You can find links to our publicly available research articles below. The accompanied code is usually given by examples that use our open-source software. Please visit our open-source site for more information on how to install our open-source Python package and documentation of its functionality.

Portfolio Management Framework for Derivative Instruments

Abstract: The investment industry lacks an unified framework for handling derivative instruments in general portfolio management. With the increased use of derivatives, there is a need for a framework that aligns fundamental terminology and concepts. The main challenges with the current practices are caused by an improper separation of exposure / notional and market value / price. This tendency is also seen in the academic literature where exposures and prices are usually treated as identical quantities, e.g., in portfolio optimization. This article proposes a simple framework that can be used for all aspects of portfolio management and has intuitive properties that align with current conventions related to portfolio return. This feature allows us to perform portfolio optimization, risk decomposition, and performance evaluation in a familiar way.

Keywords: Portfolio management, derivative instruments, leverage, portfolio optimization, performance evaluation, CVaR, tail risks, market views, stress-testing, Entropy Pooling, Kullback-Leibler divergence.


Variance for Intuition, CVaR for Optimization

Abstract: This article presents some of the pros and cons of variance and CVaR as portfolio risk measures in mean-risk optimization. While variance is the original risk measure, thoroughly studied for the past 70 years, this article argues that there are practically no reasons for continuing to use variance instead of CVaR in mean-risk optimization. However, there are several reasons for using CVaR instead of variance. Nonetheless, mean-variance can be a useful tool for illustrating fundamental investment concepts and principles. The case study illustrates that mean-variance and mean-CVaR optimization converge to the same results when demeaned Gaussian P&L is used in CVaR optimization. Hence, contrary to what seems to be the current industry standard, it is recommended to use demeaned P&L for CVaR optimization.

Keywords: Portfolio optimization, mean-variance, mean-CVaR, Monte Carlo simulation, synthetic market data generator, tail risks, convex optimization, Python Programming Language.


Sequential Entropy Pooling Heuristics

Abstract: This article introduces two sequential heuristics that are designed to overcome some of the practical limitations of the Entropy Pooling (EP) method. Both heuristics repeatedly apply the original EP method to sequentially arrive at the posterior probability and usually lead to significantly better solutions than the original approach. In some cases, the sequential heuristics coincide with the original method, while they automatically ensure logical consistency in others. They are also able to solve interesting and practically relevant problems that the original approach simply cannot. Given the benefits of the sequential heuristics, this article argues that they should become the standard for future EP applications.

Keywords: Entropy Pooling, relative entropy, Kullback-Leibler divergence, change of measure, market views, stress-tests, Monte Carlo simulation, nonlinear convex optimization, heuristic algorithms, Python Programming Language.