Note that by assumption the set is known a-priori. Consequently, we can always take , guaranteeing the existence of the necessary parameters γ, γi, Bi, θ, pi. This suggests a method for choosing z; choose z such that Lmktz achieves a target return for the market portfolio. In what follows we adopt this approach, specifying the target return as the equilibrium return on the market portfolio in the Black-Litterman model. This is done primarily for ease of comparison between the two models.
In this paper we have used techniques from inverse optimization to create a novel, richer, reformulation of the Black-Litterman (BL) framework. The major advantage of this approach is the increased flexibility for specifying views and the ability to consider more general notions of risk than the traditional mean-variance approach. We have exploited this flexibility to introduce two new BL-type estimators and their corresponding portfolios, a Mean-Variance Inverse Optimization (MV-IO) approach and a Robust Mean-Variance Inverse Optimization (RMV-IO) approach. The major distinction between the approaches is that the first allows investors to capitalize upon any private information they may have on volatility, while the second seeks to insulate investors from volatility uncertainty when they have no such information. Computational evidence suggests that these approaches provide certain benefits over the traditional BL model, especially in scenarios where views are not known precisely.
Black-Litterman HoadleyOptimalPortfolio function: The HoadleyOptimalPortfolio function has been updated to return an error code if the "Normalize" argument is set to true and it is not possible to form a portfolio where the asset weights sum to 100%. See function documentation for details.
Implied returns from portfolios which include foreign currency exposures: New function, HoadleyImpliedReturnsFX, will back-out implied returns for a portfolio which includes a mixture of domestic assets and foreign assets containing various degrees of foreign currency exposures, and hedging levels. Returns are decomposed into basic asset returns and currency returns. This new function is part of the Black-Litterman function set.
Black-Litterman Returns Estimator application: Release of a new application for estimating investment portfolio returns using the Black-Litterman model. This application, which uses the Black-Litterman components in the Add-in, is designed to facilitate an interactive/prototyping approach to investment portfolio design. .
The Black-Litterman (BL) model is a widely used asset allocation model in the financial industry. In this paper, we provide a new perspective. The key insight is to replace the statistical framework in the original approach with ideas from inverse optimization. This insight allows us to significantly expand the scope and applicability of the BL model. We provide a richer formulation that, unlike the original model, is flexible enough to incorporate investor information on volatility and market dynamics. Equally importantly, our approach allows us to move beyond the traditional mean-variance paradigm of the original model and construct “BL”-type estimators for more general notions of risk such as coherent risk measures. Computationally, we introduce and study two new “BL”-type estimators and their corresponding portfolios: a Mean Variance Inverse Optimization (MV-IO) portfolio and a Robust Mean Variance Inverse Optimization (RMV-IO) portfolio. These two approaches are motivated by ideas from arbitrage pricing theory and volatility uncertainty. Using numerical simulation and historical backtesting, we show that both methods often demonstrate a better risk-reward tradeoff than their BL counterparts and are more robust to incorrect investor views.
Litterman (2003), “The Intuition Behind Black-Litterman .An Application of the Black-Litterman Model with EGARCH-M-Derived Views for International Portfolio Management Steven L.
The Black-Litterman (BL) model is a widely used asset allocation model in the financial industry. Introduced in , the model uses an equilibrium analysis to estimate the returns of uncertain investments and employs a Bayesian methodology to “blend” these equilibrium estimates with an investor's private information, or views, about the investments. Computational experience has shown that the portfolios constructed by this method are more stable and better diversified than those constructed from the conventional mean-variance approach. Consequently, the model has found much favor with practitioners. The U.S. investment bank Goldman Sachs regularly publishes recommendations for investor allocations based on the BL model and has issued reports describing the firm's experience using the model (). A host of other firms (Zephyr Analytics, BlackRock, Neuberger Berman, etc.) also use the BL model at the core of many of their investment analytics.
Black-Litterman sample: New example added to the spreadsheet. The example compares and reconciles regression betas, and expected returns calculated with the CAPM formula, with implied betas and implied returns from reverse optimization.
It [Black and Litterman, 1992], [He and Litterman, 1999].Statistics 157 Black-Litterman Model This paper introduces the Black -Litterman model and its applications.
Implied portfolio returns from a single asset: A new function, HoadleyImpliedViews, will, given an estimate of the expected return for a single asset in an investment portfolio, back out the implied returns for all other assets (called implied views or hurdle rates in Bob Litterman's book "Modern Investment Management - an Equilibrium Approach"). The hurdle rates represent the points of indifference for the purchase or sale of any asset in the portfolio and can therefore provide some insight into which assets to buy or sell. This new function is part of the Black-Litterman function set.
Black-Litterman asset allocation model: Four new functions providing a full implementation of the Black-Litterman asset allocation model for portfolio design. apply. . View the
Black-Litterman sample spreadsheet: A new spreadsheet which replicates the results from examples contained in two of the key papers on the Black-Litterman model is available for download with the full version of the Finance Add-in for Excel.