Black-Litterman Model

What is the Black-Litterman Model?

The Black-Litterman model is an asset allocation tool that portfolio managers use to optimize investor portfolios according to their risk tolerance and market outlook. The model uses market equilibrium as a starting point and considers the investors’ subjective market views to calculate how the optimal asset weights should differ from the initial portfolio allocation. The model attempts to create more stable and efficient portfolios based on investors’ unique insight, which overcomes the issues of input sensitivity.

Black-Litterman Model Explained

The model was created as an enhancement to traditional mean-variance optimization (MVO) and addresses the following issues:

• The output (i.e. asset allocation) is highly sensitive to small changes in the inputs, making it difficult to come up with reasonable estimates for expected returns.
• Efficient portfolios that are highly concentrated in a subset of asset classes.
• MVO is not reflective of unique investment views.

Expected return is the yield investors are expecting to make on an investment. It is calculated as:

By combining investors’ unique insights and market equilibrium returns, the model provides a more robust and practical portfolio allocation strategy that overcomes the problem of input sensitivity. This approach is popular with investors seeking to blend quantitative and qualitative insights into the decision-making process.

How Does the Black-Litterman Model Work?

The starting point of the model is the global equilibrium, or “Equilibrium returns, which assumes that the aggregate of all portfolios in the market is optimal and therefore reduces the over-reliance on historical data (often an issue with traditional models). Then, it uses reverse optimisation by adding correlations and risk aversion coefficient to the asset weights that are deemed to be optimal. These are returns that would be expected if all assets were priced by the Capital Asset Pricing Model (CAPM).

The next step is the active management process, which includes investors’ unique forecasts of expected returns on the various asset classes in the portfolio that differ from the returns implied by reverse optimisation. Then, an efficient frontier graph is created to reflect these updated return expectations to determine the optimal portfolio. This is a sophisticated process that considers:

1. Each view according to its strength
2. The covariance between the view and the equilibrium
3. The covariance among the views

Black-Litterman Model Formula

After determining the assets’ implied return, the following calculation of the Black-Litterman model is applied to calculate the expected return.

Where:

E(R) = The expected return

τ = A scalar number indicating the uncertainty of the CAPM distribution

P = A matrix with investors views; each row a specific view of the market and each entry of the row represents the weights of each asset

Q = The expected returns of the portfolios from the views described in matrix P

Ω = A diagonal covariance matrix with entries of the uncertainty within each view

Σ = Covariance matrix of returns

Π = Vector of implied equilibrium expected returns

Example of Black-Litterman Model

An equity portfolio manager believes that the tech sector will outperform the market by 3%. She also has a bearish view on the consumer discretionary sector and expects it to underperform by 2%. The expected market return (market equilibrium) is 10%. After applying the Black-Litterman adjustments, the expected return for the tech sector can be revised to 13% (market consensus of 10% + 3% the subjective view of the investor), while consumer discretionary is revised to 8% (market consensus of 10% – 2% subjective forecast). The manager can then use these revised expected returns in a mean-variance optimization to determine an optimal portfolio allocation that aligns with both market equilibrium and her unique insights.

Overall, the model is expected to tilt the portfolio towards the outperforming security/asset if the view exceeds the difference between the two implied equilibrium returns.

Black-Litterman Model Excel Example

We need to find the estimated return for each asset class in a reverse optimisation framework, which then could be used to be applied as own views in a Black-Litterman model.

1. First, we calculate the total size of the market by adding up all asset classes.
2. Then we find the size of each asset class by dividing its size by the total market size (all should add up to 1, i.e. 100%).
3. Then we need to find the beta for Emerging Market equities. The beta for the Global Market should equal 1 so in that cell we use the SUMPRODUCT function and multiply together the individual weights by their relevant beta. Then we need to use the Goal Seek function to find the Emerging Markets beta (Data-What if Analysis-Goal Seek). As noted, cell E15 should equal 1 by changing cell E14.
4. Next, to calculate the risk premium, we need to use the expected return for the Global Market of 7% and deduct the risk-free rate of 3.5%. This is consistent across each asset class in the portfolio.
5. Finally, to calculate the expected return we take the risk-free rate and add the individual beta for each asset class multiplied by the risk premium. Download the full model to check your answer.

Black-Litterman Model Vs. Markowitz

Harry Markowitz is an economist and Nobel Prize laureate who introduced the groundbreaking Modern Portfolio Theory in 1952. The Black-Litterman model further develops the Markowitz model (also known as the mean variance approach) by incorporating investors’ own perspectives in determining the optimal asset allocation.

While the mean variance model is based on just factors (mean return and variance) and simply aims to strike the optimal balance between risk and return, the Black-Litterman approach is seeking more attractive returns without increasing the level of risk. The Markowitz model, while a fundamental theory in finance, is not as flexible and does not allow a reverse optimisation, which can be crucial for investors who seek a risk-averse method to maximise their returns.

Black-Litterman Model Disadvantages

While The Black-Litterman model has multiple strengths and use case applications, it also has several limitations, including:

• The model is complex and involves a number of mathematical calculations and statistical variations, making it difficult to implement correctly.
• Along with being a major strength, adding subjective views that are not carefully considered may lead to a bias and tilt the portfolio towards riskier territory.
• While the model allows investors to weight their views based on confidence levels, there isn’t a specific guideline on how to determine this, which presents another challenge.
• It assumes that the market is always in equilibrium, which may not be the case in more volatile markets. In addition, it heavily relies on the CAPM, which has its own assumptions of rational investor behaviour and markets being efficient. Should these not hold true, equilibrium and portfolio weights could be flawed.

Conclusion

The Black-Litterman model finds application across various businesses such as Quant and algorithmic investing firms, robo advisors and FinTech. Most notably, Goldman Sachs Asset Management (GSAM) is known for using it in its asset allocation strategy to create dynamic portfolios that account for market consensus and the firm’s own market views. The model is also constantly evolving, and some firms further enhance it by incorporating market trends such as alternative data, ESG factors and machine learning.

Additional Resources

• Portfolio Risk Management
• Capital Asset Pricing Model(CAPM)
• Arbitrage Pricing Theory
• Portfolio Management Certification