Entropic value at risk

In financial mathematics and stochastic optimization, the concept of risk measure is used to quantify the risk involved in a random outcome or risk position. Many risk measures have hitherto been proposed, each having certain characteristics. The entropic value at risk (EVaR) is a coherent risk measure introduced by Ahmadi-Javid, which is an upper bound for the value at risk (VaR) and the conditional value at risk (CVaR), obtained from the Chernoff inequality. The EVaR can also be represented by using the concept of relative entropy. Because of its connection with the VaR and the relative entropy, this risk measure is called "entropic value at risk". The EVaR was developed to tackle some computational inefficiencies of the CVaR. Getting inspiration from the dual representation of the EVaR, Ahmadi-Javid developed a wide class of coherent risk measures, called g-entropic risk measures. Both the CVaR and the EVaR are members of this class. Let be a probability space with a set of all simple events, a -algebra of subsets of and a probability measure on . Let be a random variable and be the set of all Borel measurable functions whose moment-generating function exists for all . The entropic value at risk (EVaR) of with confidence level is defined as follows: In finance, the random variable in the above equation, is used to model the losses of a portfolio. Consider the Chernoff inequality Solving the equation for results in By considering the equation (), we see that which shows the relationship between the EVaR and the Chernoff inequality. It is worth noting that is the entropic risk measure or exponential premium, which is a concept used in finance and insurance, respectively. Let be the set of all Borel measurable functions whose moment-generating function exists for all . The dual representation (or robust representation) of the EVaR is as follows: where and is a set of probability measures on with . Note that is the relative entropy of with respect to also called the Kullback–Leibler divergence.