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Value at risk for a mixture of normal distributions: The use of quasi-Bayesian estimation techniques

Subu Venkataraman

Rapid globalization, innovations in the design of derivative securities, and examples of spectacular losses associated with derivatives over the past decade have made firms recognize the growing importance of risk management. This increased focus on risk management has led to the development of various methods and tools to measure the risks firms face. One popular risk-measurement tool is value at risk (VaR), which is defined as the minimum loss expected on a portfolio of assets over a certain holding period at a given confidence level (probability). For example, consider a trader who is concerned about the risk, over the next ten days, associated with holding a specific portfolio of assets. A statement that, at the 95 percent confidence level, the VaR of this portfolio is $100,000 implies that 95 percent of the time, losses over the 10-day holding period should not exceed $100,000 (or losses should exceed $100,000 only 5 percent of the time). The use of value at risk techniques in risk management has exploded over the last few years. Financial institutions now routinely use VaR techniques in managing their trading risk and nonfinancial firms have started adopting the technology for their risk-management purposes as well. In addition, regulators are beginning to design new regulations around it. Examples of these regulations include the determination of bank capital standards for market risk and the reporting requirements for the risks associated with derivatives used by corporations.

Proponents of VaR argue that the ability to quantify risk exposure into a single number represents the single most powerful advantage of the technique.1 Despite its simplicity, however, the technique is only as good as the inputs into the VaR model.2 Many implementations of VaR assume that asset returns are normally distributed. This assumption simplifies the...