IDEAS home Printed from https://ideas.repec.org/p/inu/caeprp/2012004.html
   My bibliography  Save this paper

The Variance Drain and Jensen's Inequality

Author

Listed:
  • Robert Becker

    (Indiana University)

Abstract

The well-known approximation of the difference between the arithmetic average and geometric average returns as one-half of the variance of the underlying returns is reexamined using Jensen’s Inequality. The ”defect” in Jensen’s Inequality, is given an exlicit formula in terms of the variance following some ideas put forward by Holder. A new form of the AM-GM Inequality follows and is is applied to financial returns. Both exact, and approximate relations between the arithmetic average, geometric average, and variance of returns are discussed. The mathematical formulation of these relations are free of distributional assumptions governing the underlying returns process.

Suggested Citation

  • Robert Becker, 2012. "The Variance Drain and Jensen's Inequality," CAEPR Working Papers 2012-004, Center for Applied Economics and Policy Research, Department of Economics, Indiana University Bloomington.
    • Handle: RePEc:inu:caeprp:2012004
    as

    Download full text from publisher

    File URL: https://caepr.indiana.edu/RePEc/inu/caeprp/caepr2012-004.pdf
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Young, William E. & Trent, Robert H., 1969. "Geometric Mean Approximations of Individual Security and Portfolio Performance*," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 4(2), pages 179-199, June.
    2. Renshaw, Edward F., 1967. "Portfolio Balance Models in Perspective: Some Generalizations That Can Be Derived from the Two-Asset Case*," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 2(2), pages 123-149, June.
    3. Delong J. Bradford, 2008. "Stocks for the Long Run," The Economists' Voice, De Gruyter, vol. 5(7), pages 1-2, November.
    4. Goetzmann, William N. & Ibbotson, Roger G., 2006. "The Equity Risk Premium: Essays and Explorations," OUP Catalogue, Oxford University Press, number 9780195148145.
    5. Levy, H & Markowtiz, H M, 1979. "Approximating Expected Utility by a Function of Mean and Variance," American Economic Review, American Economic Association, vol. 69(3), pages 308-317, June.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Robert Becker, 2012. "The Variance Drain and Jensen's Inequality," Caepr Working Papers 2012-004, Center for Applied Economics and Policy Research, Economics Department, Indiana University Bloomington.
    2. Sonntag, Dominik, 2018. "Die Theorie der fairen geometrischen Rendite [The Theory of Fair Geometric Returns]," MPRA Paper 87082, University Library of Munich, Germany.
    3. Markowitz, Harry, 2014. "Mean–variance approximations to expected utility," European Journal of Operational Research, Elsevier, vol. 234(2), pages 346-355.
    4. Harry M. Markowitz, 2002. "Efficient Portfolios, Sparse Matrices, and Entities: A Retrospective," Operations Research, INFORMS, vol. 50(1), pages 154-160, February.
    5. Gregor Dorfleitner & Mai Nguyen, 2017. "A new approach for optimizing responsible investments dependently on the initial wealth," Journal of Asset Management, Palgrave Macmillan, vol. 18(2), pages 81-98, March.
    6. Diamond, Harvey & Gelles, Gregory, 1999. "Gaussian approximation of expected utility," Economics Letters, Elsevier, vol. 64(3), pages 301-307, September.
    7. Antonio Díaz & Carlos Esparcia, 2021. "Dynamic optimal portfolio choice under time-varying risk aversion," International Economics, CEPII research center, issue 166, pages 1-22.
    8. Li, Chenguang & Sexton, Richard J., 2009. "Impacts of Retailers’ Pricing Strategies for Produce Commodities on Farmer Welfare," 2009 Conference, August 16-22, 2009, Beijing, China 51720, International Association of Agricultural Economists.
    9. Marc Peter Radke & Manuel Rupprecht, 2021. "Household Wealth: Low-Yielding and Poorly Structured?," JRFM, MDPI, vol. 14(3), pages 1-40, March.
    10. Silva, A. Christian & Prange, Richard E., 2007. "Virtual volatility," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 376(C), pages 507-516.
    11. Semenov, Andrei, 2021. "Measuring the stock's factor beta and identifying risk factors under market inefficiency," The Quarterly Review of Economics and Finance, Elsevier, vol. 80(C), pages 635-649.
    12. Johnson, Michael & O'Connor, Ian & Malcolm, Bill, 2006. "Agribusiness Assets in Investment Portfolios," 2006 Conference (50th), February 8-10, 2006, Sydney, Australia 139794, Australian Agricultural and Resource Economics Society.
    13. Marcus Berliant & Axel H. Watanabe, 2018. "A scale‐free transportation network explains the city‐size distribution," Quantitative Economics, Econometric Society, vol. 9(3), pages 1419-1451, November.
    14. Guo, Xu & Lien, Donald & Wong, Wing-Keung, 2015. "Good Approximation of Exponential Utility Function for Optimal Futures Hedging," MPRA Paper 66841, University Library of Munich, Germany.
    15. Prakash, Arun J. & Chang, Chun-Hao & Pactwa, Therese E., 2003. "Selecting a portfolio with skewness: Recent evidence from US, European, and Latin American equity markets," Journal of Banking & Finance, Elsevier, vol. 27(7), pages 1375-1390, July.
    16. Berck, Peter, 1980. "Portfolio Theory and the Demand for Futures: theory and the case of California cotton," Department of Agricultural & Resource Economics, UC Berkeley, Working Paper Series qt58j4t4qp, Department of Agricultural & Resource Economics, UC Berkeley.
    17. Raquel M. Gaspar & Paulo M. Silva, 2023. "Investors’ perspective on portfolio insurance," Portuguese Economic Journal, Springer;Instituto Superior de Economia e Gestao, vol. 22(1), pages 49-79, January.
    18. Shi, Shasha & Yin, Yafeng & An, Qingxian & Chen, Ke, 2021. "Optimal build-operate-transfer road contracts under information asymmetry and uncertainty," Transportation Research Part B: Methodological, Elsevier, vol. 152(C), pages 65-86.
    19. Bali, Turan G. & Demirtas, K. Ozgur & Levy, Haim & Wolf, Avner, 2009. "Bonds versus stocks: Investors' age and risk taking," Journal of Monetary Economics, Elsevier, vol. 56(6), pages 817-830, September.
    20. Oyetayo Oluwatosin J & Adeyeye Patrick Olufemi, 2017. "A Robust Application of the Arbitrage Pricing Theory: Evidence from Nigeria," Journal of Economics and Behavioral Studies, AMH International, vol. 9(1), pages 141-151.

    More about this item

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:inu:caeprp:2012004. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Center for Applied Economics and Policy Research (email available below). General contact details of provider: https://edirc.repec.org/data/caeprus.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.