Investment Horizon

This tutorial explores the difference between the general procedure using different investment horizon and the simplified procedure as explained in data preparation.

Prices

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 skfolio import PerfMeasure, Population, RiskMeasure
from skfolio.datasets import load_sp500_dataset
from skfolio.optimization import MeanRisk
from skfolio.preprocessing import prices_to_returns
from skfolio.prior import EmpiricalPrior

prices = load_sp500_dataset()
prices.head()

	AAPL	AMD	BAC	BBY	CVX	GE	HD	JNJ	JPM	KO	LLY	MRK	MSFT	PEP	PFE	PG	RRC	UNH	WMT	XOM
Date
1990-01-02	0.264	4.125	4.599	0.144	4.991	14.391	1.117	3.438	3.394	2.235	6.658	4.215	0.384	4.738	1.021	3.860	3.322	0.310	3.653	4.068
1990-01-03	0.266	4.000	4.636	0.161	4.910	14.364	1.121	3.452	3.508	2.203	6.658	4.282	0.386	4.692	1.024	3.853	3.322	0.304	3.653	4.027
1990-01-04	0.267	3.938	4.537	0.159	4.847	14.283	1.128	3.459	3.522	2.192	6.621	4.215	0.397	4.646	1.041	3.777	3.322	0.301	3.634	3.987
1990-01-05	0.268	3.812	4.438	0.159	4.775	14.148	1.113	3.423	3.536	2.174	6.549	4.128	0.387	4.581	1.032	3.709	3.322	0.288	3.595	3.966
1990-01-08	0.269	3.812	4.463	0.147	4.820	14.229	1.102	3.481	3.536	2.220	6.549	4.181	0.393	4.664	1.023	3.777	3.322	0.282	3.644	4.027

Linear Returns

We transform the daily prices into daily linear returns:

X = prices_to_returns(prices)

Model

We first create a Mean-Variance model using the simplified procedure:

population = Population([])

model = MeanRisk(
    risk_measure=RiskMeasure.VARIANCE,
    efficient_frontier_size=30,
    portfolio_params=dict(name="Simplified", tag="Simplified"),
)
population.extend(model.fit_predict(X))

for tag, investment_horizon in [
    ("3M", 252 / 4),
    ("1Y", 252),
    ("10Y", 10 * 252),
]:
    model = MeanRisk(
        risk_measure=RiskMeasure.VARIANCE,
        efficient_frontier_size=30,
        prior_estimator=EmpiricalPrior(
            is_log_normal=True, investment_horizon=investment_horizon
        ),
        portfolio_params=dict(name=f"General - {tag}", tag=f"General - {tag}"),
    )
    population.extend(model.fit_predict(X))

Let’s plot the efficient frontier:

fig = population.plot_measures(
    x=RiskMeasure.ANNUALIZED_VARIANCE,
    y=PerfMeasure.ANNUALIZED_MEAN,
)
show(fig)

Let’s plot the portfolios compositions:

population.plot_composition()

We can see that the simplified procedure only start to diverge from the general one for investment horizons longer than one year.