Value at risk: value at risk theory and practice, value at risk var, institute introduction risk securities value

Value at Risk (VaR) is defined with respect to a specific portfolio of financial assets, at a specified probability and a specified time horizon. The probability that the mark-to-market loss on the portfolio over the time horizon is greater than VaR, assuming normal markets and no trading, is the specified probability level.^[1]

For example, if a portfolio of stocks has a one-day 5% VaR of $1 million, there is a 5% probability that the portfolio will decline in value by more than $1 million over the next day, assuming markets are normal and there is no trading. Such an event is termed a “VaR break.”^[2]

Details

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

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, the only thing we can say is they won't do so very often.^[6]

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 0.5.^[1]

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 $1 million implies the portfolio has a 95% chance of making $1 million or more over the next day. ^[7]

Varieties of VaR

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.^[8] 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. ^[9]

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 Information Technology breakdowns, fraud and rogue trading), computation errors (including failure to produce a VaR on time) and market movements.^[10]

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.^[1]

For risk measurement a number is needed, not a system. A Bayesian probability claim is made, that given the information and beliefs at the time, the subjective probability 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.^[7] 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.^[2]

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 today, 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 is 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 at the end of the reporting period.^[4]

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, assuming normal markets. There are many alternative risk measures in finance. Instead of mark-to-market, which uses market prices to define loss, loss is often defined 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. Or we could try to incorporate the economic cost of things not measured in daily financial statements, such as loss of market confidence or employee morale, impairment of brand names or lawsuits.^[4]

Rather than assuming a fixed portfolio over a fixed time horizon, some risk measures incorporate the effect of expected trading (such as a stop loss order) and consider the expected holding period of positions. Finally, some risk measures adjust for the possible effects of abnormal markets, rather than excluding them from the computation.^[4]

VaR risk management

Supporters of VaR-based risk management claim the first and possibly greatest benefit of VaR is the improvement in systems and modeling it forces on an institution. In 1997, Philippe Jorion wrote:^[11]

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.” ^[12]

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 you have a sum many independent observations with a left bound on the outcome. A casino doesn't 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.^[13]

Outside the VaR limit, all bets are off. Risk should be analyzed with stress testing based on long-term and broad market data.^[14] Probability statements are no longer meaningful.^[15] 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]

"A risk manager has two jobs: make people take more risk the 99% of the time it is safe to do so, and survive the other 1% of the time. VaR is the border."^[12]

VaR risk measurement

The VaR risk measure is a popular way to aggregate risk across an institution. Individual business units have risk measures such as duration for a fixed income portfolio or beta for an equity business. These cannot be combined in a meaningful way.^[1] It is also difficult to aggregate results available at different times, such as positions marked in different time zones, or a high frequency trading desk with a business holding relatively illiquid positions. But since every business contributes to profit and loss in an additive fashion, and many financial businesses mark-to-market daily, it is natural to define firm-wide risk using the distribution of possible losses at a fixed point in the future.^[4]

History of VaR

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.^[1]

The crash was so unlikely given standard statistical 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 and derivative pricing. These affected many markets at once, including ones that were usually not correlated, and seldom had discernable economic cause or warning (although after-the-fact explanations were plentiful).^[15] Much later, they were named Black Swans by Nassim Taleb and the concept extended far beyond finance.^[18]

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.^[15]^[18]^[12]

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 has proven to be true is controversial.^[15]

Abnormal markets and trading were excluded from the VaR estimate in order to make it observable.^[13] 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.^[12] Losses can also be hard to define if the risk-bearing institution fails or breaks up.^[13] A measure that depends on traders taking certain actions, and avoiding other actions, can lead to self reference.^[1]

This is risk management VaR. It was well-established in quantative trading groups at several financial institutions, notably Bankers Trust, before 1990, although neither the name nor the definition had been standardized. There was no effort to aggregate VaRs across trading desks.^[15]

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 on one page, available within 15 minutes of the market close.^[8]

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.^[8]

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, and concepts similar to VaR are used in other parts of the accord.^[1]

Mathematics

"Given some confidence level $\alpha \in (0,1)$ the VaR of the portfolio at the confidence level

α

is given by the smallest number

l

such that the probability that the loss

L

exceeds

l

is not larger than

(1 - α)

"^[3]

The left equality is a definition of VaR. The right equality assumes an underlying probability distribution, which makes it true only for parametric VaR. 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.^[6] Nassim Taleb has labeled this assumption, "charlatanism."^[19] On the other hand, many academics prefer to assume a well-defined distribution, albeit usually one with fat tails.^[1] This point has probably caused more contention among VaR theorists than any other.^[8]

Criticism

VaR has been controversial since it moved off of trading desks and into the public eye in 1994. A famous 1997 debate between Nassim Taleb and Philippe Jorion set out some of the major points of contention. Taleb claimed VaR:^[20]

A common complaint among academics is that VaR is not subadditive.^[4] That means the VaR of a combined portfolio can be larger than the sum of the VaRs of its components. To a practicing risk manager this makes sense. For example, the average bank branch in the United States is robbed about once every ten years. A single-branch bank has about 0.004% 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.^[12]

Cited references

^ ^a ^b ^c ^d ^e ^f ^g ^h ⁱ ^j ^k Philippe Jorion, Value at Risk: The New Benchmark for Managing Financial Risk, 3rd ed. McGraw-Hill (2006). ISBN 978-0071464956
^ ^a ^b Glyn Holton, Value-at-Risk: Theory and Practice, Academic Press (2003). ISBN 978-0123540102.
^ ^a ^b Alexander McNeil, Rüdiger Frey and Paul Embrechts, Quantitative Risk Management: Concepts Techniques and Tools, Princeton University Press (2005). ISBN 978-0691122557
^ ^a ^b ^c ^d ^e ^f ^g Kevin Dowd, Measuring Market Risk. John Wiley & Sons (2005) ISBN 978-0470013038.
^ Neil Pearson, Risk Budgeting: Portfolio Problem Solving with Value-at-Risk. John Wiley & Sons (2002). ISBN 978-0471405566.
^ ^a ^b ^c ^d Aaron Brown, The Unbearable Lightness of Cross-Market Risk, Wilmott Magazine, March 2004.
^ ^a ^b Michel Crouhy, Dan Galai and Robert Mark, The Essentials of Risk Management. McGraw-Hill (2001) ISBN 978-0071429665
^ ^a ^b ^c ^d ^e Joe Kolman, Michael Onak, Philippe Jorion, Nassim Taleb, Emanuel Derman, Blu Putnam, Richard Sandor, Stan Jonas, Ron Dembo, George Holt, Richard Tanenbaum, William Margrabe, Dan Mudge, James Lam and Jim Rozsypal, Roundtable: The Limits of VaR. Derivatives Strategy, April 1998.
^ Aaron Brown, The Next Ten VaR Disasters. Derivatives Strategy, March 1997.
^ Paul Wilmott, Paul Wilmott Introduces Quantitative Finance. Wiley (2007). ISBN: 978-0470319581
^ Philippe Jorion in Nassim Taleb and Philippe Jorion, The Jorion/Taleb Debate. Derivatives Strategy, April 1997.
^ ^a ^b ^c ^d ^e ^f ^g Aaron Brown, in David Einhorn and Aaron Brown, Private Profits and Socialized Risk. GARP Risk Review (June/July 2008).
^ ^a ^b ^c Espen Haug, Derivative Models on Models. John Wiley & Sons (2007). ISBN:978-0470013229
^ Ezra Zask, Taking the Stress Out of Stress Testing. Derivative Strategy, February 1999.
^ ^a ^b ^c ^d ^e Joe Kolman, Michael Onak, Philippe Jorion, Nassim Taleb, Emanuel Derman, Blu Putnam, Richard Sandor, Stan Jonas, Ron Dembo, George Holt, Richard Tanenbaum, William Margrabe, Dan Mudge, James Lam and Jim Rozsypal, Roundtable: The Limits of Models. Derivatives Strategy, April 1998.
^ Aaron Brown, On Stressing the Right Size. GARP Risk Review, December 2007.
^ Paul Glasserman, Monte Carlo Methods in Financial Engineering. Springer (2004). ISBN 978-0387004518.
^ ^a ^b Nassim Taleb, The Black Swan: The Impact of the Highly Improbable. Random House (2007). ISBN: 978-1400063512
^ Nassim Taleb, The World According to Nassim Taleb. Derivatives Strategy, December 1996/January 1997.
^ Nassim Taleb in Philippe Jorion in Nassim Taleb and Philippe Jorion, The Jorion/Taleb Debate. Derivatives Strategy, April 1997.
^ David Einhorn in David Einhorn and Aaron Brown, Private Profits and Socialized Risk. GARP Risk Review (June/July 2008).

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