends on the shape parameter; under the tail convention used in practice, a hazard frozen at the last observed age, it depends only on that terminal hazard. The framework is tested on two public registries. In nineteen US sectors (BLS Business Employment Dynamics, establishment survival, 1994 cohort followed for 31 years), every sector shows decreasing hazard (k between 0.60 and 0.74); the tail hazard projected from a curve fitted on ages 0–20 differs from the observed hazard at ages 21–31 by 0.62 percentage points on average, a broadly consistent but imprecise projection; and a constant hazard estimated over the full observed lifetime overstates the tail hazard by a factor of 1.5 to 2, depending on the estimator. In Japan (1,397,101 corporations incorporated after October 2015, National Tax Agency registry), the observed legal-exit hazard is approximately flat from the first year onward (k = 1.03), so that for legal exit the exponential approximation appears reasonable without a tail convention; the evidence does not establish a constant hazard for economic exit. Comparing five survival specifications on the same data, the paper finds that only a constant hazard or a hazard frozen at the tail gives a value at g ≥ r that is finite and stable: the pure Weibull form either keeps the singularity at g = r (k < 1) or converges to values without economic meaning (k > 1), and in Japan a change in the truncation of the first year of data switches between the two. The paper closes with a four-step procedure for valuers who meet a conditional nominal growth rate equal to or above the discount rate.
Author: Freitas, Ruben Xavier de
Language: Portuguese
Subfield: Strategy and Management
Domain: Social Sciences
Open Access status: green
OpenAlex ID: W7214438697