# Value at Risk calculations

## Measurement of potential losses in the value of the asset VaR. Approaches that are used to compute Value at Risk: the variance-covariance approach, the historical and the Monte Carlo simulations. Risk assessment value at risk as a qualitative tool.

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FINANCE UNIVERSITY

UNDER THE GOVERNMENT OF THE RUSSIAN FEDERATION

DEPARTMENT OF FOREIGN LANGUAGES - 1

ESSAY

“Value at Risk calculations”

Done by: Петухов Максим Сергеевич

Supervisor: Кантышева А.А.

Moscow 2014

Оглавление

• 1. What is VaR?
• 2. History of VaR
• 3. Measuring Value at Risk
• Variance-Covariance Method
• Historical Simulation
• Monte Carlo Simulation
• 4. Comparing Approaches
• 5. Limitations of VaR
• 6. VaR can be wrong
• 7. Criticism
• 8. Conclusion
• Bibliography

# There is no precise measure of Value at Risk, and each measure comes with its own limitations. The end-result is that the Value at Risk that we compute for an asset, portfolio or a firm can be wrong, and sometimes, the errors can be large enough to make VaR a misleading measure of risk exposure. The reasons for the errors can vary across firms and for different measures and include the following.

a. Return distributions: Every VaR measure makes assumptions about return distributions, which, if violated, result in incorrect estimates of the Value at Risk. With delta-normal estimates of VaR, we are assuming that the multivariate return distribution is the normal distribution, since the Value at Risk is based entirely on the standard deviation in returns. With Monte Carlo simulations, we get more freedom to specify different types of return distributions, but we can still be wrong when we make those judgments. Finally, with historical simulations, we are assuming that the historical return distribution (based upon past data) is representative of the distribution of returns looking forward.

b. History is not a good predictor: All measures of Value at Risk use historical data to some degree or the other. In the variance-covariance method, historical data is used to compute the variance-covariance matrix that is the basis for the computation of VaR. In historical simulations, the VaR is entirely based upon the historical data with the likelihood of value losses computed from the time series of returns. In Monte Carlo simulations, the distributions don't have to be based upon historical data but it is difficult to see how else they can be derived. In short, any Value at Risk measure will be a function of the time period over which the historical data is collected. If that time period was a relatively stable one, the computed Value at Risk will be a low number and will understate the risk looking forward. Conversely, if the time period examined was volatile, the Value at Risk will be set too high. Earlier in this chapter, we provided the example of VaR for oil price movements and concluded that VaR measures based upon the 1992-98 period, where oil prices were stable, would have been too low for the 1999-2004 period, when volatility returned to the market.

# 7. Criticism

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

VaR is claimed to:

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

More recently 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 Financial crisis of 2007-2008. After interviewing risk managers (including several of the ones cited above) the article suggests that VaR was very useful to risk experts, but nevertheless exacerbated the crisis by giving false security to bank executives and regulators. A powerful tool for professional risk managers, VaR is portrayed as both easy to misunderstand, and dangerous when misunderstood.

Taleb, in 2009, testified in Congress asking for the banning of VaR on two arguments, the first that "tail risks are non-measurable" scientifically and the second is that for anchoring reasons VaR for leading to higher risk taking.

A common complaint among academics is that VaR is not sub additive. That means the VaR of a combined portfolio can be larger than the sum of the VaRs of its components. To a practising 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 0.0004% chance of being robbed on a specific day, so the risk of robbery would not figure into one-day 1% VaR. It would not even be within an order of magnitude of that, so it is in the range where the institution should not worry about it, it should insure against it and take advice from insurers on precautions. The whole point of insurance is to aggregate risks that are beyond individual VaR limits, and bring them into a large enough portfolio to get statistical predictability. It does not pay for a one-branch bank to have a security expert on staff.

As institutions get more branches, the risk of a robbery on a specific day rises to within an order of magnitude of VaR. At that point it makes sense for the institution to run internal stress tests and analyze the risk itself. It will spend less on insurance and more on in-house expertise. For a very large banking institution, robberies are a routine daily occurrence. Losses are part of the daily VaR calculation, and tracked statistically rather than case-by-case. A sizable in-house security department is in charge of prevention and control, the general risk manager just tracks the loss like any other cost of doing business.

As portfolios or institutions get larger, specific risks change from low-probability/low-predictability/high-impact to statistically predictable losses of low individual impact. That means they move from the range of far outside VaR, to be insured, to near outside VaR, to be analyzed case-by-case, to inside VaR, to be treated statistically.

Even VaR supporters generally agree there are common abuses of VaR:

1. Referring to VaR as a "worst-case" or "maximum tolerable" loss. In fact, you expect two or three losses per year that exceed one-day 1% VaR.

2. Making VaR control or VaR reduction the central concern of risk management. It is far more important to worry about what happens when losses exceed VaR.

3. Assuming plausible losses will be less than some multiple, often three, of VaR. The entire point of VaR is that losses can be extremely large, and sometimes impossible to define, once you get beyond the VaR point. To a risk manager, VaR is the level of losses at which you stop trying to guess what will happen next, and start preparing for anything.

4. Reporting a VaR that has not passed a back test. Regardless of how VaR is computed, it should have produced the correct number of breaks (within sampling error) in the past. A common specific violation of this is to report a VaR based on the unverified assumption that everything follows a multivariate normal distribution.

# We understand why Value at Risk is a popular risk assessment tool in financial service firms, where assets are primarily marketable securities; there is limited capital at play and a regulatory overlay that emphasizes short term exposure to extreme risks. We are hard pressed to see why Value at Risk is of particular use to non-financial service firms, unless they are highly levered and risk default if cash flows or value fall below a pre-specified level. Even in those cases, it would seem to us to be more prudent to use all of the information in the probability distribution rather than a small slice of it.

## Bibliography

1. Stein, J.C., S.E. Usher, D. LaGattuta and J. Youngen, 2000, A Comparables Approach to Measuring Cashflow-at-Risk for Non-Financial Firms, Working Paper, National Economic Research Associates.

2. Larsen, N., H. Mausser and S. Ursyasev, 2001, Algorithms for Optimization of Value-at-Risk, Research Report, University of Florida.

3. Embrechts, P., 2001, Extreme Value Theory: Potential and Limitations as an Integrated Risk Management Tool, Working Paper (listed on GloriaMundi.org).

4. Basak, S. and A. Shapiro, 2001, Value-at-Risk Based Management: Optimal Policies and Asset Prices, Review of Financial Studies, v14 , 371-405.

5. Ju, X. and N.D. Pearson, 1998, Using Value-at-Risk to Control Risk Taking: How wrong can you be?, Working Paper, University of Illinois at Urbana-Champaign.

6. Hallerback, W.G. and A.J. Menkveld, 2002, Analyzing Perceived Downside Risk: the Component Value-at-Risk Framework, Working Paper.

7. Jorion, P., 2002, How informative are Value-at-Risk Disclosures?, The Accounting Review, v77, 911-932.

8. Marshall, Chris, and Michael Siegel, “Value at Risk: Implementing a Risk Measurement Standard.” Journal of Derivatives 4, No. 3 (1997), pp. 91-111. Different measures of Value at Risk are estimated using different software packages on the J.P. Morgan RiskMetrics data and methodology.

9. Berkowitz, J. and J. O'Brien, 2002, How accurate are Value at Risk Models at Commercial Banks, Journal of Finance, v57, 1093-1111.

10. Hendricks, D., 1996, Evaluation of value-at-risk models using historical data, Federal Reserve Bank of New York, Economic Policy Review, v2,, 39-70.

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