# Tracking Error This tutorial shows how to incorporate a tracking error constraint into the [`MeanRisk`](https://skfolio.org/generated/skfolio.optimization.MeanRisk.html.md#skfolio.optimization.MeanRisk) optimization. The tracking error is defined as the RMSE (root-mean-square error) of the portfolio returns compared to target returns. In this example we will create a long-short portfolio of 20 stocks that tracks the SPX Index with a tracking error constraint of 0.30% while minimizing the CVaR (Conditional Value at Risk) at 95%. ## 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 and the prices of the S&P 500 Index itself: ```Python import numpy as np from plotly.io import show from sklearn import clone from sklearn.model_selection import train_test_split from skfolio import Population, RiskMeasure from skfolio.datasets import load_sp500_dataset, load_sp500_index from skfolio.optimization import EqualWeighted, MeanRisk, ObjectiveFunction from skfolio.preprocessing import prices_to_returns prices = load_sp500_dataset() prices = prices["2014":] spx_prices = load_sp500_index() spx_prices = spx_prices["2014":] X, y = prices_to_returns(prices, spx_prices) X_train, X_test, y_train, y_test = train_test_split(X, y, test_size=0.33, shuffle=False) ``` ## Model We create two long-short models: a Minimum CVaR without tracking error and a Minimum CVaR with a 0.30% tracking error constraint versus the SPX Index. A 0.30% tracking error constraint is a constraint on the RMSE of the difference between the daily portfolio returns and the SPX Index returns. We first create the Minimum CVaR model without tracking error: ```Python model_no_tracking = MeanRisk( objective_function=ObjectiveFunction.MINIMIZE_RISK, risk_measure=RiskMeasure.CVAR, min_weights=-1, portfolio_params=dict(name="Minimum-CVaR", tag="No Tracking"), ) model_no_tracking.fit(X_train, y_train) model_no_tracking.weights_ ``` ```none array([ 0.03676195, -0.02374444, 0.02155918, 0.02117437, 0.0373302 , -0.00370324, 0.01028137, 0.0298146 , -0.07317879, 0.15980287, -0.00139632, 0.01100645, -0.0810491 , 0.20227491, 0.21580436, 0.17510456, 0.00322793, 0.06858182, 0.11093659, 0.07941073]) ``` Then we create the Minimum CVaR model with a 0.30% tracking error constraint versus the SPX Index: ```Python model_tracking = clone(model_no_tracking) model_tracking.set_params( max_tracking_error=0.003, portfolio_params=dict(name="Minimum-CVaR", tag="Tracking 0.30%"), ) model_tracking.fit(X_train, y_train) model_no_tracking.weights_ ``` ```none array([ 0.03676195, -0.02374444, 0.02155918, 0.02117437, 0.0373302 , -0.00370324, 0.01028137, 0.0298146 , -0.07317879, 0.15980287, -0.00139632, 0.01100645, -0.0810491 , 0.20227491, 0.21580436, 0.17510456, 0.00322793, 0.06858182, 0.11093659, 0.07941073]) ``` For comparison, we create a single asset Portfolio model containing the SPX Index. ```Python model_spx = EqualWeighted(portfolio_params=dict(name="SPX Index")) model_spx.fit(y_train) model_spx.weights_ ``` ```none array([1.]) ``` Now we plot both models and the SPX Index on the training set: ```Python ptf_no_tracking_train = model_no_tracking.predict(X_train) ptf_tracking_train = model_tracking.predict(X_train) spx_train = model_spx.predict(y_train) # Note that we coule have directly used: # train_spx = Portfolio(y_train, weights=[1], name="SPX Index") population_train = Population([ptf_no_tracking_train, ptf_tracking_train, spx_train]) fig = population_train.plot_cumulative_returns() show(fig) ``` [plotly figure stripped from llms output]
Let’s print the tracking error and the CVaR: ```Python for portfolio in [ptf_no_tracking_train, ptf_tracking_train]: tracking_rmse = np.sqrt(np.mean((portfolio.returns - spx_train.returns) ** 2)) print("========================") print(portfolio.tag) print("========================") print(f"Tracking RMSE: {tracking_rmse:0.2%}") print(f"CVaR at 95%: {portfolio.cvar:0.2%}") print(f"CVaR ratio: {portfolio.cvar_ratio:0.2f}") print("\n") ``` ```none ======================== No Tracking ======================== Tracking RMSE: 0.60% CVaR at 95%: 1.58% CVaR ratio: 0.02 ======================== Tracking 0.30% ======================== Tracking RMSE: 0.30% CVaR at 95%: 1.78% CVaR ratio: 0.03 ``` The model with tracking error achieved the required RMSE of 0.30% versus the SPX on the training set. The tradeoff of this constraint is the higher CVaR value versus the model without tracking error. ## Prediction Finally, we predict both models on the test set: ```Python ptf_no_tracking_test = model_no_tracking.predict(X_test) ptf_tracking_test = model_tracking.predict(X_test) spx_test = model_spx.predict(y_test) for portfolio in [ptf_no_tracking_test, ptf_tracking_test]: tracking_rmse = np.sqrt(np.mean((portfolio.returns - spx_test.returns) ** 2)) print("========================") print(portfolio.tag) print("========================") print(f"Tracking RMSE: {tracking_rmse:0.2%}") print(f"CVaR at 95%: {portfolio.cvar:0.2%}") print(f"CVaR ratio: {portfolio.cvar_ratio:0.2f}") print("\n") ``` ```none ======================== No Tracking ======================== Tracking RMSE: 1.04% CVaR at 95%: 3.08% CVaR ratio: 0.02 ======================== Tracking 0.30% ======================== Tracking RMSE: 0.58% CVaR at 95%: 3.39% CVaR ratio: 0.02 ``` As expected, the model with tracking error also achieved a lower RMSE on the test set compared to the model without tracking error. **Total running time of the script:** (0 minutes 0.880 seconds)