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.
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In financial mathematics, a risk measure is used to determine the amount of an asset or set of assets (traditionally currency) to be kept in reserve. The purpose of this reserve is to make the risks taken by financial institutions, such as banks and insurance companies, acceptable to the regulator. In recent years attention has turned towards convex and coherent risk measurement. A risk measure is defined as a mapping from a set of random variables to the real numbers. This set of random variables represents portfolio returns.
Value at risk (VaR) is a measure of the risk of loss of investment/Capital. It estimates how much a set of investments might lose (with a given probability), given normal market conditions, in a set time period such as a day. VaR is typically used by firms and regulators in the financial industry to gauge the amount of assets needed to cover possible losses. For a given portfolio, time horizon, and probability p, the p VaR can be defined informally as the maximum possible loss during that time after excluding all worse outcomes whose combined probability is at most p.
Expected shortfall (ES) is a risk measure—a concept used in the field of financial risk measurement to evaluate the market risk or credit risk of a portfolio. The "expected shortfall at q% level" is the expected return on the portfolio in the worst of cases. ES is an alternative to value at risk that is more sensitive to the shape of the tail of the loss distribution. Expected shortfall is also called conditional value at risk (CVaR), average value at risk (AVaR), expected tail loss (ETL), and superquantile.
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