Tracking Error

This tutorial shows how to incorporate a tracking error constraint into the MeanRisk optimization.

The tracking error is defined as the RMSE (root-mean-square error) of the portfolio returns compared to a 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 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:

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:

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_

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:

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_

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.

model_spx = EqualWeighted(portfolio_params=dict(name="SPX Index"))
model_spx.fit(y_train)
model_spx.weights_

array([1.])

Now we plot both models and the SPX Index on the training set:

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)

Let’s print the tracking error and the CVaR:

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")

========================
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:

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")

========================
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.