thumb|300px|The 5% Value at Risk of a hypothetical profit-and-loss probability density function

Value at risk (VaR) is a measure of the risk of loss of investment/capital. 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, and probability 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 5% VaR of $1 million, that means that there is a 0.05 probability that the portfolio will fall in value by $1 million or more 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".

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 he wins $100 if this does not happen (with probability 127/128) and he loses $12,700 if it does happen (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). However, it is a controversial risk management tool.

Important related ideas are economic capital, backtesting, stress testing, 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.

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 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 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, financial reporting and computing regulatory capital.

To a risk manager, 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 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 in time and independent of the level of VaR. This claim is validated by a backtest, 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.

SEC Rule 18f-4 requires certain funds that use derivatives to comply with a relative VaR test as part of the fund’s leverage risk management framework. A fund relying on the rule generally must comply with an outer limit on fund leverage risk based on value-at-risk, or “VaR.” This outer limit is based on a relative VaR test that compares the fund’s VaR to the VaR of a “designated reference portfolio” for that fund. A fund generally can use either an index that meets certain requirements or the fund’s own securities portfolio (excluding derivatives transactions) as its designated reference portfolio. If the fund’s derivatives risk manager reasonably determines that a designated reference portfolio would not provide an appropriate reference portfolio for purposes of the relative VaR test, the fund would be required to comply with an absolute VaR test. The fund’s VaR generally is not permitted to exceed 200% of the VaR of the fund’s designated reference portfolio under the relative VaR test or 20% of the fund’s net assets under the absolute VaR test.

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>

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. 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.

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 that declines in market price because interest 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 cost of harm not measured in daily financial statements, such as loss of market confidence or employee morale, impairment of brand names or lawsuits.<blockquote>[T]he 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.</blockquote>

Publishing a daily number, on-time and with specified statistical 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 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 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. 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.

  1. One to three times VaR are normal occurrences. Periodic VaR breaks are expected. The loss distribution typically has fat tails, 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.
  2. Three to ten times VaR is the range for stress testing. 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.
  3. 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.

Computation methods

VaR can be estimated either parametrically (for example, variance-covariance VaR or delta-gamma VaR) or nonparametrically (for examples, historical simulation VaR or resampled VaR). 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 (often applying a PCA decomposition).

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) is often used to obtain correct size properties for the tests. Backtest toolboxes are available in Matlab, or R—though only the first implements the parametric bootstrap method.

The second pillar of Basel II includes a backtesting step to validate the VaR figures.

History

The problem of risk measurement is an old one in statistics, economics and finance. 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.

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. It is not always possible to define loss if, for example, markets are closed as after 9/11, or severely illiquid, as happened several times in 2008.

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

In 2008 David Einhorn and Aaron Brown debated VaR in Global Association of Risk Professionals Review. Einhorn compared VaR to "an airbag that works all the time, except when you have a car accident". He further charged that VaR:

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

New York Times reporter Joe Nocera wrote an extensive piece Risk Mismanagement on January 4, 2009, discussing the role VaR played in the 2008 financial crisis. 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 reasons VaR leads to higher risk taking.

VaR is not subadditive:

For <math> X\in \mathbf{L}_{M^+} </math> (with <math>\mathbf{L}_{M^+} </math> the set of all Borel measurable functions 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 at . In the above equations the variable denotes the financial loss, rather than wealth as is typically the case.

See also

  • Capital Adequacy Directive
  • 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
  • Risk return ratio
  • Tail value at risk

References

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;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
  • Risk Mismanagement, 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
  • 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