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Portfolio optimization under the Value-at-Risk constraint

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  • Traian A. Pirvu

Abstract

In this paper we analyse the effects arising from imposing a Value-at-Risk constraint in an agent's portfolio selection problem. The financial market is incomplete and consists of multiple risky assets (stocks) plus a risk-free asset. The stocks are modelled as exponential Brownian motions with random drift and volatility. The risk of the trading portfolio is re-evaluated dynamically, hence the agent must satisfy the Value-at-Risk constraint continuously. We derive the optimal consumption and portfolio allocation policy in closed form for the case of logarithmic utility. The non-logarithmic CRRA utilities are considered as well, when the randomness of market coefficients is independent of the Brownian motion driving the stocks. The portfolio selection, a stochastic control problem, is reduced, in this context, to a deterministic control one, which is analysed, and a numerical treatment is proposed.

Suggested Citation

  • Traian A. Pirvu, 2007. "Portfolio optimization under the Value-at-Risk constraint," Quantitative Finance, Taylor & Francis Journals, vol. 7(2), pages 125-136.
    • Handle: RePEc:taf:quantf:v:7:y:2007:i:2:p:125-136
    DOI: 10.1080/14697680701213868
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    Citations

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    Cited by:

    1. Zhang, Qingye & Gao, Yan, 2016. "Optimal consumption—portfolio problem with CVaR constraints," Chaos, Solitons & Fractals, Elsevier, vol. 91(C), pages 516-521.
    2. Marcos Escobar-Anel & Michel Kschonnek & Rudi Zagst, 2022. "Portfolio optimization: not necessarily concave utility and constraints on wealth and allocation," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 95(1), pages 101-140, February.
    3. Nicole Bäuerle & André Mundt, 2009. "Dynamic mean-risk optimization in a binomial model," Mathematical Methods of Operations Research, Springer;Gesellschaft für Operations Research (GOR);Nederlands Genootschap voor Besliskunde (NGB), vol. 70(2), pages 219-239, October.
    4. Priscilla Serwaa Nkyira Gambrah & Traian Adrian Pirvu, 2014. "Risk Measures and Portfolio Optimization," JRFM, MDPI, vol. 7(3), pages 1-17, September.
    5. Thai Nguyen, 2016. "Optimal investment and consumption with downside risk constraint in jump-diffusion models," Papers 1604.05584, arXiv.org.
    6. Ye, Jun & Li, Tiantian, 2012. "The optimal mean–variance investment strategy under value-at-risk constraints," Insurance: Mathematics and Economics, Elsevier, vol. 51(2), pages 344-351.
    7. Santiago Moreno-Bromberg & Traian Pirvu & Anthony R'eveillac, 2011. "CRRA Utility Maximization under Risk Constraints," Papers 1106.1702, arXiv.org, revised Mar 2012.
    8. Dongchen Li & Virginia R. Young, 2020. "Maximizing expected exponential utility of consumption with a constraint on expected time in poverty," Annals of Finance, Springer, vol. 16(1), pages 63-99, March.
    9. Moreno-Bromberg, Santiago & Pirvu, Traian A. & Réveillac, Anthony, 2011. "CRRA utility maximization under risk constraints," SFB 649 Discussion Papers 2011-043, Humboldt University Berlin, Collaborative Research Center 649: Economic Risk.
    10. Castaneda, Pablo & Rudolph, Heinz P., 2011. "Upgrading investment regulations in second pillar pension systems : a proposal for Colombia," Policy Research Working Paper Series 5775, The World Bank.
    11. Redeker Imke & Wunderlich Ralf, 2018. "Portfolio optimization under dynamic risk constraints: Continuous vs. discrete time trading," Statistics & Risk Modeling, De Gruyter, vol. 35(1-2), pages 1-21, January.

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