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| '''Expected shortfall (ES)''' is a [[risk measure]], a concept used in finance (and more specifically in the field of financial risk measurement) to evaluate the [[market risk]] or [[credit risk]] of a portfolio. It is an alternative to [[value at risk]] that is more sensitive to the shape of the loss distribution in the tail of the distribution. The "expected shortfall at q% level" is the expected return on the portfolio in the worst <math>q</math>% of the cases.
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| Expected shortfall is also called '''conditional value at risk''' ('''CVaR'''), '''average value at risk''' ('''AVaR'''), and '''expected tail loss''' ('''ETL''').
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| ES evaluates the value (or risk) of an investment in a conservative way, focusing on the less profitable outcomes. For high values of <math>q</math> it ignores the most profitable but unlikely possibilities, for small values of <math>q</math> it focuses on the worst losses. On the other hand, unlike the [[discounted maximum loss]] even for lower values of <math>q</math> expected shortfall does not consider only the single most catastrophic outcome. A value of <math>q</math> often used in practice is 5%.{{Citation needed|date=February 2011}}
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| Expected shortfall is a [[Coherent risk measure|coherent]], and moreover a [[Spectral risk measure|spectral]], [[Risk measure|measure]] of financial portfolio risk. It requires a [[quantile]]-level <math>q</math>, and is defined to be the expected loss of [[Portfolio (finance)|portfolio]] value given that a loss is occurring at or below the <math>q</math>-quantile.
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| == Formal definition ==
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| If <math>X \in L^p(\mathcal{F})</math> (an [[Lp space]]) is the payoff of a portfolio at some future time and <math>0 < \alpha < 1</math> then we define the expected shortfall as <math>ES_{\alpha} = \frac{1}{\alpha}\int_0^{\alpha} VaR_{1-\gamma}(X)d\gamma</math> where <math>VaR_{\gamma}</math> is the [[Value at risk]]. This can be equivalently written as <math>ES_{\alpha} = -\frac{1}{\alpha}\left(E[X \ 1_{\{X \leq x_{\alpha}\}}] + x_{\alpha}(\alpha - P[X \leq x_{\alpha}])\right)</math> where <math>x_{\alpha} = \inf\{x \in \mathbb{R}: P(X \leq x) \geq \alpha\}</math> is the lower <math>\alpha</math>-[[quantile]] and <math>1_A(x) = \begin{cases}1 &\text{if }x \in A\\ 0 &\text{else}\end{cases}</math> is the [[indicator function]].<ref name="AcerbiTasche">{{cite journal|
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| author = Carlo Acerbi|
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| author2= Dirk Tasche|
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| title = Expected Shortfall: a natural coherent alternative to Value at Risk|
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| journal = Economic Notes|
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| year = 2002|
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| volume = 31|
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| pages = 379–388|
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| url = http://www.bis.org/bcbs/ca/acertasc.pdf|
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| format = pdf|
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| accessdate = April 25, 2012}}</ref> The dual representation is
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| :<math>ES_{\alpha} = \inf_{Q \in \mathcal{Q}_{\alpha}} E^Q[X]</math>
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| where <math>\mathcal{Q}_{\alpha}</math> is the set of [[probability measure]]s which are [[absolutely continuous]] to the physical measure <math>P</math> such that <math>\frac{dQ}{dP} \leq \alpha^{-1}</math> [[almost surely]].<ref>{{cite journal|last=Föllmer|first=H.|last2=Schied|first2=A.|year=2008|title=Convex and coherent risk measures|url=http://wws.mathematik.hu-berlin.de/~foellmer/papers/CCRM.pdf|format=pdf|accessdate=October 4, 2011}}</ref> Note that <math>\frac{dQ}{dP}</math> is the [[Radon–Nikodym derivative]] of <math>Q</math> with respect to <math>P</math>.
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| If the underlying distribution for <math>X</math> is a continuous distribution then the expected shortfall is equivalent to the [[tail conditional expectation]] defined by <math>TCE_{\alpha}(X) = E[-X\mid X \leq -VaR_{\alpha}(X)]</math>.<ref>{{cite web|url=https://statistik.ets.kit.edu/download/doc_secure1/7_StochModels.pdf|title=Average Value at Risk|format=pdf|accessdate=February 2, 2011}}</ref>
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| Informally, and non rigorously, this equation amounts to saying "in case of losses so severe that they occur only alpha percent of the time, what is our average loss".
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| Expected shortfall can also be written as a [[distortion risk measure]] given by the [[distortion function]] <math>g(x) = \begin{cases}\frac{x}{1-\alpha} & \text{if }0 \leq x < 1-\alpha,\\ 1 & \text{if }1-\alpha \leq x \leq 1.\end{cases}</math><ref name="Wirch">{{cite web|title=Distortion Risk Measures: Coherence and Stochastic Dominance|author=Julia L. Wirch|author2=Mary R. Hardy|url=http://pascal.iseg.utl.pt/~cemapre/ime2002/main_page/papers/JuliaWirch.pdf|format=pdf|accessdate=March 10, 2012}}</ref><ref name="PropertiesDRM">{{cite doi|10.1007/s11009-008-9089-z}}</ref>
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| == Examples ==
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| Example 1. If we believe our average loss on the worst 5% of the possible outcomes for our portfolio is EUR 1000, then we could say our expected shortfall is EUR 1000 for the 5% tail.
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| Example 2. Consider a portfolio that will have the following possible values at the end of the period:
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| {| class="wikitable"
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| |-
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| ! probability
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| ! ending value
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| |-
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| ! of event
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| ! of the portfolio
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| |-
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| | 10%
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| | 0
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| |-
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| | 30%
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| | 80
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| |-
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| | 40%
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| | 100
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| |-
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| | 20%
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| | 150
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| |}
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| Now assume that we paid 100 at the beginning of the period for this portfolio. Then the profit in each case is (''ending value''−100) or:
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| {| class="wikitable"
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| |-
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| ! probability
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| !
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| |-
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| ! of event
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| ! profit
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| |-
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| | 10%
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| | −100
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| |-
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| | 30%
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| | −20
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| |-
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| | 40%
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| | 0
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| |-
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| | 20%
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| | 50
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| |}
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| From this table let us calculate the expected shortfall <math>ES_q</math> for a few values of <math>q</math>:
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| {| class="wikitable"
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| |-
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| ! <math>q</math>
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| ! expected shortfall <math>ES_q</math>
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| |-
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| | 5%
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| | −100
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| |-
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| | 10%
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| | −100
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| |-
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| | 20%
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| | −60
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| |-
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| | 30%
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| | −46.<span style="text-decoration: overline;">6</span>
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| |-
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| | 40%
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| | −40
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| |-
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| | 50%
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| | −32
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| |-
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| | 60%
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| | −26.<span style="text-decoration: overline;">6</span>
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| |-
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| | 80%
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| | −20
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| |-
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| | 90%
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| | −12.<span style="text-decoration: overline;">2</span>
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| |-
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| | 100%
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| | −6
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| |}
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| To see how these values were calculated, consider the calculation of <math>ES_{0.05}</math>, the expectation in the worst 5% of cases. These cases belong to (are a [[subset]] of) row 1 in the profit table, which have a profit of −100 (total loss of the 100 invested). The expected profit for these cases is −100.
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| Now consider the calculation of <math>ES_{0.20}</math>, the expectation in the worst 20 out of 100 cases. These cases are as follows: 10 cases from row one, and 10 cases from row two (note that 10+10 equals the desired 20 cases). For row 1 there is a profit of −100, while for row 2 a profit of −20. Using the expected value formula we get
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| : <math>\frac{ \frac{10}{100}(-100)+\frac{10}{100}(-20) }{ \frac{20}{100}} = -60.</math>
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| Similarly for any value of <math>q</math>. We select as many rows starting from the top as are necessary to give a cumulative probability of <math>q</math> and then calculate an expectation over those cases. In general the last row selected may not be fully used (for example in calculating <math>ES_{0.20}</math> we used only 10 of the 30 cases per 100 provided by row 2).
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| As a final example, calculate <math>ES_1</math>. This is the expectation over all cases, or
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| : <math>0.1(-100)+0.3(-20)+0.4\cdot 0+0.2\cdot 50 = -6. \, </math>
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| The [[Value_at_risk|Value at Risk (Var)]] is given below for comparison.
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| {| class="wikitable"
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| |-
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| ! <math>q</math>
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| ! <math>\operatorname{VaR}_q</math>
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| |-
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| | 0% ≤ <math>q</math> < 10%
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| | −100
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| |-
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| | 10% ≤ <math>q</math> < 40%
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| | −20
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| |-
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| | 40% ≤ <math>q</math> < 80%
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| | 0
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| |-
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| | 80% ≤ <math>q</math> ≤ 100%
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| | 50
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| |}
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| == Properties == | |
| The expected shortfall <math>ES_q</math> increases as <math>q</math> increases.
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| The 100%-quantile expected shortfall <math>ES_{1.0}</math> equals the [[expected value]] of the portfolio.
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| For a given portfolio, the expected shortfall <math>ES_q</math> is greater than or equal to the Value at Risk <math>\operatorname{VaR}_q</math> at the same <math>q</math> level.
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| == Dynamic expected shortfall ==
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| The [[conditional risk measure|conditional]] version of the expected shortfall at the time ''t'' is defined by
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| :<math>ES_{\alpha}^t(X) = \operatorname*{ess\sup}_{Q \in \mathcal{Q}_{\alpha}^t} E^Q[-X\mid\mathcal{F}_t]</math>
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| where <math>\mathcal{Q}_{\alpha}^t = \{Q = P\,\vert_{\mathcal{F}_t}: \frac{dQ}{dP} \leq \alpha_t^{-1} \mathrm{ a.s.}\}</math>.<ref>{{cite journal|title=Conditional and dynamic convex risk measures|first1=Kai|last1=Detlefsen|first2=Giacomo|last2=Scandolo|journal=Finance
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| Stoch.|volume=9|issue=4|pages=539–561|year=2005|url=http://www.dmd.unifi.it/scandolo/pdf/Scandolo-Detlefsen-05.pdf|format=pdf|accessdate=October 11, 2011}}{{Dead link|date=January 2012}}</ref><ref>{{cite journal|title=Dynamic convex risk measures|first1=Beatrice|last1=Acciaio|first2=Irina|last2=Penner|year=2011|url=http://wws.mathematik.hu-berlin.de/~penner/Acciaio_Penner.pdf|format=pdf|accessdate=October 11, 2011}}</ref>
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| This is not a [[time-consistent]] risk measure. The time-consistent version is given by
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| :<math>\rho_{\alpha}^t(X) = \operatorname*{ess\sup}_{Q \in \tilde{\mathcal{Q}}_{\alpha}^t} E^Q[-X\mid\mathcal{F}_t]</math>
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| such that
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| :<math>\tilde{\mathcal{Q}}_{\alpha}^t = \left\{Q \ll P: \mathbb{E}\left[\frac{dQ}{dP}\mid\mathcal{F}_{\tau+1}\right] \leq \alpha_t^{-1} \mathbb{E}\left[\frac{dQ}{dP}\mid\mathcal{F}_{\tau}\right] \; \forall \tau \geq t \; \mathrm{a.s.}\right\}.</math><ref>{{cite journal|first1=Patrick|last1=Cheridito|first2=Michael|last2=Kupper|title=Composition of time-consistent dynamic monetary risk measures in discrete time|journal=International Journal of Theoretical and Applied Finance|date=May 2010|url=http://wws.mathematik.hu-berlin.de/~kupper/papers/comp2010.pdf|format=pdf|accessdate=February 4, 2011}}</ref>
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| == See also ==
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| * [[Coherent risk measure]]
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| * [[Value at risk]]
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| * [[Entropic value at risk]]
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| Methods of statistical estimation of VaR and ES can be found in
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| Embrechts et al.<ref name="Embrechts et al">Embrechts P., Kluppelberg C. and Mikosch T., Modelling Extremal Events for Insurance and Finance. Springer (1997).</ref> and Novak.<ref name="Novak">Novak S.Y., Extreme value methods with applications to finance. Chapman & Hall/CRC Press (2011). ISBN 978-1-4398-3574-6.</ref>
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| ==References==
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| * [http://www.ise.ufl.edu/uryasev/CVaR1_JOR.pdf Rockafellar, Uryasev: Optimization of conditional Value-at-Risk, 2000.]
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| * [http://arxiv.org/pdf/cond-mat/0104295%22%20/ C. Acerbi and D. Tasche: On the Coherence of Expected Shortfall, 2002.]
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| * [http://www.ise.ufl.edu/uryasev/cvar2_jbf.pdf Rockafellar, Uryasev: Conditional Value-at-Risk for general loss distributions, 2002.]
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| * [http://www.finance-and-physics.org/susinno/acerbi1.pdf Acerbi: Spectral measures of risk, 2005]
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| {{Reflist}}
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| * [https://editorialexpress.com/cgi-bin/conference/download.cgi?db_name=QMF2004&paper_id=142: Phi-Alpha optimal portfolios and extreme risk management, Best of Wilmott, 2003]
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| [[Category:Financial risk]]
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| [[Category:Actuarial science]]
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| [[Category:Mathematical finance]]
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