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Expected shortfall is considered a more useful risk measure than VaR because it is a coherent spectral measure of financial portfolio risk. It is calculated for a given quantile -level q {\displaystyle q} and is defined to be the mean loss of portfolio value given that a loss is occurring at or below the q {\displaystyle q} -quantile.
Under some formulations, it is only equivalent to expected shortfall when the underlying distribution function is continuous at (), the value at risk of level . [2] Under some other settings, TVaR is the conditional expectation of loss above a given value, whereas the expected shortfall is the product of this value with the probability of ...
However, it can be bounded by coherent risk measures like Conditional Value-at-Risk (CVaR) or entropic value at risk (EVaR). CVaR is defined by average of VaR values for confidence levels between 0 and α. However VaR, unlike CVaR, has the property of being a robust statistic. A related class of risk measures is the 'Range Value at Risk' (RVaR ...
In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value evaluated with respect to the conditional probability distribution. If the random variable can take on only a finite number of values, the "conditions" are that the variable can only take on a subset of ...
We know that there is a difference between Expected Shortfall and CVar. In this article seems to be the same. [Unsigned comment by 189.60.201.90] According to Acerbi and Tasche (2002), On the coherence of Expected Shortfall, page 5 "We will see below (Corollary 4.3) that the Expected Shortfall is in fact identical with CVaR".
In words: the variance of Y is the sum of the expected conditional variance of Y given X and the variance of the conditional expectation of Y given X. The first term captures the variation left after "using X to predict Y", while the second term captures the variation due to the mean of the prediction of Y due to the randomness of X.
Let Y be a random variable and X another random variable on the same probability space. The law of total variance can be understood by noting: The law of total variance can be understood by noting: Var ( Y ∣ X ) {\displaystyle \operatorname {Var} (Y\mid X)} measures how much Y varies around its conditional mean E [ Y ∣ X ...
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.