Editing Value at risk (section)

== Mathematical definition ==

Let <math>X</math> be a profit and loss distribution (loss negative and profit positive). The VaR at level <math>\alpha\in(0,1)</math> is the smallest number <math>y</math> such that the probability that <math>Y:=-X</math> does not exceed <math>y</math> is at least <math>1-\alpha</math>. Mathematically, <math>\operatorname{VaR}_{\alpha}(X)</math> is the <math>(1-\alpha)</math>-[[quantile]] of <math>Y</math>, i.e.,

:<math>\operatorname{VaR}_\alpha(X)=-\inf\big\{x\in\mathbb{R}:F_X(x)>\alpha\big\} = F^{-1}_Y(1-\alpha).</math><ref>{{cite journal|last1=Artzner|first1=Philippe|last2=Delbaen|first2=Freddy|last3=Eber|first3=Jean-Marc|last4=Heath|first4=David|year=1999|title=Coherent Measures of Risk|journal=Mathematical Finance|volume=9|issue=3|pages=203–228|url=http://www.math.ethz.ch/~delbaen/ftp/preprints/CoherentMF.pdf|access-date=February 3, 2011|doi=10.1111/1467-9965.00068|s2cid=6770585 }}</ref><ref>{{cite book|first1=Hans|last1=Foellmer|first2=Alexander|last2=Schied|title=Stochastic Finance|publisher=[[Walter de Gruyter]]|year=2004|isbn=978-311-0183467|series=de Gruyter Series in Mathematics|volume=27|location=Berlin|pages=177–182|mr=2169807}}</ref>

This is the most general definition of VaR and the two identities are equivalent (indeed, for any real random variable <math>X</math> its [[cumulative distribution function]] <math>F_X</math> is well defined). 
However this formula cannot be used directly for calculations unless we assume that <math>X</math> has some parametric distribution.

Risk managers typically assume that some fraction of the bad events will have undefined losses, either because markets are closed or illiquid, or because the entity bearing the loss breaks apart or loses the ability to compute accounts. Therefore, they do not accept results based on the assumption of a well-defined probability distribution.<ref name="Unbearable">{{Citation|author=Aaron Brown|title=The Unbearable Lightness of Cross-Market Risk|publisher=Wilmott Magazine|date=March 2004|author-link=Aaron Brown (financial author)}}</ref> [[Nassim Taleb]] has labeled this assumption, "charlatanism".<ref name="Taleb II">{{Citation|author=Nassim Taleb|title=The World According to Nassim Taleb|publisher=Derivatives Strategy|date=December 1996 – January 1997|url=http://www.derivativesstrategy.com/magazine/archive/1997/1296qa.asp|archive-url=https://web.archive.org/web/20000829231106/http://www.derivativesstrategy.com/magazine/archive/1997/1296qa.asp |archive-date=2000-08-29 }}</ref> On the other hand, many academics prefer to assume a well-defined distribution, albeit usually one with [[Kurtosis|fat tails]].<ref name="Jorion" /> This point has probably caused more contention among VaR theorists than any other.<ref name="Roundtable I" />

Value at risk can also be written as a [[distortion risk measure]] given by the [[distortion function]] <math>g(x) = \begin{cases}0 & \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|access-date=March 10, 2012|archive-date=July 5, 2016|archive-url=https://web.archive.org/web/20160705041252/http://pascal.iseg.utl.pt/~cemapre/ime2002/main_page/papers/JuliaWirch.pdf|url-status=dead}}</ref><ref name="PropertiesDRM">{{Cite journal | last1 = Balbás | first1 = A. | last2 = Garrido | first2 = J. | last3 = Mayoral | first3 = S. | doi = 10.1007/s11009-008-9089-z | title = Properties of Distortion Risk Measures | journal = Methodology and Computing in Applied Probability | volume = 11 | issue = 3 | pages = 385 | year = 2008 | url = https://e-archivo.uc3m.es/bitstream/10016/14071/1/properties_balbas_MCAP_2009_ps.pdf | hdl = 10016/14071 | s2cid = 53327887 | hdl-access = free }}</ref>