A symmetric LPM model for heuristic mean-semivariance analysis
While the semivariance (lower partial moment degree 2) has been variously described as being more in line with investors' attitude towards risk, implementation in a forecasting portfolio management role has been hampered by computational problems. The original formulation by Markowitz (1959) requires a laborious iterative process because the cosemivariance matrix is endogenous and a closed form solution does not exist. There have been attempts at optimizing an exogenous asymmetric cosemivariance matrix. However, this approach does not always provide a positive semi-definite matrix for which a closed form solution exists. We provide a proof that converts the exogenous asymmetric matrix to a symmetric matrix for which a closed form solution does exist. This approach allows the mean-semivariance formulation to be solved using Markowitz's critical line algorithm. Empirical results compare the cosemivariance algorithm to the covariance algorithm which is currently the best optimization proxy for the cosemivariance. We also compare our formulation to Estrada's (2008) cosemivariance formulation. The results demonstrate that the cosemivariance algorithm is robust to a 45 security universe and is still effective at increasing portfolio skewness at a 150 security universe. There are four major benefits to a usable mean-semivariance formulation: (1) managers may engineer skewness into the portfolio without resorting to option strategies, (2) managers will be able to evaluate the skewness effect of option strategies within their portfolio, (3) a workable mean-semivariance algorithm leads to a workable n-degree lower partial moment (LPM) algorithms which provides managers access to a wider variety of investor utility functions including risk averse, risk neutral, and risk seeking utility functions, and (4) a workable LPM algorithm leads to a workable UPM/LPM (upper partial moment/lower partial moment) algorithm.
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- Simkowitz, Michael A. & Beedles, William L., 1978. "Diversification in a Three-Moment World," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 13(05), pages 927-941, December.
- Elton, Edwin J & Gruber, Martin J & Padberg, Manfred W, 1976. "Simple Criteria for Optimal Portfolio Selection," Journal of Finance, American Finance Association, vol. 31(5), pages 1341-57, December.
- Harlow, W. V. & Rao, Ramesh K. S., 1989. "Asset Pricing in a Generalized Mean-Lower Partial Moment Framework: Theory and Evidence," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 24(03), pages 285-311, September.
- Andrew Ang & Joseph Chen & Yuhang Xing, 2005.
NBER Working Papers
11824, National Bureau of Economic Research, Inc.
- Bawa, Vijay S. & Lindenberg, Eric B., 1977. "Abstract: Capital Market Equilibrium in a Mean-Lower Partial Moment Framework," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 12(04), pages 635-635, November.
- Bawa, Vijay S., 1975. "Optimal rules for ordering uncertain prospects," Journal of Financial Economics, Elsevier, vol. 2(1), pages 95-121, March.
- Ang, James S., 1975. "A Note on the E, SL Portfolio Selection Model," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 10(05), pages 849-857, December.
- Grootveld, Henk & Hallerbach, Winfried, 1999. "Variance vs downside risk: Is there really that much difference?," European Journal of Operational Research, Elsevier, vol. 114(2), pages 304-319, April.
- C. James Hueng & Ruey Yau, 2006. "Investor preferences and portfolio selection: is diversification an appropriate strategy?," Quantitative Finance, Taylor & Francis Journals, vol. 6(3), pages 255-271.
- Fishburn, Peter C, 1977. "Mean-Risk Analysis with Risk Associated with Below-Target Returns," American Economic Review, American Economic Association, vol. 67(2), pages 116-26, March.
- Kroll, Yoram & Levy, Haim & Markowitz, Harry M, 1984. " Mean-Variance versus Direct Utility Maximization," Journal of Finance, American Finance Association, vol. 39(1), pages 47-61, March.
- Post, G.T. & van Vliet, P. & Lansdorp, S.D., 2009. "Sorting out Downside Beta," ERIM Report Series Research in Management ERS-2009-006-F&A, Erasmus Research Institute of Management (ERIM), ERIM is the joint research institute of the Rotterdam School of Management, Erasmus University and the Erasmus School of Economics (ESE) at Erasmus University Rotterdam.
- Hogan, William W. & Warren, James M., 1974. "Toward the Development of an Equilibrium Capital-Market Model Based on Semivariance," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 9(01), pages 1-11, January.
- Hanqing Jin & Harry Markowitz & Xun Yu Zhou, 2006. "A Note On Semivariance," Mathematical Finance, Wiley Blackwell, vol. 16(1), pages 53-61.
- Ang, James S. & Chua, Jess H., 1979. "Composite Measures for the Evaluation of Investment Performance," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 14(02), pages 361-384, June.
- Bawa, Vijay S. & Lindenberg, Eric B., 1977. "Capital market equilibrium in a mean-lower partial moment framework," Journal of Financial Economics, Elsevier, vol. 5(2), pages 189-200, November.
- Enrique Ballestero, 2005. "Mean-Semivariance Efficient Frontier: A Downside Risk Model for Portfolio Selection," Applied Mathematical Finance, Taylor & Francis Journals, vol. 12(1), pages 1-15.
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