# Risk Parity - Covariance shrinkage
This tutorial shows how to incorporate covariance shrinkage in the
[`RiskBudgeting`](https://skfolio.org/generated/skfolio.optimization.RiskBudgeting.html.md#skfolio.optimization.RiskBudgeting) optimization.
## 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 S&P 500 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.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`](https://skfolio.org/generated/skfolio.moments.ShrunkCovariance.html.md#skfolio.moments.ShrunkCovariance) as
the covariance estimator then fit it on the training set:
```Python
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_
```
```none
array([0.04774307, 0.04370215, 0.0450328 , 0.04647857, 0.05284639,
0.04907481, 0.04852994, 0.05373958, 0.04539454, 0.05360737,
0.05178488, 0.05137988, 0.0492705 , 0.05375749, 0.05112782,
0.05417595, 0.04755101, 0.04988012, 0.05199328, 0.05292985])
```
To compare this model, we use a basic risk parity without covariance shrinkage:
```Python
bench = RiskBudgeting(
risk_measure=RiskMeasure.VARIANCE,
portfolio_params=dict(name="Risk Parity - Basic"),
)
bench.fit(X_train)
bench.weights_
```
```none
array([0.04135491, 0.03210918, 0.03372616, 0.03784991, 0.0610532 ,
0.0443272 , 0.04252063, 0.06593451, 0.03451906, 0.06469335,
0.05418982, 0.05209335, 0.04535122, 0.06568127, 0.05104022,
0.06894491, 0.0404655 , 0.04667832, 0.05627135, 0.06119597])
```
## Prediction
We predict the model and the benchmark on the test set:
```Python
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`](https://skfolio.org/generated/skfolio.population.Population.html.md#skfolio.population.Population):
```Python
population = Population([ptf_model_test, ptf_bench_test])
```
Let’s plot each portfolio cumulative returns:
```Python
fig = population.plot_cumulative_returns()
show(fig)
```
[plotly figure stripped from llms output]
Finally, we print a full summary of both strategies evaluated on the test set:
```Python
population.summary()
```
|
Risk Parity - Covariance Shrinkage |
Risk Parity - Basic |
| Mean |
0.068% |
0.065% |
| Annualized Mean |
17.04% |
16.40% |
| Variance |
0.00011 |
0.00010 |
| Annualized Variance |
2.84% |
2.63% |
| Semi-Variance |
0.000058 |
0.000054 |
| Annualized Semi-Variance |
1.47% |
1.36% |
| Standard Deviation |
1.06% |
1.02% |
| Annualized Standard Deviation |
16.86% |
16.22% |
| Semi-Deviation |
0.76% |
0.73% |
| Annualized Semi-Deviation |
12.11% |
11.66% |
| Mean Absolute Deviation |
0.69% |
0.66% |
| CVaR at 95% |
2.46% |
2.37% |
| EVaR at 95% |
5.37% |
5.23% |
| Worst Realization |
10.66% |
10.34% |
| CDaR at 95% |
13.02% |
12.39% |
| MAX Drawdown |
34.58% |
34.30% |
| Average Drawdown |
2.58% |
2.40% |
| EDaR at 95% |
20.30% |
19.86% |
| First Lower Partial Moment |
0.35% |
0.33% |
| Ulcer Index |
0.043 |
0.041 |
| Gini Mean Difference |
1.04% |
1.00% |
| Value at Risk at 95% |
1.51% |
1.46% |
| Drawdown at Risk at 95% |
9.30% |
8.97% |
| Entropic Risk Measure at 95% |
3.00 |
3.00 |
| Fourth Central Moment |
0.000025% |
0.000023% |
| Fourth Lower Partial Moment |
0.000012% |
0.000011% |
| Skew |
-3.68% |
-5.68% |
| Kurtosis |
1998.27% |
2067.02% |
| Sharpe Ratio |
0.064 |
0.064 |
| Annualized Sharpe Ratio |
1.01 |
1.01 |
| Sortino Ratio |
0.089 |
0.089 |
| Annualized Sortino Ratio |
1.41 |
1.41 |
| Mean Absolute Deviation Ratio |
0.097 |
0.098 |
| First Lower Partial Moment Ratio |
0.19 |
0.20 |
| Value at Risk Ratio at 95% |
0.045 |
0.045 |
| CVaR Ratio at 95% |
0.027 |
0.027 |
| Entropic Risk Measure Ratio at 95% |
0.00023 |
0.00022 |
| EVaR Ratio at 95% |
0.013 |
0.012 |
| Worst Realization Ratio |
0.0063 |
0.0063 |
| Drawdown at Risk Ratio at 95% |
0.0073 |
0.0073 |
| CDaR Ratio at 95% |
0.0052 |
0.0053 |
| Calmar Ratio |
0.0020 |
0.0019 |
| Average Drawdown Ratio |
0.026 |
0.027 |
| EDaR Ratio at 95% |
0.0033 |
0.0033 |
| Ulcer Index Ratio |
0.016 |
0.016 |
| Gini Mean Difference Ratio |
0.065 |
0.065 |
| Effective Number of Assets |
19.921041997950763 |
18.995841455293334 |
| Assets Number |
20 |
20 |
**Total running time of the script:** (0 minutes 0.661 seconds)