Estimate of Downside Risk with Fat-Tailed and Skewed Models

JAMES X. XIONG, PhD, CFA

Senior Research Consultant, Ibbotson Associates, A Morningstar Company

Abstract: Asset returns are often not normally distributed and exhibit several stylized empirical facts: fat tails, skewness, finite variance, time scaling, and volatility clustering. Modeling the tail distribution of asset returns plays an essential role in downside risk management. The “left tail” of the distribution is where market crashes or crises occur. Downside risk can be measured in terms of conditional value-at-risk and estimated by fat-tailed and skewed models such as Lévy stable, truncated Lévy flight, skewed Student's t, mixture of normal distributions, and GARCH models. These fat-tailed and skewed models have different characteristics in describing the tail distribution of asset returns. The objective is to select appropriate ones that can accurately model the downside risk.

The financial crisis of 2008 has led many practitioners and academics to reassess the adequacy of the return distribution models, in particular, the left tail. This entry focuses on modeling the left fat tails since they reflect market crashes or crises and play an essential role in downside risk management.

The most common model of asset returns is assumed to be normally or Gaussian distributed (see Bachelier, 1900). In other words, the returns follow a random walk or Brownian motion. This model is natural if one assumes the return over ...

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