Statistical Methods in Risk Management

As noted in Chapters 3 and 11, the theory of investment involves two fundamental concepts, namely expected returns and risk. In these chapters, we have conveniently measured risk by historic or model-based volatility in considering the optimal trade-off between maximizing expected returns and minimizing risk. In this chapter, we consider other risk measures and risk management from the corporate/regulatory perspective in order to provide safeguards (via capital reserves, hedging instruments, etc.) against extreme downside price movements, thereby protecting the company and its investors should these rare events occur.

Besides outlining important types of financial risks, Section 12.1 also describes several measures of market risk, in particular Value at Risk and Expected Shortfall. Section 12.2 introduces statistical methods and models for these risk measures in the case of a portfolio of assets. Section 12.3 considers the more complicated case of nonlinear financial instruments such as derivatives and describes simulation-based approaches besides commonly used linear and quadratic approximations. The Basel Committee requires backtesting of internal market risk models, which is considered in Section 12.2.4, and stress testing, which is considered in Section 12.4. In connection with stress testing, extreme value theory and computationally tractable Monte Carlo methods are introduced in Section 12.4.