In this article, Youssef Louraoui (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022) presents an implementation of the Black-Litterman model, used to determine the expected return of a portfolio by integrating investor’s views regarding the performance of the underlying assets selected in the investment portfolio.
This article follows the following structure: first, we introduce the Black-Litterman model. We then present the mathematical foundations of this model. We conclude with an explanation of the methodology to build the spreadsheet with the results obtained. You will find in this post an Excel spreadsheet which implement the model.
Introduction
The Black-Litterman asset allocation model, established for the first time in the early 1990’s by Fischer Black and Robert Litterman, is a sophisticated strategy for dealing with unintuitive, highly concentrated, and input-sensitive portfolios. The most likely reason that more portfolio managers do not use the Markowitz model, which maximises return for a given degree of risk, is input sensitivity, a well-documented issue with mean-variance optimization.
The Black-Litterman Model employs a Bayesian technique to integrate an investor’s subjective views of expected returns on one or more assets with the market equilibrium vector (prior distribution) of expected returns to obtain a new, mixed estimate of expected returns. The new vector of returns (the posterior distribution) is a weighted complex average of the investor’s views and market equilibrium.
Mathematical foundation
The idea of the Black Litterman estimates is not to find the optimum portfolio weights as in the Markowitz optimization framework, but instead to find the expected return that would be used as an input to compute the optimum portfolio weights. This approach is referred as reversion portfolio optimization technique. The idea behind is that optimum weights are already observed in the market and captured in the market portfolio. We can approach the reasoning by maximizing the following utility function adjusted to the risk:
- wT = transposed of portfolio weights
- Π = Implied equilibrium excess return vector
- A = price of risk or risk aversion factor
- Σ = variance-covariance matrix
We take the partial derivative of U in terms of weights (w) and we derive the following expression:
By setting the partial derivative equal to zero, we can maximize the utility function in term of weights. The proposed approach in the Black Litterman approach is that instead of seeking the optimal weights, which are incorporated in the market portfolio and thus computable via the market capitalization of the equities in the portfolio, we’ll isolate the Π (implied equilibrium excess return) to obtain the optimal expected returns for the portfolio:
We can deconstruct the Black-Litterman model as
- τ= scalar
- P = Linking matrix
- ∑ = Variance-covariance matrix
- Π= implied equilibrium excess return
- A = Price of risk
- w = weight vector
- Ω = uncertainty of views
The first term of the formula is introduced in order to respect the constraint that the portfolio weights should be equal to one:
The second term of the formula is to compute a weighted average of the implied equilibrium excess return adjusted to the uncertainty of the returns by the view vector weighted with the uncertainty of views:
The final output E(R) is a vector of return n x 1 that represent the equilibrium returns of the market adjusted to investors views.
Implementation of the Black-Litterman asset allocation model in practice
To model a Black-Litterman portfolio allocation, we obtained a large time series to obtain useful results by downloading the equivalent of 23 years of market data from a data provider (in this example, Bloomberg). We generate the variance-covariance matrix after obtaining all necessary statistical data, which includes the expected return and volatility indicated by the standard deviation of the returns for each stock during the provided period.
The data is derived from the Bloomberg terminal. The first step is to calculate the logarithmic returns and excess returns on the selected assets (returns minus the risk-free rate). After calculating the logarithmic returns on each asset, we can estimate the capital asset pricing model’s returns (CAPM) expected returns. This information will be used to calculate the Black-Litterman expected returns on a comparative basis.
1. The first input for the model is the price of risk A, which represents the risk aversion of investor and is obtained by subtracting the expected return of the market the risk-free rate and divided by the variance of the market:
- E(rm)= expected market returns
- rf = risk-free rate
- σ2m = variance of market
2. We extract the respective market capitalization of each security to obtain their market weights in the portfolio. Given that our investable universe is made of five stocks, we can retrieve their respective market capitalization and compute the weights of each stock in relation to the sum of total market-capitalization in the portfolio.
Table 1 depicts the optimal weights obtained from their respective market capitalisation, coupled with the respective expected return and volatility.
Table 1. Asset characteristics of Apple, Amazon, Microsoft, Goldman Sachs, and Pfizer.
Source: computation by the author.
3. We compute the variance-covariance matrix of logarithmic returns using the data analysis tool pack available in Excel (Table 2).
Table 2. Variance-covariance matrix of asset returns
Source: computation by the author.
4. We compute the implied equilibrium excess return (Π) as the matrix calculation of the price of risk (A) times the matrix multiplication of the weights computed in step 4 times the variance-covariance matrix computed in step 3.
- Π= implied equilibrium excess return
- A = Price of risk
- w = weight vector
5. The views are incorporated into the model. To achieve this, we provide three views to include into the model. If there are no views, the values will correspond to the market portfolio. The investment manager’s views for the expected return on certain of the portfolio’s assets regularly diverge from the Implied Equilibrium Return Vector (), which serves as the market-neutral starting point for the Black-Litterman model that quantifies the uncertainty associated with each view. The Black-Litterman Model can be used to depict such views in absolute or relative terms. As an illustration, let us suppose that the real and simulated portfolio will have the same views:
- View 1: Apple will outperform Microsoft by .05 percent
- View 2: Amazon will outperform Microsoft by .1 percent
- View 3: Apple will outperform Amazon by .05 percent
To incorporate the vector Q of views, we create a link matrix P where the rows sum to zero. Figure 3 depicts the workings done in the spreadsheet.
Table 3. Views vector and Link Matrix (P)
Source: computation by the author.
6. We compute omega to determine the degree of uncertainty associated with the views. While Black-Litterman paper used a value of tau equal to 0.25, an important number of academics went for calculating the tau equal to one. For the sake of simplifying the model, consider tau to be equal to one. This input is obtained by multiplying the Linking matrix by the variance-covariance matrix and transposing the Linking matrix (P).
- τ= scalar
- P = Linking matrix
- ∑ = Variance-covariance matrix
7. We integrate all the values computed previously in the Black-Litterman model. Table 4 depicts the results obtained via the Black-Litterman allocation model.
Table 4. Results of the Black-Litterman allocation
Source: computation by the author.
We can see that the results converge slightly to those from CAPM. Additionally, we can see that the views are reflected in the Black-Litterman expected returns. As a result, we can determine whether or not each view is satisfied. Indeed, Apple outperforms Amazon and Microsoft, while Amazon outperforms Microsoft.
You can download an Excel file to help you construct a portfolio via the Black-Litterman allocation model.
Why should I be interested in this post?
Modern Portfolio Theory is at the heart of modern finance, shaping the modern investing landscape. MPT has become the cornerstone of current financial theory and practice. MPT’s thesis is that winning the market is difficult and requires diversification and taking higher-than-average risks.
MPT has been around for nearly sixty years and shows no signs of slowing down. His theoretical contributions paved the way for more portfolio theory study. But Markowitz’s portfolio theory is sensitive to and depends on further ‘probabilistic’ expansion. This paper expanded on Markowitz’s previous work by incorporating investor views into the asset allocation process.
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▶ Youssef LOURAOUI Portfolio
▶ Youssef LOURAOUI Alpha
▶ Youssef LOURAOUI Factor Investing
▶ Jayati WALIA Capital Asset Pricing Model (CAPM)
Useful resources
Academic research
Black, F. and Litterman, R. 1990. Asset Allocation: Combining Investors Views with Market Equilibrium. Goldman Sachs Fixed Income Research working paper
Black, F. and Litterman, R. 1991. Global Asset Allocation with Equities, Bonds, and Currencies. Goldman Sachs Fixed Income Research working paper
Black, F. and Litterman, R. 1992. Global Portfolio Optimization.Financial Analysts Journal, 28-43.
Idzorek, T.M. 2002. A step-by-step guide to Black-Litterman model. Incorporating user-specified confidence levels. Working Paper, 2-11.
Markowitz, H., 1952. Portfolio Selection. The Journal of Finance, 7(1): 77-91.
About the author
The article was written in Mars 2022 by Youssef LOURAOUI (Bayes Business School, MSc. Energy, Trade & Finance, 2021-2022).