The distribution of the returns on investment depends on the rules in the economic system. The article reviews various return distributions, ranging from equity securities in equilibrium, to antiques bought at auction, to debt instruments with uncertain payouts. A general methodology is provided to construct distributions of returns.
Markowitz, H. (1952) Portfolio Selection. The Journal of Finance, 7, 77-91.
Mandelbrot, B. (1963) The Variation of Certain Speculative Prices. The Journal of Business, 36, 394-419. https://doi.org/10.1086/294632
Fama, E.F. (1963) Mandelbrot and the Stable Paretian Hypothesis. Journal of Business, 36, 420-429. https://doi.org/10.1086/294633
Fama, E. (1965) The Behavior of Stock Market Prices. Journal of Business, 38, 34-105. https://doi.org/10.1086/294743
Fama, E.F. and Roll, R. (1968) Some Properties of Symmetric Stable Distributions. Journal of the American Statistical Association, 63, 817-836.
Fama, E.F. and Roll, R. (1971) Parameter Estimates for Symmetric Stable Distributions. Journal of the American Statistical Association, 66, 331-338. https://doi.org/10.1080/01621459.1971.10482264
Fama, E.F. and MacBeth, J.D. (1973) Risk, Return, and Equilibrium: Empirical Tests. The Journal of Political Economy, 81, 607-636. https://doi.org/10.1086/260061
Fama, E.F. and French, K.R. (2008) Dissecting Anomalies. The Journal of Finance, 63, 1653-1678. https://doi.org/10.1111/j.1540-6261.2008.01371.x
Yilmaz, B.Z. (2010) Completion, Pricing and Calibration in a Levy Market Model. Master’s Thesis, The Institute of Applied Mathematics of Middle East Technical University, Ankara.
Fama, E.F. and French, K.R. (1992) The Cross-Section of Expected Stock. The Journal of Finance, 47, 427-465. https://doi.org/10.1111/j.1540-6261.1992.tb04398.x
Harris, D. (2015) A Population Test of Distribution Assumptions in Mean-Variance Models for the Years 1925-2013. Working Paper at SSRN, Amsterdam.
Harris, D. (2016) Why Practitioners Should Use Bayesian Statistics. Working Paper at SSRN, Amsterdam.
Parmigiani, G. and Inoue, L. (2009) Decision Theory: Principles and Approaches. Wiley Series in Probability and Statistics, Chichester, 155-171. https://doi.org/10.1002/9780470746684
Jaynes, E.T. (2003) Probability Theory: The Language of Science. Cambridge University Press, Cambridge, 205-207. https://doi.org/10.1017/CBO9780511790423
Stigler, S.M. (1974) Studies in the History of Probability and Statistics. Xxxiii: Cauchy and the Witch of Agnesi: An Historical Note on the Cauchy Distribution. Biometrika, 61, 375-380. https://doi.org/10.1093/biomet/61.2.375
Koopman, B.O. (1936) On Distributions Admitting a Sufficient Statistic. Transactions of the American Mathematical Society, 39, 399-409. https://doi.org/10.1090/S0002-9947-1936-1501854-3
Rothenberg, T.J., Fisher, F.M. and Tilanus, C.B. (1964) A Note on Estimation from a Cauchy Sample. Journal of the American Statistical Association, 59, 460-463. https://doi.org/10.1080/01621459.1964.10482170
Bayes, T. (1764) An Essay towards solving a Problem in the Doctrine of Chances. Philosophical Transactions, 53, 370-418. https://doi.org/10.1098/rstl.1763.0053
Savage, L.J. (1954) The Foundations of Statistics. John Wiley & Sons, New York.
Cox, R.T. (1961) The Algebra of Probable Inference. Johns Hopkins University Press, Baltimore.
Markowitz, H. and Usmen, N. (1996) The Likelihood of Various Stock Market Return Distributions, Part 1: Principles of Inference. Journal of Risk and Uncertainty, 13, 207-219. https://doi.org/10.1007/BF00056153
Curtiss, J.H. (1941) On the Distribution of the Quotient of Two Chance Variables. Annals of Mathematical Statistics, 12, 409-421. https://doi.org/10.1214/aoms/1177731679
Gurland, J. (1948) Inversion Formulae for the Distribution of Ratios. The Annals of Mathematical Statistics, 19, 228-237. https://doi.org/10.1214/aoms/1177730247
Marsaglia, G. (1965) Ratios of Normal Variables and Ratios of Sums of Uniform Variables. Journal of the American Statistical Association, 60, 193-204. https://doi.org/10.1080/01621459.1965.10480783
Marsaglia, G. (2006) Ratios of Normal Variables. Journal of Statistical Software, 16, 1-10. https://doi.org/10.18637/jss.v016.i04
Mann, H. and Wald, A. (1943) On the Statistical Treatment of Linear Stochastic Difference Equations. Econometrica, 11, 173-200. https://doi.org/10.2307/1905674
White, J.S. (1958) The Limiting Distribution of the Serial Correlation Coefficient in the Explosive Case. The Annals of Mathematical Statistics, 29, 1188-1197. https://doi.org/10.1214/aoms/1177706450
Rao, M.M. (1961) Consistency and Limit Distributions of Estimators of Parameters in Explosive Stochastic Difference Equations. The Annals of Mathematical Statistics, 32, 195-218. https://doi.org/10.1214/aoms/1177705151
NIST (2012) Nist/Sematech E-Handbook of Statistical Methods. http://www.itl.nist.gov/div898/handbook/
Beckmann, P. (1964) Rayleigh Distribution and Its Generalizations. Radio Science Journal of Research, 68D, 92-932. https://doi.org/10.6028/jres.068D.092
Gull, S.F. (1988) Bayesian Inductive Inference and Maximum Entropy. Kluwer Academic Publishers, Berlin. https://doi.org/10.1007/978-94-009-3049-0_4
McCullaugh, P. (1992) Conditional Inference and Cauchy Models. Biometrika, 79, 547-559.
Burdzy, K. (1984) Excursions of Complex Brownian Motion. University of California, Berkeley.
Berger, J.O. (1980) Statistical Decision Theory, Foundations, Concepts, and Methods. Springer Series in Statistics, New York. https://doi.org/10.1007/978-1-4757-1727-3
Graham, B. (1973) The Intelligent Investor: A Book of Practical Counsel. 4th Edition, Harper and Row, New York, 18-22.
Landon, V.D. (1941) The Distribution of Amplitude with Time in Fluctuation Noise. Proceedings IRE, 29, 50-54. https://doi.org/10.1109/JRPROC.1941.233980
Dickey, D.A. and Fuller, W.A. (1979) Distribution of the Estimators for Autoregressive Time Series with a Unit Root. Journal of the American Statistical Association, 74, 427-431.
Cramer, H. (1946) Mathematical Methods in Statistics. Princeton University Press, Princeton.
Ding, P. (2014) Three Occurrences of the Hyperbolic-Secant Distribution. The American Statistician, 68, 32-35. https://doi.org/10.1080/00031305.2013.867902
Richardson, C.J. (2005) How the Loss of Property Rights Caused Zimbabwe’s Collapse. Cato Institute Economic Development Bulletin, 4, 1-4.
Abbott, A. (2009) Valuation Handbook: Measures of Discount for Lack of Marketability and Liquidity. Wiley Finance, Hoboken, 474-507.