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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 q % {\displaystyle q\%} of cases.
The average value at risk (sometimes called expected shortfall or conditional value-at-risk or ) is a coherent risk measure, even though it is derived from Value at Risk which is not. The domain can be extended for more general Orlitz Hearts from the more typical Lp spaces .
For example, the distribution of could be inferred from historical data if one assumes that the distribution does not significantly change over the considered period of time. Also, the empirical distribution of the sample could be used as an approximation to the distribution of the future values of ξ {\displaystyle \xi } .
Since there are three risk measures covered by RiskMetrics, there are three incremental risk measures: Incremental VaR (IVaR), Incremental Expected Shortfall (IES), and Incremental Standard Deviation (ISD).
For example, if a portfolio of stocks has a one-day 5% VaR of $1 million, that means that there is a 0.05 probability that the portfolio will fall in value by more than $1 million over a one-day period if there is no trading. Informally, a loss of $1 million or more on this portfolio is expected on 1 day out of 20 days (because of 5% probability).
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 ...
The expected returns of GBM are independent of the value of the process (stock price), which agrees with what we would expect in reality. [4] A GBM process only assumes positive values, just like real stock prices. A GBM process shows the same kind of 'roughness' in its paths as we see in real stock prices.
For example, the expression ′, (,) might denote the action of sampling from the generative model where and are the current state and action, and ′ and are the new state and reward. Compared to an episodic simulator, a generative model has the advantage that it can yield data from any state, not only those encountered in a trajectory.