Titulo:
Teoría moderna de portafolio: desarrollos fundamentales, extensiones y enfoques robustos
.
Sumario:
En este trabajo se presentan los principales desarrollos teóricos de la teoría moderna de portafolios. Inicialmente, se introducen los elementos fundamentales del modelo media-varianza (MV) de Markowitz, su formulación y solución del problema de optimización, así como sus limitaciones. Luego, se presentan diferentes extensiones del MV al introducir medidas alternativas de riesgo, así como los ajustes del modelo de construcción de portafolios. En este ámbito, se expone el enfoque de downside risk. Finalmente, se introducen los enfoques robustos de portafolio teniendo en cuenta los enfoques: bayesiano, de optimización robusta y de paridad de riesgo. Desde estos nuevos enfoques se resaltan aquellos ajustes que permiten superar las principales... Ver más
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Carlos Andrés Zapata Quimbayo - 2023
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Teoría moderna de portafolio: desarrollos fundamentales, extensiones y enfoques robustos Modern Portfolio Theory: Fundamental Developments, Extensions, and Robust Approaches En este trabajo se presentan los principales desarrollos teóricos de la teoría moderna de portafolios. Inicialmente, se introducen los elementos fundamentales del modelo media-varianza (MV) de Markowitz, su formulación y solución del problema de optimización, así como sus limitaciones. Luego, se presentan diferentes extensiones del MV al introducir medidas alternativas de riesgo, así como los ajustes del modelo de construcción de portafolios. En este ámbito, se expone el enfoque de downside risk. Finalmente, se introducen los enfoques robustos de portafolio teniendo en cuenta los enfoques: bayesiano, de optimización robusta y de paridad de riesgo. Desde estos nuevos enfoques se resaltan aquellos ajustes que permiten superar las principales limitaciones del modelo MV. También, se introducen desarrollos recientes que extienden las formulaciones originales del modelo de portafolio para tratar nuevos desafíos y problemáticas actuales. This paper presents the main theoretical developments of modern portfolio theory. At first, the fundamental elements of the Markowitz mean-variance model (MV), its formulation and solution of the optimization problem, as well as its limitations, are introduced. Then, different extensions of the MV model are presented by introducing alternative risk measures, as well as the adjustments of the portfolio construction model. In that sense, the downside risk approach is presented. Finally, robust portfolio approaches are introduced considering Bayesian, robust optimization, and risk-parity approaches. From these novel approaches, the adjustments that allow us overcoming some of the limitations of the MV model are highlighted. Also, recent developments that extend the original formulations of the portfolio model to address new challenges and current issues are introduced. Zapata Quimbayo, Carlos Andrés Portfolio theory; risk measures; robust portfolios teoría de portafolio; medidas de riesgo; portafolios robustos 24 Núm. 24 , Año 2023 : Enero-Junio Artículo de revista Journal article 2023-11-30T09:55:17Z 2023-11-30T09:55:17Z 2023-11-30 application/pdf text/html Universidad Externado de Colombia ODEON 1794-1113 2346-2140 https://revistas.uexternado.edu.co/index.php/odeon/article/view/9075 10.18601/17941113.n24.06 https://doi.org/10.18601/17941113.n24.06 spa http://creativecommons.org/licenses/by-nc-sa/4.0 Carlos Andrés Zapata Quimbayo - 2023 Esta obra está bajo una licencia internacional Creative Commons Atribución-NoComercial-CompartirIgual 4.0. 93 118 Acerbi, C. y Tasche, D. (2002). On the coherence of expected shortfall. Journal of Banking and Finance, 26(7), 1487-1503. https://doi.org/10.1016/S0378-4266(02)00283-2 Artzner, P., Delbaen, F., Eber, J. y Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9(3), 203-228. https://doi.org/10.1111/1467-9965.00068 Bellman, R. (1957). A Markovian decision process. Journal of Mathematics and Mechanics, 679-684. Ben-Tal, A. y Nemirovski, A. (1998). Robust convex optimization. Mathematics of Operations Research, 23(4), 769-805. https://doi.org/10.1287/moor.23.4.769 Berstein, P. (1992). Capital Ideas: The Improbable Origins of Modern Wall Street. The Free Press. Best, M. y Grauer, R. (1991). Sensitivity analysis for mean-variance portfolio problems. Management Science, 37(8), 980-989. https://doi.org/10.1287/mnsc.37.8.980 Bertsimas, D., Brown, D. y Caramanis, C. (2011). Theory and applications of robust optimization. SIAM Review, 53(3), 464-501. https://doi.org/10.1137/080734510 Black, F. y Litterman, R. (1991). Global Asset Allocation with Equities, Bonds, and Currencies. Goldman, Sachs & Co Fixed Income Research, 1-44. Black, F. y Litterman, R. (1992). Global portfolio optimization. Financial Analysts Journal, 48(5), 28-43. https://doi.org/10.2469/faj.v48.n5.28 Carmona, D. y Gamboa, J. (2022). Optimización robusta de portafolio empleando métodos Bayesianos. ODEON, 21, 81-104. https://doi.org/10.18601/17941113.n21.05 Cesarone, F., Martino, M. y Carleo, A. (2022). Does ESG impact really enhance portfolio profitability? Sustainability, 14(4), 20-50. https://doi.org/10.3390/su14042050 Chopra, V. y Ziemba, W. (1993). The effects of errors in means, variances, and covariances on optimal portfolio choice. Journal of Portfolio Management, 19(2), -11. https://doi.org/ 10.1142/9789814417358_0021 Choueifaty, Y. y Coignard, Y. (2008). Toward maximum diversification. The Journal of Portfolio Management, 35(1), 40-51. Constantinides, G. y Malliaris, A. (1995). Portfolio theory. Handbooks in Operations Research and Management Science, 9(1), 1-30. https://doi.org/10.1016/S0927-0507(05)80045-3 Coqueret, G. (2022). Perspectives in Sustainable Equity Investing. CRC Press. De Finetti, B. (1940). The problem of ‘full-risk insurances’. Journal of Investment Management, 4(1), 19-43. El Ghaoui, L. y Lebret, H. (1997). Robust solutions to least-squares problems with uncertain data. SIAM Journal on Matrix Analysis and Applications, 18(4), 1035-1064. https://doi.org/10.1137/S0895479896298130 El Ghaoui, L., Oustry, F. y Lebret, H. (1998). Robust solutions to uncertain semidefinite programs. SIAM Journal on Optimization, 9(1), 33-52. https://doi.org/10.1137/S1052623496305717 Elton, E., Gruber, M. y Padberg, M. (1976). Simple criteria for optimal portfolio selection. Journal of Finance, 11(5), 1341-1357. https://doi.org/10.2307/2326684 Fisher, I. (1907). The Rate of Interest: Its nature, determination and relation to economic phenomena. MacMillan. Fabozzi, F., Focardi, S., Kolm, P. y Pachamanova, D. (2007). Robust portfolio optimization and management. John Wiley & Sons. Gasser, S. M., Rammerstorfer, M. y Weinmayer, K. (2017). Markowitz revisited: Social portfolio engineering. European Journal of Operational Research, 258(3), 1181-1190. https://doi.org/10.1016/j.ejor.2016.10.043 Georgantas, A., Doumpos, M. y Zopounidis, C. (2021). Robust optimization approaches for portfolio selection: A comparative analysis. Annals of Operations Research, 1-17. https://doi.org/10.1007/s10479-021-04177-y Goldfarb, D., Iyengar, G. (2003). Robust portfolio selection problems. Mathematics of Operations Research, 28(1), 1-38. https://doi.org/10.1287/moor.28.1.1.14260 He, G. y Litterman, R. (1999). The intuition behind Black-Litterman model portfolios. Goldman Sachs - Investment Management Research, Technical report, 1-18. Henriksson, R., Livnat, J., Pfeifer, P. y Stumpp, M. (2019). Integrating ESG in portfolio construction. The Journal of Portfolio Management, 45(4), 67-81. https://doi.org/10.3905/ jpm.2019.45.4.067 Hirschberger, M., Steuer, R., Utz, S., Wimmer, M. y Qi, Y. (2013). Computing the nondominated surface in tri-criterion portfolio selection. Operations Research, 61(1), 169-183. https://doi.org/10.1287/opre.1120.1140 James, W., y Stein, C. (1961). Estimation with quadratic loss. Proceedings Fourth Berkeley Symposium of Math. Statis. Prob., 1, 361-380. Kim, J. H., Kim, W. C., Kwon, D. G. y Fabozzi, F. J. (2018). Robust equity portfolio performance. Annals of Operations Research, 266(1), 293-312. https://doi.org/10.1007/ s10479-017-2739-1 Kolm, P., Tütüncü, R., y Fabozzi, F. (2014). 60 years of portfolio optimization: Practical challenges and current trends. European Journal of Operational Research, 234(2), 356-371. https://doi.org/10.1016/j.ejor.2013.10.060 Krokhmal, P., Palmquist, J. y Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of Risk, 4(1), 43-68. Ledoit, O. y Wolf, M. (2003). Improved estimation of the covariance matrix of stock returns with an application to portfolio selection. Journal of Empirical Finance, 10(5), 603-621. https://doi.org/10.1016/S0927-5398(03)00007-0 Ledoit, O. y Wolf, M. (2004). A well-conditioned estimator for large-dimensional covariance matrices. Journal of Multivariate Analysis, 88(2), 365-411. https://doi.org/10.1016/S0047-259X(03)00096-4 León, B. y Zapata, C. (2023). Gestión moderna de portafolio: una guía cuantitativa con aplicaciones en R y Python. Colegio de Estudios Superiores de Administración (CESA). Lintner, J. (1965). Security prices, risk, and maximal gains from diversification. The Journal of Finance, 20(4), 587-615. https://doi.org/10.2307/2977249 Maillard, S., Roncalli, T. y Teiletche, J. (2010). The properties of equally weighted risk contribution portfolios. Journal of Portfolio Management, 36(4), 60–70. https://doi.org/10.3905/jpm.2010.36.4.060 Markowitz, H. (1952). Portfolio selection. The Journal of Finance, 7(1), 77-91. Markowitz, H. (1959). Portfolio selection: Efficient diversification of investments. Yale University Press. Marschak, J. (1938). Money and the Theory of Assets. Econometrica, Journal of the Econometric Society, 311-325. https://doi.org/10.2307/1905409 Michaud, R. (1989). The Markowitz optimization enigma: Is optimization optimal? Financial Analysts Journal, 45(1), 31-42. Michaud, R. (1998). Efficient asset management: A practical guide to stock portfolio optimization and asset allocation. Oxford University Press. Mossin, J. (1966). Equilibrium in a capital asset market. Econometrica, 34(4), 768-783. https://doi.org/10.2307/1910098 Pachamanova, D. y Fabozzi, F. (2012). Equity portfolio selection models in practice. Encyclopedia of Financial Models, 1(1), 61-87. https://doi.org/10.1002/9781118182635.efm0046 Qian, E. (2005). Risk parity portfolios: Efficient portfolios through true diversification. Panagora Asset Management, Technical Report. Qian, E. (2006). On the financial interpretation of risk contribution: Risk budgets do add up. Journal of Investment Management, 4(4), 1-11. Qian, E. (2011). Risk parity and diversification. The Journal of Investing, 20(1), 119-127. Rockefellar, R. y Uryasev, S. (2000). Optimization of conditional value-at-risk. Journal of Risk, 2(3), 21-41. https://doi.org/10.21314/JOR.2000.038 Rockafellar, R. y Uryasev, S. (2002). Conditional value-at-risk for general loss distributions. Journal of Banking & Finance, 26(7), 1443-1471. https://doi.org/10.1016/S0378-4266(02)00271-6 Romero, C. (2010). La Teoría Moderna de Portafolio: un ensayo sobre sus formulaciones originales y sus repercusiones contemporáneas. ODEON, 5, 103-118. Roncalli, T. (2014). Introduction to Risk Parity and Budgeting. CRC Press. Roy, A. (1952). Safety first and the holding of assets. Journal of the Econometric Society, 20(3), 431-449. https://doi.org/10.2307/1907413 Rubinstein, M. (2006). Bruno de Finetti and mean-variance portfolio selection. Journal of Investment Management, 4(3),1-19. Sharpe, W. (1963). A simplified model for portfolio analysis. Management science, 9(2), 277-293. https://doi.org/10.1287/mnsc.9.2.277 Sharpe, W. (1964). Capital Asset Prices: A theory of market equilibrium under conditions of risk. Journal of Finance, 19(3), 425-442. https://doi.org/10.1111/j.1540-6261.1964.tb02865.x Sharpe, W. (1971). A linear programming approximation for the general portfolio analysis problem. Journal of Financial and Quantitative Analysis, 6(5), 1263-1275. https://doi.org/10.2307/2329860 Sortino, F. y Price, L. (1994). Performance measurement in a downside risk framework. Journal of Investing, 3(3), 59-64. https://doi.org/10.3905/joi.3.3.59 Tobin, J. (1958). Liquidity preference as behavior towards risk. Review of Economic Studies, 25(1), 68-85. Treynor, J. (1961). Toward a theory of market value of risky assets. Working paper. Tütüncü, R. y Koenig, M. (2004). Robust asset allocation. Annals of Operations Research, 132(1), 157-187. https://doi.org/10.1023/B:ANOR.0000045281.41041.ed Uryasev, S. y Rockafellar, R. T. (2001). Conditional value-at-risk: Optimization approach. Stochastic optimization: Algorithms and Applications, 411-435. https://doi.org/10.1007/978-1-4757-6594-6_17 Utz, S., Wimmer, M., Hirschberger, M. y Steuer, R. (2014). Tri-criterion inverse portfolio optimization with application to socially responsible mutual funds. European Journal of Operational Research, 234(2), 491-498. https://doi.org/10.1016/j.ejor.2013.07.024 Von Neumann, J. y Morgenstern, O. (1944). Theory of Games and Economic Behavior. Princeton University Press. Zapata, C. (2021a). Optimización robusta de portafolios: conjuntos de incertidumbre y contrapartes robustas. ODEON, 20, 93-121. https://doi.org/10.18601/17941113.n20.04 Zapata, C. (2021b). Modelo Media-Varianza y criterios ASG: de Markowitz al portafolio socialmente responsable. ODEON, 21, 55-79. https://doi.org/10.18601/17941113.n21.04 https://revistas.uexternado.edu.co/index.php/odeon/article/download/9075/16487 https://revistas.uexternado.edu.co/index.php/odeon/article/download/9075/16488 info:eu-repo/semantics/article http://purl.org/coar/resource_type/c_6501 http://purl.org/redcol/resource_type/ARTREF info:eu-repo/semantics/publishedVersion http://purl.org/coar/version/c_970fb48d4fbd8a85 info:eu-repo/semantics/openAccess http://purl.org/coar/access_right/c_abf2 Text Publication |
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UNIVERSIDAD EXTERNADO DE COLOMBIA |
| thumbnail |
https://nuevo.metarevistas.org/UNIVERSIDADEXTERNADODECOLOMBIA/logo.png |
| country_str |
Colombia |
| collection |
Revista ODEON |
| title |
Teoría moderna de portafolio: desarrollos fundamentales, extensiones y enfoques robustos |
| spellingShingle |
Teoría moderna de portafolio: desarrollos fundamentales, extensiones y enfoques robustos Zapata Quimbayo, Carlos Andrés Portfolio theory; risk measures; robust portfolios teoría de portafolio; medidas de riesgo; portafolios robustos |
| title_short |
Teoría moderna de portafolio: desarrollos fundamentales, extensiones y enfoques robustos |
| title_full |
Teoría moderna de portafolio: desarrollos fundamentales, extensiones y enfoques robustos |
| title_fullStr |
Teoría moderna de portafolio: desarrollos fundamentales, extensiones y enfoques robustos |
| title_full_unstemmed |
Teoría moderna de portafolio: desarrollos fundamentales, extensiones y enfoques robustos |
| title_sort |
teoría moderna de portafolio: desarrollos fundamentales, extensiones y enfoques robustos |
| title_eng |
Modern Portfolio Theory: Fundamental Developments, Extensions, and Robust Approaches |
| description |
En este trabajo se presentan los principales desarrollos teóricos de la teoría moderna de portafolios. Inicialmente, se introducen los elementos fundamentales del modelo media-varianza (MV) de Markowitz, su formulación y solución del problema de optimización, así como sus limitaciones. Luego, se presentan diferentes extensiones del MV al introducir medidas alternativas de riesgo, así como los ajustes del modelo de construcción de portafolios. En este ámbito, se expone el enfoque de downside risk. Finalmente, se introducen los enfoques robustos de portafolio teniendo en cuenta los enfoques: bayesiano, de optimización robusta y de paridad de riesgo. Desde estos nuevos enfoques se resaltan aquellos ajustes que permiten superar las principales limitaciones del modelo MV. También, se introducen desarrollos recientes que extienden las formulaciones originales del modelo de portafolio para tratar nuevos desafíos y problemáticas actuales.
|
| description_eng |
This paper presents the main theoretical developments of modern portfolio theory. At first, the fundamental elements of the Markowitz mean-variance model (MV), its formulation and solution of the optimization problem, as well as its limitations, are introduced. Then, different extensions of the MV model are presented by introducing alternative risk measures, as well as the adjustments of the portfolio construction model. In that sense, the downside risk approach is presented. Finally, robust portfolio approaches are introduced considering Bayesian, robust optimization, and risk-parity approaches. From these novel approaches, the adjustments that allow us overcoming some of the limitations of the MV model are highlighted. Also, recent developments that extend the original formulations of the portfolio model to address new challenges and current issues are introduced.
|
| author |
Zapata Quimbayo, Carlos Andrés |
| author_facet |
Zapata Quimbayo, Carlos Andrés |
| topic |
Portfolio theory; risk measures; robust portfolios teoría de portafolio; medidas de riesgo; portafolios robustos |
| topic_facet |
Portfolio theory; risk measures; robust portfolios teoría de portafolio; medidas de riesgo; portafolios robustos |
| topicspa_str_mv |
teoría de portafolio; medidas de riesgo; portafolios robustos |
| citationissue |
24 |
| citationedition |
Núm. 24 , Año 2023 : Enero-Junio |
| publisher |
Universidad Externado de Colombia |
| ispartofjournal |
ODEON |
| source |
https://revistas.uexternado.edu.co/index.php/odeon/article/view/9075 |
| language |
spa |
| format |
Article |
| rights |
http://creativecommons.org/licenses/by-nc-sa/4.0 Carlos Andrés Zapata Quimbayo - 2023 Esta obra está bajo una licencia internacional Creative Commons Atribución-NoComercial-CompartirIgual 4.0. info:eu-repo/semantics/openAccess http://purl.org/coar/access_right/c_abf2 |
| references |
Acerbi, C. y Tasche, D. (2002). On the coherence of expected shortfall. Journal of Banking and Finance, 26(7), 1487-1503. https://doi.org/10.1016/S0378-4266(02)00283-2 Artzner, P., Delbaen, F., Eber, J. y Heath, D. (1999). Coherent measures of risk. Mathematical Finance, 9(3), 203-228. https://doi.org/10.1111/1467-9965.00068 Bellman, R. (1957). A Markovian decision process. Journal of Mathematics and Mechanics, 679-684. Ben-Tal, A. y Nemirovski, A. (1998). Robust convex optimization. Mathematics of Operations Research, 23(4), 769-805. https://doi.org/10.1287/moor.23.4.769 Berstein, P. (1992). Capital Ideas: The Improbable Origins of Modern Wall Street. The Free Press. Best, M. y Grauer, R. (1991). Sensitivity analysis for mean-variance portfolio problems. Management Science, 37(8), 980-989. https://doi.org/10.1287/mnsc.37.8.980 Bertsimas, D., Brown, D. y Caramanis, C. (2011). Theory and applications of robust optimization. SIAM Review, 53(3), 464-501. https://doi.org/10.1137/080734510 Black, F. y Litterman, R. (1991). Global Asset Allocation with Equities, Bonds, and Currencies. Goldman, Sachs & Co Fixed Income Research, 1-44. Black, F. y Litterman, R. (1992). Global portfolio optimization. Financial Analysts Journal, 48(5), 28-43. https://doi.org/10.2469/faj.v48.n5.28 Carmona, D. y Gamboa, J. (2022). Optimización robusta de portafolio empleando métodos Bayesianos. ODEON, 21, 81-104. https://doi.org/10.18601/17941113.n21.05 Cesarone, F., Martino, M. y Carleo, A. (2022). Does ESG impact really enhance portfolio profitability? Sustainability, 14(4), 20-50. https://doi.org/10.3390/su14042050 Chopra, V. y Ziemba, W. (1993). The effects of errors in means, variances, and covariances on optimal portfolio choice. Journal of Portfolio Management, 19(2), -11. https://doi.org/ 10.1142/9789814417358_0021 Choueifaty, Y. y Coignard, Y. (2008). Toward maximum diversification. The Journal of Portfolio Management, 35(1), 40-51. Constantinides, G. y Malliaris, A. (1995). Portfolio theory. Handbooks in Operations Research and Management Science, 9(1), 1-30. https://doi.org/10.1016/S0927-0507(05)80045-3 Coqueret, G. (2022). Perspectives in Sustainable Equity Investing. CRC Press. De Finetti, B. (1940). The problem of ‘full-risk insurances’. Journal of Investment Management, 4(1), 19-43. El Ghaoui, L. y Lebret, H. (1997). Robust solutions to least-squares problems with uncertain data. SIAM Journal on Matrix Analysis and Applications, 18(4), 1035-1064. https://doi.org/10.1137/S0895479896298130 El Ghaoui, L., Oustry, F. y Lebret, H. (1998). Robust solutions to uncertain semidefinite programs. SIAM Journal on Optimization, 9(1), 33-52. https://doi.org/10.1137/S1052623496305717 Elton, E., Gruber, M. y Padberg, M. (1976). Simple criteria for optimal portfolio selection. Journal of Finance, 11(5), 1341-1357. https://doi.org/10.2307/2326684 Fisher, I. (1907). The Rate of Interest: Its nature, determination and relation to economic phenomena. MacMillan. Fabozzi, F., Focardi, S., Kolm, P. y Pachamanova, D. (2007). Robust portfolio optimization and management. John Wiley & Sons. Gasser, S. M., Rammerstorfer, M. y Weinmayer, K. (2017). Markowitz revisited: Social portfolio engineering. European Journal of Operational Research, 258(3), 1181-1190. https://doi.org/10.1016/j.ejor.2016.10.043 Georgantas, A., Doumpos, M. y Zopounidis, C. (2021). Robust optimization approaches for portfolio selection: A comparative analysis. Annals of Operations Research, 1-17. https://doi.org/10.1007/s10479-021-04177-y Goldfarb, D., Iyengar, G. (2003). Robust portfolio selection problems. Mathematics of Operations Research, 28(1), 1-38. https://doi.org/10.1287/moor.28.1.1.14260 He, G. y Litterman, R. (1999). The intuition behind Black-Litterman model portfolios. Goldman Sachs - Investment Management Research, Technical report, 1-18. Henriksson, R., Livnat, J., Pfeifer, P. y Stumpp, M. (2019). Integrating ESG in portfolio construction. The Journal of Portfolio Management, 45(4), 67-81. https://doi.org/10.3905/ jpm.2019.45.4.067 Hirschberger, M., Steuer, R., Utz, S., Wimmer, M. y Qi, Y. (2013). Computing the nondominated surface in tri-criterion portfolio selection. Operations Research, 61(1), 169-183. https://doi.org/10.1287/opre.1120.1140 James, W., y Stein, C. (1961). Estimation with quadratic loss. Proceedings Fourth Berkeley Symposium of Math. Statis. Prob., 1, 361-380. Kim, J. H., Kim, W. C., Kwon, D. G. y Fabozzi, F. J. (2018). Robust equity portfolio performance. Annals of Operations Research, 266(1), 293-312. https://doi.org/10.1007/ s10479-017-2739-1 Kolm, P., Tütüncü, R., y Fabozzi, F. (2014). 60 years of portfolio optimization: Practical challenges and current trends. European Journal of Operational Research, 234(2), 356-371. https://doi.org/10.1016/j.ejor.2013.10.060 Krokhmal, P., Palmquist, J. y Uryasev, S. (2002). Portfolio optimization with conditional value-at-risk objective and constraints. Journal of Risk, 4(1), 43-68. Ledoit, O. y Wolf, M. (2003). Improved estimation of the covariance matrix of stock returns with an application to portfolio selection. Journal of Empirical Finance, 10(5), 603-621. https://doi.org/10.1016/S0927-5398(03)00007-0 Ledoit, O. y Wolf, M. (2004). A well-conditioned estimator for large-dimensional covariance matrices. Journal of Multivariate Analysis, 88(2), 365-411. https://doi.org/10.1016/S0047-259X(03)00096-4 León, B. y Zapata, C. (2023). Gestión moderna de portafolio: una guía cuantitativa con aplicaciones en R y Python. Colegio de Estudios Superiores de Administración (CESA). Lintner, J. (1965). Security prices, risk, and maximal gains from diversification. The Journal of Finance, 20(4), 587-615. https://doi.org/10.2307/2977249 Maillard, S., Roncalli, T. y Teiletche, J. (2010). The properties of equally weighted risk contribution portfolios. Journal of Portfolio Management, 36(4), 60–70. https://doi.org/10.3905/jpm.2010.36.4.060 Markowitz, H. (1952). Portfolio selection. The Journal of Finance, 7(1), 77-91. Markowitz, H. (1959). Portfolio selection: Efficient diversification of investments. Yale University Press. Marschak, J. (1938). Money and the Theory of Assets. Econometrica, Journal of the Econometric Society, 311-325. https://doi.org/10.2307/1905409 Michaud, R. (1989). The Markowitz optimization enigma: Is optimization optimal? Financial Analysts Journal, 45(1), 31-42. Michaud, R. (1998). Efficient asset management: A practical guide to stock portfolio optimization and asset allocation. Oxford University Press. Mossin, J. (1966). Equilibrium in a capital asset market. 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