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Risk management with multiple VaR constraints

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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.

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  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. 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.


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Correspondence to Thai Nguyen.

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

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