Advances in Applied Probability

Modeling growth stocks via birth-death processes

S. C. Kou and S. G. Kou

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The inability to predict the future growth rates and earnings of growth stocks (such as biotechnology and internet stocks) leads to the high volatility of share prices and difficulty in applying the traditional valuation methods. This paper attempts to demonstrate that the high volatility of share prices can nevertheless be used in building a model that leads to a particular cross-sectional size distribution. The model focuses on both transient and steady-state behavior of the market capitalization of the stock, which in turn is modeled as a birth-death process. Numerical illustrations of the cross-sectional size distribution are also presented.

Article information

Adv. in Appl. Probab. Volume 35, Number 3 (2003), 641-664.

First available in Project Euclid: 29 July 2003

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60H30: Applications of stochastic analysis (to PDE, etc.) 91B70: Stochastic models

Biotechnology and internet stocks asset pricing convergence rate volatility power-type distribution Zipf's law Pareto distribution regression


Kou, S. C.; Kou, S. G. Modeling growth stocks via birth-death processes. Adv. in Appl. Probab. 35 (2003), no. 3, 641--664. doi:10.1239/aap/1059486822.

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  • Abramowitz, M. and Stegun, I. A. (eds) (1972). Handbook of Mathematical Functions. National Bureau of Standards, Washington, DC.
  • Adler, R. J., Feldman, R. E. and Taqqu, M. S. (eds) (1998). A Practical Guide to Heavy Tails. Birkhäuser, Boston, MA.
  • Axtell, R. L. (2001). Zipf distribution of U.S. firm sizes. Science 293, 1818--1820.
  • Chen, W. C. (1980). On the weak form of Zipf's law. J. Appl. Prob. 17, 611--622.
  • Drees, H., de Haan, L. and Resnick, S. (2000). How to make a Hill plot. Ann. Statist. 28, 254--274.
  • Embrechts, P., Klüppelberg, C. and Mikosch, T. (1997). Modelling Extremal Events. Springer, Berlin.
  • Erdélyi, A. et al. (1953). High Transcendental Functions (Bateman Manuscript Project), Vol. 1. McGraw-Hill, New York.
  • Feenberg, D. R. and Poterba, J. M. (1993). Income inequality and the incomes of very high-income taxpayers: evidence from tax returns. In Tax Policy and the Economy, Vol. 7, ed. J. M. Poterba, MIT Press, Cambridge, MA, pp. 145--177.
  • Gabaix, X. (1999). Zipf's law for cities: an explanation. Quart. J. Econom. 154, 739--767.
  • Gibrat, R. (1931). Les Inégalités Économiques. Recueil Sirey, Paris.
  • Glaeser, E., Scheinkman, J. and Shleifer, A. (1995). Economic growth in a cross-section of cities. J. Monetary Econom. 36, 117--143.
  • Heyde, C. C. and Kou, S. G. (2002). On the controversy over tailweight of distributions. Preprint, Columbia University.
  • Hill, B. M. (1975). A simple general approach to inference about the tail of a distribution. Ann. Statist. 3, 1163--1174.
  • Ijiri, Y. and Simon, H. A. (1977). Skew Distributions and the Sizes of Business Firms. North-Holland, Amsterdam.
  • Karlin, S. and McGregor, J. (1958). Linear growth, birth and death processes. J. Math. Mech. 7, 643--662.
  • Karlin, S. and Taylor, H. M. (1975). A First Course in Stochastic Processes, 2nd edn. Academic Press, New York.
  • Kerins, F., Smith, J. K. and Smith, R. (2001). New venture opportunity cost of capital and financial contracting. Preprint, Washington State University.
  • Kijima, M. (1997). Markov Processes for Stochastic Modeling. Chapman and Hall, London.
  • Kou, S. C. and Kou, S. G. (2002). A tale of two growths: modeling stochastic endogenous growth and growth stocks. Preprint, Harvard University and Columbia University.
  • Krugman, P. (1996a). Confronting the urban mystery. J. Japanese Internat. Econom. 10, 399--418.
  • Krugman, P. (1996b). The Self-Organizing Economy. Blackwell, Cambridge, MA.
  • Lo, G. S. (1986). Asymptotic behavior of Hill's estimator and applications. J. Appl. Prob. 23, 922--936.
  • Lucas, R. (1978). On the size distribution of business firms. Bell J. Econom. 9, 508--523.
  • Mandelbrot, B. B. (1960). The Pareto--Lévy law and the distribution of income. Internat. Econom. Rev. 1, 79--106.
  • Mandelbrot, B. B. (1997). Fractals and Scaling in Finance. Springer, New York.
  • Mauboussin, M. J. and Schay, A. (2000). Still powerful: the internet's hidden order. Equity Res. Rep., Credit Suisse First Boston Corporation.
  • Meyn, S. P. and Tweedie, R. (1993). Markov Chains and Stochastic Stability. Springer, London.
  • Pareto, V. (1896). Cours d'Économie Politique. Rouge, Lausanne.
  • Resnick, S. (1997). Heavy tail modelling and teletraffic data. Ann. Statist. 25, 1805--1869.
  • Rutherford, R. (1955). Income distributions: a new model. Econometrica 23, 425--440.
  • Shorrocks, A. F. (1975). On stochastic models of size distributions. Rev. Econom. Studies 42, 631--641.
  • Simon, H. A. (1955). On a class of skew distribution functions. Biometrika 52, 425--440.
  • Simon, H. A. and Bonini, C. P. (1958). The size distribution of business firms. Amer. Econom. Rev. 48, 607--617.
  • Steindl, J. (1965). Random Processes and the Growth of Firms. Hafner, New York.
  • Woodroofe, M. and Hill, B. M. (1975). On Zipf's law. J. Appl. Prob. 12, 425--434.
  • Yule, G. U. (1924). A mathematical theory of evolution, based on the conclusions of Dr. J. R. Willis. Phil. Trans. R. Soc. London B 213, 21--83.
  • Yule, G. U. (1944). The Statistical Study of Literary Vocabulary. Cambridge University Press.
  • Zipf, G. (1949). Human Behavior and the Principle of Least Effort. Addison-Wesley, Cambridge, MA.