January  2016, 1: 1 doi: 10.1186/s41546-016-0001-4

Portfolio theory for squared returns correlated across time

1 University of Freiburg, Freiburg im Breisgau, Germany;

2 University of Maryland, College Park, USA

Received  January 12, 2016 Revised  June 10, 2016

Allowing for correlated squared returns across two consecutive periods, portfolio theory for two periods is developed. This correlation makes it necessary to work with non-Gaussian models. The two-period conic portfolio problem is formulated and implemented. This development leads to a mean ask price frontier, where the latter employs concave distortions. The modeling permits access to skewness via randomized drifts. Optimal portfolios maximize a conservative market value seen as a bid price for the portfolio. On the mean ask price frontier we observe a tradeoff between the deterministic and random drifts and the volatility costs of increasing the deterministic drift. From a historical perspective, we also implement a mean-variance analysis. The resulting mean-variance frontier is three-dimensional expressing the minimal variance as a function of the targeted levels for the deterministic and random drift.
Citation: Ernst Eberlein, Dilip B. Madan. Portfolio theory for squared returns correlated across time. Probability, Uncertainty and Quantitative Risk, doi: 10.1186/s41546-016-0001-4
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Ait-Sahalia, Y, Brandt, MW:Variable selection for portfolio choice. J. Financ 56, 1297-1351 (2001),

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Barndorff-Nielsen, OE, Shephard, N:Econometric analysis of realized volatility and its use in estimating stochastic volatility models. J. R. Stat. Soc. B 64, 253-280 (2002),

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Bielecki, T, Jin, H, Pliska, SR, Zhou, XY:Continuous-time mean-variance portfolio selection with bankruptcy prohibition. Math. Financ 15, 213-244 (2005),

[8]

Brandt, MW, Goyal, A, Santa-Clara, P, Stroud, JR:A simulation approach to dynamic portfolio choice with an application to learning about predictability. Rev. Financ. Stud 18, 831-873 (2005),

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Brandt, MW:Portfolio Choice Problems. In:Ait-Sahalia, Y, Hansen, LP (eds.) Handbook of Financial Econometrics, Chapter 5, pp. 269-336. Elsevier, Amsterdam (2009),

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Brandt, MW, Santa-Clara, P:Dynamic portfolio selection by augmenting the asset space. J. Financ 61, 2187-2218 (2006),

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Campbell, JY, Viceira, LM:Strategic Asset Allocation:Portfolio Choice for Long Term Investors. Oxford University Press, Oxford (2002),

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Cherny, A, Madan, D:New measures for performance evaluation. Rev. Financ. Stud 22, 2571-2606 (2009),

[13]

Cochrane, JHL:A mean-variance benchmark for intertemporal portfolio theory. J. Financ 69, 1-49 (2014),

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Cvitanic, J, Lazrak, A, Wang, T:Implications of the Sharpe ratio as a performance in multi-period settings.J. Econ. Dyn. Control 32, 1622-1649 (2008),

[15]

Cvitanic, J, Zapatero, F:Introduction to the Economics and Mathematics of Financial Markets. MIT Press, Cambridge, MA (2004),

[16]

Duffie, D, Richardson, H:Mean-variance hedging in continuous time. Ann. Appl. Probab 1, 1-15 (1991),

[17]

Hong, H, Scheinkman, J, Xiong, W:Asset float and speculative bubbles. J. Financ 61, 1073-1117 (2006),

[18]

Jagannathan, R, Ma, T:Risk reduction in large portfolios:why imposing the wrong constraints helps. J.Financ 58, 1651-1683 (2003),

[19]

Kusuoka, S:On law invariant coherent risk measures. Adv. Math. Econ 3, 83-95 (2001),

[20]

Leippold, M, Trojani, F, Vanini, P:Geometric approach to multiperiod mean variance optimization of assets and liabilities. J. Econ. Dyn. Control 28, 1079-1113 (2004),

[21]

Lim, AEB, Zhou, XY:Mean-variance portfolio selection with random parameters in a complete market.Math. Oper. Res 27, 101-120 (2002),

[22]

MacLean, LC, Zhao, Y, Ziemba, WT:Mean-variance versus expected utility in dynamic investment analysis. Comput. Manag. Sci 8, 3-22 (2011),

[23]

Madan, DB:Conic portfolio theory. Int. J. Theor. Appl. Financ (2016). doi:10.1142/SO219024916500199,

[24]

Madan, DB, Pistorius, M, Stadje, M:On Dynamic Spectral Risk Measures and a Limit Theorem. Working Paper, Imperial College, London (2015),

[25]

Markowitz, HM:Portfolio selection. J. Financ 7, 77-91 (1952),

[26]

Markowitz, HM:Foundations of portfolio theory. J. Financ 46, 469-477 (1991),

[27]

Skiadas, C:Asset Pricing Theory. Princeton University Press, Princeton, NJ (2009),

[28]

Strotz, RH:Myopia and inconsistency in dynamic utility maximization. Rev. Econ. Stud 23, 165-180 (1956),

[29]

Zhou, XY, Li, D:Continuous-time mean-variance portfolio selection:a stochastic LQ framework. Appl.Math. Optim 42, 19-33 (2000),

show all references

References:
[1]

Acharya, VV, Pedersen, LH:Asset pricing with liquidity risk. J. Financ. Econ 77, 375-410 (2005),

[2]

Ait-Sahalia, Y, Brandt, MW:Variable selection for portfolio choice. J. Financ 56, 1297-1351 (2001),

[3]

Bajeux-Besnainou, I, Portait, R:Dynamic asset allocation in a mean-variance framework. Manag. Sci 44, 79-95 (1998),

[4]

Bansal, R, Dahlquist, M, Harvey, CR:Dynamic Trading Strategies and Portfolio Choice. Working Paper, Duke University (2004),

[5]

Barndorff-Nielsen, OE, Shephard, N:Econometric analysis of realized volatility and its use in estimating stochastic volatility models. J. R. Stat. Soc. B 64, 253-280 (2002),

[6]

Basak, S, Chabakauri, G:Dynamic mean-variance asset allocation. Rev. Financ. Stud 23, 2970-3016 (2010),

[7]

Bielecki, T, Jin, H, Pliska, SR, Zhou, XY:Continuous-time mean-variance portfolio selection with bankruptcy prohibition. Math. Financ 15, 213-244 (2005),

[8]

Brandt, MW, Goyal, A, Santa-Clara, P, Stroud, JR:A simulation approach to dynamic portfolio choice with an application to learning about predictability. Rev. Financ. Stud 18, 831-873 (2005),

[9]

Brandt, MW:Portfolio Choice Problems. In:Ait-Sahalia, Y, Hansen, LP (eds.) Handbook of Financial Econometrics, Chapter 5, pp. 269-336. Elsevier, Amsterdam (2009),

[10]

Brandt, MW, Santa-Clara, P:Dynamic portfolio selection by augmenting the asset space. J. Financ 61, 2187-2218 (2006),

[11]

Campbell, JY, Viceira, LM:Strategic Asset Allocation:Portfolio Choice for Long Term Investors. Oxford University Press, Oxford (2002),

[12]

Cherny, A, Madan, D:New measures for performance evaluation. Rev. Financ. Stud 22, 2571-2606 (2009),

[13]

Cochrane, JHL:A mean-variance benchmark for intertemporal portfolio theory. J. Financ 69, 1-49 (2014),

[14]

Cvitanic, J, Lazrak, A, Wang, T:Implications of the Sharpe ratio as a performance in multi-period settings.J. Econ. Dyn. Control 32, 1622-1649 (2008),

[15]

Cvitanic, J, Zapatero, F:Introduction to the Economics and Mathematics of Financial Markets. MIT Press, Cambridge, MA (2004),

[16]

Duffie, D, Richardson, H:Mean-variance hedging in continuous time. Ann. Appl. Probab 1, 1-15 (1991),

[17]

Hong, H, Scheinkman, J, Xiong, W:Asset float and speculative bubbles. J. Financ 61, 1073-1117 (2006),

[18]

Jagannathan, R, Ma, T:Risk reduction in large portfolios:why imposing the wrong constraints helps. J.Financ 58, 1651-1683 (2003),

[19]

Kusuoka, S:On law invariant coherent risk measures. Adv. Math. Econ 3, 83-95 (2001),

[20]

Leippold, M, Trojani, F, Vanini, P:Geometric approach to multiperiod mean variance optimization of assets and liabilities. J. Econ. Dyn. Control 28, 1079-1113 (2004),

[21]

Lim, AEB, Zhou, XY:Mean-variance portfolio selection with random parameters in a complete market.Math. Oper. Res 27, 101-120 (2002),

[22]

MacLean, LC, Zhao, Y, Ziemba, WT:Mean-variance versus expected utility in dynamic investment analysis. Comput. Manag. Sci 8, 3-22 (2011),

[23]

Madan, DB:Conic portfolio theory. Int. J. Theor. Appl. Financ (2016). doi:10.1142/SO219024916500199,

[24]

Madan, DB, Pistorius, M, Stadje, M:On Dynamic Spectral Risk Measures and a Limit Theorem. Working Paper, Imperial College, London (2015),

[25]

Markowitz, HM:Portfolio selection. J. Financ 7, 77-91 (1952),

[26]

Markowitz, HM:Foundations of portfolio theory. J. Financ 46, 469-477 (1991),

[27]

Skiadas, C:Asset Pricing Theory. Princeton University Press, Princeton, NJ (2009),

[28]

Strotz, RH:Myopia and inconsistency in dynamic utility maximization. Rev. Econ. Stud 23, 165-180 (1956),

[29]

Zhou, XY, Li, D:Continuous-time mean-variance portfolio selection:a stochastic LQ framework. Appl.Math. Optim 42, 19-33 (2000),

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