Portfolio Modeling with Heavy Tailed Random Vectors

Since the work of Mandelbrot in the 1960’s there has accumulated a great deal of empirical evidence for heavy tailed models in finance. In these models, the probability of a large fluctuation falls off like a power law. The generalized central limit theorem shows that these heavy-tailed fluctuations accumulate to a stable probability distribution. If the tails are not too heavy then the variance is finite and we find the familiar normal limit, a special case of stable distributions. Otherwise the limit is a nonnormal stable distribution, whose bell-shaped density may be skewed, and whose probability tails fall off like a power law. The most important model parameter for such distributions is the tail thickness α, which governs the rate at which the probability of large fluctuations diminishes. A smaller value of α means that the probability tails are fatter, implying more volatility. In fact, when α < 2 the theoretical variance is infinite. A portfolio can be modeled using random vectors, where each entry of the vector represents a different asset. The tail parameter α usually depends on the coordinate. The wrong coordinate system can mask variations in α, since the heaviest tail tends to dominate. A judicious choice of coordinate system is given by the eigenvectors of the sample covariance matrix. This isolates the heaviest tails, associated with the largest eigenvalues, and allows a more faithful representation of the dependence between assets.