American Journal of Theoretical and Applied Statistics

| Peer-Reviewed |

Linear Scale Dilation of Asset Returns

Received: 06 March 2013    Accepted:     Published: 02 April 2013
Views:       Downloads:

Share This Article

Abstract

Comparing the order statistics of daily returns of the S&P 500 index from 03.01.1950 to 04.03.2013 with the corresponding rankits, a linear scale dilation is observed. This observation is used to derive a five-parameter density function for the parsimonious description of the unconditional distribution of stock returns. The typical graph of this density function looks like a wizard's hat. Its signature feature is the discontinuity at zero.

DOI 10.11648/j.ajtas.20130202.15
Published in American Journal of Theoretical and Applied Statistics (Volume 2, Issue 2, March 2013)
Page(s) 38-41
Creative Commons

This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.

Copyright

Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Discontinuity, Rankits, Stock Returns, Unconditional Distribution

References
[1] A. Azzalini, A class of distributions which includes the normal ones, Scandinavian Journal of Statistics 12, pp. 171-178, 1985.
[2] A. Azzalini, Further results on a class of distributions which includes the normal ones, Statistica 46, pp. 199-208, 1986.
[3] A. Azzalini, The skew-normal distribution and related multivariate families, Scandinavian Journal of Statistics 32, pp. 159-188, 2005.
[4] F. M. Aparicio and J. Estrada, Empirical distributions of stock returns: European securities markets, 1990-95, The European Journal of Finance 7, pp. 1-21, 2001.
[5] T. Bollerslev, Generalized autoregressive conditional heteroskedasticity, Journal of Econometrics 31, pp. 307-327, 1986.
[6] R. F. Engle, Autoregressive conditional heteroscedasticity with estimates of the variance of United Kingdom inflation, Econometrica 50, pp. 987-1007, 1982.
[7] E. F. Fama, The behavior of stock market prices, Journal of Business 38, pp. 34-105, 1965.
[8] C. Fernandez and M. F. Steel, On Bayesian modeling of fat tails and skewness, Journal of the American Statistical Association 93, pp. 359–371, 1998.
[9] S. J. Kon, Models of stock returns - a comparison, The Journal of Finance 39, pp. 147-165, 1984.
[10] F. M. Longin, The choice of the distribution of asset returns: how extreme value theory can help?, Journal of Banking and Finance 29, pp. 1017–1035, 2005.
[11] B. Mandelbrot, The variation of certain speculative prices, Journal of Business 34, pp. 392-417, 1963.
[12] S. Nadarajah, A generalized normal distribution, Journal of Applied Statistics 32, pp. 685–694, 2005.
[13] R. R. Officer, The distribution of stock returns, Journal of the American Statistical Association 67, pp. 807-812, 1972.
[14] H. T. Robertson and D. B. Allison, A novel generalized normal distribution for human longevity and other negatively skewed data, PLoS ONE 7, e37025, 2012.
[15] J. P. Royston, Expected normal order statistics (exact and approximate), Journal of the Royal Statistical Society Series C (Applied Statistics) 31, pp. 161-165, 1982.
Author Information
  • Department of Statistics and Operations Research, University of Vienna, Vienna, Austria

Cite This Article
  • APA Style

    E. Reschenhofer. (2013). Linear Scale Dilation of Asset Returns. American Journal of Theoretical and Applied Statistics, 2(2), 38-41. https://doi.org/10.11648/j.ajtas.20130202.15

    Copy | Download

    ACS Style

    E. Reschenhofer. Linear Scale Dilation of Asset Returns. Am. J. Theor. Appl. Stat. 2013, 2(2), 38-41. doi: 10.11648/j.ajtas.20130202.15

    Copy | Download

    AMA Style

    E. Reschenhofer. Linear Scale Dilation of Asset Returns. Am J Theor Appl Stat. 2013;2(2):38-41. doi: 10.11648/j.ajtas.20130202.15

    Copy | Download

  • @article{10.11648/j.ajtas.20130202.15,
      author = {E. Reschenhofer},
      title = {Linear Scale Dilation of Asset Returns},
      journal = {American Journal of Theoretical and Applied Statistics},
      volume = {2},
      number = {2},
      pages = {38-41},
      doi = {10.11648/j.ajtas.20130202.15},
      url = {https://doi.org/10.11648/j.ajtas.20130202.15},
      eprint = {https://download.sciencepg.com/pdf/10.11648.j.ajtas.20130202.15},
      abstract = {Comparing the order statistics of daily returns of the S&P 500 index from 03.01.1950 to 04.03.2013 with the corresponding rankits, a linear scale dilation is observed. This observation is used to derive a five-parameter density function for the parsimonious description of the unconditional distribution of stock returns. The typical graph of this density function looks like a wizard's hat. Its signature feature is the discontinuity at zero.},
     year = {2013}
    }
    

    Copy | Download

  • TY  - JOUR
    T1  - Linear Scale Dilation of Asset Returns
    AU  - E. Reschenhofer
    Y1  - 2013/04/02
    PY  - 2013
    N1  - https://doi.org/10.11648/j.ajtas.20130202.15
    DO  - 10.11648/j.ajtas.20130202.15
    T2  - American Journal of Theoretical and Applied Statistics
    JF  - American Journal of Theoretical and Applied Statistics
    JO  - American Journal of Theoretical and Applied Statistics
    SP  - 38
    EP  - 41
    PB  - Science Publishing Group
    SN  - 2326-9006
    UR  - https://doi.org/10.11648/j.ajtas.20130202.15
    AB  - Comparing the order statistics of daily returns of the S&P 500 index from 03.01.1950 to 04.03.2013 with the corresponding rankits, a linear scale dilation is observed. This observation is used to derive a five-parameter density function for the parsimonious description of the unconditional distribution of stock returns. The typical graph of this density function looks like a wizard's hat. Its signature feature is the discontinuity at zero.
    VL  - 2
    IS  - 2
    ER  - 

    Copy | Download

  • Sections