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Traditional VaR methods, however, tend to underestimate the likelihood of extreme events because they usually assume that returns are normally or lognormally distributed. In reality, the empirical distribution of asset returns shows that extreme events, referred to as “fat tails,” are more likely to occur than normal distribution patterns imply. Assuming a normal distribution may underestimate the likelihood and magnitude of extreme losses. Risk estimates acceptable during normal periods are prone to fail during severe downturns. This problem of fat tails is not unique to financial markets and has received much attention in other disciplines, such as hydrology and structural engineering. Researchers in these disciplines approach this problem using extreme value theory, which focuses on the distribution of returns at the tails. This is precisely what is required for analyzing market crises, since these are extreme or tail events. The framework of normal distribution is only adequate as a reflection of average returns, not extreme ones. In extreme value theory, the distribution of the tail values instead follows the generalized extreme value (GEV) distribution. Once the tail distribution is determined, the VaR can then be computed as in the normal distribution.
Figure 4 provides an example, extended from Goldberg et al. [2008], of how extreme value theory can provide a better reflection of the downside risk. We compare the relative robustness of the traditional VaR and the extreme-value VaR for a variety of portfolios composed of U.S. equities. Daily returns are taken from December 1996 to October 2007, a period that covers major crises that include the Asian crisis, LTCM, Tech Bubble, Sept. 11 and the Quant Meltdown in August 2007. VaR figures are generated using two methods: the traditional way in which returns are assumed to be normally distributed and exponentially weighted across time, and through using extreme value theory. We choose a confidence level of 99 percent and a time horizon of one day, so that the resultant VaR figures should represent the maximum daily loss that would be incurred with 99 percent probability.
To compare the two measures over a variety of different portfolios, we evaluated 74 factor-tilted portfolios. It is important to note that the VaR numbers generated here are forecast values. In Figure 4, the horizontal axis is divided into interval ranges that denote the percentage of days in which actual losses are greater than the VaR, while the vertical axis displays the number of portfolios (out of the possible 74 in our sample) within each interval. Ideally, all portfolios should be in the expected range, left of the broken line. This is true for the majority of the portfolios when using the extreme-value VaR, but not in the case of the traditional VaR. While about 80 percent of the portfolios meet this criterion under the extreme-value measure, only 3 percent do so in the case of the traditional measure. Clearly, the estimates of risk generated using VaR are starkly different depending on whether the distribution is normal or not.
In addition to better modeling of fat tails, extreme value theory also introduces a tail-risk measure that provides a more complete reflection of the expected loss in a worst-case scenario. While VaR tells an investor his worst expected loss in 99 percent of the trading days, it does not indicate how severe the loss would be in the remaining 1 percent. Expected shortfall measures the expected loss within that worst 1 percent. Goldberg et al. [2009] have demonstrated how this new concept can be integrated in the standard tool kit used to measure portfolio risk, and show that different portfolios have distinct downside characteristics.
Options To Manage Tail Risk
BCP is a standard risk control practice for organizations, enabling critical processes to stay in operation even if a disaster strikes. This practice is well-established in many industries, including finance, as well as in nonbusiness organizations such as the military. We suggest that an analogous concept, which will be referred to as Portfolio BCP, would be relevant for portfolio management.
Generally, Portfolio BCP would require the following steps to be implemented. Firstly, the definition of an extreme event has to be determined. This could be based on returns, volatility, tracking error, VaR, drawdown or a combination of these measures as captured in a scenario. The probability and severity of the extreme events can then be quantified, as was carried out in Bhansali [2008].
Secondly, thresholds have to be decided upon, so that the conditions that trigger BCP are clear to everyone involved. Thirdly, scenarios should be elaborated to cover the most likely current threats. In order to rehearse these potential extreme events, stress tests should be performed to simulate the performance of portfolios under such situations. Finally, portfolio trades reflecting the mitigating decisions should be prepared and preapproved by the various investment committees for fast and consistent execution, should those extreme events happen.
In this context, the importance of stress-testing under extreme conditions should be emphasized, since this would help determine how much tail risk should be hedged away. In recent years, stress-testing has attracted the attention both of regulators and practitioners as an important measure that complements traditional risk measures such as volatility, tracking error and VaR. Increasingly, regulators such as the Basel Committee and the EU Commission (UCITS III Directive) require that practitioners incorporate stress-testing into their regular risk management practice. Stress-testing is particularly important because it helps to mitigate the overdependence on recent historical data. Multi-asset class risk systems often include dozens of predefined stress-testing scenarios.
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