Management Fees

This tutorial shows how to incorporate management fees (MF) into the MeanRisk optimization.

By using The management_fees parameter, you can add linear MF to the optimization problem:

\[total\_fee = \sum_{i=1}^{N} f_{i} \times w_{i}\]

with \(f_{i}\) the management fee of asset i and \(w_{i}\) its weight. The float \(total\_fee\) is impacting the portfolio expected return in the optimization:

\[expected\_return = \mu^{T} \cdot w - total\_fee\]

with \(\mu\) the vector af assets expected returns and \(w\) the vector of assets weights.

The management_fees parameter can be a float, a dictionary or an array-like of shape (n_assets, ). If a float is provided, it is applied to each asset. If a dictionary is provided, its (key/value) pair must be the (asset name/asset MF) and the input X of the fit method must be a DataFrame with the assets names in columns. The default is 0.0 (no management fees).

Note

Another approach is to direcly impact the MF to the input X in order to express the returns net of fee. However, when estimating the \(\mu\) parameter using, for example, Shrinkage estimators, this approach would mix a deterministic amount with an uncertain one leading to unwanted bias in the management fees.

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. We select only 3 assets to make the example more readable, which are Apple (AAPL), General Electric (GE) and JPMorgan (JPM).

import numpy as np
from plotly.io import show

from skfolio import Population
from skfolio.datasets import load_sp500_dataset
from skfolio.model_selection import WalkForward, cross_val_predict
from skfolio.optimization import MeanRisk, ObjectiveFunction
from skfolio.preprocessing import prices_to_returns

prices = load_sp500_dataset()
prices = prices[["AAPL", "GE", "JPM"]]

X = prices_to_returns(prices)

Model

In this tutorial, we will use the Maximum Mean-Variance Utility model with a risk aversion of 1.0:

model = MeanRisk(objective_function=ObjectiveFunction.MAXIMIZE_UTILITY)
model.fit(X)
model.weights_

array([6.17733231e-01, 3.78775169e-09, 3.82266765e-01])

Management Fees

Management fees are usually used in assets under management but for this example we will assume that it also applies for the below stocks:

• Apple: 3% p.a.

• General Electric: 6% p.a.

• JPMorgan: 1% p.a.

The MF are expressed in per annum, so we need to convert them in daily MF. We suppose 252 trading days in a year:

management_fees = {"AAPL": 0.03 / 252, "GE": 0.06 / 252, "JPM": 0.01 / 252}
# Same as management_fees = np.array([0.03, 0.06, 0.01]) / 252

model_mf = MeanRisk(
    objective_function=ObjectiveFunction.MAXIMIZE_UTILITY,
    management_fees=management_fees,
)
model_mf.fit(X)
model_mf.weights_

array([5.74787861e-01, 1.43028265e-08, 4.25212125e-01])

The higher MF of Apple induced a change of weights toward JPMorgan:

model_mf.weights_ - model.weights_

array([-4.29453703e-02,  1.05150748e-08,  4.29453598e-02])

Multi-period portfolio

Let’s assume that we want to rebalance our portfolio every 60 days by re-fitting the model on the latest 60 days. We test the impact of MF using Walk Forward Analysis:

holding_period = 60
fitting_period = 60
cv = WalkForward(train_size=fitting_period, test_size=holding_period)

As explained above, we transform the yearly MF into a daily MF:

management_fees = np.array([0.03, 0.06, 0.01]) / 252

First, we train the model without MF and test it with MF. Note that portfolio_params are parameters passed to the Portfolio during predict and not during fit:

model = MeanRisk(
    objective_function=ObjectiveFunction.MAXIMIZE_UTILITY,
    portfolio_params=dict(management_fees=management_fees),
)
# pred1 is a MultiPeriodPortfolio
pred1 = cross_val_predict(model, X, cv=cv, n_jobs=-1)
pred1.name = "pred1"

Then, we train and test the model with MF:

model.set_params(management_fees=management_fees)
pred2 = cross_val_predict(model, X, cv=cv, n_jobs=-1)
pred2.name = "pred2"

We visualize the results by plotting the cumulative returns of the successive test periods:

population = Population([pred1, pred2])
fig = population.plot_cumulative_returns()
show(fig)

We notice that the model fitted with MF outperform the model fitted without MF.