Black & Litterman

This tutorial shows how to use the BlackLitterman estimator in the MeanRisk optimization.

A Prior Estimator in skfolio fits a ReturnDistribution containing your pre-optimization inputs (\(\mu\), \(\Sigma\), returns, sample weight, Cholesky decomposition).

The term “prior” is used in a general optimization sense, not confined to Bayesian priors. It denotes any a priori assumption or estimation method for the return distribution before optimization, unifying both Frequentist, Bayesian and Information-theoretic approaches into a single cohesive framework:

1. Frequentist:

   • EmpiricalPrior

   • FactorModel

   • SyntheticData

2. Bayesian:

   • BlackLitterman

3. Information-theoretic:

   • EntropyPooling

   • OpinionPooling

In skfolio’s API, all such methods share the same interface and adhere to scikit-learn’s estimator API: the fit method accepts X (the asset returns) and stores the resulting ReturnDistribution in its return_distribution_ attribute.

The ReturnDistribution is a dataclass containing:

• mu: Estimated expected returns of shape (n_assets,)

• covariance: Estimated covariance matrix of shape (n_assets, n_assets)

• returns: (Estimated) asset returns of shape (n_observations, n_assets)

• sample_weight : Sample weight for each observation of shape (n_observations,) (optional)

• cholesky : Lower-triangular Cholesky factor of the covariance (optional)

The BlackLitterman estimator estimates the ReturnDistribution using the Black & Litterman model. It takes a Bayesian approach by starting from a prior estimate of the assets’ expected returns and covariance matrix, then updating them with the analyst’s views to obtain the posterior estimates.

In this tutorial we will build a Maximum Sharpe Ratio portfolio using the BlackLitterman estimator.

Data

We load the S&P 500 dataset composed of the daily prices of 20 assets from the SPX Index composition starting from 1990-01-02 up to 2022-12-28:

from plotly.io import show
from sklearn.model_selection import train_test_split

from skfolio import Population, RiskMeasure
from skfolio.datasets import load_sp500_dataset
from skfolio.optimization import MeanRisk, ObjectiveFunction
from skfolio.preprocessing import prices_to_returns
from skfolio.prior import BlackLitterman

prices = load_sp500_dataset()
X = prices_to_returns(prices)
X_train, X_test = train_test_split(X, test_size=0.33, shuffle=False)

Analyst views

Let’s assume we are able to accurately estimate views about future realization of the market. We estimate that Apple will have an expected return of 25% p.a. (absolute view) and will outperform General Electric by 22% p.a. (relative view). We also estimate that JPMorgan will outperform General Electric by 15% p.a (relative view). By converting these annualized estimates into daily estimates to be homogenous with the input X, we get:

analyst_views = [
    "AAPL == 0.00098",
    "AAPL - GE == 0.00086",
    "JPM - GE == 0.00059",
]

Black & Litterman Model

We create a Maximum Sharpe Ratio model using the Black & Litterman estimator that we fit on the training set:

model_bl = MeanRisk(
    risk_measure=RiskMeasure.VARIANCE,
    objective_function=ObjectiveFunction.MAXIMIZE_RATIO,
    prior_estimator=BlackLitterman(views=analyst_views),
    portfolio_params=dict(name="Black & Litterman"),
)
model_bl.fit(X_train)
model_bl.weights_

array([4.73688339e-01, 2.66641803e-02, 2.20094198e-07, 2.02229897e-02,
       7.84166754e-03, 3.01723522e-08, 5.20800181e-07, 2.47479547e-03,
       3.27262165e-01, 4.77426684e-03, 1.73055734e-02, 3.34208578e-02,
       2.10139601e-03, 1.65511656e-02, 2.20704187e-06, 8.58902957e-07,
       3.40428153e-02, 3.36417527e-02, 6.20216084e-07, 3.57834054e-06])

Empirical Model

For comparison, we also create a Maximum Sharpe Ratio model using the default Empirical estimator:

model_empirical = MeanRisk(
    risk_measure=RiskMeasure.VARIANCE,
    objective_function=ObjectiveFunction.MAXIMIZE_RATIO,
    portfolio_params=dict(name="Empirical"),
)
model_empirical.fit(X_train)
model_empirical.weights_

array([9.43631399e-02, 1.13184579e-06, 5.04970598e-07, 1.20834667e-01,
       3.18126275e-02, 8.57806907e-07, 7.11596802e-04, 1.24104939e-01,
       9.49223801e-07, 2.77547553e-02, 1.23409042e-06, 1.37593860e-06,
       1.16299875e-01, 5.73516411e-02, 9.58498589e-06, 1.09493919e-01,
       8.64761638e-02, 1.83992252e-01, 1.32350165e-02, 3.35537683e-02])

Prediction

We predict both models on the test set:

pred_bl = model_bl.predict(X_test)
pred_empirical = model_empirical.predict(X_test)

population = Population([pred_bl, pred_empirical])

population.plot_cumulative_returns()

Because our views were accurate, the Black & Litterman model outperformed the Empirical model on the test set. From the below composition, we can see that Apple and JPMorgan were allocated more weights:

fig = population.plot_composition()
show(fig)