On Some Class of Distance Functions for Measuring Portfolio Efficiency

ABSTRACT

Morey and Morey [1] have developed an approach for gauging portfolio efficiencies in the context of the Markowitz model. Following some recent contributions [2,3], this paper analyzes the axiomatic properties of distance functions extending an earlier approach proposed by Morey and Morey. The paper also focusses on the hyperbolic measure and the McFadden gauge function [4]. Among other things, overall, allocative and portfolio improvements possibilities (in term of return expansion or/and risk contraction) based upon the indirect mean-variance utility function are analyzed. Along this line, duality results are established in each case. This enables us to calculate the degree of risk aversion maximizing the investor indirect mean-variance utility function in either return expansion or risk contraction. An empirical illustration is provided and reveal ranking of preferred risks aversion for some “CAC40” assets.

Morey and Morey [1] have developed an approach for gauging portfolio efficiencies in the context of the Markowitz model. Following some recent contributions [2,3], this paper analyzes the axiomatic properties of distance functions extending an earlier approach proposed by Morey and Morey. The paper also focusses on the hyperbolic measure and the McFadden gauge function [4]. Among other things, overall, allocative and portfolio improvements possibilities (in term of return expansion or/and risk contraction) based upon the indirect mean-variance utility function are analyzed. Along this line, duality results are established in each case. This enables us to calculate the degree of risk aversion maximizing the investor indirect mean-variance utility function in either return expansion or risk contraction. An empirical illustration is provided and reveal ranking of preferred risks aversion for some “CAC40” assets.

Cite this paper

nullC. Barros, W. Briec and H. Ratsimbanierana, "On Some Class of Distance Functions for Measuring Portfolio Efficiency,"*Journal of Mathematical Finance*, Vol. 1 No. 2, 2011, pp. 15-27. doi: 10.4236/jmf.2011.12003.

nullC. Barros, W. Briec and H. Ratsimbanierana, "On Some Class of Distance Functions for Measuring Portfolio Efficiency,"

References

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[2] W. Briec, K. Kerstens and J.-B. Lesourd, “Single Period Markowitz Portfolio Selection, Performance Gauging and Duality: A Variation on Luenberger’s Shortage Function,” Journal of Optimization Theory and Applications, Vol. 120, No. 1, 2004, pp. 1-27. doi:10.1023/B:JOTA.0000012730.36740.bb

[3] W. Briec, K. Kerstens and O. Jokung, “Mean-Variance- Skewness Portfolio Performance Gauging: A General Shortage Function and Dual Approach,” Management Science, Vol. 53, No. 1, 2007, pp. 135-149. doi:10.1287/mnsc.1060.0596

[4] D. McFadden, “Cost, Revenue, and Profit Functions,” In: M. Fuss and D. McFadden, Eds., Production Economics: A Dual Approach to Theory and Applications, Amsterdam, North-Holland, Vol. 1, 1978, pp. 3-109.

[5] R. W. Shephard, “Cost and Production Functions,” Princeton University Press, Princeton, 1953.

[6] H. Markowitz, “Portfolio Selection,” Journal of Finance, Vol. 7, No. 1, 1952, pp. 77-91. doi:10.2307/2975974

[7] H. Markowitz, “Portfolio Selection: Efficient Diversification of Investments,” John Wiley, New York, 1959.

[8] W. Sharpe, “A Simplified Model for Portfolio Analysis,” Management Science, Vol. 9, No. 2, 1963, pp. 277-293. doi:10.1287/mnsc.9.2.277

[9] W. Sharpe, “Capital Asset Prices: A Theory of Market Equilibrium under Condition of Risk,” Journal of Finance, Vol. 19, No. 3, 1964, pp. 425-442. doi:10.2307/2977928

[10] J. Lintner, “The Valuation of Risk Assets and the Selection of Risky Investment in Stock Portfolios and Capital Budgets,” Review of Economics and Statistics, Vol. 47, No. 1, 1965, pp. 13-37. doi:10.2307/1924119

[11] H. Markowitz, “CAPM Investors do Not Get Paid for Bearing Risk: A Linear Relation does Not Imply Payment for Risk,” Journal of Portfolio Management, Vol. 34, No. 2, 2008, pp. 91-96.

[12] D. G. Luenberger, “Microeconomic Theory,” McGraw Hill, Boston, 1995.

[13] M. Farrell, “The Measurement of Productive Efficiency,” Journal of the Royal Statistical Society, Vol. 120, No. 3, 1957, pp. 253-281. doi:10.2307/2343100

[14] J. A. Pogue, “An Extension of the Markowitz Portfolio Selection Model to include Variable Transaction Cost, short Sales, Leverage Polycies and Taxes,” Journal of Finance, Vol. 25, 1970, pp. 1005-1028. doi:10.2307/2325576

[15] A. Rudd and B. Rosenberg, “A Realistic Portfolio Selection Model,” In: E. J. Elton and M. J. Grubber, Eds., Portfolio Theory-Lectures in Management Science, Vol. II, North Holland, Amsterdam, 1979.

[16] A. Charnes, W. W. Cooper and E. Rhodes, “Measuring the Efficiency of Decision-Making Units,” European Journal of Operational Research, Vol. 3, No. 6, 1978, pp. 429-444. doi:10.1016/0377-2217(78)90138-8

[17] J. K. Sengupta, “Nonparametric Tests of Efficiency of Portfolio Investment,” Journal of Economics, Vol. 50, No. 1, 1989, pp. 1-15. doi:10.1007/BF01227605

[18] R. F?re, S. Grosskopf, C. A. K. Lovell, “The Measurement of Efficiency of Production,” Kluwer, Boston, 1985.

[19] W. Briec and J.-B. Lesourd, “The Efficiency of Investment Fund Management: A Stochastic Frontier Model,” In: Dunis, Eds., Advances in Quantitative Asset Management, Klwer Academic Publishers, Norwell. 2000, pp. 41-58.

[20] A. Ruszczynski and R. Vanderbei, “Frontiers of Stochastically Nondominated Portfolios,” Econometrica, Vol. 71, No. 4, 2003, pp. 1287-1297. doi:10.1111/1468-0262.t01-1-00448

[21] A. V. Fiacco and G. P. McGormick, “Nonlinear Programming: Sequential Uncontrained Minimization Techniques,” John Wiley, New York, 1968.

[1] M. R. Morey and R. C. Morey, “Mutual Fund Performance Appraisals: A Multi-horizon Perspective with Endogenous Benchmarking,” Omega, Vol. 27, No. 2, 1999, pp. 241-258. doi:10.1016/S0305-0483(98)00043-7

[2] W. Briec, K. Kerstens and J.-B. Lesourd, “Single Period Markowitz Portfolio Selection, Performance Gauging and Duality: A Variation on Luenberger’s Shortage Function,” Journal of Optimization Theory and Applications, Vol. 120, No. 1, 2004, pp. 1-27. doi:10.1023/B:JOTA.0000012730.36740.bb

[3] W. Briec, K. Kerstens and O. Jokung, “Mean-Variance- Skewness Portfolio Performance Gauging: A General Shortage Function and Dual Approach,” Management Science, Vol. 53, No. 1, 2007, pp. 135-149. doi:10.1287/mnsc.1060.0596

[4] D. McFadden, “Cost, Revenue, and Profit Functions,” In: M. Fuss and D. McFadden, Eds., Production Economics: A Dual Approach to Theory and Applications, Amsterdam, North-Holland, Vol. 1, 1978, pp. 3-109.

[5] R. W. Shephard, “Cost and Production Functions,” Princeton University Press, Princeton, 1953.

[6] H. Markowitz, “Portfolio Selection,” Journal of Finance, Vol. 7, No. 1, 1952, pp. 77-91. doi:10.2307/2975974

[7] H. Markowitz, “Portfolio Selection: Efficient Diversification of Investments,” John Wiley, New York, 1959.

[8] W. Sharpe, “A Simplified Model for Portfolio Analysis,” Management Science, Vol. 9, No. 2, 1963, pp. 277-293. doi:10.1287/mnsc.9.2.277

[9] W. Sharpe, “Capital Asset Prices: A Theory of Market Equilibrium under Condition of Risk,” Journal of Finance, Vol. 19, No. 3, 1964, pp. 425-442. doi:10.2307/2977928

[10] J. Lintner, “The Valuation of Risk Assets and the Selection of Risky Investment in Stock Portfolios and Capital Budgets,” Review of Economics and Statistics, Vol. 47, No. 1, 1965, pp. 13-37. doi:10.2307/1924119

[11] H. Markowitz, “CAPM Investors do Not Get Paid for Bearing Risk: A Linear Relation does Not Imply Payment for Risk,” Journal of Portfolio Management, Vol. 34, No. 2, 2008, pp. 91-96.

[12] D. G. Luenberger, “Microeconomic Theory,” McGraw Hill, Boston, 1995.

[13] M. Farrell, “The Measurement of Productive Efficiency,” Journal of the Royal Statistical Society, Vol. 120, No. 3, 1957, pp. 253-281. doi:10.2307/2343100

[14] J. A. Pogue, “An Extension of the Markowitz Portfolio Selection Model to include Variable Transaction Cost, short Sales, Leverage Polycies and Taxes,” Journal of Finance, Vol. 25, 1970, pp. 1005-1028. doi:10.2307/2325576

[15] A. Rudd and B. Rosenberg, “A Realistic Portfolio Selection Model,” In: E. J. Elton and M. J. Grubber, Eds., Portfolio Theory-Lectures in Management Science, Vol. II, North Holland, Amsterdam, 1979.

[16] A. Charnes, W. W. Cooper and E. Rhodes, “Measuring the Efficiency of Decision-Making Units,” European Journal of Operational Research, Vol. 3, No. 6, 1978, pp. 429-444. doi:10.1016/0377-2217(78)90138-8

[17] J. K. Sengupta, “Nonparametric Tests of Efficiency of Portfolio Investment,” Journal of Economics, Vol. 50, No. 1, 1989, pp. 1-15. doi:10.1007/BF01227605

[18] R. F?re, S. Grosskopf, C. A. K. Lovell, “The Measurement of Efficiency of Production,” Kluwer, Boston, 1985.

[19] W. Briec and J.-B. Lesourd, “The Efficiency of Investment Fund Management: A Stochastic Frontier Model,” In: Dunis, Eds., Advances in Quantitative Asset Management, Klwer Academic Publishers, Norwell. 2000, pp. 41-58.

[20] A. Ruszczynski and R. Vanderbei, “Frontiers of Stochastically Nondominated Portfolios,” Econometrica, Vol. 71, No. 4, 2003, pp. 1287-1297. doi:10.1111/1468-0262.t01-1-00448

[21] A. V. Fiacco and G. P. McGormick, “Nonlinear Programming: Sequential Uncontrained Minimization Techniques,” John Wiley, New York, 1968.