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Portfolio Selection: A Review

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Abstract

This paper reviews portfolio selection models and provides perspective on some open issues. It starts with a review of the classic Markowitz mean-variance framework. It then presents the intertemporal portfolio choice approach developed by Merton and the fundamental notion of dynamic hedging. Martingale methods and resulting portfolio formulas are also reviewed. Their usefulness for economic insights and numerical implementations is illustrated. Areas of future research are outlined.

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Notes

  1. Brennan, Schwartz and Lagnado [9] provide numerical results for a class of low dimensional problems when utilities are constant relative risk averse. Explicit solutions are available in a few special cases (e.g., Wachter [10]).

  2. Merton [16] provides an analytical formula for the efficient frontier (the set of efficient portfolios) and derives a mutual fund decomposition of optimal portfolios.

  3. Maccheroni, Marinacci, Rustichini and Taboga [17] propose a monotone extension of mean-variance preferences and examine its implications for portfolio selection.

  4. Rothschild and Stiglitz [18] show that an increase in risk in the second order stochastic dominance sense can be compatible with a decrease in variance. A risk averse individual may therefore be led to prefer (resp. reject) certain gambles with higher (resp. lower) variance.

  5. Mean-variance preferences, extended to a dynamic setting, do not have a recursive structure. This precludes the straightforward application of dynamic programming methods and leads to the problem of time inconsistency. A resolution of this problem is proposed by Basak and Chabakauri [20] and extended by Bjork, Murgoci and Zhou [21].

  6. Results can be generalized to stochastic time-preferences.

  7. An alternative is to apply a change of variables studied by Doss [26] in order to stabilize the volatility coefficients of the processes to be simulated (see Detemple, Garcia and Rindisbacher [8, 27]). The portfolio formula can be rewritten in terms of the transformed state variables. Portfolio components can then be calculated by Monte Carlo simulation based on the transformed state variables.

  8. Simulation-based methods have also been proposed by Cvitanic, Goukassian and Zapatero [28], Brandt, Goyal and Santa-Clara [29] and others. A review of simulation methods for portfolio choice appears in Detemple, Garcia and Rindisbacher [30]. Asymptotic properties of simulation-based portfolio estimators are derived in Detemple, Garcia and Rindisbacher [31]. A comparison of methods is carried out in Detemple, Garcia and Rindisbacher [32].

  9. As the HARA utility fails to satisfy the Inada condition at 0, this implementation is based on an extension of Theorem 4.1 (see [8]).

  10. Models seeking to explain the demand for bonds include Wachter [39] and Lioui [36].

  11. Solutions with non-zero shadow price have been found for special cases (e.g., Detemple and Rindisbacher [15, 54]).

  12. Various applications of portfolio models with jumps can be found in the literature. Examples include Ahn and Thompson [55], who focus on implications for the term structure of interest rates, Liu, Longstaff and Pan [56], who study the effects of rare events affecting prices and volatility, and Das and Uppal [57], who examine the impact of systemic risk on international portfolios.

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Acknowledgements

I would like to thank the referee for useful comments. I am also pleased to acknowledge the comments of the editor, Franco Giannessi.

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  1. Boston University School of Management, Boston, USA

    Jérôme Detemple

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Correspondence to Jérôme Detemple.

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This review article is based on a keynote lecture given at the 2011 Conference on Applied Financial Economics.

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Detemple, J. Portfolio Selection: A Review. J Optim Theory Appl 161, 1–21 (2014). https://doi.org/10.1007/s10957-012-0208-1

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