Value at Risk and Expected Tail Loss

Value at Risk and Expected Tail Loss - Value at Risk and...

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Value at Risk and Expected Tail Loss Conditional value at risk (CVaR) is also referred to as expected tail loss (ETL) [see Acerbi and Tasche (2001)]. It has gained significant importance in the risk management field and is our choice of downside risk measure. Let ( ) p Xt be the value of an investment at time p t which assumed to be known and let ( ) ( ) 1 pp XX t X t + =− be the future uncertain value of the investment profit or loss. Let ( ) X f x and ( ) X Fx be the probability density and distribution functions of this future uncertain investment value. Given a confidence level q ( 01 q < < ) and time period duration 1 tt + (i.e. month or quarter), value at risk (VaR) is that quantity ( ) * Xq such that. ( ) ( ) { } ( ) ** 1 pX PXt X q F X q q + ⎡⎤ ≤= = ⎣⎦ (5.1) Let ( ) ( ) * | X FxX Xq < be the conditional distribution for the future investment value given that ( ) 1 p + it is less than ( ) * . From basic probability theory, this conditional distribution is related to the unconditional distribution ( ) X in the following way.

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Value at Risk and Expected Tail Loss - Value at Risk and...

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