We measure the contributions to risk of a set of factors, strategies, or investments, based on "Minimum-Torsion Bets", namely a set of uncorrelated factors, optimized to closely track the factors used to allocate the portfolio. We then introduce a novel definition of contributions to risk, which...

We introduce "factors on demand", a modular, multi-asset-class return decomposition framework that extends beyond the standard systematic-plus-idiosyncratic approach. This framework, which rests on the conditional link between flexible bottom-up estimation factor models and flexible top-down...

In the Flexible Probabilities approach, given the historical distribution (histogram) of the returns of a portfolio, we can stress-test the portfolio under different time periods and market environments, by adjusting the relative weights (Flexible Probabilities) of the historical returns in the...

Using the Bayesian posterior distribution of the market parameters we define self-adjusting uncertainty regions for the robust mean-variance problem. Under a normal-inverse-Wishart conjugate assumption for the market, the ensuing robust Bayesian mean-variance optimal portfolios are shrunk by the...

We review the main processes used to model financial variables. We emphasize the parallel between discrete-time processes, mainly used by econometricians for risk- and portfolio-management, and their continuous-time counterparts, mainly used by mathematicians to price derivatives. We highlight...

The Black-Litterman and related approaches modify the return distribution of a normally distributed market according to views or stress-test scenarios. We discuss how to broaden the range of applications of these approaches significantly by letting them act on the risk factors underlying the...

We describe a simple recursive routine to estimate by maximum likelihood the correlation matrix and the degrees of freedom of the t-copula, when structure needs to be imposed on the eigenvalues for dimensionality issues

We walk the reader through the Black-Litterman approach, providing all the proofs. We show how minor modifications of the original model greatly improve its range of applications. We discuss full generalizations of this and related models. MATLAB code is available through MATLAB Central