Editing Value at risk (section)

== Criticism ==

VaR has been controversial since it moved from trading desks into the public eye in 1994. A famous 1997 [https://web.archive.org/web/20170706074507/http://www.derivativesstrategy.com/magazine/archive/1997/0497fea2.asp debate] between [[Nassim Nicholas Taleb|Nassim Taleb]] and Philippe Jorion set out some of the major points of contention. Taleb claimed VaR:<ref name="Taleb Criticism">{{Citation|author=Nassim Taleb|title=The Jorion-Taleb Debate|publisher=Derivatives Strategy|date=April 1997}}</ref>

# Ignored 2,500 years of experience in favor of untested models built by non-traders
# Was charlatanism because it claimed to estimate the risks of rare events, which is impossible
# Gave false confidence
# Would be exploited by traders

In 2008 [[David Einhorn (hedge fund manager)|David Einhorn]] and [[Aaron Brown (financial author)|Aaron Brown]] debated VaR in [https://web.archive.org/web/20110226150454/http://www.garpdigitallibrary.org/download/GRR/2012.pdf Global Association of Risk Professionals Review].<ref name="Einhorn I" /><ref name="Einhorn II">{{Citation|author=[[David Einhorn (hedge fund manager)|Einhorn, David]]|url=https://www.tilsonfunds.com/Einhorn-4-08.pdf|archive-url=https://web.archive.org/web/20160426114310/http://www.tilsonfunds.com/Einhorn-4-08.pdf|archive-date=April 26, 2016|url-status=live|title=Private Profits and Socialized Risk|journal=[[Global Association of Risk Professionals#History|GARP Risk Review]]|date=June–July 2008}}</ref> Einhorn compared VaR to "an [[airbag]] that works all the time, except when you have a car accident". He further charged that VaR:

# Led to excessive risk-taking and leverage at financial institutions
# Focused on the manageable risks near the center of the distribution and ignored the tails
# Created an incentive to take "excessive but remote risks"
# Was "potentially catastrophic when its use creates a false sense of security among senior executives and watchdogs."

[[The New York Times|New York Times]] reporter [[Joseph Nocera|Joe Nocera]] wrote an extensive piece [https://www.nytimes.com/2009/01/04/magazine/04risk-t.html?pagewanted=1&_r=1 Risk Mismanagement]<ref name="Nocera">{{Citation|author=Nocera, Joe|title=Risk Mismanagement|publisher=[[The New York Times]] Magazine|date=January 4, 2009|author-link=Joe Nocera}}</ref> on January 4, 2009, discussing the role VaR played in the [[Financial crisis of 2007–2008]]. After interviewing risk managers (including several of the ones cited above) the article suggests that VaR was very useful to risk experts, but nevertheless exacerbated the crisis by giving false security to bank executives and regulators. A powerful tool for professional risk managers, VaR is portrayed as both easy to misunderstand, and dangerous when misunderstood.

Taleb in 2009 testified in Congress asking for the banning of VaR for a number of reasons. One was that tail risks are non-measurable. Another was that for [[Anchoring_(cognitive_bias)|anchoring]] reasons VaR leads to higher risk taking.<ref>{{cite web|last1=Nassim Taleb|title=Report on The Risks of Financial Modeling, VaR and the Economic Breakdown|url=http://gop.science.house.gov/Media/hearings/oversight09/sept10/taleb.pdf|publisher=U.S. House of Representatives|archive-url=https://web.archive.org/web/20091104013038/http://gop.science.house.gov/Media/hearings/oversight09/sept10/taleb.pdf|archive-date=November 4, 2009|date=Sep 10, 2009}}</ref>

VaR is not [[Subadditivity#Finance|subadditive]]:<ref name="Dowd" /> VaR of a combined portfolio can be larger than the sum of the VaRs of its components.

For example, the average bank branch in the United States is robbed about once every ten years. A single-branch bank has about 0.0004% chance of being robbed on a specific day, so the risk of robbery would not figure into one-day 1% VaR. It would not even be within an order of magnitude of that, so it is in the range where the institution should not worry about it, it should insure against it and take advice from insurers on precautions. The whole point of insurance is to aggregate risks that are beyond individual VaR limits, and bring them into a large enough portfolio to get statistical predictability. It does not pay for a one-branch bank to have a security expert on staff.

As institutions get more branches, the risk of a robbery on a specific day rises to within an order of magnitude of VaR. At that point it makes sense for the institution to run internal stress tests and analyze the risk itself. It will spend less on insurance and more on in-house expertise. For a very large banking institution, robberies are a routine daily occurrence. Losses are part of the daily VaR calculation, and tracked statistically rather than case-by-case. A sizable in-house security department is in charge of prevention and control, the general risk manager just tracks the loss like any other cost of doing business.
As portfolios or institutions get larger, specific risks change from low-probability/low-predictability/high-impact to statistically predictable losses of low individual impact. That means they move from the range of far outside VaR, to be insured, to near outside VaR, to be analyzed case-by-case, to inside VaR, to be treated statistically.<ref name="Einhorn I" />

VaR is a static measure of risk. By definition, VaR is a particular characteristic of the probability distribution of the underlying (namely, VaR is essentially a quantile). For a dynamic measure of risk, see Novak,<ref name="Novak"/> ch. 10.

There are common abuses of VaR:<ref name="Unbearable" /><ref name="Roundtable I" />

# Assuming that plausible losses will be less than some multiple (often three) of VaR. Losses can be extremely large. 
# Reporting a VaR that has not passed a [[backtesting|backtest]]. Regardless of how VaR is computed, it should have produced the correct number of breaks (within [[sampling error]]) in the past. A common violation of common sense is to estimate a VaR based on the unverified assumption that everything follows a [[multivariate normal distribution]].

===VaR, CVaR, RVaR and EVaR===

The VaR is not a [[coherent risk measure]] since it violates the sub-additivity property, which is

:<math>\mathrm{If}\; X,Y \in \mathbf{L} ,\; \mathrm{then}\; \rho(X + Y) \leq \rho(X) + \rho(Y).</math>

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 {{mvar|&alpha;}}.

However VaR, unlike CVaR, has the property of being a [[robust statistics|robust statistic]].
A related class of risk measures is the 'Range Value at Risk' (RVaR), which is a robust version of CVaR.<ref name="RVaR">{{cite journal|last1=Cont|first1=Rama|last2=Deguest|first2=Romain|last3=Giacomo|first3=Giacomo|title=Robustness and Sensitivity Analysis of Risk Measurement Procedures|journal=Quantitative Finance|volume=10|year=2010|issue=6|pages=593–606|doi=10.1080/14697681003685597|s2cid=158678050|url=https://hal.archives-ouvertes.fr/hal-00413729/file/robustriskarxiv.pdf}}</ref>

For <math> X\in \mathbf{L}_{M^+} </math> (with <math>\mathbf{L}_{M^+} </math> the set of all [[Borel measure|Borel]] [[measurable function]]s whose [[moment-generating function]] exists for all positive real values) we have

:<math>\text{VaR}_{1-\alpha}(X)\leq \text{RVaR}_{\alpha,\beta}(X) \leq \text{CVaR}_{1-\alpha}(X)\leq\text{EVaR}_{1-\alpha}(X),</math>

where

:<math>
\begin{align}
&\text{VaR}_{1-\alpha}(X):=\inf_{t\in\mathbf{R}}\{t:\text{Pr}(X\leq t)\geq 1-\alpha\},\\
&\text{CVaR}_{1-\alpha}(X) := \frac{1}{\alpha}\int_0^{\alpha} \text{VaR}_{1-\gamma}(X)d\gamma,\\
&\text{RVaR}_{\alpha,\beta}(X) := \frac{1}{\beta-\alpha}\int_{\alpha}^{\beta} \text{VaR}_{1-\gamma}(X)d\gamma,\\
&\text{EVaR}_{1-\alpha}(X):=\inf_{z>0}\{z^{-1}\ln(M_X(z)/\alpha)\},
\end{align}
</math>

in which <math> M_X(z) </math> is the moment-generating function of {{mvar|X}} at {{mvar|z}}. In the above equations the variable {{mvar|X}} denotes the financial loss, rather than wealth as is typically the case.