Random Variables and Expectations

The rapid globalization of financial and product markets, innovations in the design of derivative instruments, and the spectacular investor losses associated with derivatives over the past two decades have made financial institutions recognize the growing importance of risk management. A primary tool for financial risk assessment is the value-at-risk (VaR) measure, which is defined as the potential loss in value a portfolio of risky assets over a certain holding period at a given confidence level (probability). The use of VaR and its variants in risk management has exploded over the past decade because of its endorsement by bank regulators. Financial institutions now routinely use VaR techniques in managing their trading risk. Many implementations of VaR assume that asset returns are normally distributed. This assumption simplifies the computation of VaR considerably. However, it is inconsistent with the empirical evidence of asset returns, which finds that the distribution of asset returns is skewed and fat tailed. This implies that extreme events are much more likely to occur in practice than would be predicted by the symmetric thinner-tailed normal distribution. This also suggests that the normality assumption can produce VaR numbers that are inappropriate measures of the true risk faced by financial institutions. In addition to departures from normality, there is substantial evidence for time-varying probability distributions ...

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