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Optimizing 3-objective portfolio selection with equality constraints and analyzing the effect of varying constraints on the efficient sets
1. | China Academy of Corporate Governance & Department of Financial Management, Business School, Nankai University, 94 Weijin Road, Tianjin 300071, China |
2. | Department of Financial Management, Business School, Nankai University, 94 Weijin Road, Tianjin 300071, China |
Markowitz proposes portfolio selection as a 2-objective model and emphasizes computing (whole) efficient sets and nondominated sets. Computing the sets has long been a topic in multiple-objective optimization. Researchers have gradually recognized other criteria in addition to variance and expected return. To formulate the additional criteria, researchers propose multiple-objective portfolio selection. However, computing the corresponding efficient set and nondominated set is not fully achieved. Moreover, discovering the sets' properties and utilizing the properties remain typically unanswered.
In this paper, we extend Sharpe's and Merton's model by adding a general linear objective and imposing equality constraints. To optimize the model, we analytically derive the minimum-variance surface (defined later), prove it as a nondegenerate paraboloid, and prove the nondominated set as a paraboloidal segment. We also analytically derive the efficient set and prove it as a 2-dimensional translated cone. We then prove that the set subsumes the efficient set of the corresponding traditional model, so the efficient set expands as the general linear objective is added. Furthermore, constraints can be changed or added. We utilize the translated-cone properties and readily compute the changing constraints' effect on the efficient sets by formulae or linear-equation systems.
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show all references
References:
[1] |
V. V. Acharya and L. H. Pedersen,
Asset pricing with liquidity risk, J. Financial Economics, 77 (2005), 375-410.
doi: 10.3386/w10814. |
[2] |
A. Alankar, P. Blaustein and M. S. Scholes, The cost of constraints: Risk management, agency theory and asset prices, work in progress, Stanford University, Graduate School of Business, 2014.
doi: 10.2139/ssrn.2337797. |
[3] |
A. Almazan, K. C. Brown, M. Carlson and D. A. Chapman,
Why constrain your mutual fund manager?, J. Financial Economics, 73 (2004), 289-321.
doi: 10.1016/j.jfineco.2003.05.007. |
[4] |
Y. Amihud and H. Mendelson, Asset pricing and the bid-ask spread, in Market Liquidity, Cambridge University Press, 2012, 9-46.
doi: 10.1017/CBO9780511844393.003. |
[5] |
M. Ammann, G. Coqueret and J.-P. Schade,
Characteristics-based portfolio choice with leverage constraints, J. Banking & Finance, 70 (2016), 23-37.
doi: 10.2139/ssrn.2736324. |
[6] |
B. Aouni, M. Doumpos, B. Pérez-Gladish and R. E. Steuer,
On the increasing importance of multiple criteria decision aid methods for portfolio selection, J. Oper. Research Society, 69 (2018), 1525-1542.
doi: 10.1080/01605682.2018.1475118. |
[7] |
F. D. Arditti,
Risk and the required return on equity, J. Finance, 22 (1967), 19-36.
doi: 10.1111/j.1540-6261.1967.tb01651.x. |
[8] |
C. A. Bana e Costa and J. O. Soares, Multicriteria approaches for portfolio selection: An overview, Rev. Financial Markets, 4 (2001), 19-26. Google Scholar |
[9] |
P. Behr, A. Guettler and F. Truebenbach,
Using industry momentum to improve portfolio performance, J. Banking & Finance, 36 (2012), 1414-1423.
doi: 10.1016/j.jbankfin.2011.12.007. |
[10] |
M. J. Best, An algorithm for the solution of the parametric quadratic programming problem, in Applied Mathematics and Parallel Computing, Physica, Heidelberg, 1996, 57–76.
doi: 10.1007/978-3-642-99789-1_5. |
[11] |
A. Bilbao-Terol, M. Arenas-Parra, V. Cañal-Fernández and C. Bilbao-Terol, Selection of socially responsible portfolios using hedonic prices, in Operations Research Proceedings 2012, Operations Research Proceedings, Springer, Cham, 2014.
doi: 10.1007/978-3-319-00795-3_8. |
[12] |
F. Black,
Capital market equilibrium with restricted borrowing, J. Business, 45 (1972), 444-455.
doi: 10.1086/295472. |
[13] |
Z. Bodie, A. Kane and A. J. Marcus, Investments, McGraw-Hill Education, New York, 2018. Google Scholar |
[14] |
M. W. Brandt, P. Santa-Clara and R. Valkanov,
Parametric portfolio policies: Exploiting characteristics in the cross-section of equity returns, Rev. Financial Studies, 22 (2009), 3411-3447.
doi: 10.3386/w10996. |
[15] |
P. J. Brockwell and R. A. Davis, Time Series: Theory and Methods, Springer Series in Statistics, Springer-Verlag, New York, 1987.
doi: 10.1007/978-1-4899-0004-3. |
[16] |
O. Bunn, A. Staal, J. Zhuang, A. Lazanas, C. Ural and R. Shiller,
Es-cape-ing from overvalued sectors: Sector selection based on the cyclically adjusted price-earnings (CAPE) ratio, J. Portfolio Management, 41 (2014), 16-33.
doi: 10.3905/jpm.2014.41.1.016. |
[17] |
G. Capelle-Blancard and S. Monjon,
The performance of socially responsible funds: Does the screening process matter?, European Financial Management, 20 (2014), 494-520.
doi: 10.1111/j.1468-036X.2012.00643.x. |
[18] |
L. K. Chan, J. Karceski and J. Lakonishok,
The risk and return from factors, J. Financial Quantitative Anal., 33 (1998), 159-188.
doi: 10.3386/w6098. |
[19] |
G. Chow,
Portfolio selection based on return, risk, and relative performance, Financial Analysts Journal, 51 (1995), 54-60.
doi: 10.2469/faj.v51.n2.1881. |
[20] |
T. Chow, E. Kose and F. Li,
The impact of constraints on minimum-variance portfolios, Financial Analysts Journal, 72 (2016), 52-70.
doi: 10.2469/faj.v72.n2.5. |
[21] |
V. DeMiguel, L. Garlappi, F. J. Nogales and R. Uppal, A generalized approach to portfolio optimization: Improving performance by constraining portfolio norms, Management Science, 55 (2009), 798-812. Google Scholar |
[22] |
V. DeMiguel, L. Garlappi and R. Uppal,
Optimal versus naive diversification: How inefficient is the 1/N portfolio strategy?, Rev. Financial Studies, 22 (2009), 1915-1953.
doi: 10.1093/acprof:oso/9780199744282.003.0034. |
[23] |
G. Dorfleitner, M. Leidl and J. Reeder,
Theory of social returns in portfolio choice with application to microfinance, J. Asset Management, 13 (2012), 384-400.
doi: 10.1057/jam.2012.18. |
[24] |
P. H. Dybvig, H. K. Farnsworth and J. N. Carpenter,
Portfolio performance and agency, Rev. Financial Studies, 23 (2010), 1-23.
doi: 10.1093/rfs/hhp056. |
[25] |
M. Ehrgott, K. Klamroth and C. Schwehm,
An MCDM approach to portfolio optimization, European J. Oper. Res., 155 (2004), 752-770.
doi: 10.1016/S0377-2217(02)00881-0. |
[26] |
E. J. Elton, M. J. Gruber, S. J. Brown and W. N. Goetzmann, Modern Portfolio Theory and Investment Analysis, John Wiley & Sons, New York, 2014. Google Scholar |
[27] |
F. J. Fabozzi, S. Focardi and C. Jonas,
Trends in quantitative equity management: Survey results, Quantitative Finance, 7 (2007), 115-122.
doi: 10.1080/14697680701195941. |
[28] |
E. F. Fama, Foundations of Finance: Portfolio Decisions and Securities Prices, Basic Books, Inc., New York, 1976.
doi: 10.2307/2553407. |
[29] |
E. F. Fama and K. R. French,
The cross-section of expected stock returns, J. Finance, 47 (1992), 427-465.
doi: 10.1111/j.1540-6261.1992.tb04398.x. |
[30] |
E. F. Fama and K. R. French,
International tests of a five-factor asset pricing model, J. Financial Economics, 123 (2017), 441-463.
doi: 10.1016/j.jfineco.2016.11.004. |
[31] |
M. A. Ferreira and P. Matos,
The colors of investors' money: The role of institutional investors around the world, J. Financial Economics, 88 (2008), 499-533.
doi: 10.1016/j.jfineco.2007.07.003. |
[32] |
A. M. Geoffrion,
Proper efficiency and the theory of vector maximization, J. Math. Anal. Appl., 22 (1968), 618-630.
doi: 10.1016/0022-247X(68)90201-1. |
[33] |
R. R. Grauer and F. C. Shen,
Do constraints improve portfolio performance?, J. Banking & Finance, 24 (2000), 1253-1274.
doi: 10.1016/S0378-4266(99)00069-2. |
[34] |
J. B. Guerard and A. Mark,
The optimization of efficient portfolios: The case for an R & D quadratic term, Research in Finance, 20 (2003), 217-247.
doi: 10.1016/S0196-3821(03)20011-3. |
[35] |
C. R. Harvey, J. C. Liechty, M. W. Liechty and P. Müller,
Portfolio selection with higher moments, Quant. Finance, 10 (2010), 469-485.
doi: 10.1080/14697681003756877. |
[36] |
M. Hirschberger, Y. Qi and R. E. Steuer,
Large-scale MV efficient frontier computation via a procedure of parametric quadratic programming, European J. Oper. Res., 204 (2010), 581-588.
doi: 10.1016/j.ejor.2009.11.016. |
[37] |
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For (24) | For (25) | For (26) | For (27) |
mean for |
mean for |
mean for |
mean for |
mean for |
mean for |
mean for |
mean for |
p-value: 0.9585 | p-value: 0.0130 | p-value: 0.9268 | p-value: 0.0085 |
accept |
reject |
accept |
reject |
For (24) | For (25) | For (26) | For (27) |
mean for |
mean for |
mean for |
mean for |
mean for |
mean for |
mean for |
mean for |
p-value: 0.9585 | p-value: 0.0130 | p-value: 0.9268 | p-value: 0.0085 |
accept |
reject |
accept |
reject |
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