At its core, portfolio optimisation is a statistical tool designed to maximise the trade-off between risk and return. A crucial factor in its success is the understanding of the return behaviour of broad asset classes, including equities (representing risk capital), fixed income (present value of future cashflows), cash (involving interest), and commodities. These asset classes exhibit diverse behaviours, particularly in response to different economic regimes, and their returns often deviate from normal distributions, especially during stressed periods.
In Markowitz days, conventional investing wisdom centered on the notion of becoming a skilled stock-picker or hiring one. This approach resulted in portfolios characterised by a select few names, with little consideration given to the diversification of risks associated with individual stocks.
Markowitz's pioneering work in 1952 marked a transformative moment in investment theory, introducing the concept of Mean-Variance optimisation and reshaping the way portfolios are constructed. He laid the foundation for the practical application of diversification in his Ph.D dissertation over 70 years ago now known as Mean-Variance optimisation.
The central tenet of his mathematical model to calculate portfolio efficiency was the correlation-covariance matrix of the assets involved. It seeks to capture and optimise the relationship between the returns of these assets, aiming for the best return per unit of risk across varying levels of risk on the x-axis.
Markowitz's discovery triggered decades of scholarly research by luminaries such as William Sharpe, Robert Merton, Eugene Fama, and James Tobin. Sharpe, notable for the Capital Asset Pricing Model (CAPM), introduced the concept of a "market portfolio" devoid of idiosyncratic risk. This portfolio, by definition, represents a cap-weighted index fund of all underlying securities. The CAPM world is stunningly simple (with all its assumptions) where investors only need to make one decision: what % of their portfolio to be the index portfolio or beta.
The contributions of these scholars transformed the landscape of investing from a realm of stock tips and intuition to a mechanical application of statistical principles, including mean, variance, and correlation.
However, the real-world application of these discoveries is not without its challenges. These include forecasting returns, managing risk, estimating correlation matrices, and dealing with statistical estimation problems. As the paper suggests, delving into more detail on these challenges is essential.
For now, the key takeaway is the significance of risk diversification and an understanding of asset class return behaviour in constructing resilient and efficient investment portfolios.
Asset Allocation models Asset Allocation, a crucial element in Investment Management, involves selecting the appropriate mix of asset classes to achieve investment goals.
The two broad categories of models are 1) Constant-Weight models (modular) and 2) Optimisation models.
The Constant Weight model is straightforward, offering a simplistic approach without complex mathematical modelling. It entails assigning a fixed percentage allocation to broad asset classes, ranging from risk-free to high-risk, typically measured in standard deviation terms. This "fixed asset allocation" model generates portfolios with varying equity percentages across the risk curve, for instance, 20%, 40%, up to 100%. The model assumes that underlying assets will consistently behave normally, and the combination of different asset classes will mitigate individual risks. However, during periods of stress, cross-asset correlations may turn positive, especially in response to exogenous stimuli, leading to a material shift in the risk-return frontier. As this model does not optimise volatility, portfolios exhibit higher volatility during stressed periods. The frontier shifts down and right on the return/risk scale.
The second type of models are quantitative models. Drawing from our discussion on asset class behaviour and diversification, these models are dependent on variance-covariance matrix.
The first one to discuss here is Mean-Variance Optimisation (MVO).
These models leverage mathematical frameworks, particularly the variance-covariance matrix, to optimise asset allocations. MVO, in existence for over seven decades, is known for its simplicity in application but faces challenges related to forecasting returns, risk, and the variance-covariance matrix of all involved assets.
One significant limitation of traditional MVO lies in its assumption of normally distributed returns, disregarding fat-tailed or non-normal return distributions. If we were to assume that returns are normally distributed, a one-day move of >7% in Dow Jones Index should occur once every 300,000 years. Beniot B Mendebrot and Richard L Hudson report 48 such moves occurred in 20th century. We have seen these fat tailed events as late as 2022, 2020 and before that 2008. This empirical evidence refutes the notion of random walk.
The model's reliance on historical data for returns, risk, and correlation matrices introduces further challenges. Practitioners often use historical averages, but the choice of which historical period to model can be arbitrary, considering that regimes change, affecting underlying correlations.
MVO's sensitivity to input changes is another concern, aptly termed as "error maximiser."
Small adjustments to input parameters can lead to significant shifts in output, impacting the reliability of the model's results. Then there is a problem with the outputs – linear and have extreme allocations as shown in the chart below. The risk assets allocation increases linearly as you travel right on the risk axis. Practitioners allow for constraints around the inputs to generate a more “appropriate” output. The output from this process is generally used as a Strategic Asset Allocation (one time period model) with tactical changes around it during the period. Which then questions the reliability of initial data used.
Source: Morningstar, 2024 Mean Variance optimisation (MVO), in our opinion was the first step towards understanding the statistical nature of the asset returns and their behaviours collectively. Asset Allocation requires a nuanced understanding of these models and a consideration of their limitations in the context of dynamic market conditions.
The Black-Litterman model, introduced by Fischer Black and Robert Litterman in 1990, represents a Bayesian approach to optimisation, combining prior market knowledge with practitioners' future expectations. This model essentially extends the concept of the global Capital Asset Pricing Model (CAPM) equilibrium. The key technical aspect to note is that the definition of the "global market" in this model is contingent upon the practitioner's selection of asset classes.
In simpler terms, the Black-Litterman model allows investors to integrate their own insights and expectations about future returns into the optimisation process. It provides a mechanism to blend these practitioner-specific views with the equilibrium market expectations, and probability of achieving these returns. By allowing practitioners to express their views it model offers a dynamic and flexible framework. This addresses some of the limitations of Mean-Variance optimisation.
The stylised version of the Black-Litterman model :
As with all optimisation models, the variance-covariance matrix remains the key and encompasses asset’s behaviour with each other over a period. The equilibrium is defined by the choice of assets in our market portfolio, which in turn are dependent on the asset class behaviour. From a practitioner’s perspective is important to understand how asset class returns behave. For example, in our model we would class government debt in three categories instead of one.
The starting point of an equilibrium portfolio of the defined market allows for a deviation based on views and probability. The model doesn't mandate expected returns and risk estimates for all asset classes, allowing practitioners to focus on areas where they hold strong views. These views can be an output from a Machine Learning model or based on technical estimates. In our case our views are based on a 4-D model or a scorecard.
By starting with an equilibrium point and introducing probabilistic views, the Black-Litterman model proves less sensitive to inputs, enhancing its realism and adaptability in capturing practitioner expectations and market dynamics. It also addresses issues seen in mean variance optimisation, avoiding extreme and linear portfolio outcomes as shown in the chart below.
Source: Morningstar, 2024
The Black-Litterman model, while valuable, is not immune to criticism. Like Mean-Variance Optimisation, it relies on the variance-covariance matrix. This proves challenging in exogenous, fat-tail events, as observed in stress periods where positive correlations contradict the normal behaviour of non-correlated or negatively correlated assets. This limitation underscores the difficulty in accurately capturing extreme market dynamics and emphasises the need for supplementary risk management strategies in such unprecedented scenarios. As shown below, in periods of stress, most asset correlations turn positive.
Source: Morningstar, 2024
The Black-Litterman model enhances traditional Mean-Variance Optimisation by addressing its weaknesses, offering more intuitive portfolios, reduced sensitivity to inputs, and minimizing estimation error. While an improvement, it's not a standalone solution.
Investment Management is a dynamic undertaking, especially where exogenous shocks can affect the investment outcomes.
Understanding the precise nature of optimisation model’s shortfalls enables us to devise a process to address them.
Our approach to Dynamic Asset Allocation enables us to adjust the asset mix in response to evolving views and market conditions, utilising proprietary tools for a flexible and adaptive portfolio construction that aligns with changing investment landscapes.
We have established that view formation is pivotal, after defining the market and regime. Practitioners face the challenge of discerning key asset classes and predicting future returns. Successful view formation is crucial for the model's effectiveness in portfolio optimisation and decision-making.
A robust view formation technique is needed to weed out any erroneous assumption. At AlphaBeta Partners we apply a scorecard approach that analyse data from macro, fundamental, technical, and geopolitical variables in order to derive at least one asset class view with an assigned probability.
In this paper, we have looked at optimisation techniques, their application, and constraints from a practitioner’s point of view, our preference and how we use it within a wider Investment Process.
Whilst we have pointed out limitations of price forecasting – we do believe this is one area where Machine Learning can improve our forecasting abilities, especially related to exogenous shocks or black swan events. Machine Learning (ML) in finance involves adapting model parameters with data, automating information processing. Key ML techniques of interest are genetic algorithms, LASSO, and natural language processing(NLP), widely used for automation in asset management and finance. The following gives a hint of things to come :
(Source : CFA Institute – Artificial Intelligence in Asset Management)
We are particularly interested in the use case of NLP and LASSO regression framework to enhance our view formation ability. Without going into details, we believe the ability to capture factors that explain the highest explanatory power of returns to form a large set of predictive signals and finding lead-lag relationships between asset groups may be a game changer for the Asset Managers.
To conclude, asset allocation models continue to evolve with better and robust statistical models, computer power and algorithms to answer one question – how to better diversify risk. These models are one part of the Investment toolkit. The Investment Process and feedback loop determine the use of these tools to exact better outcomes.
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