Black-Litterman Research Articles

Dynamic Black–Litterman Portfolios Incorporating Asymmetric Fractal Uncertainty

This study investigates the profitability of portfolios that integrate asymmetric fractality within the Black–Litterman (BL) framework. It predicts 10-day-ahead exchange-traded fund (ETF) prices using recurrent neural networks (RNNs) based on historical price information and technical indicators; these predictions are utilized as BL views. While constructing the BL portfolio, the Hurst exponent obtained from the asymmetric multifractal detrended fluctuation analysis is employed to determine the uncertainty associated with the views. The Hurst exponent describes the long-range persistence in time-series data, which can also be interpreted as the uncertainty in time-series predictions. Additionally, uncertainty is measured using asymmetric fractality to account for the financial time series’ asymmetric characteristics. Then, backtesting is conducted on portfolios comprising 10 countries’ ETFs, rebalanced on a 10-day basis. While benchmarking to a Markowitz portfolio and the MSCI world index, profitability is assessed using the Sharpe ratio, maximum drawdown, and sub-period analysis. The results reveal that the proposed model enhances the overall portfolio return and demonstrates particularly strong performance during negative trends. Moreover, it identifies ongoing investment opportunities, even in recent periods. These findings underscore the potential of fractality in adjusting uncertainty for diverse portfolio optimization applications.

ON THE SOLUTION UNIQUENESS IN PORTFOLIO OPTIMIZATION AND RISK ANALYSIS

In this paper, we consider the issue of solution uniqueness of the mean-deviation portfolio optimization problem and its inverse for asset returns distributed over a finite number of scenarios. Due to the asymmetry of returns, the risk is assessed by a general deviation measure introduced by Rockafellar et al. [(2006b) Optimality conditions in portfolio analysis with general deviation measures, Mathematical Programming 108, 515–540] instead of the standard deviation as in the classical Markowitz optimization problem. We demonstrate that, in general, one cannot expect the uniqueness of Pareto-optimal profit sharing in cooperative investment and the uniqueness of solutions in the mean-deviation Black–Litterman asset allocation model. For a large class of deviation measures, we provide a resolution of the above nonuniqueness issues based on the principle of law-invariance. We provide several examples illustrating the nonuniqueness and the law-invariant solution.

When the tide wanes: A study of post systemic collapse portfolio management

This study applies the Black–Litterman model to portfolio management in a post-crisis scenario, where standard parametric models often fail due to market irrationality and investor overreactions. The research focuses on the performance of surviving firms, identifying that over-performing stocks tend to outperform in the medium- to long-term, while under-performing stocks exhibit stronger short-term returns as the market corrects its initial overreaction. The study specifically adjusts views within the Black–Litterman framework using only the top and bottom quartiles of stocks, which experience the most significant shifts during a crisis. By controlling for firm size and book-to-market ratios, consistent with the Fama-French three-factor model, the research demonstrates that adjusting portfolio weights with the Black–Litterman model can achieve substantially higher returns with lower downside risk during recovery periods. These findings have significant implications for portfolio managers seeking to optimize returns in volatile, post-crisis markets.

Combining transformer based deep reinforcement learning with Black-Litterman model for portfolio optimization

Combining transformer based deep reinforcement learning with Black-Litterman model for portfolio optimization

Portfolio Optimization with Sector Return Prediction Models

We analyze return predictability for U.S. sectors based on fundamental, macroeconomic, and technical indicators and analyze whether return predictions improve tactical asset allocation decisions. We study the out-of-sample predictive power of individual variables for forecasting sector returns and analyze multivariate predictive regression models, including OLS, regularized regressions, principal component regressions, the three-pass regression filter, and forecast combinations. Using an out-of-sample Black–Litterman portfolio optimization framework and employing predicted returns as investors’ ‘views’, we evaluate the benefits of sector return forecasts for investors. We find that portfolio optimization with sector return prediction models significantly outperforms portfolios using historical averages as well as passive benchmark portfolios.

Decomposing Countries Consumption in Active Portfolio Management: A Black–Litterman Application

Decomposing Countries Consumption in Active Portfolio Management: A Black–Litterman Application

Fuzzy Portfolio with a Novel Power Membership Function Based on GARCH and Black–Litterman Model

Fuzzy Portfolio with a Novel Power Membership Function Based on GARCH and Black–Litterman Model

Optimization of Portfolio Management Models with Indexed Stocks on the Lima Stock Exchange

Investment fund managers are limited by the fact that Latin American financial markets offer very few investment possibilities, which forces them to carry out operations at a global level. The objective is to optimize the portfolio management models with indexed stocks in the Lima Stock Exchange (27 stocks considering 2082 days from January 02, 2014 to April 13, 2022) applying the Markowitz models that determine the portfolios of the frontier of investment possibilities. The Sharpe model (CAPM), which calculates the expected return considering systemic risks, and the Sharpe index, which measures the stock return with total risk. Finally, the Black-Litterman (BL) model adjusts the ex-post expected return with expert opinion. By comparing the BL/CAPM ratio, an index is obtained that improves the predictability of expected returns and it is observed that the efficiency of this indicator is greater than the Sharpe index of subsequent expected returns (Sharpe BL). Therefore, the hypothesis that optimizing the portfolio management models improves the predictability of the expected returns of the indexed shares of the Lima Stock Exchange is accepted.
 
 Received: 23 July 2023 / Accepted: 10 January 2024 / Published: 5 March 2024

Constructing factor-bearing portfolios based on the Black-Litterman model

Constructing factor-bearing portfolios based on the Black-Litterman model

Investment Portfolio Optimization Using Black-Litterman Model in Smart Carbon Economy Transition

An optimal investment portfolio needs to be formed before an investor invests because it can help investors determine which financial instruments are suitable to choose in order to get the maximum return or profit and the minimum level of risk. In the current situation, where there is an economic transition to a smart carbon economy or low carbon economy, it is necessary to form the optimal portfolio of stocks to facilitate investors in making investments. The purpose of this study is to form the optimal investment portfolio using the Black-Litterman model in a smart carbon economy. The data used is stock data from 24 companies listed on the LQ45 Low Carbon Leaders index for the period 2022-2023. Based on the research results, the Black-Litterman model generates the optimal portfolio with a 0.1% expected return. Thus, the optimal portfolio results with the Black-Litterman model are estimated to generate a profit of 0.1% for smart carbon stock data listed on the LQ45 Low Carbon Leaders index for the 2022-2023 period.

A novel integration of the Fama–French and Black–Litterman models to enhance portfolio management

We propose a novel portfolio model integrating the Fama–French three-factor model into the Black–Litterman framework, enabling efficient investment strategies. The model surpasses traditional benchmarks, significantly increasing alpha, minimizing estimation error, and improving diversification. Performance improvements are shown by a tripled Sharpe ratio and doubled Certainty Equivalent Return compared to standard models. It maintains stability across different parameters and economic climates, leveraging improved weight adjustment to reduce estimation errors and withstand market volatility. It provides a new perspective for portfolio construction, leveraging long-term insights from asset pricing theory with significant implications.

Forecasting Method for Optimal Diversification

Forecasting is a technique that uses historical data as inputs to make estimates that are predictive in determining the direction of future trends. The goal of investors is to make optimal choice that leads to minimization of risk and maximization of returns, but the method that leads to these objectives has been a challenge for investor. In this study, Black-Litterman model (BLM) is adopted and two forecasting methods; EGARCH and GARCH methods are used for two parameters of BLM; investor views and level of uncertainty. The aim of this paper is to investigate the best forecasting method to estimate BLM that would lead to minimum risk and maximum returns. The analysis of this paper shows that EGARCH method gives maximum expected returns and minimum risk.

基于Black-Litterman模型的基金投资组合策略研究

基于Black-Litterman模型的基金投资组合策略研究

Risk-Adjusted Portfolio Optimization: Monte Carlo Simulation and Rebalancing

This study evaluates the risk-adjusted performance of a diversified portfolio in the Indian financial market from 2011 to 2021, incorporating Nifty 50 stocks and new-age assets. Leveraging Monte Carlo simulations and mathematical optimization, the research identifies an optimal portfolio on the efficient frontier. Integration of the Black-Litterman model provides a comparative analysis, emphasizing the impact of investor views. Despite transaction costs, optimized portfolios outperform the Nifty 50 index, with the rebalanced portfolio demonstrating higher cumulative returns. Key findings include TCS. NS is a leader in share price, HDFCBANK.NS showcasing stability and alternative assets exhibit higher volatility but have the potential for amplified returns. This research offers valuable insights for investors seeking resilient strategies in the Indian financial landscape.

Enhancing investment performance of Black-Litterman model with AI hybrid system: Can it be done?

The B-L (Black-Litterman) model combines the subjective views of investors with the market's historical equilibrium return through the Bayesian theorem. This approach somewhat compensates for the limitations of the Mean-Variance model, making the asset allocation model attract significant attention. However, due to many investors lacking ample investment experience, they might struggle to provide appropriate subjective views. To address this issue, an Iaah (Industry Asset Allocation Hybrid) system based on the B-L model is proposed. First, a combined model is constructed, which simulates view return vectors and solves the problem of inaccurate view quantification. Then, the Iaah system introduces a dynamic condition correlation algorithm to optimize inherent parameters and explores asset allocation results in the time-varying effects. Next, it expands the types of objective functions and improves the deficiency of only considering the return and variance while ignoring the downside risk. Finally, based on a variety of performance metrics, the Iaah system is evaluated from multiple perspectives to test its practical value. The experiment results with the daily data of 10 SSE (Shanghai Stock Exchange) industry indexes show that the proposed system not only realizes the diversified allocation of industry assets and enhances the investment performance, but also exhibits strong robustness against changes in subjective parameters. It is conducive to managing personal wealth and achieving preservation and appreciation.

Black–Litterman Portfolio Management Using the Investor’s Views Generated by Recurrent Neural Networks and Support Vector Regression

Black–Litterman Portfolio Management Using the Investor’s Views Generated by Recurrent Neural Networks and Support Vector Regression

Stock and Structured Warrant Portfolio Optimization Using Black-Litterman Model and Binomial Method

In recent years, the number of Indonesian investors has rapidly increased during the COVID-19 pandemic which happened all around the world. There have been a massive number of influencers in social media who were promoting investment. Although stocks and warrants are interesting choices, mutual funds still become the main ones for beginners. Therefore, this research focuses on the development of a stock portfolio model using the Black-Litterman method which involves the investor’s views towards the stock returns. The research refers to one of the largest equity funds in Indonesia, that is Sucorinvest Equity Fund, by using the top ten of its stocks that are majority in the fund (as of April 28, 2023). Furthermore, this research also constructs a structured warrant portfolio, but it is separated from the initially constructed stock portfolio. Structured warrants could be an appropriate choice for low-budget investors. It was newly introduced in Indonesia in September 2022 so it is interesting to be observed. Based on the results and the implemented assumptions, the return obtained from the stock portfolio is superior to the observed fund’s return. Meanwhile, call structured warrant portfolio using the existing product in the market yields a negative return, because the exercise price and warrant offered price were too high. Thus, structured warrants could be considered overpriced at the moment, so the chance of obtaining profit is extremely small. Due to its similar properties to call and put options, we propose the warrant pricing and use it in simulations, so in the future, structured warrants may become an attractive instrument for the investors.

Using fear, greed and machine learning for optimizing global portfolios: A Black-Litterman approach

We introduce a new dimension in constructing relative investor views for the Black-Litterman model by incorporating fear/greed technical indicator predictions as a proxy for investor sentiment in the portfolio construction process. We apply a hybrid CEEMDAN-GRU deep learning model to forecast this indicator and the XGBoost ensemble learning algorithm to forecast returns for ten country ETFs and create relative views for the Black-Litterman model. These models beat several benchmark forecasting models. Our empirical results show that the proposed approach outperforms the Markowitz, minimum-variance, equally-weighted and risk-parity strategies along with four other Black-Litterman approaches from the literature for six investment periods.

STOCK PORTFOLIO OPTIMIZATION IN BULLISH AND BEARISH CONDITIONS USING THE BLACK-LITTERMAN MODEL

Bullish and bearish phenomena characterize the development of the capital market. Therefore, this study aimed to identify and analyze bullish and bearish conditions in the Indonesian capital market to formulate an optimal portfolio. The sample consisted of 20 selected companies based on their substantial market capitali- zation. The results showed that from January 2011 to December 2020, the capital market experienced 77 bullish and 43 bearish months. The transition probability from bullish to bearish and bearish to bullish state was 15.67% and 56.14%. Furthermore, employing the Markov-switching model for determining market conditions and using the Black-Litterman model for portfolio construction proved advantageous for investors' financial forecasting techniques and their potential to generate valuable insights to create a well-informed portfolio.

A derivation of the Black–Litterman formula and its symmetry property

I present a novel derivation of the Black–Litterman formula by using Theil’s mixed estimator. Additionally, I derive the Black–Litterman formula’s symmetry property. This latter derivation can be used to provide guidance on the estimate of τ. I develop a method to do so and provide a toy asset allocation example.