Bayesian Portfolio Optimisation: Introducing the Black-Litterman Model

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The Black-Litterman (BL) model is one of the many successfully used portfolio allocation models out there. Developed by Fischer Black and Robert Litterman at Goldman Sachs, it combines Capital Asset Pricing Theory (CAPM) with Bayesian statistics and Markowitz’s modern portfolio theory (Mean-Variance Optimisation) to produce efficient estimates of the portfolio weights.

Before getting into the nitty-gritty of the algorithm it is important to understand the motivations behind developing it and why is it favored by practitioners in the industry. For a long while, investors worked under the assumption that the risk and return relationship of a portfolio was linear, meaning that if an investor wanted higher returns, they would have to take on a higher level of risk.

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Portfolio Optimisation with PortfolioLab: Mean-Variance Optimisation

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For a long while, investors worked under the assumption that the risk and return relationship of a portfolio was linear, meaning that if an investor wanted higher returns, they would have to take on a higher level of risk. This assumption changed when in 1952, Harry Markowitz introduced Modern Portfolio Theory (MPT). MPT introduced the notion that the diversification of a portfolio can inherently decrease the risk of a portfolio. Markowitz’s work on MPT was groundbreaking in the world of asset allocation, eventually earning him a Nobel prize for his work in 1990. Throughout this blog post, we will explore Markowitz’s Modern Portfolio Theory and work through a full implementation in the MlFinLab library.

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Portfolio Optimisation with PortfolioLab: Estimation of Risk

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Risk has always played a very large role in the world of finance with the performance of a large number of investment and trading strategies being dependent on the efficient estimation of underlying market risk. The covariance matrix is one of the most popular and widely used estimator of risk but due to its sensitivity to market conditions and dependence on historical data, it produces an unreliable estimation of true market risk. In this post, we go over some important methods of estimating covariance matrices which can be used in practice to remove noise from empirical estimates and produce better and reliable risk estimations.

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Portfolio Optimisation with PortfolioLab: Hierarchical Equal Risk Contribution

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In 2018, Thomas Raffinot developed the Hierarchical Equal Risk Contribution (HERC) algorithm, combining the machine learning approach of the Hierarchical Clustering based Asset Allocation (HCAA) algorithm with the recursive bisection approach from Hierarchical Risk Parity. The HERC algorithm aims to diversify capital and risk allocations and generate robust risk-adjusted portfolios which outperform out-of-sample.

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Portfolio Optimisation with PortfolioLab: Theory-Implied Correlation Matrix

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For over half a century, most asset managers have used historical correlation matrices (empirical or factor-based) to develop investment strategies and build diversified portfolios. The Theory-Implied Correlation matrix combines external market views with emprirical values to generate new correlations which are less noisy and in sync with the economic theory.

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Beyond Risk Parity: The Hierarchical Equal Risk Contribution Algorithm

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As diversification is the only free lunch in finance, the Hierarchical Equal Risk Contribution Portfolio (HERC) aims at diversifying capital allocation and risk allocation. Briefly, the principle is to retain the correlations that really matter and once the assets are hierarchically clustered, a capital allocation is estimated. HERC allocates capital within and across the “right” number of clusters of assets at multiple hierarchical levels.

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Portfolio Optimisation with PortfolioLab: Hierarchical Risk Parity

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Throughout this post, we will explore the intuition behind Hierarchical Risk Parity and also learn to apply it using the MlFinLab library.

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