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A coherent risk measure is a function that satisfies properties of monotonicity, sub-additivity, homogeneity, and translational invariance. Properties.
2.1 Risk Measure to Acceptance Set. 2.2 Acceptance Set to Risk Measure. 3 Examples. ... if and only if the corresponding risk measure is convex (coherent).
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 .
The former definition may not be a coherent risk measure in general, however it is coherent if the underlying distribution is continuous. [4] The latter definition is a coherent risk measure. [3] TVaR accounts for the severity of the failure, not only the chance of failure. The TVaR is a measure of the expectation only in the tail of the ...
A coherent risk measure satisfies the following four properties: 1. Subadditivity. A risk measure is subadditive if for any portfolios A and B, the risk of A+B is never greater than the risk of A plus the risk of B. In other words, the risk of the sum of subportfolios is smaller than or equal to the sum of their individual risks.
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, [1] [2] which is an upper bound for the value at risk (VaR) and the conditional value at risk (CVaR), obtained from the Chernoff inequality.
A risk measure can be thought of as a conditional risk measure on the trivial sigma algebra. A dynamic risk measure is a risk measure that deals with the question of how evaluations of risk at different times are related. It can be interpreted as a sequence of conditional risk measures. [1]
In measure theory, Carathéodory's extension theorem (named after the mathematician Constantin Carathéodory) states that any pre-measure defined on a given ring of subsets R of a given set Ω can be extended to a measure on the σ-ring generated by R, and this extension is unique if the pre-measure is σ-finite.