Skip to main content

Risk management with multiple VaR constraints


We study a utility maximization problem under multiple Value-at-Risk (VaR)-type constraints. The optimization framework is particularly important for financial institutions which have to follow short-time VaR-type regulations under some realistic regulatory frameworks like Solvency II, but need to serve long-term liabilities. Deriving closed-form solutions, we show that risk management using multiple VaR constraints is more useful for loss prevention at intertemporal time instances compared with the well-known result of the one-VaR problem in Basak and Shapiro (Rev Financ Stud 14:371–405, 2001), confirming the numerical analysis of Shi and Werker (J Bank Finance 36(12):3227–3238, 2012). In addition, the multiple-VaR solution at maturity on average dominates the one-VaR solution in a wide range of intermediate market scenarios, but performs worse in good and very bad market scenarios. The range of these very bad market scenarios is however rather limited. Finally, we show that it is preferable to reach a fixed terminal state through insured intertemporal states rather than through extreme up and down movements, showing that a multiple-VaR framework induces a preference for less volatility.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7


  1. 1.

    For instance, according to Solvency II, the solvency capital requirement for the insurance companies is based on an annual VaR-measure calibrated to a 99.5% confidence level (c.f. Article 100 of Solvency II directive).

  2. 2.

    While finishing this paper we found out that the problem of an one-VaR constraint with possible stochastic liability level \(L_T\) has been considered in Boyle and Tian (2007). We would like to thank Monique Jeanblanc for pointing out this important reference.


  1. Alexander GJ, Baptista AM, Yan S (2012) When more is less: using multiple constraints to reduce tail risk. J Bank Finance 36(10):2693–2716

    Article  Google Scholar 

  2. Basak S, Shapiro A (2001) Value-at-risk-based risk management: optimal policies and asset prices. Rev Financ Stud 14:371–405

    Article  Google Scholar 

  3. Basak S, Shapiro A, Teplá L (2006) Risk management with benchmarking. Manag Sci 52(4):542–557

    Article  Google Scholar 

  4. Boyle P, Tian W (2007) Portfolio management with constraints. Math Finance 17(3):319–343

    MathSciNet  Article  Google Scholar 

  5. Chen A, Nguyen T, Stadje M (2018) Optimal investment under var-regulation and minimum insurance. Insur Math Econ 79:194–209

    MathSciNet  Article  Google Scholar 

  6. Cuoco D (1997) Optimal consumption and equilibrium prices with portfolio constraints and stochastic income. J Econ Theory 71(1):33–73

    MathSciNet  Article  Google Scholar 

  7. Cuoco D, He H, Issaenko S (2008) Optimal dynamic trading strategies with risk limits. Oper Res 56:358–368

    MathSciNet  Article  Google Scholar 

  8. Cuoco D, Liu H (2006) An analysis of var-based capital requirements. J Financ Intermed 15(3):362–394

    Article  Google Scholar 

  9. Cvitanic J, Karatzas I (1992) Convex duality in constrained portfolio optimization. Ann Appl Probab 2(4):767–818

    MathSciNet  Article  Google Scholar 

  10. Gandy R (2005) Portfolio optimization with risk constraints. PhD thesis, University of Ulm

  11. Grossman SJ, Zhou Z (1996) Equilibrium analysis of portfolio insurance. J Finance 51(4):1379–1403

    Article  Google Scholar 

  12. Gundel A, Weber S (2008) Utility maximization under a shortfall risk constraint. J Math Econ 44(11):1126–1151

    MathSciNet  Article  Google Scholar 

  13. Jang BG, Park S (2016) Ambiguity and optimal portfolio choice with value-at-risk constraint. Finance Res Lett 18((Supplement C)):158–176

    Article  Google Scholar 

  14. Kraft H, Steffensen M (2013) A dynamic programming approach to constrained portfolios. Eur J Oper Res 229(2):453–461

    MathSciNet  Article  Google Scholar 

  15. MacLean LC, Zhao Y, Ziemba WT (2016) Optimal capital growth with convex shortfall penalties. Quant Finance 16(1):101–117

    MathSciNet  Article  Google Scholar 

  16. Santos AAP, Nogales FJ, Ruiz E, Dijk DV (2012) Optimal portfolios with minimum capital requirements. J Bank Finance 36(7):1928–1942

    Article  Google Scholar 

  17. Sass J, Wunderlich R (2010) Optimal portfolio policies under bounded expected loss and partial information. Math Methods Oper Res 72(1):25–61

    MathSciNet  Article  Google Scholar 

  18. Schyns M, Crama Y, Hübner G (2010) Optimal selection of a portfolio of options under value-at-risk constraints: a scenario approach. Ann Oper Res 181(1):683–708

    MathSciNet  Article  Google Scholar 

  19. Shi Z, Werker JMB (2012) Short-horizon regulation for long-term investors. J Bank Finance 36(12):3227–3238

    Article  Google Scholar 

  20. Xu L, Gao C, Kou G, Liu Q (2017) Comonotonic approximation to periodic investment problems under stochastic drift. Eur J Oper Res 262(1):251–261

    MathSciNet  Article  Google Scholar 

  21. Yiu KFC (2004) Optimal portfolios under a value-at-risk constraint. J Econ Dyn Control 28(7):1317–1334

    MathSciNet  Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Thai Nguyen.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Chen, A., Nguyen, T. & Stadje, M. Risk management with multiple VaR constraints. Math Meth Oper Res 88, 297–337 (2018).

Download citation


  • Value at Risk
  • Optimal portfolio
  • Multiple risk constraints
  • Risk management
  • Solvency II regulation

JEL Classification

  • C61
  • G11
  • G18
  • G31