Hostname: page-component-7dd5485656-7jgsp Total loading time: 0 Render date: 2025-10-31T02:36:01.361Z Has data issue: false hasContentIssue false
  • English
  • Français

Modeling growth stocks via birth-death processes

Part of: Mathematical economics Stochastic analysis

Published online by Cambridge University Press:  01 July 2016

S. C. Kou  and
S. G. Kou
S. C. Kou*
Affiliation:
Harvard University
S. G. Kou*
Affiliation:
Columbia University
*
∗ Postal address: Department of Statistics, Science Center 603, Harvard University, Cambridge, MA 02138, USA.
∗∗ Postal address: Department of IEOR, 312 Mudd Building, Columbia University, New York, NY 10027, USA. Email address: sk75@columbia.edu
Get access Rights & Permissions [Opens in a new window]

Abstract

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.

Keywords

Biotechnology and internet stocksasset pricingconvergence ratevolatilitypower-type distributionZipf's lawPareto distributionregression

MSC classification

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

Information

Type
General Applied Probability
Information
Advances in Applied Probability , Volume 35 , Issue 3 , September 2003 , pp. 641 - 664
Copyright
Copyright © Applied Probability Trust 2003 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable

References

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