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Proof That Properly Discounted Present Values of Assets Vibrate Randomly

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  • Paul A. Samuelson

Abstract

Even the best investors seem to find it hard to do better than the comprehensive common-stock averages, or better on the average than random selection among stocks of comparable variability. Examination of historical samples of percentage changes in a stock's price show that, when these relative price changes are properly adjusted for expected dividends paid out, they are more or less indistinguishable from white noise, or, at the least, their expected percentage movements constitute a driftless random walk (or random walk with mean drift specifiable in terms of an interest factor appropriate to the stock's variability or riskiness). The present contribution shows that such observable patterns can be deduced rigorously from a model which hypothesizes that a stock's present price is set at the expected discounted value of its future dividends, where the future dividends are supposed to be random variables generated according to any general (but known) stochastic process. This fundamental theorem follows by an easy superposition applied to the 1965 Samuelson theorem that properly anticipated futures prices fluctuate randomly -- i.e., constitute a martingale sequence, or a generalized martingale with specifiable mean drift. Examples demonstrate that even when the economy is not free to wander randomly, intelligent speculation is able to whiten the spectrum of observed stock-price changes. A subset of investors might have better information or modes of analysis and get above average gains in the random-walk model; and the model's underlying probabilities could be shaped by fundamentalists' economic forces.

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Bibliographic Info

Article provided by The RAND Corporation in its journal Bell Journal of Economics.

Volume (Year): 4 (1973)
Issue (Month): 2 (Autumn)
Pages: 369-374

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Handle: RePEc:rje:bellje:v:4:y:1973:i:autumn:p:369-374

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Cited by:
  1. Andrea GAMBA & Nicola FUSARI, 2008. "Valuing modularity as a real option," Swiss Finance Institute Research Paper Series 08-20, Swiss Finance Institute.
  2. Wachter, Jessica A. & Warusawitharana, Missaka, 2009. "Predictable returns and asset allocation: Should a skeptical investor time the market?," Journal of Econometrics, Elsevier, vol. 148(2), pages 162-178, February.
  3. Guglielmo D’Amico, 2013. "A semi-Markov approach to the stock valuation problem," Annals of Finance, Springer, vol. 9(4), pages 589-610, November.
  4. Keller, Andreas, 2010. "Competition effects of mergers: An event study of the German electricity market," Energy Policy, Elsevier, vol. 38(9), pages 5264-5271, September.
  5. Brandouy, Olivier & Delahaye, Jean-Paul & Ma, Lin & Zenil, Hector, 2014. "Algorithmic complexity of financial motions," Research in International Business and Finance, Elsevier, vol. 30(C), pages 336-347.
  6. Lucy Ackert & William Hunter, 2001. "An Empirical Examination of the Price-Dividend Relation with Dividend Management," Journal of Financial Services Research, Springer, vol. 19(2), pages 115-129, April.
  7. Ehnts, Dirk & Carrión Álvarez, Miguel, 2013. "The theory of reflexivity: A non-stochastic randomness theory for business schools only?," IPE Working Papers 28/2013, Berlin School of Economics and Law, Institute for International Political Economy (IPE).
  8. D. Sornette, 2014. "Physics and Financial Economics (1776-2014): Puzzles, Ising and Agent-Based models," Papers 1404.0243, arXiv.org.
  9. Merton, Robert C., 1977. "On the microeconomic theory of investment under uncertainty," Working papers 958-77., Massachusetts Institute of Technology (MIT), Sloan School of Management.
  10. Juho Kanniainen, 2009. "Can properly discounted projects follow geometric Brownian motion?," Computational Statistics, Springer, vol. 70(3), pages 435-450, December.
  11. P. Van Cauwenberge & I. De Beelde, 2006. "Does The Comprehensive Income Matrix Make Unsophisticated Users Overemphasise Fair Value Income?," Working Papers of Faculty of Economics and Business Administration, Ghent University, Belgium 06/400, Ghent University, Faculty of Economics and Business Administration.
  12. Rita De Siano, 2000. "Financial Variables As Leading Indicators: An Application To The G7 Countries," Working Papers 6_2000, D.E.S. (Department of Economic Studies), University of Naples "Parthenope", Italy.
  13. Cherubini, Umberto & Mulinacci, Sabrina & Romagnoli, Silvia, 2011. "A copula-based model of speculative price dynamics in discrete time," Journal of Multivariate Analysis, Elsevier, vol. 102(6), pages 1047-1063, July.

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