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Mean-Variance and Expected Utility: The Borch Paradox


  • David Johnstone
  • Dennis Lindley


The model of rational decision-making in most of economics and statistics is expected utility theory (EU) axiomatised by von Neumann and Morgenstern, Savage and others. This is less the case, however, in financial economics and mathematical finance, where investment decisions are commonly based on the methods of mean-variance (MV) introduced in the 1950s by Markowitz. Under the MV framework, each available investment opportunity ("asset") or portfolio is represented in just two dimensions by the ex ante mean and standard deviation $(\mu,\sigma)$ of the financial return anticipated from that investment. Utility adherents consider that in general MV methods are logically incoherent. Most famously, Norwegian insurance theorist Borch presented a proof suggesting that two-dimensional MV indifference curves cannot represent the preferences of a rational investor (he claimed that MV indifference curves "do not exist"). This is known as Borch's paradox and gave rise to an important but generally little-known philosophical literature relating MV to EU. We examine the main early contributions to this literature, focussing on Borch's logic and the arguments by which it has been set aside.

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  • David Johnstone & Dennis Lindley, 2013. "Mean-Variance and Expected Utility: The Borch Paradox," Papers 1306.2728,
  • Handle: RePEc:arx:papers:1306.2728

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    References listed on IDEAS

    1. Baron, David P, 1977. "On the Utility Theoretic Foundations of Mean-Variance Analysis," Journal of Finance, American Finance Association, vol. 32(5), pages 1683-1697, December.
    2. Borch, Karl, 1978. "Portfolio theory is for risk lovers," Journal of Banking & Finance, Elsevier, vol. 2(2), pages 179-181, August.
    3. M. S. Feldstein, 1969. "Mean-Variance Analysis in the Theory of Liquidity Preference and Portfolio Selection," Review of Economic Studies, Oxford University Press, vol. 36(1), pages 5-12.
    4. D. J. Johnstone, 2012. "Log-optimal economic evaluation of probability forecasts," Journal of the Royal Statistical Society Series A, Royal Statistical Society, vol. 175(3), pages 661-689, July.
    5. Meyer, Jack, 1977. "Choice among distributions," Journal of Economic Theory, Elsevier, vol. 14(2), pages 326-336, April.
    6. Björn Hagströmer & Richard G. Anderson & Jane M. Binner & Thomas Elger & Birger Nilsson, 2008. "Mean–Variance Versus Full‐Scale Optimization: Broad Evidence For The Uk," Manchester School, University of Manchester, vol. 76(s1), pages 134-156, September.
    7. Liu, Liping, 2004. "A new foundation for the mean-variance analysis," European Journal of Operational Research, Elsevier, vol. 158(1), pages 229-242, October.
    8. L. Eeckhoudt & C. Gollier & H. Schlesinger, 2005. "Economic and financial decisions under risk," Post-Print hal-00325882, HAL.
    9. Borch, Karl, 1973. "Expected utility expressed in terms of moments," Omega, Elsevier, vol. 1(3), pages 331-343, June.
    10. Barone, Luca, 2008. "Bruno de Finetti and the case of the critical line's last segment," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 359-377, February.
    11. Markowitz, Harry M, 1991. "Foundations of Portfolio Theory," Journal of Finance, American Finance Association, vol. 46(2), pages 469-477, June.
    12. Sarnat, Marshall, 1974. "A Note on the Implications of Quadratic Utility for Portfolio Theory," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 9(4), pages 687-689, September.
    13. Björn Hagströmer & Richard G. Anderson & Jane M. Binner & Thomas Elger & Birger Nilsson, 2007. "Mean-variance vs. full-scale optimization: broad evidence for the U.K," Working Papers 2007-016, Federal Reserve Bank of St. Louis.
    14. Borch, Karl, 1979. "Equilibrium in capital markets," Economics Letters, Elsevier, vol. 2(2), pages 175-179.
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    Cited by:

    1. Grant, Andrew & Johnstone, David & Kwon, Oh Kang, 2019. "The cost of capital in a prediction market," International Journal of Forecasting, Elsevier, vol. 35(1), pages 313-320.
    2. Z. Landsman & U. Makov & T. Shushi, 2020. "Portfolio Optimization by a Bivariate Functional of the Mean and Variance," Journal of Optimization Theory and Applications, Springer, vol. 185(2), pages 622-651, May.

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