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The Black-Litterman (BL) model [1] takes a Bayesian approach to asset allocation.Specifically, it combines a prior estimate of returns (for example, the market-impliedreturns) with views on certain assets, to produce a posterior estimate of expectedreturns. The advantages of this are:

Black Litterman Model Black Litterman Overview In principle, Modern Portfolio Theory (the mean-variance approach of Markowitz) offers a solution to this problem once the expected returns and covariances of the assets are known. He and Litterman 1999 discuss three simple intuitive properties of the BL model, which can be easily seen from (6) and (9): The unconstrained optimal portfolio is the market equilibrium portfolio plus a weighted sum of portfolios representing an investor’s views.' This is a direct implication of (9).

• You can provide views on only a subset of assets and BL will meaningfully propagate it,taking into account the covariance with other assets.
• You can provide confidence in your views.
• Using Black-Litterman posterior returns results in much more stable portfolios thanusing mean-historical return.

Essentially, Black-Litterman treats the vector of expected returns itself as a quantity tobe estimated. The Black-Litterman formula is given below:

• (E(R)) is a Nx1 vector of expected returns, where N is the number of assets.
• (Q) is a Kx1 vector of views.
• (P) is the KxN picking matrix which maps views to the universe of assets.Essentially, it tells the model which view corresponds to which asset(s).
• (Omega) is the KxK uncertainty matrix of views.
• (Pi) is the Nx1 vector of prior expected returns.
• (Sigma) is the NxN covariance matrix of asset returns (as always)
• (tau) is a scalar tuning constant.

Though the formula appears to be quite unwieldy, it turns out that the formula simply representsa weighted average between the prior estimate of returns and the views, where the weightingis determined by the confidence in the views and the parameter (tau).

Similarly, we can calculate a posterior estimate of the covariance matrix:

Though the algorithm is relatively simple, BL proved to be a challenge from a softwareengineering perspective because it’s not quite clear how best to fit it into PyPortfolioOpt’sAPI. The full discussion can be found on a Github issue thread,but I ultimately decided that though BL is not technically an optimizer, it didn’t make sense tosplit up its methods into expected_returns or risk_models. I have thus made it an independentmodule and owing to the comparatively extensive theory, have given it a dedicated documentation page.I’d like to thank Felipe Schneider for his multiplecontributions to the Black-Litterman implementation. A full example of its usage, including the acquistionof market cap data for free, please refer to the cookbook recipe.

Tip

Thomas Kirschenmann has built a neat interactive Black-Litterman toolon top of PyPortfolioOpt, which allows you to visualise BL outputs and compare optimization objectives.

Priors¶

You can think of the prior as the “default” estimate, in the absence of any information.Black and Litterman (1991) [2] provide the insight that a natural choice for this prioris the market’s estimate of the return, which is embedded into the market capitalisationof the asset.

Every asset in the market portfolio contributes a certain amount of risk to the portfolio.Standard theory suggests that investors must be compensated for the risk that they take, sowe can attribute to each asset an expected compensation (i.e prior estimate of returns). Thisis quantified by the market-implied risk premium, which is the market’s excess return dividedby its variance:

To calculate the market-implied returns, we then use the following formula:

Here, (w_{mkt}) denotes the market-cap weights. This formula is calculating the totalamount of risk contributed by an asset and multiplying it with the market price of risk,resulting in the market-implied returns vector (Pi). We can use PyPortfolioOpt to calculatethis as follows:

There is nothing stopping you from using any prior you see fit (but it must have the same dimensionality as the universe).If you think that the mean historical returns are a good prior,you could go with that. But a significant body of research shows that mean historical returns are a completely uninformativeprior.

Note

You don’t technically have to provide a prior estimate to the Black-Litterman model. This is particularly usefulif your views (and confidences) were generated by some proprietary model, in which case BL is essentially a clever wayof mixing your views.

Views¶

In the Black-Litterman model, users can either provide absolute or relative views. Absolute views are statements like:“AAPL will return 10%” or “XOM will drop 40%”. Relative views, on the other hand, are statements like “GOOG will outperform FB by 3%”.

These views must be specified in the vector (Q) and mapped to the asset universe via the picking matrix (P). A briefexample of this is shown below, though a comprehensive guide is given by Idzorek.Let’s say that our universe is defined by the ordered list: SBUX, GOOG, FB, AAPL, BAC, JPM, T, GE, MSFT, XOM. We want to representfour views on these 10 assets, two absolute and two relative:

1. SBUX will drop 20% (absolute)
2. MSFT will rise by 5% (absolute)
3. GOOG outperforms FB by 10%
4. BAC and JPM will outperform T and GE by 15%

The corresponding views vector is formed by taking the numbers above and putting them into a column:

The picking matrix is more interesting. Remember that its role is to link the views (which mention 8 assets) to the universe of 10assets. Arguably, this is the most important part of the model because it is what allows us to propagate our expectations (andconfidences in expectations) into the model:

A brief explanation of the above:

• Each view has a corresponding row in the picking matrix (the order matters)
• Absolute views have a single 1 in the column corresponding to the ticker’s order in the universe.
• Relative views have a positive number in the nominally outperforming asset columns and a negative numberin the nominally underperforming asset columns. The numbers in each row should sum up to 0.

PyPortfolioOpt provides a helper method for inputting absolute views as either a dict or pd.Series –if you have relative views, you must build your picking matrix manually:

Confidence matrix and tau¶

The confidence matrix is a diagonal covariance matrix containing the variances of each view. One heuristic for calculating(Omega) is to say that is proportional to the variance of the priors. This is reasonable - quantities that movearound a lot are harder to forecast! Hence PyPortfolioOpt does not require you to input a confidence matrix, and defaults to:

Alternatively, we provide an implementation of Idzorek’s method [1]. This allows you to specify your view uncertainties aspercentage confidences. To use this, choose omega='idzorek' and pass a list of confidences (from 0 to 1) into the view_confidencesparameter.

You are of course welcome to provide your own estimate. This is particularly applicable if your views are the outputof some statistical model, which may also provide the view uncertainty.

Another parameter that controls the relative weighting of the priors views is (tau). There is a lot to be said about tuningthis parameter, with many contradictory rules of thumb. Indeed, there has been an entire paper written on it [3]. We choosethe sensible default (tau = 0.05).

Note

If you use the default estimate of (Omega), or omega='idzorek', it turns out that the value of (tau) does not matter. Thisis a consequence of the mathematics: the (tau) cancels in the matrix multiplications.

Output of the BL model¶

The BL model outputs posterior estimates of the returns and covariance matrix. The default suggestion in the literature is tothen input these into an optimizer (see General Efficient Frontier). A quick alternative, which is quite useful for debugging, isto calculate the weights implied by the returns vector [4]. It is actually the reverse of the procedure we used to calculate thereturns implied by the market weights.

In PyPortfolioOpt, this is available under BlackLittermanModel.bl_weights(). Because the BlackLittermanModel classinherits from BaseOptimizer, this follows the same API as the EfficientFrontier objects:

Documentation reference¶

The black_litterman module houses the BlackLittermanModel class, whichgenerates posterior estimates of expected returns given a prior estimate and user-suppliedviews. In addition, two utility functions are defined, which calculate:

• market-implied prior estimate of returns
• market-implied risk-aversion parameter

A BlackLittermanModel object (inheriting from BaseOptimizer) contains requiresa specific input format, specifying the prior, the views, the uncertainty in views,and a picking matrix to map views to the asset universe. We can then computeposterior estimates of returns and covariance. Helper methods have been providedto supply defaults where possible.

Black Litterman Model Excel Download

Instance variables:

• Inputs:

  • cov_matrix - np.ndarray
  • n_assets - int
  • tickers - str list
  • Q - np.ndarray
  • P - np.ndarray
  • pi - np.ndarray
  • omega - np.ndarray
  • tau - float

• Output:

  • posterior_rets - pd.Series
  • posterior_cov - pd.DataFrame
  • weights - np.ndarray

Public methods:

• default_omega() - view uncertainty proportional to asset variance
• idzorek_method() - convert views specified as percentages into BL uncertainties
• bl_returns() - posterior estimate of returns
• bl_cov() - posterior estimate of covariance
• bl_weights() - weights implied by posterior returns
• portfolio_performance() calculates the expected return, volatilityand Sharpe ratio for the allocated portfolio.
• set_weights() creates self.weights (np.ndarray) from a weights dict
• clean_weights() rounds the weights and clips near-zeros.
• save_weights_to_file() saves the weights to csv, json, or txt.

Caution

You must specify the covariance matrix and either absolute views or both Q and P, except in the special casewhere you provide exactly one view per asset, in which case P is inferred.

Calculate the posterior estimate of the covariance matrix,given views on some assets. Based on He and Litterman (2002).It is assumed that omega is diagonal. If this is not the case,please manually set omega_inv.

Calculate the posterior estimate of the returns vector,given views on some assets.

Compute the weights implied by the posterior returns, given themarket price of risk. Technically this can be applied to anyestimate of the expected returns, and is in fact a special caseof mean-variance optimization

If the uncertainty matrix omega is not provided, we calculate using the method ofHe and Litterman (1999), such that the ratio omega/tau is proportional to thevariance of the view portfolio.

Use Idzorek’s method to create the uncertainty matrix given user-specifiedpercentage confidences. We use the closed-form solution described byJay Walters in The Black-Litterman Model in Detail (2014).

Alias for bl_weights for consistency with other methods.

After optimising, calculate (and optionally print) the performance of the optimalportfolio. Currently calculates expected return, volatility, and the Sharpe ratio.This method uses the BL posterior returns and covariance matrix.

Compute the prior estimate of returns implied by the market weights.In other words, given each asset’s contribution to the risk of the marketportfolio, how much are we expecting to be compensated?

Calculate the market-implied risk-aversion parameter (i.e market price of risk)based on market prices. For example, if the market has excess returns of 10% a yearwith 5% variance, the risk-aversion parameter is 2, i.e you have to be compensated 2xthe variance.

References¶

Black Litterman Model Excel

Black Litterman Model Excel Book