Value at risk - Knowledge (XXG)

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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.

2300: 182:. 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. 1872: 320:

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

2295:{\displaystyle {\begin{aligned}&{\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{aligned}}} 1450:

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.

4379: 1432:. 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. 38: 52:) 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. 3267: 248:, 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 113:

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

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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

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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

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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,

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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

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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

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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

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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

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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

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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

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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 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

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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

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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

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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

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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

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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

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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

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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,

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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

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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

1204:. 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 208:

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

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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.

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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

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Kolman, Joe; Onak, Michael; Jorion, Philippe; Taleb, Nassim; Derman, Emanuel; Putnam, Blu; Sandor, Richard; Jonas, Stan; Dembo, Ron; Holt, George; Tanenbaum, Richard; Margrabe, William; Mudge, Dan; Lam, James; Rozsypal, Jim (April 1998). "Roundtable: The Limits of Models".

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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

1088:. 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. 1259:

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.

1607: 1271:, 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 1081:, 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. 2703:

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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

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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

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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.

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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

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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, or

585: 275:, 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. 820: 1309:, 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 1877: 521: 1674: 1267:

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

1077:, 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 1710: 303:

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

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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

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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.

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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

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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

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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.

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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.

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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

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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

948:. This sometimes leads to confusion. Sources earlier than 1995 usually emphasize the risk measure, later sources are more likely to emphasize the metric. 3432: 2887: 3518: 3201: 803:

has labeled this assumption, "charlatanism". On the other hand, many academics prefer to assume a well-defined distribution, albeit usually one with

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VaR can be defined informally as the maximum possible loss during that time after excluding all worse outcomes whose combined probability is at most

3714: 2518: 1856:{\displaystyle {\text{VaR}}_{1-\alpha }(X)\leq {\text{RVaR}}_{\alpha ,\beta }(X)\leq {\text{CVaR}}_{1-\alpha }(X)\leq {\text{EVaR}}_{1-\alpha }(X),} 979:

Rather than assuming a static portfolio over a fixed time horizon, some risk measures incorporate the dynamic effect of expected trading (such as a

3680: 1360: 1020: 4414: 2381: 1321: 2866: 1286:, before 1990, although neither the name nor the definition had been standardized. There was no effort to aggregate VaRs across trading desks. 4161: 3464: 3147: 3027: 2749: 2663: 2610: 2585: 2549: 2469: 1240:, and seldom had discernible economic cause or warning (although after-the-fact explanations were plentiful). Much later, they were named " 229:

and the other primarily for risk measurement. The distinction is not sharp, however, and hybrid versions are typically used in financial

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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

3407: 3108: 2506: 1385: 715:{\displaystyle \operatorname {VaR} _{\alpha }(X)=-\inf {\big \{}x\in \mathbb {R} :F_{X}(x)>\alpha {\big \}}=F_{Y}^{-1}(1-\alpha ).} 1429: 1393: 1476:) in the past. A common violation of common sense is to estimate a VaR based on the unverified assumption that everything follows a 1313:. Two years later, the methodology was spun off into an independent for-profit business now part of RiskMetrics Group (now part of 4404: 1477: 746: 171:

Common parameters for VaR are 1% and 5% probabilities and one day and two week horizons, although other combinations are in use.

4029: 1256: 1137: 926:{\displaystyle g(x)={\begin{cases}0&{\text{if }}0\leq x<1-\alpha \\1&{\text{if }}1-\alpha \leq x\leq 1\end{cases}}.} 4308: 4111: 3083: 2995: 2719: 2633: 1389: 1141: 4383: 4369: 3665: 2523: 3707: 1145: 160: 3368:"Monte carlo tests with nuisance parameters: A general approach to finite-sample inference and nonstandard asymptotics" 2920: 725:

This is the most general definition of VaR and the two identities are equivalent (indeed, for any real random variable

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One to three times VaR are normal occurrences. Periodic VaR breaks are expected. The loss distribution typically has

144:. VaR is sometimes used in non-financial applications as well. However, it is a controversial risk management tool. 1413:

Was "potentially catastrophic when its use creates a false sense of security among senior executives and watchdogs."

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Assuming that plausible losses will be less than some multiple (often three) of VaR. Losses can be extremely large.

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Supporters of VaR-based risk management claim the first and possibly greatest benefit of VaR is the improvement in

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The second claimed benefit of VaR is that it separates risk into two regimes. Inside the VaR limit, conventional

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businesses and aggregated without strong assumptions, it was the natural choice for reporting firmwide risk.

4313: 4034: 4019: 3961: 3700: 1130: 811: 152: 3525: 1631:. A related class of risk measures is the 'Range Value at Risk' (RVaR), which is a robust version of CVaR. 4009: 3227: 2387: 1679: 1617: 316: 249: 3617: 3543: 4351: 4253: 4226: 4211: 3979: 3834: 3755: 3669: 3297:

Christoffersen, Peter; Pelletier, Denis (2004). "Backtesting Value-at-Risk: A Duration-Based Approach".

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VaR has been controversial since it moved from trading desks into the public eye in 1994. A famous 1997

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is well defined). However this formula cannot be used directly for calculations unless we assume that

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The reason for assuming normal markets and no trading, and to restricting loss to things measured in

137: 844: 4346: 4303: 3814: 3232: 1716: 1295: 1205: 815: 238: 141: 56: 1472:. Regardless of how VaR is computed, it should have produced the correct number of breaks (within 4323: 4318: 4298: 4186: 3989: 3947: 3844: 3770: 3566: 3503: 3245: 2953: 2803: 2417: 2371: 1417: 1157: 995: 526: 218: 156: 2780: 1170: 3692: 976:, such as loss of market confidence or employee morale, impairment of brand names or lawsuits. 461: 4201: 4191: 4181: 4136: 4131: 4085: 4081: 4054: 3984: 3901: 3775: 3765: 3642: 3460: 3426: 3143: 3104: 3023: 2837: 2833: 2745: 2659: 2606: 2581: 2545: 2465: 2412: 2407: 2308: 1628: 1241: 991: 1374:

Was charlatanism because it claimed to estimate the risks of rare events, which is impossible

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that works all the time, except when you have a car accident". He further charged that VaR:

1225: 1122: 1118: 980: 304: 300: 3677:, Simon Benninga and Zvi Wiener. (Mathematica in Education and Research Vol. 7 No. 4 1998.) 3411: 2851: 1332:

and dealers chose to implement the rule by including VaR information in the notes to their

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Foreseeable events should not cause losses beyond ten times VaR. If they do they should be

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because they happen frequently, and not enough about what might happen on the worst days.

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Focused on the manageable risks near the center of the distribution and ignored the tails

994:, a point with a specified probability of greater losses. A common alternative metric is 260:), computation errors (including failure to produce a VaR on time) and market movements. 3686: 1446:: VaR of a combined portfolio can be larger than the sum of the VaRs of its components. 351:

be a profit and loss distribution (loss negative and profit positive). The VaR at level

4221: 4077: 4073: 4064: 3932: 3911: 3859: 3849: 3839: 3829: 3798: 3780: 3723: 3164: 2422: 2402: 2397: 1620:(EVaR). CVaR is defined by average of VaR values for confidence levels between 0 and 1473: 1367:

and Philippe Jorion set out some of the major points of contention. Taleb claimed VaR:

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Risk measurement VaR was developed for this purpose. Development was most extensive at

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A McKinsey report published in May 2012 estimated that 85% of large banks were using

807:. This point has probably caused more contention among VaR theorists than any other. 179: 2957: 4283: 4268: 4044: 3906: 3867: 3612: 3456: 2807: 2737: 1236:

pricing. These affected many markets at once, including ones that were usually not

1208:. This was the first major financial crisis in which a lot of academically-trained 941: 800: 257: 60: 41:

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

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Ignored 2,500 years of experience in favor of untested models built by non-traders

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ruled that public corporations must disclose quantitative information about their

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Jose A. Lopez (September 1996). "Regulatory Evaluation of Value-at-Risk Models".

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VaR is defined such that the probability of a loss greater than VaR is (at most)

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The Pricing and Hedging of Interest Rate Derivatives: A Practical Guide to Swaps

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end-of-day valuation was the only reliable number so the former became the

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Artzner, Philippe; Delbaen, Freddy; Eber, Jean-Marc; Heath, David (1999).

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in time and independent of the level of VaR. This claim is validated by a

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Estimated potential loss for an investment under a given set of conditions

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Haas, M. (2006). "Improved duration-based backtesting of value-at-risk".

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denotes the financial loss, rather than wealth as is typically the case.

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medium term and strategic decisions for the future. When VaR is used for

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Outside the VaR limit, all bets are off. Risk should be analyzed with

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claim is made that given the information and beliefs at the time, the

4243: 3283: 1397: 1351:, and concepts similar to VaR are used in other parts of the accord. 3544:"Robustness and Sensitivity Analysis of Risk Measurement Procedures" 3392: 3351:

Tokpavi, S. "Backtesting Value-at-Risk: A GMM Duration-Based Test".

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Led to excessive risk-taking and leverage at financial institutions

1173:—though only the first implements the parametric bootstrap method. 101:. A loss which exceeds the VaR threshold is termed a "VaR breach". 3655: 3169:"Value-at-Risk Prediction: A Comparison of Alternative Strategies" 253: 1212:

were in high enough positions to worry about firm-wide survival.

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definition is the most common both in theory and practice today.

4258: 1329: 1314: 1299: 961: 3696: 3602:"Perfect Storms" – Beautiful & True Lies In Risk Management 3218:

Christoffersen, Peter (1998). "Evaluating interval forecasts".

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Cyber risk quantification based on cyber value-at-risk or CyVaR

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on one page, available within 15 minutes of the market close.

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Risk Budgeting: Portfolio Problem Solving with Value-at-Risk

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Value at Risk: The New Benchmark for Managing Financial Risk

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while the probability of a loss less than VaR is (at least)

3268:"Backtesting Value-at-Risk: A Generalized Markov Framework" 2542:

Quantitative Risk Management: Concepts Techniques and Tools

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famously called for a "4:15 report" that combined all firm

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compute VaR might escape disaster, but an institution that

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of getting to VAR may be as important as the number itself.

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implies the portfolio has a 95% chance of making more than

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McNeil, Alexander; Frey, Rüdiger; Embrechts, Paul (2005).

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However, it can be bounded by coherent risk measures like

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on January 4, 2009, discussing the role VaR played in the

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it should incorporate elements of both. For example, if a

2998:(June–July 2008). "Private Profits and Socialized Risk". 1410:

Created an incentive to take "excessive but remote risks"

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VaR can be estimated either parametrically (for example,

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For risk measurement a number is needed, not a system. A

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since it violates the sub-additivity property, which is

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This is risk management VaR. It was well established in

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Cont, Rama; Deguest, Romain; Giacomo, Giacomo (2010).

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Nonparametric analysis of univariate heavy-tailed data

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However VaR, unlike CVaR, has the property of being a

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Publishing a daily number, on-time and with specified

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on 1 day out of 20 days (because of 5% probability).

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VaR) or nonparametrically (for examples, historical

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The Black Swan: The Impact of the Highly Improbable

1016:and modeling it forces on an institution. In 1997, 3681:Derivatives Strategy Magazine. "Inside D. E. Shaw" 3140:Extreme value methods with applications to finance 2333: 2294: 1855: 1704: 1668: 1601: 1282:groups at several financial institutions, notably 925: 787: 767: 737: 714: 571: 547: 515: 476: 450: 430: 401: 381: 343: 2654:Crouhy, Michel; Galai, Dan; Mark, Robert (2001). 1192:The problem of risk measurement is an old one in 516:{\displaystyle \operatorname {VaR} _{\alpha }(X)} 3086:(December 2007). "On Stressing the Right Size". 3062: 3060: 3058: 3056: 3054: 2928:Methodology and Computing in Applied Probability 2212: 1912: 617: 3524:. U.S. House of Representatives. Archived from 2698: 2696: 2694: 2692: 2690: 2683:. Wharton Financial Institutions Center: 96–51. 1394:Global Association of Risk Professionals Review 147:Important related ideas are economic capital, 3708: 3444: 3442: 2919:Balbás, A.; Garrido, J.; Mayoral, S. (2008). 2865:Nassim Taleb (December 1996 – January 1997), 2638:The Unbearable Lightness of Cross-Market Risk 1723:exists for all positive real values) we have 668: 622: 8: 3202:"McKinsey Working Papers on Risk, Number 32" 2990: 2988: 2986: 2984: 2982: 2742:Paul Wilmott Introduces Quantitative Finance 2628: 2626: 2624: 2622: 2501: 2499: 2282: 2227: 1970: 1929: 1101:portfolio to reduce, and minimise, the VaR. 1219:models, that it called the entire basis of 3738: 3715: 3701: 3693: 2828:Foellmer, Hans; Schied, Alexander (2004). 1547: 1532: 1511: 82:pricing, and no trading in the portfolio. 3391: 3231: 3184: 2947: 2316: 2310: 2271: 2256: 2234: 2215: 2187: 2182: 2146: 2141: 2134: 2129: 2107: 2083: 2078: 2042: 2037: 2030: 2025: 2011: 1987: 1982: 1938: 1922: 1915: 1887: 1882: 1876: 1874: 1829: 1824: 1799: 1794: 1769: 1764: 1739: 1734: 1731: 1694: 1689: 1684: 1681: 1669:{\displaystyle X\in \mathbf {L} _{M^{+}}} 1658: 1653: 1648: 1639: 1533: 1524: 1503: 1501: 1439:reasons VaR leads to higher risk taking. 1215:The crash was so unlikely given standard 960:. For example, if an institution holds a 887: 852: 839: 822: 799:a well-defined probability distribution. 780: 759: 753: 730: 685: 680: 667: 666: 645: 634: 633: 621: 620: 593: 587: 564: 528: 495: 489: 463: 443: 414: 394: 356: 336: 3618:"Gloria Mundi" – All About Value at Risk 3133: 3131: 2921:"Properties of Distortion Risk Measures" 2649: 2647: 2483: 2481: 2455: 2453: 2451: 2449: 2447: 2445: 2443: 2441: 2439: 2437: 2393:Financial risk management § Banking 1084:Three to ten times VaR is the range for 1069:One specific system uses three regimes. 1008:Financial risk management § Banking 36: 4374: 3045:Taking the Stress Out of Stress Testing 3013: 3011: 3009: 2433: 1322:U.S. Securities and Exchange Commission 810:Value at risk can also be written as a 3431:: CS1 maint: archived copy as title ( 3424: 2871:, Derivatives Strategy, archived from 2571: 2569: 2567: 2565: 2563: 2561: 2349:. In the above equations the variable 1468:Reporting a VaR that has not passed a 964:that declines in market price because 2511:"Private Profits and Socialized Risk" 2341:is the moment-generating function of 951:The VaR risk measure defines risk as 7: 1705:{\displaystyle \mathbf {L} _{M^{+}}} 1248:and the concept extended far beyond 307:at the end of the reporting period. 2868:The World According to Nassim Taleb 2529:from the original on April 26, 2016 986:The VaR risk metric summarizes the 972:cost of harm not measured in daily 2490:Value-at-Risk: Theory and Practice 1543: 1540: 1537: 1534: 1507: 1504: 1184:step to validate the VaR figures. 940:The term "VaR" is used both for a 795:has some parametric distribution. 25: 3687:Simulate Historical Value at Risk 3353:Journal of Financial Econometrics 3299:Journal of Financial Econometrics 3173:Journal of Financial Econometrics 2656:The Essentials of Risk Management 1255:If these events were included in 4377: 3990:Conditional Value-at-Risk (CVaR) 3683:Trading and Risk Management 1998 3142:. Chapman & Hall/CRC Press. 1923: 1685: 1649: 1525: 1478:multivariate normal distribution 1461:There are common abuses of VaR: 747:cumulative distribution function 382:{\displaystyle \alpha \in (0,1)} 3666:Online real-time VaR calculator 3449:Taleb, Nassim Nicholas (2007). 2971:Jorion, Philippe (April 1997). 2886:Julia L. Wirch; Mary R. Hardy. 1160:is the availability of several 409:such that the probability that 4415:Monte Carlo methods in finance 4309:Strategic financial management 4112:Asset and liability management 2544:. Princeton University Press. 2382:EMP for stochastic programming 2328: 2322: 2279: 2268: 2262: 2249: 2205: 2199: 2164: 2158: 2101: 2095: 2060: 2054: 2005: 1999: 1955: 1943: 1905: 1899: 1847: 1841: 1817: 1811: 1787: 1781: 1757: 1751: 1593: 1587: 1578: 1572: 1563: 1551: 1396:. Einhorn compared VaR to "an 833: 827: 706: 694: 657: 651: 608: 602: 542: 530: 510: 504: 376: 364: 1: 3637:"VaR Doesn't Have To Be Hard" 3517:Nassim Taleb (Sep 10, 2009). 3384:10.1016/j.jeconom.2005.06.007 3220:International Economic Review 2706:Roundtable: The Limits of VaR 2464:(3rd ed.). McGraw-Hill. 1430:Financial crisis of 2007–2008 1380:Would be exploited by traders 322:Best Practices in Governance. 2767:Best Practices in Governance 936:Risk measure and risk metric 161:tail conditional expectation 3887:Operational risk management 3643:"Coherent measures of Risk" 3480:Nassim Taleb (April 1997), 3043:Ezra Zask (February 1999), 3020:Derivative Models on Models 2781:"Coherent Measures of Risk" 548:{\displaystyle (1-\alpha )} 315:VaR can also be applied to 4436: 4059:Proportional hazards model 4010:Interest rate immunization 3103:, J H M Darbyshire, 2016, 2724:The Next Ten VaR Disasters 2363:Capital Adequacy Directive 1721:moment-generating function 1339:Worldwide adoption of the 1005: 124:VaR has four main uses in 29: 4342: 3728:financial risk management 3563:10.1080/14697681003685597 3167:; Paolella, Marc (2006). 3022:. John Wiley & Sons. 2940:10.1007/s11009-008-9089-z 2605:. John Wiley & Sons. 2580:. John Wiley & Sons. 2460:Jorion, Philippe (2006). 2368:Conditional value-at-risk 1614:Conditional Value-at-Risk 1424:wrote an extensive piece 1066:survive the loss if not. 477:{\displaystyle 1-\alpha } 217:The definition of VaR is 186:certainly less than 50%. 4005:First-hitting-time model 3970:Arbitrage pricing theory 3200:McKinsey & Company. 2487:Holton, Glyn A. (2014). 2334:{\displaystyle M_{X}(z)} 1485:VaR, CVaR, RVaR and EVaR 990:of possible losses by a 30:Not to be confused with 4405:Financial risk modeling 4314:Stress test (financial) 4020:Modern portfolio theory 3482:The Jorion-Taleb Debate 3372:Journal of Econometrics 2975:. Derivatives Strategy. 2973:The Jorion-Taleb Debate 2800:10.1111/1467-9965.00068 2708:. Derivatives Strategy. 2493:second edition, e-book. 1045:compute VaR will not." 812:distortion risk measure 389:is the smallest number 327:Mathematical definition 4384:Business and economics 3484:, Derivatives Strategy 3311:10.1093/jjfinec/nbh004 3272:Journal of Forecasting 3266:Pajhede, Thor (2017). 3186:10.1093/jjfinec/nbj002 3120:Markovich, N. (2007), 2765:Lawrence York (2009), 2726:, Derivatives Strategy 2601:Pearson, Neil (2002). 2388:Entropic value at risk 2335: 2296: 1857: 1706: 1670: 1618:entropic value at risk 1603: 1030: 927: 789: 769: 739: 716: 573: 549: 517: 478: 452: 432: 403: 383: 345: 284:subjective probability 178:, is to make the loss 42: 4352:Investment management 4254:Investment management 3980:Replicating portfolio 3756:Sovereign credit risk 3670:University of Alabama 3338:10.21314/JOR.2006.128 3047:, Derivative Strategy 2578:Measuring Market Risk 2336: 2297: 1858: 1707: 1671: 1604: 1491:coherent risk measure 1377:Gave false confidence 1257:quantitative analysis 1230:investment management 1176:The second pillar of 1140:. The other 15% used 1138:historical simulation 1025: 1006:Further information: 928: 790: 770: 768:{\displaystyle F_{X}} 740: 717: 574: 550: 518: 479: 453: 433: 431:{\displaystyle Y:=-X} 404: 384: 346: 40: 4357:Mathematical finance 4289:Risk-return spectrum 4279:Mathematical finance 4234:Fundamental analysis 4167:Exchange traded fund 3751:Consumer credit risk 3551:Quantitative Finance 3531:on November 4, 2009. 3366:Dufour, J-M (2006). 3138:Novak, S.Y. (2011). 3070:Derivatives Strategy 2836:. pp. 177–182. 2788:Mathematical Finance 2576:Dowd, Kevin (2005). 2309: 1873: 1730: 1717:measurable functions 1680: 1638: 1500: 1334:financial statements 1280:quantitative trading 974:financial statements 821: 779: 752: 729: 586: 563: 527: 488: 462: 442: 413: 393: 355: 335: 280:Bayesian probability 119:p ≤ 0.78125% (1/128) 4347:Financial economics 4304:Statistical finance 4070:Value-at-Risk (VaR) 3975:Black–Scholes model 3815:Holding period risk 3675:Value-at-Risk (VaR) 3668:, Razvan Pascalau, 3498:(January 4, 2009), 3018:Espen Haug (2007). 2139: 2035: 1296:Dennis Weatherstone 1146:a PCA decomposition 1142:Monte Carlo methods 1105:Computation methods 1002:VaR risk management 816:distortion function 693: 297:financial reporting 235:financial reporting 198:over the next day. 138:financial reporting 4324:Structured product 4319:Structured finance 4299:Speculative attack 3985:Cash flow matching 3948:Non-financial risk 3845:Interest rate risk 3771:Concentration risk 3624:Risk Mismanagement 3607:2009-03-25 at the 3504:The New York Times 3500:Risk Mismanagement 2830:Stochastic Finance 2640:, Wilmott Magazine 2509:(June–July 2008), 2418:Tail value at risk 2372:Expected shortfall 2331: 2292: 2290: 2226: 2125: 2021: 1928: 1853: 1702: 1666: 1599: 1426:Risk Mismanagement 1158:expected shortfall 996:expected shortfall 923: 915: 785: 765: 735: 712: 676: 569: 545: 513: 484:. Mathematically, 474: 448: 428: 399: 379: 341: 239:regulatory capital 157:expected shortfall 142:regulatory capital 43: 4400:Actuarial science 4365: 4364: 4137:Corporate finance 4132:Capital structure 4086:Cash flow at risk 4082:Liquidity at risk 4055:Survival analysis 3956: 3955: 3902:Reputational risk 3776:Credit derivative 3689:Online Calculator 3620:, Barry Schachter 3466:978-1-4000-6351-2 3149:978-1-4398-3574-6 3029:978-0-470-01322-9 2834:Walter de Gruyter 2751:978-0-470-31958-1 2665:978-0-07-142966-5 2612:978-0-471-40556-6 2587:978-0-470-01303-8 2551:978-0-691-12255-7 2471:978-0-07-146495-6 2413:Risk return ratio 2408:Liquidity at risk 2211: 2185: 2144: 2123: 2081: 2040: 2019: 1985: 1941: 1911: 1885: 1827: 1797: 1767: 1737: 1489:The VaR is not a 958:fundamental value 890: 855: 788:{\displaystyle X} 738:{\displaystyle X} 572:{\displaystyle Y} 451:{\displaystyle y} 402:{\displaystyle y} 344:{\displaystyle X} 293:financial control 221:; it specifies a 16:(Redirected from 4427: 4382: 4381: 4373: 4239:Growth investing 4157:Enterprise value 4107:Asset allocation 4090:Earnings at risk 4072:and extensions ( 4015:Market portfolio 3879:Operational risk 3864:Refinancing risk 3739: 3717: 3710: 3703: 3694: 3660:J H M Darbyshire 3639:, Rich Tanenbaum 3575: 3574: 3548: 3539: 3533: 3532: 3530: 3523: 3514: 3508: 3507: 3492: 3486: 3485: 3477: 3471: 3470: 3446: 3437: 3436: 3430: 3422: 3420: 3419: 3410:. Archived from 3404: 3398: 3397: 3395: 3363: 3357: 3356: 3348: 3342: 3341: 3321: 3315: 3314: 3294: 3288: 3287: 3284:10.1002/for.2456 3263: 3254: 3253: 3235: 3215: 3209: 3208: 3206: 3197: 3191: 3190: 3188: 3163:Kuester, Keith; 3160: 3154: 3153: 3135: 3126: 3125: 3117: 3111: 3098: 3092: 3091: 3088:GARP Risk Review 3080: 3074: 3073: 3064: 3049: 3048: 3040: 3034: 3033: 3015: 3004: 3003: 3000:GARP Risk Review 2992: 2977: 2976: 2968: 2962: 2961: 2951: 2925: 2916: 2910: 2909: 2907: 2905: 2899: 2893:. Archived from 2892: 2883: 2877: 2876: 2862: 2856: 2855: 2825: 2819: 2818: 2816: 2814: 2785: 2776: 2770: 2769: 2762: 2756: 2755: 2734: 2728: 2727: 2716: 2710: 2709: 2700: 2685: 2684: 2676: 2670: 2669: 2651: 2642: 2641: 2630: 2617: 2616: 2598: 2592: 2591: 2573: 2556: 2555: 2537: 2531: 2530: 2528: 2519:GARP Risk Review 2515: 2503: 2494: 2485: 2476: 2475: 2457: 2352: 2348: 2344: 2340: 2338: 2337: 2332: 2321: 2320: 2301: 2299: 2298: 2293: 2291: 2275: 2261: 2260: 2242: 2241: 2225: 2198: 2197: 2186: 2183: 2179: 2157: 2156: 2145: 2142: 2138: 2133: 2124: 2122: 2108: 2094: 2093: 2082: 2079: 2075: 2053: 2052: 2041: 2038: 2034: 2029: 2020: 2012: 1998: 1997: 1986: 1983: 1979: 1942: 1939: 1927: 1926: 1898: 1897: 1886: 1883: 1879: 1862: 1860: 1859: 1854: 1840: 1839: 1828: 1825: 1810: 1809: 1798: 1795: 1780: 1779: 1768: 1765: 1750: 1749: 1738: 1735: 1711: 1709: 1708: 1703: 1701: 1700: 1699: 1698: 1688: 1675: 1673: 1672: 1667: 1665: 1664: 1663: 1662: 1652: 1629:robust statistic 1623: 1608: 1606: 1605: 1600: 1546: 1528: 1510: 1328:activity. Major 1144:(often applying 932: 930: 929: 924: 919: 918: 891: 888: 856: 853: 794: 792: 791: 786: 774: 772: 771: 766: 764: 763: 744: 742: 741: 736: 721: 719: 718: 713: 692: 684: 672: 671: 650: 649: 637: 626: 625: 598: 597: 578: 576: 575: 570: 554: 552: 551: 546: 522: 520: 519: 514: 500: 499: 483: 481: 480: 475: 457: 455: 454: 449: 438:does not exceed 437: 435: 434: 429: 408: 406: 405: 400: 388: 386: 385: 380: 350: 348: 347: 342: 197: 193: 21: 4435: 4434: 4430: 4429: 4428: 4426: 4425: 4424: 4390: 4389: 4388: 4376: 4368: 4366: 4361: 4338: 4274:Systematic risk 4172:Expected return 4152:Economic bubble 4147:Diversification 4142:Cost of capital 4095: 3952: 3921: 3873: 3855:Volatility risk 3819:Price area risk 3785: 3761:Settlement risk 3730: 3721: 3609:Wayback Machine 3596:Ben Sopranzetti 3592:"Value At Risk" 3583: 3578: 3546: 3541: 3540: 3536: 3528: 3521: 3516: 3515: 3511: 3494: 3493: 3489: 3479: 3478: 3474: 3467: 3448: 3447: 3440: 3423: 3417: 3415: 3408:"Archived copy" 3406: 3405: 3401: 3365: 3364: 3360: 3350: 3349: 3345: 3326:Journal of Risk 3323: 3322: 3318: 3296: 3295: 3291: 3265: 3264: 3257: 3242:10.2307/2527341 3217: 3216: 3212: 3204: 3199: 3198: 3194: 3165:Mittnik, Stefan 3162: 3161: 3157: 3150: 3137: 3136: 3129: 3119: 3118: 3114: 3099: 3095: 3082: 3081: 3077: 3066: 3065: 3052: 3042: 3041: 3037: 3030: 3017: 3016: 3007: 2994: 2993: 2980: 2970: 2969: 2965: 2923: 2918: 2917: 2913: 2903: 2901: 2900:on July 5, 2016 2897: 2890: 2885: 2884: 2880: 2864: 2863: 2859: 2844: 2843:978-311-0183467 2827: 2826: 2822: 2812: 2810: 2783: 2778: 2777: 2773: 2764: 2763: 2759: 2752: 2736: 2735: 2731: 2718: 2717: 2713: 2702: 2701: 2688: 2678: 2677: 2673: 2666: 2658:. McGraw-Hill. 2653: 2652: 2645: 2632: 2631: 2620: 2613: 2600: 2599: 2595: 2588: 2575: 2574: 2559: 2552: 2539: 2538: 2534: 2526: 2513: 2505: 2504: 2497: 2486: 2479: 2472: 2459: 2458: 2435: 2431: 2359: 2350: 2346: 2342: 2312: 2307: 2306: 2289: 2288: 2252: 2230: 2181: 2177: 2176: 2140: 2112: 2077: 2073: 2072: 2036: 1981: 1977: 1976: 1881: 1871: 1870: 1823: 1793: 1763: 1733: 1728: 1727: 1712:the set of all 1690: 1683: 1678: 1677: 1654: 1647: 1636: 1635: 1621: 1498: 1497: 1487: 1392:debated VaR in 1357: 1341:Basel II Accord 1190: 1154: 1107: 1018:Philippe Jorion 1010: 1004: 981:stop loss order 938: 914: 913: 885: 879: 878: 850: 840: 819: 818: 777: 776: 755: 750: 749: 727: 726: 641: 589: 584: 583: 561: 560: 525: 524: 491: 486: 485: 460: 459: 440: 439: 411: 410: 391: 390: 353: 352: 333: 332: 329: 313: 227:risk management 219:nonconstructive 215: 195: 191: 169: 130:risk management 89:More formally, 78:. This assumes 35: 28: 23: 22: 15: 12: 11: 5: 4433: 4431: 4423: 4422: 4417: 4412: 4407: 4402: 4392: 4391: 4387: 4386: 4363: 4362: 4360: 4359: 4354: 4349: 4343: 4340: 4339: 4337: 4336: 4331: 4326: 4321: 4316: 4311: 4306: 4301: 4296: 4291: 4286: 4281: 4276: 4271: 4266: 4261: 4256: 4251: 4246: 4241: 4236: 4231: 4230: 4229: 4224: 4219: 4214: 4209: 4204: 4199: 4194: 4189: 4184: 4174: 4169: 4164: 4159: 4154: 4149: 4144: 4139: 4134: 4129: 4124: 4119: 4114: 4109: 4103: 4101: 4100:Basic concepts 4097: 4096: 4094: 4093: 4078:Margin at risk 4074:Profit at risk 4067: 4065:Tracking error 4062: 4052: 4047: 4042: 4037: 4035:Risk-free rate 4032: 4027: 4022: 4017: 4012: 4007: 4002: 3997: 3992: 3987: 3982: 3977: 3972: 3966: 3964: 3958: 3957: 3954: 3953: 3951: 3950: 3945: 3940: 3935: 3933:Execution risk 3929: 3927: 3923: 3922: 3920: 3919: 3914: 3912:Political risk 3909: 3904: 3899: 3894: 3889: 3883: 3881: 3875: 3874: 3872: 3871: 3860:Liquidity risk 3857: 3852: 3850:Inflation risk 3847: 3842: 3840:Margining risk 3837: 3832: 3830:Valuation risk 3827: 3822: 3799:Commodity risk 3795: 3793: 3787: 3786: 3784: 3783: 3781:Securitization 3778: 3773: 3768: 3763: 3758: 3753: 3747: 3745: 3736: 3732: 3731: 3724:Financial risk 3722: 3720: 3719: 3712: 3705: 3697: 3691: 3690: 3684: 3678: 3672: 3663: 3652: 3651: 3647: 3646: 3640: 3634: 3621: 3615: 3599: 3588: 3587: 3582: 3581:External links 3579: 3577: 3576: 3557:(6): 593–606. 3534: 3509: 3487: 3472: 3465: 3438: 3399: 3378:(2): 443–477. 3358: 3343: 3316: 3289: 3278:(5): 597–613. 3255: 3233:10.1.1.41.8009 3210: 3192: 3155: 3148: 3127: 3112: 3109:978-0995455511 3093: 3075: 3050: 3035: 3028: 3005: 2978: 2963: 2911: 2878: 2857: 2842: 2820: 2794:(3): 203–228. 2771: 2757: 2750: 2729: 2722:(March 1997), 2711: 2686: 2681:Working Papers 2671: 2664: 2643: 2636:(March 2004), 2618: 2611: 2593: 2586: 2557: 2550: 2532: 2507:Einhorn, David 2495: 2477: 2470: 2432: 2430: 2427: 2426: 2425: 2423:Valuation risk 2420: 2415: 2410: 2405: 2403:Margin at risk 2400: 2398:Profit at risk 2395: 2390: 2385: 2379: 2374: 2365: 2358: 2355: 2330: 2327: 2324: 2319: 2315: 2303: 2302: 2287: 2284: 2281: 2278: 2274: 2270: 2267: 2264: 2259: 2255: 2251: 2248: 2245: 2240: 2237: 2233: 2229: 2224: 2221: 2218: 2214: 2210: 2207: 2204: 2201: 2196: 2193: 2190: 2180: 2178: 2175: 2172: 2169: 2166: 2163: 2160: 2155: 2152: 2149: 2137: 2132: 2128: 2121: 2118: 2115: 2111: 2106: 2103: 2100: 2097: 2092: 2089: 2086: 2076: 2074: 2071: 2068: 2065: 2062: 2059: 2056: 2051: 2048: 2045: 2033: 2028: 2024: 2018: 2015: 2010: 2007: 2004: 2001: 1996: 1993: 1990: 1980: 1978: 1975: 1972: 1969: 1966: 1963: 1960: 1957: 1954: 1951: 1948: 1945: 1937: 1934: 1931: 1925: 1921: 1918: 1914: 1910: 1907: 1904: 1901: 1896: 1893: 1890: 1880: 1878: 1864: 1863: 1852: 1849: 1846: 1843: 1838: 1835: 1832: 1822: 1819: 1816: 1813: 1808: 1805: 1802: 1792: 1789: 1786: 1783: 1778: 1775: 1772: 1762: 1759: 1756: 1753: 1748: 1745: 1742: 1697: 1693: 1687: 1661: 1657: 1651: 1646: 1643: 1610: 1609: 1598: 1595: 1592: 1589: 1586: 1583: 1580: 1577: 1574: 1571: 1568: 1565: 1562: 1559: 1556: 1553: 1550: 1545: 1542: 1539: 1536: 1531: 1527: 1523: 1520: 1517: 1514: 1509: 1506: 1486: 1483: 1482: 1481: 1474:sampling error 1466: 1418:New York Times 1415: 1414: 1411: 1408: 1405: 1382: 1381: 1378: 1375: 1372: 1356: 1353: 1273:self reference 1189: 1186: 1153: 1150: 1106: 1103: 1098: 1097: 1089: 1086:stress testing 1082: 1063:stress testing 1003: 1000: 953:mark-to-market 937: 934: 922: 917: 912: 909: 906: 903: 900: 897: 894: 886: 884: 881: 880: 877: 874: 871: 868: 865: 862: 859: 851: 849: 846: 845: 843: 838: 835: 832: 829: 826: 784: 762: 758: 734: 723: 722: 711: 708: 705: 702: 699: 696: 691: 688: 683: 679: 675: 670: 665: 662: 659: 656: 653: 648: 644: 640: 636: 632: 629: 624: 619: 616: 613: 610: 607: 604: 601: 596: 592: 568: 544: 541: 538: 535: 532: 512: 509: 506: 503: 498: 494: 473: 470: 467: 447: 427: 424: 421: 418: 398: 378: 375: 372: 369: 366: 363: 360: 340: 328: 325: 312: 309: 237:and computing 214: 211: 176:daily accounts 168: 165: 153:stress testing 140:and computing 80:mark-to-market 32:Valuation risk 26: 24: 14: 13: 10: 9: 6: 4: 3: 2: 4432: 4421: 4418: 4416: 4413: 4411: 4408: 4406: 4403: 4401: 4398: 4397: 4395: 4385: 4380: 4375: 4371: 4358: 4355: 4353: 4350: 4348: 4345: 4344: 4341: 4335: 4332: 4330: 4329:Systemic risk 4327: 4325: 4322: 4320: 4317: 4315: 4312: 4310: 4307: 4305: 4302: 4300: 4297: 4295: 4292: 4290: 4287: 4285: 4282: 4280: 4277: 4275: 4272: 4270: 4267: 4265: 4262: 4260: 4257: 4255: 4252: 4250: 4247: 4245: 4242: 4240: 4237: 4235: 4232: 4228: 4225: 4223: 4220: 4218: 4215: 4213: 4210: 4208: 4205: 4203: 4200: 4198: 4195: 4193: 4190: 4188: 4185: 4183: 4180: 4179: 4178: 4175: 4173: 4170: 4168: 4165: 4163: 4160: 4158: 4155: 4153: 4150: 4148: 4145: 4143: 4140: 4138: 4135: 4133: 4130: 4128: 4127:Capital asset 4125: 4123: 4120: 4118: 4117:Asset pricing 4115: 4113: 4110: 4108: 4105: 4104: 4102: 4098: 4091: 4087: 4083: 4079: 4075: 4071: 4068: 4066: 4063: 4060: 4056: 4053: 4051: 4050:Sortino ratio 4048: 4046: 4043: 4041: 4038: 4036: 4033: 4031: 4028: 4026: 4023: 4021: 4018: 4016: 4013: 4011: 4008: 4006: 4003: 4001: 3998: 3996: 3993: 3991: 3988: 3986: 3983: 3981: 3978: 3976: 3973: 3971: 3968: 3967: 3965: 3963: 3959: 3949: 3946: 3944: 3943:Systemic risk 3941: 3939: 3936: 3934: 3931: 3930: 3928: 3924: 3918: 3915: 3913: 3910: 3908: 3905: 3903: 3900: 3898: 3895: 3893: 3892:Business risk 3890: 3888: 3885: 3884: 3882: 3880: 3876: 3869: 3865: 3861: 3858: 3856: 3853: 3851: 3848: 3846: 3843: 3841: 3838: 3836: 3833: 3831: 3828: 3826: 3823: 3820: 3816: 3812: 3808: 3804: 3800: 3797: 3796: 3794: 3792: 3788: 3782: 3779: 3777: 3774: 3772: 3769: 3767: 3764: 3762: 3759: 3757: 3754: 3752: 3749: 3748: 3746: 3744: 3740: 3737: 3733: 3729: 3725: 3718: 3713: 3711: 3706: 3704: 3699: 3698: 3695: 3688: 3685: 3682: 3679: 3676: 3673: 3671: 3667: 3664: 3661: 3657: 3654: 3653: 3649: 3648: 3644: 3641: 3638: 3635: 3632: 3629: 3625: 3622: 3619: 3616: 3614: 3610: 3606: 3603: 3600: 3597: 3593: 3590: 3589: 3585: 3584: 3580: 3572: 3568: 3564: 3560: 3556: 3552: 3545: 3538: 3535: 3527: 3520: 3513: 3510: 3505: 3501: 3497: 3491: 3488: 3483: 3476: 3473: 3468: 3462: 3458: 3454: 3453: 3445: 3443: 3439: 3434: 3428: 3414:on 2017-10-02 3413: 3409: 3403: 3400: 3394: 3389: 3385: 3381: 3377: 3373: 3369: 3362: 3359: 3354: 3347: 3344: 3339: 3335: 3331: 3327: 3320: 3317: 3312: 3308: 3304: 3300: 3293: 3290: 3285: 3281: 3277: 3273: 3269: 3262: 3260: 3256: 3251: 3247: 3243: 3239: 3234: 3229: 3226:(4): 841–62. 3225: 3221: 3214: 3211: 3203: 3196: 3193: 3187: 3182: 3178: 3174: 3170: 3166: 3159: 3156: 3151: 3145: 3141: 3134: 3132: 3128: 3123: 3116: 3113: 3110: 3106: 3102: 3097: 3094: 3089: 3085: 3079: 3076: 3071: 3063: 3061: 3059: 3057: 3055: 3051: 3046: 3039: 3036: 3031: 3025: 3021: 3014: 3012: 3010: 3006: 3001: 2997: 2991: 2989: 2987: 2985: 2983: 2979: 2974: 2967: 2964: 2959: 2955: 2950: 2945: 2941: 2937: 2933: 2929: 2922: 2915: 2912: 2896: 2889: 2882: 2879: 2875:on 2000-08-29 2874: 2870: 2869: 2861: 2858: 2853: 2849: 2845: 2839: 2835: 2831: 2824: 2821: 2809: 2805: 2801: 2797: 2793: 2789: 2782: 2775: 2772: 2768: 2761: 2758: 2753: 2747: 2743: 2739: 2738:Wilmott, Paul 2733: 2730: 2725: 2721: 2715: 2712: 2707: 2699: 2697: 2695: 2693: 2691: 2687: 2682: 2675: 2672: 2667: 2661: 2657: 2650: 2648: 2644: 2639: 2635: 2629: 2627: 2625: 2623: 2619: 2614: 2608: 2604: 2597: 2594: 2589: 2583: 2579: 2572: 2570: 2568: 2566: 2564: 2562: 2558: 2553: 2547: 2543: 2536: 2533: 2525: 2521: 2520: 2512: 2508: 2502: 2500: 2496: 2492: 2491: 2484: 2482: 2478: 2473: 2467: 2463: 2456: 2454: 2452: 2450: 2448: 2446: 2444: 2442: 2440: 2438: 2434: 2428: 2424: 2421: 2419: 2416: 2414: 2411: 2409: 2406: 2404: 2401: 2399: 2396: 2394: 2391: 2389: 2386: 2383: 2380: 2378: 2375: 2373: 2369: 2366: 2364: 2361: 2360: 2356: 2354: 2325: 2317: 2313: 2285: 2276: 2272: 2265: 2257: 2253: 2246: 2243: 2238: 2235: 2231: 2222: 2219: 2216: 2208: 2202: 2194: 2191: 2188: 2173: 2170: 2167: 2161: 2153: 2150: 2147: 2135: 2130: 2126: 2119: 2116: 2113: 2109: 2104: 2098: 2090: 2087: 2084: 2069: 2066: 2063: 2057: 2049: 2046: 2043: 2031: 2026: 2022: 2016: 2013: 2008: 2002: 1994: 1991: 1988: 1973: 1967: 1964: 1961: 1958: 1952: 1949: 1946: 1935: 1932: 1919: 1916: 1908: 1902: 1894: 1891: 1888: 1869: 1868: 1867: 1850: 1844: 1836: 1833: 1830: 1820: 1814: 1806: 1803: 1800: 1790: 1784: 1776: 1773: 1770: 1760: 1754: 1746: 1743: 1740: 1726: 1725: 1724: 1722: 1718: 1715: 1695: 1691: 1659: 1655: 1644: 1641: 1632: 1630: 1625: 1619: 1615: 1596: 1590: 1584: 1581: 1575: 1569: 1566: 1560: 1557: 1554: 1548: 1529: 1521: 1518: 1515: 1512: 1496: 1495: 1494: 1492: 1484: 1479: 1475: 1471: 1467: 1464: 1463: 1462: 1459: 1455: 1451: 1447: 1445: 1440: 1438: 1433: 1431: 1427: 1423: 1419: 1412: 1409: 1406: 1403: 1402: 1401: 1399: 1395: 1391: 1387: 1386:David Einhorn 1379: 1376: 1373: 1370: 1369: 1368: 1366: 1362: 1354: 1352: 1350: 1346: 1342: 1337: 1335: 1331: 1327: 1323: 1320:In 1997, the 1318: 1316: 1312: 1308: 1303: 1301: 1297: 1293: 1287: 1285: 1284:Bankers Trust 1281: 1276: 1274: 1270: 1265: 1261: 1258: 1253: 1251: 1247: 1243: 1239: 1235: 1231: 1227: 1222: 1218: 1213: 1211: 1207: 1206:crash of 1987 1203: 1199: 1195: 1187: 1185: 1183: 1179: 1174: 1172: 1166: 1163: 1159: 1151: 1149: 1147: 1143: 1139: 1134: 1132: 1128: 1124: 1120: 1116: 1112: 1104: 1102: 1094: 1090: 1087: 1083: 1080: 1076: 1072: 1071: 1070: 1067: 1064: 1059: 1056: 1051: 1046: 1044: 1040: 1035: 1029: 1024: 1022: 1019: 1015: 1009: 1001: 999: 997: 993: 989: 984: 982: 977: 975: 971: 967: 963: 959: 954: 949: 947: 943: 935: 933: 920: 910: 907: 904: 901: 898: 895: 892: 882: 875: 872: 869: 866: 863: 860: 857: 847: 841: 836: 830: 824: 817: 814:given by the 813: 808: 806: 802: 796: 782: 760: 756: 748: 732: 709: 703: 700: 697: 689: 686: 681: 677: 673: 663: 660: 654: 646: 642: 638: 630: 627: 614: 611: 605: 599: 594: 590: 582: 581: 580: 566: 558: 539: 536: 533: 507: 501: 496: 492: 471: 468: 465: 445: 425: 422: 419: 416: 396: 373: 370: 367: 361: 358: 338: 326: 324: 323: 318: 311:In governance 310: 308: 306: 302: 298: 294: 288: 285: 281: 276: 274: 270: 266: 261: 259: 258:rogue trading 255: 251: 247: 242: 240: 236: 232: 228: 224: 220: 212: 210: 207: 206: 199: 187: 183: 181: 177: 172: 166: 164: 162: 158: 154: 150: 145: 143: 139: 135: 131: 127: 122: 120: 116: 111: 107: 102: 100: 96: 92: 87: 83: 81: 77: 73: 69: 66: 62: 58: 53: 51: 47: 46:Value at risk 39: 33: 19: 18:Value-at-Risk 4284:Moral hazard 4269:Risk of ruin 4069: 4045:Sharpe ratio 3907:Country risk 3868:Deposit risk 3766:Default risk 3613:Satyajit Das 3598:, Ph.D., CPA 3554: 3550: 3537: 3526:the original 3512: 3499: 3490: 3481: 3475: 3457:Random House 3455:. New York: 3451: 3416:. Retrieved 3412:the original 3402: 3375: 3371: 3361: 3352: 3346: 3332:(2): 17–38. 3329: 3325: 3319: 3302: 3298: 3292: 3275: 3271: 3223: 3219: 3213: 3195: 3176: 3172: 3158: 3139: 3121: 3115: 3096: 3087: 3078: 3069: 3044: 3038: 3019: 2999: 2972: 2966: 2931: 2927: 2914: 2902:. Retrieved 2895:the original 2881: 2873:the original 2867: 2860: 2829: 2823: 2811:. Retrieved 2791: 2787: 2774: 2766: 2760: 2741: 2732: 2723: 2714: 2705: 2680: 2674: 2655: 2637: 2602: 2596: 2577: 2541: 2535: 2517: 2488: 2461: 2304: 1865: 1633: 1626: 1611: 1488: 1460: 1456: 1452: 1448: 1441: 1434: 1416: 1383: 1365:Nassim Taleb 1358: 1338: 1319: 1307:J. P. Morgan 1304: 1292:J. P. 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