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Differential equations connecting VaR and CVaR

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  • Balbás, Beatriz
  • Balbás, Raquel

Abstract

The Value at Risk (VaR) is a very important risk measure for practitioners, supervisors and researchers. Many practitioners draw on VaR as a critical instrument in Risk Management and other Actuarial/Financial problems, while super- visors and regulators must deal with VaR due to the Basel Accords and Solvency II, among other reasons. From a theoretical point of view VaR presents some drawbacks overcome by other risk measures such as the Conditional Value at Risk (CVaR). VaR is neither differentiable nor sub-additive because it is neither continuous nor convex. On the contrary, CVaR satis es all of these properties, and this simpli es many ana- lytical studies if VaR is replaced by CVaR. In this paper several differential equations connecting both VaR and CVaR will be presented. They will allow us to address several important issues involving VaR with the help of the CVaR properties. This new methodology seems to be very efficient. In particular, a new VaR Representation Theorem may be found, and optimization problems involving VaR or probabilistic constraints always have an equivalent differentiable optimization problem. Applications in VaR, marginal VaR, CVaR and marginal CVaR estimates will be addressed as well. An illustrative actuarial numerical example will be given.

Suggested Citation

  • Balbás, Beatriz & Balbás, Raquel, 2017. "Differential equations connecting VaR and CVaR," IC3JM - Estudios = Working Papers 24017, Instituto Mixto Carlos III - Juan March de Ciencias Sociales (IC3JM).
  • Handle: RePEc:cte:imrepe:24017
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    References listed on IDEAS

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    More about this item

    Keywords

    Differential Equations;

    JEL classification:

    • G11 - Financial Economics - - General Financial Markets - - - Portfolio Choice; Investment Decisions
    • G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
    • G22 - Financial Economics - - Financial Institutions and Services - - - Insurance; Insurance Companies; Actuarial Studies
    • C65 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Miscellaneous Mathematical Tools

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