Research
You can find our publicly available research articles below. The accompanying code is usually given by examples that use our open-source Python package.
Please visit our open-source site for more information on how to install our open-source Python package and documentation of its functionality.
The best way to learn about our investment framework and methods is to study the the Portfolio Construction and Risk Management book as well as complete the Quantitative Investment Management course.
Conditional Maximum Loss Portfolio Optimization
Abstract: This article introduces a new investment risk measure designed for the Fully General Investment Framework (FGIF), which is centered around fully general Monte Carlo simulation paths and their as sociated probabilities. The new risk measure is coined Conditional Maximum Loss (CML). It draws inspiration from drawdowns but adjusts for their practical shortcomings. The CML risk measure focuses on the expected loss over the entire horizon between portfolio rebalancing times, not just the losses at the end of the rebalancing horizon. CML portfolio optimization problems can be solved using linear programming, similar to the popular Conditional Value-at-Risk (CVaR). In fact, we can formulate joint portfolio optimization problems where both CML and CVaR are minimized, while the portfolio’s expected return is maximized. This allows investment managers to both optimize the distribution at the end of the investment horizon as well as the portfolio’s path, making the portfolio more robust and minimizing the probability of experiencing stop losses or client withdrawals. The Python case study includes several practical examples of how the new investment risk measure can be optimized in combination with Sequential Entropy Pooling (SeqEP) views and stress testing.
Keywords: Conditional Maximum Loss, Expected Maximum Loss, CML, EML, CVaR, CDaR, portfolio optimization, investment risk measure, portfolio construction, risk management, linear programming, Fully Flexible Resampling, FFR, Sequential Entropy Pooling, SeqEP, Python Programming Language.
The Normal Distribution Myth
Abstract: This article busts the myth about empirical asset returns following normal distributions. Such arguments are sometimes made by finance and economics academics based on a misuse of some undefined central limit theorem (CLT). The case study performs several formal tests for normality of 10 US equity index returns on various horizons, illustrating that the assumptions underlying the CLT apparently do not hold for real-world investment returns. Hence, the overall conclusion based on a CLT argument cannot be made. The article concludes that simple empirical analysis of historical investment returns shows that markets do not follow normal distributions, or even more generally elliptical ones. Building investment portfolios based on such oversimplifying assumptions, effectively ignoring the empirically observed fat left tails and risk-off dependencies, can lead to disastrous outcomes and should therefore be avoided in practice.
Keywords: Normal distribution, central limit theorem, Aggregational Gaussianity, mean-variance, investment returns, normality tests, Shapiro–Wilk, D’Agostino’s K-squared, Anderson–Darling, stocks, fat tails, skewness, kurtosis, Sequential Entropy Pooling, CVaR, Python Programming Language.
Time- and State-Dependent Resampling
Abstract: This article introduces Time- and State-Dependent Resampling methods that allow us to perform resampling of historical investment time series to generate new synthetic paths conditional on arbitrarily complex state variables and time decay. The article also presents a mathematical analysis of the Fully Flexible Resampling (FFR) method, which is an instance of the new resampling class recently introduced in the Portfolio Construction and Risk Management book. We prove that the new approach generates stationary simulations while giving us additional flexibility to capture the time series dependencies compared to other traditional resampling methods. The case study illustrates that the FFR method also allows us to perform state variable stress testing, for example, assessing the effect of a sudden volatility spike on various time horizons. The appendices contain all the necessary proofs for the new class of Time- and State-Dependent Resampling methods.
Keywords: Time- and State-Dependent Resampling, Fully Flexible Resampling, Markov chain, Markov chain Monte Carlo, Monte Carlo simulation, market simulation, synthetic market data, Entropy Pooling, relative entropy, Kullback–Leibler divergence, stationary transformations, Python Programming Language.
Derivatives Portfolio Optimization and Parameter Uncertainty
Abstract: Portfolio optimization in practice almost always needs to account for market model uncertainty. Resampled portfolio optimization is a common heuristic to tackle the parameter uncertainty issue. The recently introduced Resampled Portfolio Stacking approach makes the resampled approach even more attractive. While resampled optimization of cash portfolios is straightforward and well known, resampled optimization of portfolios containing derivatives is a largely unexplored area. Derivatives introduce an additional layer of complexity, because there needs to be a logical consistency between the parameter uncertainty of the underlying and risk factors such as implied volatilities as well as the derivative instrument’s P&L. This article presents an elegant solution to the problem and introduces a new class of portfolio optimization with fully general risk factor parameter uncertainty.
Keywords: Portfolio optimization, parameter uncertainty, derivatives, risk factors, Exposure Stacking, Resampled Portfolio Stacking, mean-CVaR, tail risk, efficient portfolio, efficient frontier, mean squared error, bias-variance trade-off, stacked generalization, quadratic programming, convex optimization, Python Programming Language.
Portfolio Optimization and Parameter Uncertainty
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. Resampled optimization is a popular mathematical heuristic to tackle the parameter uncertainty issue. It 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. 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.
Causal and Predictive Market Views and Stress Testing
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, Python Programming Language.
Portfolio Management Framework for Derivative Instruments
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 portfolio return conventions. Hence, 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.
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. Although mean-CVaR is computationally more complex, its analytical benefits significantly 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 in cases where the mean-variance assumptions are satisfied. Hence, nothing is lost from using CVaR in the idealized textbook case, while CVaR gives much more meaningful results for realistic investment distributions and avoids the undesirable upside penalization inherent to mean-variance.
Keywords: Portfolio optimization, mean-variance, mean-CVaR, tail risks, convex optimization, risk budgeting, Monte Carlo simulation, synthetic market data generator, 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 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. The heuristics are also able to solve interesting and practically relevant problems that the original approach simply cannot. The new approach is coined Sequential Entropy Pooling (SeqEP). Given the benefits of the new method, this article argues that it 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.