Differential equations connecting VaR and CVaR Articles uri icon

publication date

  • December 2017

start page

  • 247

end page

  • 267


  • 326

International Standard Serial Number (ISSN)

  • 0377-0427

Electronic International Standard Serial Number (EISSN)

  • 1879-1778


  • 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 supervisors 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 satisfies all of these properties, and this simplifies many analytical studies if VaR is replaced by CVaR. In this paper several differential equations connecting both VaR and CVaR will be given. 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. Illustrative actuarial and financial examples will be likewise presented. (C) 2017 Elsevier B.V. All rights reserved.


  • var and cvar; differential equations; var representation theorem; risk optimization and probabilistic constraints; risk and marginal risk estimation