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Value at risk (VaR) is a measure of the risk of loss for investments. It estimates how much a set of investments might lose (with a given probability), given normal market conditions, in a set time period such as a day. VaR is typically used by firms and regulators in the financial industry to gauge the amount of assets needed to cover possible losses. For a given portfolio,
time horizon Time is the continued sequence of existence and events that occurs in an apparently irreversible succession from the past, through the present, into the future. It is a component quantity of various measurements used to sequence events, to co ...
, and
probability Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking, ...
''p'', the ''p'' VaR can be defined informally as the maximum possible loss during that time after excluding all worse outcomes whose combined probability is at most ''p''. This assumes mark-to-market pricing, and no trading in the portfolio. For example, if a portfolio of stocks has a one-day 95% 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). More formally, ''p'' VaR is defined such that the probability of a loss greater than VaR is (at most) ''(1-p)'' while the probability of a loss less than VaR is (at least) ''p''. A loss which exceeds the VaR threshold is termed a "VaR breach".Holton, Glyn A. (2014).
Value-at-Risk: Theory and Practice
' second edition, e-book.
It is important to note that, for a fixed ''p'', the ''p'' VaR does not assess the magnitude of loss when a VaR breach occurs and therefore is considered by some to be a questionable metric for risk management. For instance, assume someone makes a bet that flipping a coin seven times will not give seven heads. The terms are that they win $100 if this does not happen (with probability 127/128) and lose $12,700 if it does (with probability 1/128). That is, the possible loss amounts are $0 or $12,700. The 1% VaR is then $0, because the probability of any loss at all is 1/128 which is less than 1%. They are, however, exposed to a possible loss of $12,700 which can be expressed as the ''p'' VaR for any ''p ≤ 0.78125% (1/128)''. VaR has four main uses in
finance Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of production, distribution, and consumption of money, assets, goods and services (the discipline of f ...
: risk management, financial
control Control may refer to: Basic meanings Economics and business * Control (management), an element of management * Control, an element of management accounting * Comptroller (or controller), a senior financial officer in an organization * Controlli ...
, financial reporting and computing regulatory capital. VaR is sometimes used in non-financial applications as well. However, it is a controversial risk management tool. Important related ideas are
economic capital In finance, mainly for financial services firms, economic capital (ecap) is the amount of risk capital, assessed on a realistic basis, which a firm requires to cover the risks that it is running or collecting as a going concern, such as market ...
, backtesting,
stress testing Stress testing (sometimes called torture testing) is a form of deliberately intense or thorough testing used to determine the stability of a given system, critical infrastructure or entity. It involves testing beyond normal operational capacity, ...
, expected shortfall, and tail conditional expectation.


Details

Common parameters for VaR are 1% and 5% probabilities and one day and two week horizons, although other combinations are in use. The reason for assuming normal markets and no trading, and to restricting loss to things measured in daily accounts, is to make the loss observable. In some extreme financial events it can be impossible to determine losses, either because market prices are unavailable or because the loss-bearing institution breaks up. Some longer-term consequences of disasters, such as lawsuits, loss of market confidence and employee morale and impairment of brand names can take a long time to play out, and may be hard to allocate among specific prior decisions. VaR marks the boundary between normal days and extreme events. Institutions can lose far more than the VaR amount; all that can be said is that they will not do so very often. The probability level is about equally often specified as one minus the probability of a VaR break, so that the VaR in the example above would be called a one-day 95% VaR instead of one-day 5% VaR. This generally does not lead to confusion because the probability of VaR breaks is almost always small, certainly less than 50%. Although it virtually always represents a loss, VaR is conventionally reported as a positive number. A negative VaR would imply the portfolio has a high probability of making a profit, for example a one-day 5% VaR of negative implies the portfolio has a 95% chance of making more than over the next day. Another inconsistency is that VaR is sometimes taken to refer to profit-and-loss at the end of the period, and sometimes as the maximum loss at any point during the period. The original definition was the latter, but in the early 1990s when VaR was aggregated across trading desks and time zones, end-of-day valuation was the only reliable number so the former became the ''
de facto ''De facto'' ( ; , "in fact") describes practices that exist in reality, whether or not they are officially recognized by laws or other formal norms. It is commonly used to refer to what happens in practice, in contrast with '' de jure'' ("by l ...
'' definition. As people began using multiday VaRs in the second half of the 1990s, they almost always estimated the distribution at the end of the period only. It is also easier theoretically to deal with a point-in-time estimate versus a maximum over an interval. Therefore, the end-of-period definition is the most common both in theory and practice today.


Varieties

The definition of VaR is nonconstructive; it specifies a
property Property is a system of rights that gives people legal control of valuable things, and also refers to the valuable things themselves. Depending on the nature of the property, an owner of property may have the right to consume, alter, share, r ...
VaR must have, but not how to compute VaR. Moreover, there is wide scope for interpretation in the definition. This has led to two broad types of VaR, one used primarily in risk management and the other primarily for risk measurement. The distinction is not sharp, however, and hybrid versions are typically used in financial
control Control may refer to: Basic meanings Economics and business * Control (management), an element of management * Control, an element of management accounting * Comptroller (or controller), a senior financial officer in an organization * Controlli ...
, financial reporting and computing regulatory capital. To a
risk manager In simple terms, risk is the possibility of something bad happening. Risk involves uncertainty about the effects/implications of an activity with respect to something that humans value (such as health, well-being, wealth, property or the environm ...
, VaR is a system, not a number. The system is run periodically (usually daily) and the published number is compared to the computed price movement in opening positions over the time horizon. There is never any subsequent adjustment to the published VaR, and there is no distinction between VaR breaks caused by input errors (including IT breakdowns,
fraud In law, fraud is intentional deception to secure unfair or unlawful gain, or to deprive a victim of a legal right. Fraud can violate civil law (e.g., a fraud victim may sue the fraud perpetrator to avoid the fraud or recover monetary compen ...
and rogue trading), computation errors (including failure to produce a VaR on time) and market movements. A frequentist claim is made that the long-term frequency of VaR breaks will equal the specified probability, within the limits of sampling error, and that the VaR breaks will be
independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independe ...
in time and independent of the level of VaR. This claim is validated by a
backtest Backtesting is a term used in modeling to refer to testing a predictive model on historical data. Backtesting is a type of retrodiction, and a special type of cross-validation applied to previous time period(s). Financial analysis In a tradin ...
, a comparison of published VaRs to actual price movements. In this interpretation, many different systems could produce VaRs with equally good backtests, but wide disagreements on daily VaR values. For risk measurement a number is needed, not a system. A
Bayesian probability Bayesian probability is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation representing a state of knowledge or as quantification ...
claim is made that given the information and beliefs at the time, the
subjective probability Bayesian probability is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation representing a state of knowledge or as quantification o ...
of a VaR break was the specified level. VaR is adjusted after the fact to correct errors in inputs and computation, but not to incorporate information unavailable at the time of computation. In this context, "backtest" has a different meaning. Rather than comparing published VaRs to actual market movements over the period of time the system has been in operation, VaR is retroactively computed on scrubbed data over as long a period as data are available and deemed relevant. The same position data and pricing models are used for computing the VaR as determining the price movements. Although some of the sources listed here treat only one kind of VaR as legitimate, most of the recent ones seem to agree that risk management VaR is superior for making short-term and tactical decisions in the present, while risk measurement VaR should be used for understanding the past, and making medium term and strategic decisions for the future. When VaR is used for financial control or financial reporting it should incorporate elements of both. For example, if a trading desk is held to a VaR limit, that is both a risk-management rule for deciding what risks to allow today, and an input into the risk measurement computation of the desk's risk-adjusted
return Return may refer to: In business, economics, and finance * Return on investment (ROI), the financial gain after an expense. * Rate of return, the financial term for the profit or loss derived from an investment * Tax return, a blank document or t ...
at the end of the reporting period.


In governance

VaR can also be applied to
governance Governance is the process of interactions through the laws, norms, power or language of an organized society over a social system ( family, tribe, formal or informal organization, a territory or across territories). It is done by the g ...
of endowments, trusts, and pension plans. Essentially, trustees adopt portfolio Values-at-Risk metrics for the entire pooled account and the diversified parts individually managed. Instead of probability estimates they simply define maximum levels of acceptable loss for each. Doing so provides an easy metric for oversight and adds accountability as managers are then directed to manage, but with the additional constraint to avoid losses within a defined risk parameter. VaR utilized in this manner adds relevance as well as an easy way to monitor risk measurement control far more intuitive than Standard Deviation of Return. Use of VaR in this context, as well as a worthwhile critique on board governance practices as it relates to investment management oversight in general can be found in ''Best Practices in Governance.''


Mathematical definition

Let X be a profit and loss distribution (loss negative and profit positive). The VaR at level \alpha\in(0,1) is the smallest number y such that the probability that Y:=-X does not exceed y is at least 1-\alpha. Mathematically, \operatorname_(X) is the (1-\alpha)-
quantile In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities, or dividing the observations in a sample in the same way. There is one fewer quantile th ...
of Y, i.e., :\operatorname_\alpha(X)=-\inf\big\ = F^_Y(1-\alpha). This is the most general definition of VaR and the two identities are equivalent (indeed, for any real random variable X its
cumulative distribution function In probability theory and statistics, the cumulative distribution function (CDF) of a real-valued random variable X, or just distribution function of X, evaluated at x, is the probability that X will take a value less than or equal to x. Eve ...
F_X is well defined). However this formula cannot be used directly for calculations unless we assume that X 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. Nassim Taleb has labeled this assumption, "charlatanism". On the other hand, many academics prefer to assume a well-defined distribution, albeit usually one with
fat tails A fat-tailed distribution is a probability distribution that exhibits a large skewness or kurtosis, relative to that of either a normal distribution or an exponential distribution. In common usage, the terms fat-tailed and heavy-tailed are somet ...
. This point has probably caused more contention among VaR theorists than any other. Value of Risks can also be written as a distortion risk measure given by the distortion function g(x) = \begin0 & \text0 \leq x < 1-\alpha\\ 1 & \text1-\alpha \leq x \leq 1\end.


Risk measure and risk metric

The term "VaR" is used both for a risk measure and a risk metric. This sometimes leads to confusion. Sources earlier than 1995 usually emphasize the risk measure, later sources are more likely to emphasize the metric. The VaR risk measure defines risk as mark-to-market loss on a fixed portfolio over a fixed time horizon. There are many alternative risk measures in finance. Given the inability to use mark-to-market (which uses market prices to define loss) for future performance, loss is often defined (as a substitute) as change in fundamental value. For example, if an institution holds a
loan In finance, a loan is the lending of money by one or more individuals, organizations, or other entities to other individuals, organizations, etc. The recipient (i.e., the borrower) incurs a debt and is usually liable to pay interest on that ...
that declines in market price because
interest In finance and economics, interest is payment from a borrower or deposit-taking financial institution to a lender or depositor of an amount above repayment of the principal sum (that is, the amount borrowed), at a particular rate. It is distin ...
rates go up, but has no change in cash flows or credit quality, some systems do not recognize a loss. Also some try to incorporate the
economic An economy is an area of the production, distribution and trade, as well as consumption of goods and services. In general, it is defined as a social domain that emphasize the practices, discourses, and material expressions associated with th ...
cost of harm not measured in daily financial statements, such as loss of market confidence or employee morale, impairment of brand names or lawsuits. Rather than assuming a static portfolio over a fixed time horizon, some risk measures incorporate the dynamic effect of expected trading (such as a
stop loss order An order is an instruction to buy or sell on a trading venue such as a stock market, bond market, commodity market, financial derivative market or cryptocurrency exchange. These instructions can be simple or complicated, and can be sent to either ...
) and consider the expected holding period of positions. The VaR risk metric summarizes the
distribution Distribution may refer to: Mathematics * Distribution (mathematics), generalized functions used to formulate solutions of partial differential equations *Probability distribution, the probability of a particular value or value range of a vari ...
of possible losses by a
quantile In statistics and probability, quantiles are cut points dividing the range of a probability distribution into continuous intervals with equal probabilities, or dividing the observations in a sample in the same way. There is one fewer quantile th ...
, a point with a specified probability of greater losses. A common alternative metrics is expected shortfall.


VaR risk management

Supporters of VaR-based risk management claim the first and possibly greatest benefit of VaR is the improvement in
systems A system is a group of interacting or interrelated elements that act according to a set of rules to form a unified whole. A system, surrounded and influenced by its environment, is described by its boundaries, structure and purpose and expresse ...
and modeling it forces on an institution. In 1997, Philippe Jorionbr>wrote
e greatest benefit of VAR lies in the imposition of a structured methodology for critically thinking about risk. Institutions that go through the process of computing their VAR are forced to confront their exposure to financial risks and to set up a proper risk management function. Thus the process of getting to VAR may be as important as the number itself.
Publishing a daily number, on-time and with specified
statistical Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industr ...
properties holds every part of a trading organization to a high objective standard. Robust backup systems and default assumptions must be implemented. Positions that are reported, modeled or priced incorrectly stand out, as do data feeds that are inaccurate or late and systems that are too-frequently down. Anything that affects profit and loss that is left out of other reports will show up either in inflated VaR or excessive VaR breaks. "A risk-taking institution that ''does not'' compute VaR might escape disaster, but an institution that ''cannot'' compute VaR will not." The second claimed benefit of VaR is that it separates risk into two regimes. Inside the VaR limit, conventional
statistical Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industr ...
methods are reliable. Relatively short-term and specific data can be used for analysis. Probability estimates are meaningful because there are enough data to test them. In a sense, there is no true risk because these are a sum of many
independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s * Independe ...
observations with a left bound on the outcome. For example, a casino does not worry about whether red or black will come up on the next roulette spin. Risk managers encourage productive risk-taking in this regime, because there is little true cost. People tend to worry too much about these risks because they happen frequently, and not enough about what might happen on the worst days. Outside the VaR limit, all bets are off. Risk should be analyzed with
stress testing Stress testing (sometimes called torture testing) is a form of deliberately intense or thorough testing used to determine the stability of a given system, critical infrastructure or entity. It involves testing beyond normal operational capacity, ...
based on long-term and broad market data. Probability statements are no longer meaningful. Knowing the distribution of losses beyond the VaR point is both impossible and useless. The risk manager should concentrate instead on making sure good plans are in place to limit the loss if possible, and to survive the loss if not. One specific system uses three regimes. # One to three times VaR are normal occurrences. Periodic VaR breaks are expected. The loss distribution typically has
fat tails A fat-tailed distribution is a probability distribution that exhibits a large skewness or kurtosis, relative to that of either a normal distribution or an exponential distribution. In common usage, the terms fat-tailed and heavy-tailed are somet ...
, and there might be more than one break in a short period of time. Moreover, markets may be abnormal and trading may exacerbate losses, and losses taken may not be measured in daily marks, such as lawsuits, loss of employee morale and market confidence and impairment of brand names. An institution that cannot deal with three times VaR losses as routine events probably will not survive long enough to put a VaR system in place. # Three to ten times VaR is the range for
stress testing Stress testing (sometimes called torture testing) is a form of deliberately intense or thorough testing used to determine the stability of a given system, critical infrastructure or entity. It involves testing beyond normal operational capacity, ...
. Institutions should be confident they have examined all the foreseeable events that will cause losses in this range, and are prepared to survive them. These events are too rare to estimate probabilities reliably, so risk/return calculations are useless. # Foreseeable events should not cause losses beyond ten times VaR. If they do they should be hedged or insured, or the business plan should be changed to avoid them, or VaR should be increased. It is hard to run a business if foreseeable losses are orders of magnitude larger than very large everyday losses. It is hard to plan for these events because they are out of scale with daily experience. Another reason VaR is useful as a metric is due to its ability to compress the riskiness of a portfolio to a single number, making it comparable across different portfolios (of different assets). Within any portfolio it is also possible to isolate specific positions that might better hedge the portfolio to reduce, and minimise, the VaR.The Pricing and Hedging of Interest Rate Derivatives: A Practical Guide to Swaps
J H M Darbyshire, 2016,


Computation methods

VaR can be estimated either parametrically (for example,
variance In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
-
covariance In probability theory and statistics, covariance is a measure of the joint variability of two random variables. If the greater values of one variable mainly correspond with the greater values of the other variable, and the same holds for the le ...
VaR or delta-
gamma Gamma (uppercase , lowercase ; ''gámma'') is the third letter of the Greek alphabet. In the system of Greek numerals it has a value of 3. In Ancient Greek, the letter gamma represented a voiced velar stop . In Modern Greek, this letter r ...
VaR) or nonparametrically (for examples, historical
simulation A simulation is the imitation of the operation of a real-world process or system over time. Simulations require the use of Conceptual model, models; the model represents the key characteristics or behaviors of the selected system or proc ...
VaR or resampled VaR). Nonparametric methods of VaR estimation are discussed in Markovich and Novak. A comparison of a number of strategies for VaR prediction is given in Kuester et al. A McKinsey report published in May 2012 estimated that 85% of large banks were using historical simulation. The other 15% used Monte Carlo methods.


Backtesting

Backtesting is the process to determine the accuracy of VaR forecasts vs. actual portfolio profit and losses. A key advantage to VaR over most other measures of risk such as expected shortfall is the availability of several backtesting procedures for validating a set of VaR forecasts. Early examples of backtests can be found in Christoffersen (1998), later generalized by Pajhede (2017), which models a "hit-sequence" of losses greater than the VaR and proceed to tests for these "hits" to be independent from one another and with a correct probability of occurring. E.g. a 5% probability of a loss greater than VaR should be observed over time when using a 95% VaR, these hits should occur independently. A number of other backtests are available which model the time between hits in the hit-sequence, see Christoffersen and Pelletier (2004), Haas (2006), Tokpavi et al. (2014). and Pajhede (2017) As pointed out in several of the papers, the asymptotic distribution is often poor when considering high levels of coverage, e.g. a 99% VaR, therefore the parametric bootstrap method of Dufour (2006) is often used to obtain correct size properties for the tests. Backtest toolboxes are available in Matlab, o
R
��though only the first implements the parametric bootstrap method. The second pillar of
Basel II Basel II is the second of the Basel Accords, which are recommendations on banking laws and regulations issued by the Basel Committee on Banking Supervision. It is now extended and partially superseded by Basel III. The Basel II Accord was publi ...
includes a backtesting step to validate the VaR figures.


History

The problem of risk measurement is an old one in
statistics Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, indust ...
,
economics Economics () is the social science that studies the production, distribution, and consumption of goods and services. Economics focuses on the behaviour and interactions of economic agents and how economies work. Microeconomics anal ...
and
finance Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of production, distribution, and consumption of money, assets, goods and services (the discipline of f ...
. Financial risk management has been a concern of regulators and financial executives for a long time as well. Retrospective analysis has found some VaR-like concepts in this history. But VaR did not emerge as a distinct concept until the late 1980s. The triggering event was the stock market crash of 1987. This was the first major financial crisis in which a lot of academically-trained quants were in high enough positions to worry about firm-wide survival. The crash was so unlikely given standard
statistical Statistics (from German: '' Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industr ...
models, that it called the entire basis of quant finance into question. A reconsideration of history led some quants to decide there were recurring crises, about one or two per decade, that overwhelmed the statistical assumptions embedded in models used for trading,
investment management Investment management is the professional asset management of various securities, including shareholdings, bonds, and other assets, such as real estate, to meet specified investment goals for the benefit of investors. Investors may be instit ...
and
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
pricing. These affected many markets at once, including ones that were usually not
correlated In statistics, correlation or dependence is any statistical relationship, whether causal or not, between two random variables or bivariate data. Although in the broadest sense, "correlation" may indicate any type of association, in statistic ...
, and seldom had discernible economic cause or warning (although after-the-fact explanations were plentiful). Much later, they were named "
Black Swans Black swan is the common name for ''Cygnus atratus'', an Australasian waterfowl. (The) Black Swan(s) may also refer to: Film and television * ''The Black Swan'' (film), a 1942 swashbuckler film * ''Black Swans'' (film), a 2005 Dutch drama film * ' ...
" by Nassim Taleb and the concept extended far beyond
finance Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of production, distribution, and consumption of money, assets, goods and services (the discipline of f ...
. If these events were included in quantitative analysis they dominated results and led to strategies that did not work day to day. If these events were excluded, the profits made in between "Black Swans" could be much smaller than the losses suffered in the crisis. Institutions could fail as a result. VaR was developed as a systematic way to segregate extreme events, which are studied qualitatively over long-term history and broad market events, from everyday price movements, which are studied quantitatively using short-term data in specific markets. It was hoped that "Black Swans" would be preceded by increases in estimated VaR or increased frequency of VaR breaks, in at least some markets. The extent to which this has proven to be true is controversial. Abnormal markets and trading were excluded from the VaR estimate in order to make it observable. It is not always possible to define loss if, for example, markets are closed as after
9/11 The September 11 attacks, commonly known as 9/11, were four coordinated suicide terrorist attacks carried out by al-Qaeda against the United States on Tuesday, September 11, 2001. That morning, nineteen terrorists hijacked four commerci ...
, or severely illiquid, as happened several times in 2008. Losses can also be hard to define if the risk-bearing institution fails or breaks up. A measure that depends on traders taking certain actions, and avoiding other actions, can lead to
self reference Self-reference occurs in natural or formal languages when a sentence, idea or formula refers to itself. The reference may be expressed either directly—through some intermediate sentence or formula—or by means of some encoding. In philosop ...
. This is risk management VaR. It was well established in quantitative trading groups at several financial institutions, notably
Bankers Trust Bankers Trust was a historic American banking organization. The bank merged with Alex. Brown & Sons in 1997 before being acquired by Deutsche Bank in 1999. Deutsche Bank sold the Trust and Custody division of Bankers Trust to State Street Corp ...
, before 1990, although neither the name nor the definition had been standardized. There was no effort to aggregate VaRs across trading desks. The financial events of the early 1990s found many firms in trouble because the same underlying bet had been made at many places in the firm, in non-obvious ways. Since many trading desks already computed risk management VaR, and it was the only common risk measure that could be both defined for all businesses and aggregated without strong assumptions, it was the natural choice for reporting firmwide risk. J. P. Morgan CEO Dennis Weatherstone famously called for a "4:15 report" that combined all firm
risk In simple terms, risk is the possibility of something bad happening. Risk involves uncertainty about the effects/implications of an activity with respect to something that humans value (such as health, well-being, wealth, property or the environm ...
on one page, available within 15 minutes of the market close. Risk measurement VaR was developed for this purpose. Development was most extensive at J. P. Morgan, which published the methodology and gave free access to estimates of the necessary underlying parameters in 1994. This was the first time VaR had been exposed beyond a relatively small group of quants. Two years later, the methodology was spun off into an independent for-profit business now part of RiskMetrics Group (now part of
MSCI MSCI Inc. is an American finance company headquartered in New York City. MSCI is a global provider of equity, fixed income, real estate indexes, multi-asset portfolio analysis tools, ESG and climate products. It operates the MSCI World, MSCI ...
). In 1997, the U.S. Securities and Exchange Commission ruled that public corporations must disclose quantitative information about their derivatives activity. Major
bank A bank is a financial institution that accepts Deposit account, deposits from the public and creates a demand deposit while simultaneously making loans. Lending activities can be directly performed by the bank or indirectly through capital m ...
s and dealers chose to implement the rule by including VaR information in the notes to their financial statements. Worldwide adoption of the Basel II Accord, beginning in 1999 and nearing completion today, gave further impetus to the use of VaR. VaR is the preferred measure of
market risk Market risk is the risk of losses in positions arising from movements in market variables like prices and volatility. There is no unique classification as each classification may refer to different aspects of market risk. Nevertheless, the most ...
, and concepts similar to VaR are used in other parts of the accord.


Criticism

VaR has been controversial since it moved from trading desks into the public eye in 1994. A famous 199
debate
between Nassim Taleb and Philippe Jorion set out some of the major points of contention. Taleb claimed VaR: # 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 and Aaron Brown debated VaR i
Global Association of Risk Professionals Review
ref name="Einhorn I" /> 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."
New York Times ''The New York Times'' (''the Times'', ''NYT'', or the Gray Lady) is a daily newspaper based in New York City with a worldwide readership reported in 2020 to comprise a declining 840,000 paid print subscribers, and a growing 6 million paid ...
reporter Joe Nocera wrote an extensive piec
Risk Mismanagement
ref name="Nocera">
on January 4, 2009, discussing the role VaR played in the
Financial crisis of 2007–2008 Finance is the study and discipline of money, currency and capital assets. It is related to, but not synonymous with economics, the study of production, distribution, and consumption of money, assets, goods and services (the discipline of ...
. 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 An anchor is a device, normally made of metal , used to secure a vessel to the bed of a body of water to prevent the craft from drifting due to wind or current. The word derives from Latin ''ancora'', which itself comes from the Greek � ...
reasons VaR leads to higher risk taking. VaR is not
subadditive In mathematics, subadditivity is a property of a function that states, roughly, that evaluating the function for the sum of two elements of the domain always returns something less than or equal to the sum of the function's values at each element. ...
: 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. 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, ch. 10. There are common abuses of VaR: # 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
backtest Backtesting is a term used in modeling to refer to testing a predictive model on historical data. Backtesting is a type of retrodiction, and a special type of cross-validation applied to previous time period(s). Financial analysis In a tradin ...
. 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 In probability theory and statistics, the multivariate normal distribution, multivariate Gaussian distribution, or joint normal distribution is a generalization of the one-dimensional ( univariate) normal distribution to higher dimensions. One ...
.


VaR, CVaR, RVaR and EVaR

The VaR is not a coherent risk measure since it violates the sub-additivity property, which is :\mathrm\; X,Y \in \mathbf ,\; \mathrm\; \rho(X + Y) \leq \rho(X) + \rho(Y). 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 . However VaR, unlike CVaR, has the property of being a robust statistic. A related class of risk measures is the 'Range Value at Risk' (RVaR), which is a robust version of CVaR. For X\in \mathbf_ (with \mathbf_ the set of all Borel
measurable function In mathematics and in particular measure theory, a measurable function is a function between the underlying sets of two measurable spaces that preserves the structure of the spaces: the preimage of any measurable set is measurable. This is in ...
s whose moment-generating function exists for all positive real values) we have :\text_(X)\leq \text_(X) \leq \text_(X)\leq\text_(X), where : \begin &\text_(X):=\inf_\,\\ &\text_(X) := \frac\int_0^ \text_(X)d\gamma,\\ &\text_(X) := \frac\int_^ \text_(X)d\gamma,\\ &\text_(X):=\inf_\, \end in which M_X(z) is the moment-generating function of at . In the above equations the variable denotes the financial loss, rather than wealth as is typically the case.


See also

*
Capital Adequacy Directive The Capital Adequacy Directive was a European directive that aimed to establish uniform capital requirements for both banking firms and non-bank securities firms, first issued in 1993 and revised in 1998. These was superseded by the Capital Requ ...
* Conditional value-at-risk / Expected shortfall * Cyber risk quantification based on cyber value-at-risk or CyVaR * EMP for stochastic programming— solution technology for optimization problems involving VaR and CVaR * Entropic value at risk * * Profit at risk * Margin at risk *
Liquidity at risk The Liquidity-at-Risk (short: LaR) is a measure of the liquidity risk exposure of a financial portfolio. It may be defined as the net liquidity drain which can occur in the portfolio in a given risk scenario. If the Liquidity at Risk is greater ...
* Risk return ratio * Tail value at risk * Valuation risk


References


External links

;Discussion
"Value At Risk"
Ben Sopranzetti, Ph.D., CPA
"Perfect Storms" – Beautiful & True Lies In Risk Management
, Satyajit Das
"Gloria Mundi" – All About Value at Risk
Barry Schachter

Joe Nocera NY Times article.
"VaR Doesn't Have To Be Hard"
Rich Tanenbaum
"Coherent measures of Risk"
Philippe Artzner, Freddy Delbaen, Jean-Marc Eber, and David Heath ;Tools
"The Pricing and Trading of Interest Rate Derivatives"
J H M Darbyshire, MSc.
Online real-time VaR calculator
Razvan Pascalau,
University of Alabama The University of Alabama (informally known as Alabama, UA, or Bama) is a public research university in Tuscaloosa, Alabama. Established in 1820 and opened to students in 1831, the University of Alabama is the oldest and largest of the publ ...

Value-at-Risk (VaR)
Simon Benninga and Zvi Wiener. (Mathematica in Education and Research Vol. 7 No. 4 1998.)
Derivatives Strategy Magazine. "Inside D. E. Shaw"
Trading and Risk Management 1998
Simulate Historical Value at Risk
Online Calculator {{Authority control Actuarial science Financial risk modeling Market risk Monte Carlo methods in finance Credit risk