Portfolio Mathematics with General Linear and Quadratic Constraints
- 1 Department of Finance, Ohio University, Athens, Ohio, USA
Abstract
This paper explores the mathematics behind optimal portfolio construction when relative utility and risk are considered together in a general sense. I derive the portfolio optimization problems when subject to both a general liner constraint and a constraint to tracking error (a quadratic constraint), the most pervasive constraint placed on delegated portfolio managers. This unifies three very influential papers from the evolution of optimal portfolio theory. In addition, I also analyze the general linear constraint when applied to Sharpe Ratio maximization. When applied together, these formulations can allow principals and agents to better analyze alternatives and negotiate contracting in order to ensure that the constraints generate proper utility maximization.
- Markowitz, H. (1952) Portfolio Selection. The Journal of Finance, 7, 77-91. https://doi.org/10.1111/j.1540-6261.1952.tb01525.x
- Merton, R.C. (1972) An Analytic Derivation of the Efficient Portfolio Frontier. Journal of Financial and Quantitative Analysis, 7, 1851-1872. https://doi.org/10.2307/2329621
- Best, M.J. and Grauer, R.R. (1990) The Efficient Set Mathematics When Mean-Variance Problems Are Subject to General Linear Constraints. Journal of Economics and Business, 42, 105-120. https://doi.org/10.1016/0148-6195(90)90027-A
- Binsbergen, J., Brandt, M. and Koijen, R. (2008) Optimal Decentralized Investment Management. Journal of Finance, 63, 1849-1895. https://doi.org/10.1111/j.1540-6261.2008.01376.x
- Blake, D.A., Rossi, G., Timmermann, A., Tonks, I. and Wermers, R. (2013) Decentralized Investment Management: Evidence from the Pension Fund Industry. The Journal of Finance, 68, 1133-1178. https://doi.org/10.1111/jofi.12024
- Cuoco, D. and Kaniel, R. (2011) Equilibrium Prices in the Presence of Delegated Portfolio Management. Journal of Financial Economics, 101, 264-296. https://doi.org/10.1016/j.jfineco.2011.02.012
- Roll, R. (1992) A Mean/Variance Analysis of Tracking Error. The Journal of Portfolio Management, 18, 13-22. https://doi.org/10.3905/jpm.1992.701922
- Jorion, P. (2003) Portfolio Optimization with Tracking-Error Constraints. Financial Analysts Journal, 59, 70-82. https://doi.org/10.2469/faj.v59.n5.2565
- Bertrand, P. (2010) Another Look at Portfolio Optimization under Tracking Error Constraints. Financial Analysts Journal, 66, 78-90. https://doi.org/10.2469/faj.v66.n3.2
- Chen, L., Peng, J., Zhang, B. and Rosyida, I. (2017) Diversified Models for Portfolio Selection Based on Uncertain Semivariance. International Journal of Systems Science, 48, 637-648. https://doi.org/10.1080/00207721.2016.1206985
- Kar, M.B., Kar, S., Guo, S., Li, X. and Majumder, S. (2019) A New Bi-Objective Fuzzy Portfolio Selection Model and Its Solution through Evolutionary Algorithms. Soft Computing, 23, 4367-4381. https://doi.org/10.1007/s00500-018-3094-0
- Sharpe, W.F. (1966) Mutual Fund Performance. The Journal of Business, 39, 119-138. https://doi.org/10.1086/294846