 Academic journal article Journal of Real Estate Portfolio Management

# Mathematical Derivations and Practical Implications for the Use of the Black-Litterman Model

Academic journal article Journal of Real Estate Portfolio Management

# Mathematical Derivations and Practical Implications for the Use of the Black-Litterman Model

## Article excerpt

Executive Summary.

In this article, the financial portfolio model often referred to as the Black-Litterman model is described, and then mathematically derived, using a sampling theoretical approach. This approach generates a new interpretation of the model and gives an interpretable formula for the mystical parameter, τ, the weight-on-views. The practical implications of the model are discussed, along with how portfolio fund managers should arrive at model input values and what consideration must be weighted beforehand.

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In 1952, Markowitz published the article Portfolio Selection, which is the origin of modern portfolio theory. Portfolio models are tools intended to help portfolio managers determine the weights of the assets within a fund or portfolio. Markowitz's ideas have had a great impact on portfolio theory and have withstood the test of time. However, in practical portfolio management the use of Markowitz's model has not had the same impact as it has had in academia. Many fund and portfolio managers consider the composition of the portfolio given by the Markowitz model as unintuitive (Michaud, 1989; Black and Litterman, 1992). The practical problems in using the Markowitz model motivated Black and Litterman (1992) to develop a new model in the early 1990s. The model, often referred to as the Black-Litterman model (hereafter the BL model), builds on Markowitz's model and aims at handling some of its practical problems. While optimization in the Markowitz model begins from the null portfolio, the optimization in the B-L model begins from, what Black and Litterman refer to as, the equilibrium portfolio (often assessed as the benchmark weights of the assets in the portfolio). "Bets" or deviations from the equilibrium portfolio are then taken on assets to which the investor has assigned views. The manager assigns a level of confidence to each view indicating how sure he/she is of that particular view. The level of confidence affects how much the weight ofthat particular asset in the B-L portfolio differs from the weights of the equilibrium portfolio.

The purpose of this article is to (1) carefully and methodologically describe and mathematically derive the B-L model, (2) review the relevant literature to discuss the model's implications to practical implementation of its usage, (3) present and discuss theoretical starting points for future research, and (4) establish a foundation for the discussion of Mankert and Seiler (2011).

The Markowitz Model

Markowitz (1952) focuses on a portfolio as a whole, instead of an individual security selection when identifying an optimal portfolio. Previously, little research concerning the mathematical relations within portfolios of assets had been carried out. Markowitz began from John Burr Williams' Theory of Investment Value. Williams (1938) claimed that the value of a security should be the same as the net present value of future dividends. Since the future dividends of most securities are unknown, Markowitz claimed that the value of a security should be the net present value of expected future returns. Markowitz claims that it is not enough to consider the characteristics of individual assets when forming a portfolio of financial securities. Investors should take into account the co-movements represented by covariances of assets. If investors take covariances into consideration when forming portfolios, Markowitz argues they can construct portfolios that generate higher expected return at the same level of risk or a lower level of risk with the same level of expected return than portfolios ignoring the co-movements of asset returns. Risk, in Markowitz' model (as well as in many other quantitative financial models) is assessed as the variance of the portfolio. The variance of a portfolio in turn depends on the variance of the assets in the portfolio and on the covariances between the assets. …

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