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# Risk Parity - Covariance shrinkage#

This tutorial shows how to incorporate covariance shrinkage in the
`RiskBudgeting`

optimization.

## Data#

We load the S&P 500 dataset composed of the daily prices of 20 assets from the S&P 500 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.moments import ShrunkCovariance
from skfolio.optimization import RiskBudgeting
from skfolio.preprocessing import prices_to_returns
from skfolio.prior import EmpiricalPrior
prices = load_sp500_dataset()
X = prices_to_returns(prices)
X_train, X_test = train_test_split(X, test_size=0.33, shuffle=False)
```

## Model#

We create a risk parity model by using `ShrunkCovariance`

as
the covariance estimator then fit it on the training set:

```
model = RiskBudgeting(
risk_measure=RiskMeasure.VARIANCE,
prior_estimator=EmpiricalPrior(
covariance_estimator=ShrunkCovariance(shrinkage=0.9)
),
portfolio_params=dict(name="Risk Parity - Covariance Shrinkage"),
)
model.fit(X_train)
model.weights_
```

```
array([0.04774432, 0.0437015 , 0.04503263, 0.04647771, 0.0528462 ,
0.04907541, 0.04852969, 0.0537393 , 0.0453914 , 0.05360544,
0.0517859 , 0.05137979, 0.04927078, 0.05375933, 0.05112798,
0.05417573, 0.04755329, 0.04988135, 0.05199345, 0.0529288 ])
```

To compare this model, we use a basic risk parity without covariance shrinkage:

```
bench = RiskBudgeting(
risk_measure=RiskMeasure.VARIANCE,
portfolio_params=dict(name="Risk Parity - Basic"),
)
bench.fit(X_train)
bench.weights_
```

```
array([0.0413538 , 0.03210848, 0.03372647, 0.03785094, 0.06105331,
0.04432756, 0.0425223 , 0.06593521, 0.03451801, 0.06469243,
0.05418786, 0.0520937 , 0.04535355, 0.06568282, 0.05103838,
0.06894686, 0.04046603, 0.04667582, 0.05627071, 0.06119574])
```

## Prediction#

We predict the model and the benchmark on the test set:

```
ptf_model_test = model.predict(X_test)
ptf_bench_test = bench.predict(X_test)
```

## Analysis#

For improved analysis, it’s possible to load both predicted portfolios into a
`Population`

:

```
population = Population([ptf_model_test, ptf_bench_test])
```

Let’s plot each portfolio cumulative returns:

```
fig = population.plot_cumulative_returns()
show(fig)
```

Finally, we print a full summary of both strategies evaluated on the test set:

```
population.summary()
```

**Total running time of the script:** (0 minutes 1.212 seconds)