# Black & Litterman
This tutorial shows how to use the [`BlackLitterman`](https://skfolio.org/generated/skfolio.prior.BlackLitterman.html.md#skfolio.prior.BlackLitterman) estimator in
the [`MeanRisk`](https://skfolio.org/generated/skfolio.optimization.MeanRisk.html.md#skfolio.optimization.MeanRisk) optimization.
A [Prior Estimator](https://skfolio.org/user_guide/prior.html.md#prior) 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 **Frequentist**, **Bayesian** and
**Information-theoretic** approaches into a single cohesive framework:
1. Frequentist:
: * [`EmpiricalPrior`](https://skfolio.org/generated/skfolio.prior.EmpiricalPrior.html.md#skfolio.prior.EmpiricalPrior)
* [`TimeSeriesFactorModel`](https://skfolio.org/generated/skfolio.prior.TimeSeriesFactorModel.html.md#skfolio.prior.TimeSeriesFactorModel)
* [`SyntheticData`](https://skfolio.org/generated/skfolio.prior.SyntheticData.html.md#skfolio.prior.SyntheticData)
2. Bayesian:
: * [`BlackLitterman`](https://skfolio.org/generated/skfolio.prior.BlackLitterman.html.md#skfolio.prior.BlackLitterman)
3. Information-theoretic:
: * [`EntropyPooling`](https://skfolio.org/generated/skfolio.prior.EntropyPooling.html.md#skfolio.prior.EntropyPooling)
* [`OpinionPooling`](https://skfolio.org/generated/skfolio.prior.OpinionPooling.html.md#skfolio.prior.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`](https://skfolio.org/generated/skfolio.prior.ReturnDistribution.html.md#skfolio.prior.ReturnDistribution) in its `return_distribution_`
attribute.
The [`ReturnDistribution`](https://skfolio.org/generated/skfolio.prior.ReturnDistribution.html.md#skfolio.prior.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](https://skfolio.org/user_guide/datasets.html.md#datasets) composed of the daily prices of 20
assets from the SPX Index composition starting from 1990-01-02 up to 2022-12-28:
```Python
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 homogeneous with
the input `X`, we get:
```Python
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:
```Python
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_
```
```none
array([4.73688339e-01, 2.66641803e-02, 2.20094200e-07, 2.02229897e-02,
7.84166754e-03, 3.01723524e-08, 5.20800179e-07, 2.47479547e-03,
3.27262165e-01, 4.77426684e-03, 1.73055734e-02, 3.34208578e-02,
2.10139601e-03, 1.65511656e-02, 2.20704194e-06, 8.58902956e-07,
3.40428153e-02, 3.36417527e-02, 6.20216082e-07, 3.57834061e-06])
```
## Empirical Model
For comparison, we also create a Maximum Sharpe Ratio model using the default
Empirical estimator:
```Python
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_
```
```none
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, 9.58498590e-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:
```Python
pred_bl = model_bl.predict(X_test)
pred_empirical = model_empirical.predict(X_test)
population = Population([pred_bl, pred_empirical])
population.plot_cumulative_returns()
```
[plotly figure stripped from llms output]
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:
```Python
fig = population.plot_composition()
show(fig)
```
[plotly figure stripped from llms output]
**Total running time of the script:** (0 minutes 2.370 seconds)