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Abstract: Portfolio optimization has a mixed reputation among investment managers, with some being so skeptical that they believe it is almost useless due to the inherent parameter uncertainty. It is undeniable that portfolio optimization problems are sensitive to parameter estimates, especially the expected returns that are arguably also the hardest parameters to estimate. However, most practitioners still attempt to build mean-risk optimal portfolios, albeit in implicit ways. A popular mathematical heuristic to tackle the parameter uncertainty issue is called resampled optimization, which computes optimal portfolios using sampled parameter estimates and calculates a simple average of the portfolio exposures across samples. The unsatisfactory aspect of the resampled approach is that there is no mathematical justification for using the average of portfolio exposures, it just works well in practice. This article provides perspectives for understanding the resampling approach by analyzing the portfolio exposure estimation process from a bias-variance trade-off perspective. We show that the traditional resampled optimization corresponds to a naive version of stacked generalization. Finally, we introduce a stacked generalization approach that can be used to handle both parameter uncertainty and combine optimization methods in full generality. We coin the new method Exposure Stacking.
Keywords: Portfolio optimization, parameter uncertainty, Exposure Stacking, mean-CVaR, tail risk, mean-variance, efficient portfolio, efficient frontier, mean squared error, bias-variance trade-off, stacked generalization, quadratic programming, convex optimization, Python Programming Language.
Abstract: This article introduces a very flexible framework for causal and predictive market views and stress-testing. The framework elegantly combines Bayesian networks (BNs) and Entropy Pooling (EP). In the new framework, BNs are used to generate a finite set of joint causal views / stress-tests for the relevant factors of a market, while EP is used to project each of these views / stress-tests over market simulations. To tie it all together, the joint view / stress-test probabilities from BNs are naturally used as weights for the associated EP probability vectors to compute a single posterior probability distribution. The new framework allows us to implement market views and perform stress-tests conditional on realizations of relevant market variables in a truly causal and predictive way.
Keywords: Bayesian networks, minimum relative entropy, Entropy Pooling, market views, stress-testing, causality, predictiveness, Monte Carlo simulation, synthetic market generator.
Abstract: The investment industry lacks a 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. The framework 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.
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. Although mean-CVaR is computationally more complex, the analytical benefits strongly outweigh mean-variance. Mean-variance can still be a useful tool for illustrating fundamental investment concepts, but it should be avoided for investment management in practice. 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, to make CVaR more comparable to other risk measures, it is recommended to use demeaned P&L for CVaR optimization.
Keywords: Portfolio optimization, mean-variance, mean-CVaR, tail risks, convex optimization, risk budgeting, Monte Carlo simulation, synthetic market data generator, Python Programming Language.
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 EP 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.