Ex Post Efficient Set Mathematics

Author(s)
Christopher Adcock

ABSTRACT

This paper considers efficient set mathematics for the case where the covariance matrix of asset returns is assumed known but *ex ante* the vector of expected returns is replaced by an estimated or forecast value. It is shown that the *ex post *mean and variance differ from the standard results. Consequently the maximum Sharpe ratio portfolio also differs from the standard result. However, even with uncertainty about the vector of expected returns, subject to the assumptions made about the joint distribution of actual returns and estimated mean returns, *ex post *Sharpe ratio maximisers hold the *ex post *market portfolio. The properties of the zero beta portfolio are similar to the standard results leading to a capital market line. The *ex post C*apital Asset Pricing Model incorporates an intercept and the betas are not the same as those computed *ex ante. *The results are illustrated with an example.

Cite this paper

C. Adcock, "Ex Post Efficient Set Mathematics,"*Journal of Mathematical Finance*, Vol. 3 No. 1, 2013, pp. 201-210. doi: 10.4236/jmf.2013.31A019.

C. Adcock, "Ex Post Efficient Set Mathematics,"

References

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[2] C. M. Stein, “Estimation of the Mean of a Multivariate Normal Distribution,” Annals of Statistics, Vol. 9, No. 6, 1981, pp. 1135-1151. doi:10.1214/aos/1176345632

[3] J. S. Liu, “Siegel’s Formula via Stein’s Identities,” Statistics and Probability Letters, Vol. 21, No. 3, 1994, pp. 247-251. doi:10.1016/0167-7152(94)90121-X

[4] Z. Landsman and J. Neslehová, “Stein’s Lemma for Elliptical Random Vectors,” Journal of Multivariate Analysis, Vol. 99, No. 5, 2008, pp. 912-927. doi:10.1016/j.jmva.2007.05.006

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[7] C. J. Adcock, “Predicting Portfolio Returns Using The Distributions of Efficient Set Portfolios,” In S. E. Satchell and A Scowcroft, Eds., Advances in Portfolio Construction and Implementation, Butterworth Heinemann, Oxford, 2003, pp. 342-355.

[8] R. Kan and G. Zhou, “Optimal Portfolio Choice with Parameter Uncertainty,” Journal of Financial and Quantitative Analysis, Vol. 42, No. 3, 2007, pp. 621-656. doi:10.1017/S0022109000004129

[9] R. O. Michaud, “The Markowitz Optimization Enigma: Is Optimized Optimal?” Financial Analysts Journal, 1989, pp. 31-42.

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[13] R. Merton, “An Analytical Derivation of the Efficient Portfolio Frontier,” Journal of Financial and Quantitative Analysis, Vol. 7, No. 4, 1972, pp. 1851-1872. doi:10.2307/2329621

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[16] M. Britten-Jones, “The Sampling Error in Estimates of Mean-Variance Efficient Portfolio Weights”. Journal of Finance, Vol. 54, No. 2, 1999, pp. 655-672. doi:10.1111/0022-1082.00120

[17] R. Kan, and D. R. Smith, “The Distribution of the Sample Minimum-Variance Frontier,” Management Science, Vol. 54, No. 7, 2008, pp. 1364-1360. doi:10.1287/mnsc.1070.0852

[18] J. Knight, and S. E. Satchell, “Exact Properties of Measures of Optimal Investment for Benchmarked Portfolios,” Quantitative Finance, Vol. 10, No. 5, 2010, pp. 495-502. doi:10.1080/14697680903061412

[19] T. Bodnar, and W. Schmid, “A Test for the Weights of the Global Minimum Variance Portfolio in an Elliptical Model,” Metrika, Vol. 67, No. 2, 2008, pp. 127-143. doi:10.1007/s00184-007-0126-7

[20] T. Bodnar and W. Schmid “Estimation of Optimal Portfolio Compositions for Gaussian Returns,” Statistics & Decisions, Vol. 26, No. 3, 2008, pp. 179-201. doi:10.1524/stnd.2008.0918

[21] T. Bodnar and W. Schmid “Econometrical Analysis of the Sample Efficient Frontier,” The European Journal of Finance, Vol. 15, No. 3, 2009, pp. 317-335. doi:10.1080/13518470802423478

[22] G. H. Hillier and S. E. Satchell, “Some Exact Results for Efficient Portfolios with Given Returns,” In S. E. Satchell and A Scowcroft, Eds., Advances in Portfolio Construction and Implementation, Butterworth Heinemann, Oxford, 2003, pp. 310-325.

[23] Y. Okhrin and W. Schmid, “Distributional Properties of Portfolio Weights,” Journal of Econometrics, Vol. 134, No. 1, 2006, pp. 235-256. doi:10.1016/j.jeconom.2005.06.022

[24] C. J. Adcock, “The Statistical Properties of Optimised Portfolios,” Proceedings of the 1996 Chemical Bank— Imperial College Conference on Forecasting Financial Markets, London, 1996.

[25] C. J. Adcock, “Dynamic Control of Risk in Optimised Portfolios,” The IMA Journal of Mathematics Applied in Business and Industry, Vol. 11, No. 1, 2000, pp. 27-138.

[26] M. Mathai and S. B. Prevost, “Quadratic Forms in Random Variables,” Springer, Heidelberg, 1992.

[27] C. J. Adcock, M. C. Cortez, M. R. Armada and F. Silva “Time Varying Betas and the Unconditional Distribution of Asset Returns,” Quantitative Finance, Vol. 12, No. 6, 2012, pp. 951-967. doi:10.1080/14697688.2010.544667

[28] J. Tu and G. Zhou “Data-Generating Process Uncertainty: What Difference Does It Make in Portfolio Decisions?” Journal of Financial Economics, Vol. 72, No. 2, 2004, pp. 385-421. doi:10.1016/j.jfineco.2003.05.003

[29] J. P. Imhof, “Computing the Distribution of Quadratic Forms in Normal Variables,” Biometrika, Vol. 48, No. 3, 1961, pp. 419-426.

[30] C.-F. Huang and R. H. Litzenberger, “Foundations for Financial Economics,” Prentice Hall, Englewood Cliffs, 1988.

[31] Cedilnik, K Kosmelj and A. Blejec, “The Distribution of the Ratio of Jointly Normal Variables,” Metodoloski Zvezki, Vol. 1, No. 1, 2004, pp. 99-108.

[1] H. Markowitz, “Portfolio Selection,” Journal of Finance, Vol. 7, No. 1, 1952, pp. 77-91.

[2] C. M. Stein, “Estimation of the Mean of a Multivariate Normal Distribution,” Annals of Statistics, Vol. 9, No. 6, 1981, pp. 1135-1151. doi:10.1214/aos/1176345632

[3] J. S. Liu, “Siegel’s Formula via Stein’s Identities,” Statistics and Probability Letters, Vol. 21, No. 3, 1994, pp. 247-251. doi:10.1016/0167-7152(94)90121-X

[4] Z. Landsman and J. Neslehová, “Stein’s Lemma for Elliptical Random Vectors,” Journal of Multivariate Analysis, Vol. 99, No. 5, 2008, pp. 912-927. doi:10.1016/j.jmva.2007.05.006

[5] M. J. Best and R. R. Grauer, “On the Sensitivity of MeanVariance-Efficient Portfolios to Changes in Asset Means: Some Analytical and Computational Results,” Review of Financial Studies, Vol. 4, No. 2, 1991, pp. 315-342. doi:10.1093/rfs/4.2.315

[6] V. Chopra, and W. T. Ziemba, “The Effect of Errors in Means, Variances and Covariances on Optimal Portfolio Choice,” Journal of Portfolio Management, Vol. 19, No. 2, 1993, pp. 6-11. doi:10.3905/jpm.1993.409440

[7] C. J. Adcock, “Predicting Portfolio Returns Using The Distributions of Efficient Set Portfolios,” In S. E. Satchell and A Scowcroft, Eds., Advances in Portfolio Construction and Implementation, Butterworth Heinemann, Oxford, 2003, pp. 342-355.

[8] R. Kan and G. Zhou, “Optimal Portfolio Choice with Parameter Uncertainty,” Journal of Financial and Quantitative Analysis, Vol. 42, No. 3, 2007, pp. 621-656. doi:10.1017/S0022109000004129

[9] R. O. Michaud, “The Markowitz Optimization Enigma: Is Optimized Optimal?” Financial Analysts Journal, 1989, pp. 31-42.

[10] R. O. Michaud, “Efficient Asset Management,” Harvard Business School Press, Boston, 1998.

[11] V. Bawa, S. J. Brown and R. Klein, “Estimation Risk and Optimal Portfolio Choice,” Studies in Bayesian Econometrics, North Holland, Amsterdam, Vol. 3, 1979.

[12] J. D. Jobson and B. Korkie, “Estimation for Markowitz Efficient Portfolios,” Journal of the American Statistical Association, Vol. 75, No. 371, 1980, pp. 544-554. doi:10.1080/01621459.1980.10477507

[13] R. Merton, “An Analytical Derivation of the Efficient Portfolio Frontier,” Journal of Financial and Quantitative Analysis, Vol. 7, No. 4, 1972, pp. 1851-1872. doi:10.2307/2329621

[14] M. R. Gibbons, S. A. Ross and J. Shanken, “A Test of the Efficiency of a Given Portfolio,” Econometrica, Vol. 57, No. 5, 1989, pp. 1121-1152. doi:10.2307/1913625

[15] G. Huberman and S. Kandel, “Mean-Variance Spanning,” The Journal of Finance, Vol. 42, No. 4, 1987, pp. 873-888. doi:10.1111/j.1540-6261.1987.tb03917.x

[16] M. Britten-Jones, “The Sampling Error in Estimates of Mean-Variance Efficient Portfolio Weights”. Journal of Finance, Vol. 54, No. 2, 1999, pp. 655-672. doi:10.1111/0022-1082.00120

[17] R. Kan, and D. R. Smith, “The Distribution of the Sample Minimum-Variance Frontier,” Management Science, Vol. 54, No. 7, 2008, pp. 1364-1360. doi:10.1287/mnsc.1070.0852

[18] J. Knight, and S. E. Satchell, “Exact Properties of Measures of Optimal Investment for Benchmarked Portfolios,” Quantitative Finance, Vol. 10, No. 5, 2010, pp. 495-502. doi:10.1080/14697680903061412

[19] T. Bodnar, and W. Schmid, “A Test for the Weights of the Global Minimum Variance Portfolio in an Elliptical Model,” Metrika, Vol. 67, No. 2, 2008, pp. 127-143. doi:10.1007/s00184-007-0126-7

[20] T. Bodnar and W. Schmid “Estimation of Optimal Portfolio Compositions for Gaussian Returns,” Statistics & Decisions, Vol. 26, No. 3, 2008, pp. 179-201. doi:10.1524/stnd.2008.0918

[21] T. Bodnar and W. Schmid “Econometrical Analysis of the Sample Efficient Frontier,” The European Journal of Finance, Vol. 15, No. 3, 2009, pp. 317-335. doi:10.1080/13518470802423478

[22] G. H. Hillier and S. E. Satchell, “Some Exact Results for Efficient Portfolios with Given Returns,” In S. E. Satchell and A Scowcroft, Eds., Advances in Portfolio Construction and Implementation, Butterworth Heinemann, Oxford, 2003, pp. 310-325.

[23] Y. Okhrin and W. Schmid, “Distributional Properties of Portfolio Weights,” Journal of Econometrics, Vol. 134, No. 1, 2006, pp. 235-256. doi:10.1016/j.jeconom.2005.06.022

[24] C. J. Adcock, “The Statistical Properties of Optimised Portfolios,” Proceedings of the 1996 Chemical Bank— Imperial College Conference on Forecasting Financial Markets, London, 1996.

[25] C. J. Adcock, “Dynamic Control of Risk in Optimised Portfolios,” The IMA Journal of Mathematics Applied in Business and Industry, Vol. 11, No. 1, 2000, pp. 27-138.

[26] M. Mathai and S. B. Prevost, “Quadratic Forms in Random Variables,” Springer, Heidelberg, 1992.

[27] C. J. Adcock, M. C. Cortez, M. R. Armada and F. Silva “Time Varying Betas and the Unconditional Distribution of Asset Returns,” Quantitative Finance, Vol. 12, No. 6, 2012, pp. 951-967. doi:10.1080/14697688.2010.544667

[28] J. Tu and G. Zhou “Data-Generating Process Uncertainty: What Difference Does It Make in Portfolio Decisions?” Journal of Financial Economics, Vol. 72, No. 2, 2004, pp. 385-421. doi:10.1016/j.jfineco.2003.05.003

[29] J. P. Imhof, “Computing the Distribution of Quadratic Forms in Normal Variables,” Biometrika, Vol. 48, No. 3, 1961, pp. 419-426.

[30] C.-F. Huang and R. H. Litzenberger, “Foundations for Financial Economics,” Prentice Hall, Englewood Cliffs, 1988.

[31] Cedilnik, K Kosmelj and A. Blejec, “The Distribution of the Ratio of Jointly Normal Variables,” Metodoloski Zvezki, Vol. 1, No. 1, 2004, pp. 99-108.