Farid AitSahlia

Warrington College of Business Administration, University of Florida

Recent stress tests performed on some of the largest banks appear to support the observation that they are better capitalized. While it is too early to assess the impact of the financial regulations just passed in the US, it is clear that trading risk still remains under the purview of financial institutions. As the past two years have clearly illustrated, accurate evaluation of correlated tail outcomes is paramount in risk management. This issue contains papers that highlight the variety of perspectives one may have in the pursuit of this goal. They include identifying the correct dynamic model, selecting the appropriate parameter estimator and setting the right downside target for a given risk profile.

The first paper, by Eberlein and Madan, presents a return distribution modeling view. In this context, Lévy processes have gained significant acceptance in capturing dynamics involving jumps and fat tails. They better fit financial time series data, which commonly exhibit high skewness and kurtosis, and better model volatility smile effects observed in option markets. Although most of the applications are univariate, their paper develops a simple framework for modeling multi-asset dependence. By exploiting the property that a Lévy process is a time-changed Brownian motion, they build dependence by correlating Brownian motions. They observe that for such processes the actual correlation between the returns will be below the Brownian correlations but show how to recover the latter from the knowledge of marginal laws.

The second paper, by Weiß, is focused on estimation issues arising in the copula approach. Weiß presents a comprehensive simulation study of the finite-sample properties of several copula parameter estimators. His results show that, in most settings, the canonical maximum likelihood method yields smaller estimation biases with less computational effort than any of the minimum distance estimators based on copula goodness-of-fit tests. There exist, however, cases (especially when the sample size increases) where minimum distance estimators based on the empirical copula process are superior to the maximum likelihood estimator. Minimum distance estimators based on Kendall’s transform, on the other hand, yield only suboptimal results in all configurations of the simulation study. The estimates for these risk measures differ considerably depending on the choice of parameter estimator. Weiß stresses, therefore, the need for carefully choosing the parameter estimators in contrast to focusing all attention on choosing the parametric copula model. For practical applications, these results are of high relevance as copula models can vastly improve correlation-based models if calibrated correctly. At the same time, model misspecification and biased estimation can lead to woefully wrong estimates of risk measures thus partly justifying the criticism, especially of the Gaussian copula model, during the subprime crisis.

In the third paper of this issue, Mansouri et al conduct an empirical study comparing a number of value-at-risk forecasts from distributions arising from various long-memory GARCH-based volatility models. Using daily data from different countries covering more than 11 years, they find that the long-memory, asymmetric FIAPARCH model generally performs best. Their results are in line with similar studies and illustrate the generalized nature of FIAPARCH, which can fold into several other long-memory GARCH-based models.

While the previous three papers do not make explicit reference to an investor’s risk attitude, the fourth, by Olmo, directly addresses risk preferences through a utility function that captures downside risk in an alternative form to disappointment and loss aversion forms. In this model investors are not only averse to downside movements of stocks but also to the variance of their investments. In this framework, Olmo develops an asset pricing equilibrium formula in which the risk premium on a risky asset is given by a weighted sum of the regular beta capital asset pricing model and a market portfolio downside risk beta. An appealing feature of this model is its statistical tractability. In particular, Olmo develops a simple econometric model to test for the significance of the correlation between market and stock returns under distress and, more importantly, he develops a statistical device to test for the numerical value and statistical significance of the target return. This threshold value is considered endogenous to the agent or market and not exogenously given, as is done in most of the related literature so far. This new model is competitive against the benchmark three-factor model proposed in the literature when tested with real data.