Efficient Frontier

This tutorial uses the MeanRisk optimization to find an ensemble of portfolios belonging to the Mean-Variance efficient frontier (pareto font).

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:

import numpy as np
from plotly.io import show
from sklearn.model_selection import train_test_split

from skfolio import PerfMeasure, RatioMeasure, RiskMeasure
from skfolio.datasets import load_sp500_dataset
from skfolio.optimization import MeanRisk
from skfolio.preprocessing import prices_to_returns

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 the Mean-Variance model and then fit it on the training set. The parameter efficient_frontier_size=30 is used to find 30 portfolios on the entire efficient frontier:

model = MeanRisk(
    risk_measure=RiskMeasure.VARIANCE,
    efficient_frontier_size=30,
    portfolio_params=dict(name="Variance"),
)
model.fit(X_train)
print(model.weights_.shape)

(30, 20)

Prediction

We predict this model on both the training set and the test set. The predict method returns the Population of 30 Portfolio:

population_train = model.predict(X_train)
population_test = model.predict(X_test)

Analysis

For improved analysis, we add a “Train” and “Test” tag to the portfolios and concatenate the training and the test populations:

population_train.set_portfolio_params(tag="Train")
population_test.set_portfolio_params(tag="Test")

population = population_train + population_test

fig = population.plot_measures(
    x=RiskMeasure.ANNUALIZED_STANDARD_DEVIATION,
    y=PerfMeasure.ANNUALIZED_MEAN,
    color_scale=RatioMeasure.ANNUALIZED_SHARPE_RATIO,
    hover_measures=[RiskMeasure.MAX_DRAWDOWN, RatioMeasure.ANNUALIZED_SORTINO_RATIO],
)
show(fig)

Let’s plot the composition of the 30 portfolios:

population_train.plot_composition()

Let’s print the Sharpe Ratio of the 30 portfolios on the test set:

population_test.measures(measure=RatioMeasure.ANNUALIZED_SHARPE_RATIO)

array([0.91785162, 0.93000375, 0.9401232 , 0.95027284, 0.96861483,
       0.98463685, 0.9983657 , 1.00992555, 1.01943433, 1.02702131,
       1.032813  , 1.03704654, 1.03993049, 1.04204356, 1.0430764 ,
       1.04289017, 1.04158213, 1.03774863, 1.02943678, 1.02092595,
       1.01241175, 1.00356473, 0.97964062, 0.93749217, 0.87986871,
       0.78090544, 0.68550154, 0.59858003, 0.53398775, 0.55455742])

Finally, we can show a full summary of the 30 portfolios evaluated on the test set:

population.summary()

	ptf0 - Variance	ptf1 - Variance	ptf2 - Variance	ptf3 - Variance	ptf4 - Variance	ptf5 - Variance	ptf6 - Variance	ptf7 - Variance	ptf8 - Variance	ptf9 - Variance	ptf10 - Variance	ptf11 - Variance	ptf12 - Variance	ptf13 - Variance	ptf14 - Variance	ptf15 - Variance	ptf16 - Variance	ptf17 - Variance	ptf18 - Variance	ptf19 - Variance	ptf20 - Variance	ptf21 - Variance	ptf22 - Variance	ptf23 - Variance	ptf24 - Variance	ptf25 - Variance	ptf26 - Variance	ptf27 - Variance	ptf28 - Variance	ptf29 - Variance	ptf0 - Variance	ptf1 - Variance	ptf2 - Variance	ptf3 - Variance	ptf4 - Variance	ptf5 - Variance	ptf6 - Variance	ptf7 - Variance	ptf8 - Variance	ptf9 - Variance	ptf10 - Variance	ptf11 - Variance	ptf12 - Variance	ptf13 - Variance	ptf14 - Variance	ptf15 - Variance	ptf16 - Variance	ptf17 - Variance	ptf18 - Variance	ptf19 - Variance	ptf20 - Variance	ptf21 - Variance	ptf22 - Variance	ptf23 - Variance	ptf24 - Variance	ptf25 - Variance	ptf26 - Variance	ptf27 - Variance	ptf28 - Variance	ptf29 - Variance
Mean	0.062%	0.065%	0.068%	0.071%	0.074%	0.076%	0.079%	0.082%	0.085%	0.088%	0.091%	0.094%	0.096%	0.099%	0.10%	0.11%	0.11%	0.11%	0.11%	0.12%	0.12%	0.12%	0.13%	0.13%	0.13%	0.13%	0.14%	0.14%	0.14%	0.15%	0.054%	0.055%	0.056%	0.057%	0.059%	0.061%	0.063%	0.064%	0.066%	0.068%	0.070%	0.071%	0.073%	0.075%	0.077%	0.078%	0.080%	0.082%	0.083%	0.085%	0.086%	0.088%	0.088%	0.087%	0.086%	0.083%	0.081%	0.079%	0.078%	0.090%
Annualized Mean	15.66%	16.38%	17.10%	17.82%	18.54%	19.26%	19.99%	20.71%	21.43%	22.15%	22.87%	23.59%	24.32%	25.04%	25.76%	26.48%	27.20%	27.92%	28.65%	29.37%	30.09%	30.81%	31.53%	32.25%	32.98%	33.70%	34.42%	35.14%	35.86%	36.58%	13.68%	13.93%	14.18%	14.46%	14.90%	15.34%	15.78%	16.22%	16.66%	17.10%	17.53%	17.97%	18.41%	18.86%	19.31%	19.74%	20.16%	20.58%	20.95%	21.32%	21.71%	22.10%	22.16%	21.92%	21.59%	20.98%	20.37%	19.78%	19.56%	22.73%
Variance	0.011%	0.011%	0.011%	0.011%	0.012%	0.012%	0.012%	0.013%	0.014%	0.014%	0.015%	0.016%	0.017%	0.018%	0.019%	0.020%	0.022%	0.023%	0.025%	0.026%	0.028%	0.030%	0.032%	0.034%	0.038%	0.044%	0.053%	0.066%	0.084%	0.12%	0.0088%	0.0089%	0.0090%	0.0092%	0.0094%	0.0096%	0.0099%	0.010%	0.011%	0.011%	0.011%	0.012%	0.012%	0.013%	0.014%	0.014%	0.015%	0.016%	0.016%	0.017%	0.018%	0.019%	0.020%	0.022%	0.024%	0.029%	0.035%	0.043%	0.053%	0.067%
Annualized Variance	2.76%	2.78%	2.81%	2.86%	2.94%	3.03%	3.15%	3.28%	3.44%	3.62%	3.82%	4.04%	4.29%	4.55%	4.84%	5.14%	5.47%	5.82%	6.20%	6.60%	7.03%	7.48%	7.98%	8.63%	9.48%	11.00%	13.38%	16.70%	21.22%	29.72%	2.22%	2.24%	2.28%	2.32%	2.37%	2.43%	2.50%	2.58%	2.67%	2.77%	2.88%	3.00%	3.13%	3.28%	3.43%	3.58%	3.75%	3.93%	4.14%	4.36%	4.60%	4.85%	5.12%	5.47%	6.02%	7.22%	8.83%	10.92%	13.41%	16.80%
Semi-Variance	0.0054%	0.0055%	0.0055%	0.0056%	0.0058%	0.0060%	0.0062%	0.0065%	0.0068%	0.0072%	0.0076%	0.0080%	0.0085%	0.0090%	0.0096%	0.010%	0.011%	0.012%	0.012%	0.013%	0.014%	0.015%	0.016%	0.017%	0.019%	0.021%	0.026%	0.032%	0.041%	0.058%	0.0045%	0.0046%	0.0046%	0.0047%	0.0048%	0.0049%	0.0051%	0.0052%	0.0054%	0.0056%	0.0058%	0.0060%	0.0063%	0.0065%	0.0068%	0.0071%	0.0074%	0.0077%	0.0082%	0.0086%	0.0091%	0.0096%	0.010%	0.011%	0.012%	0.014%	0.017%	0.021%	0.026%	0.034%
Annualized Semi-Variance	1.37%	1.37%	1.39%	1.42%	1.46%	1.50%	1.56%	1.63%	1.71%	1.80%	1.90%	2.02%	2.14%	2.27%	2.41%	2.56%	2.73%	2.90%	3.09%	3.28%	3.50%	3.72%	3.97%	4.29%	4.69%	5.41%	6.56%	8.17%	10.39%	14.54%	1.14%	1.16%	1.17%	1.19%	1.22%	1.25%	1.28%	1.32%	1.36%	1.41%	1.46%	1.52%	1.58%	1.65%	1.72%	1.79%	1.87%	1.95%	2.06%	2.16%	2.28%	2.41%	2.54%	2.70%	2.96%	3.52%	4.29%	5.30%	6.55%	8.50%
Standard Deviation	1.05%	1.05%	1.06%	1.07%	1.08%	1.10%	1.12%	1.14%	1.17%	1.20%	1.23%	1.27%	1.30%	1.34%	1.39%	1.43%	1.47%	1.52%	1.57%	1.62%	1.67%	1.72%	1.78%	1.85%	1.94%	2.09%	2.30%	2.57%	2.90%	3.43%	0.94%	0.94%	0.95%	0.96%	0.97%	0.98%	1.00%	1.01%	1.03%	1.05%	1.07%	1.09%	1.12%	1.14%	1.17%	1.19%	1.22%	1.25%	1.28%	1.32%	1.35%	1.39%	1.43%	1.47%	1.55%	1.69%	1.87%	2.08%	2.31%	2.58%
Annualized Standard Deviation	16.62%	16.67%	16.76%	16.92%	17.13%	17.41%	17.74%	18.12%	18.55%	19.03%	19.55%	20.11%	20.70%	21.33%	21.99%	22.68%	23.39%	24.13%	24.90%	25.69%	26.51%	27.35%	28.25%	29.38%	30.80%	33.17%	36.58%	40.87%	46.06%	54.52%	14.91%	14.98%	15.09%	15.22%	15.39%	15.58%	15.81%	16.06%	16.34%	16.65%	16.98%	17.33%	17.71%	18.10%	18.51%	18.93%	19.36%	19.83%	20.35%	20.89%	21.45%	22.02%	22.63%	23.38%	24.53%	26.87%	29.72%	33.05%	36.63%	40.99%
Semi-Deviation	0.74%	0.74%	0.74%	0.75%	0.76%	0.77%	0.79%	0.81%	0.82%	0.85%	0.87%	0.89%	0.92%	0.95%	0.98%	1.01%	1.04%	1.07%	1.11%	1.14%	1.18%	1.22%	1.25%	1.30%	1.36%	1.47%	1.61%	1.80%	2.03%	2.40%	0.67%	0.68%	0.68%	0.69%	0.69%	0.70%	0.71%	0.72%	0.73%	0.75%	0.76%	0.78%	0.79%	0.81%	0.83%	0.84%	0.86%	0.88%	0.90%	0.93%	0.95%	0.98%	1.00%	1.03%	1.08%	1.18%	1.30%	1.45%	1.61%	1.84%
Annualized Semi-Deviation	11.69%	11.72%	11.79%	11.91%	12.07%	12.27%	12.51%	12.78%	13.09%	13.43%	13.80%	14.20%	14.62%	15.06%	15.53%	16.01%	16.51%	17.03%	17.57%	18.12%	18.70%	19.29%	19.92%	20.70%	21.67%	23.27%	25.61%	28.58%	32.23%	38.13%	10.69%	10.75%	10.82%	10.92%	11.03%	11.16%	11.31%	11.48%	11.66%	11.87%	12.08%	12.32%	12.57%	12.83%	13.10%	13.37%	13.66%	13.97%	14.34%	14.71%	15.11%	15.52%	15.92%	16.42%	17.19%	18.77%	20.71%	23.02%	25.59%	29.16%
Mean Absolute Deviation	0.74%	0.74%	0.74%	0.75%	0.76%	0.77%	0.79%	0.81%	0.83%	0.85%	0.87%	0.90%	0.93%	0.96%	0.99%	1.02%	1.06%	1.09%	1.13%	1.16%	1.20%	1.24%	1.28%	1.33%	1.39%	1.49%	1.62%	1.79%	2.00%	2.33%	0.61%	0.61%	0.61%	0.62%	0.62%	0.63%	0.64%	0.66%	0.67%	0.69%	0.70%	0.72%	0.74%	0.76%	0.78%	0.80%	0.82%	0.85%	0.87%	0.90%	0.93%	0.96%	0.99%	1.03%	1.09%	1.19%	1.31%	1.45%	1.59%	1.71%
CVaR at 95%	2.31%	2.32%	2.33%	2.35%	2.38%	2.42%	2.46%	2.51%	2.57%	2.63%	2.70%	2.78%	2.86%	2.94%	3.03%	3.13%	3.23%	3.33%	3.43%	3.53%	3.64%	3.76%	3.88%	4.04%	4.24%	4.57%	5.04%	5.63%	6.34%	7.53%	2.16%	2.17%	2.19%	2.21%	2.23%	2.26%	2.29%	2.32%	2.36%	2.40%	2.44%	2.49%	2.54%	2.59%	2.64%	2.69%	2.74%	2.80%	2.87%	2.94%	3.01%	3.09%	3.17%	3.25%	3.40%	3.69%	4.07%	4.54%	5.06%	5.91%
EVaR at 95%	4.23%	4.19%	4.16%	4.14%	4.14%	4.16%	4.20%	4.27%	4.36%	4.47%	4.59%	4.73%	4.88%	5.03%	5.20%	5.37%	5.55%	5.74%	5.95%	6.16%	6.37%	6.59%	6.80%	7.03%	7.29%	8.36%	10.21%	12.35%	14.79%	18.69%	4.63%	4.66%	4.68%	4.71%	4.75%	4.79%	4.83%	4.87%	4.92%	4.96%	5.01%	5.06%	5.11%	5.17%	5.24%	5.28%	5.34%	5.42%	5.51%	5.61%	5.74%	5.88%	5.88%	5.81%	5.88%	6.72%	7.97%	9.35%	10.93%	13.85%
Worst Realization	8.35%	8.18%	8.00%	7.84%	7.71%	7.58%	7.79%	8.16%	8.53%	8.90%	9.28%	9.65%	10.02%	10.39%	10.76%	11.13%	11.50%	11.91%	12.36%	12.81%	13.26%	13.71%	14.06%	14.33%	14.21%	15.83%	20.22%	24.87%	29.84%	38.78%	8.61%	8.73%	8.84%	8.96%	9.08%	9.20%	9.33%	9.45%	9.57%	9.69%	9.81%	9.93%	10.06%	10.20%	10.34%	10.45%	10.57%	10.73%	10.91%	11.11%	11.38%	11.65%	11.55%	11.08%	10.44%	13.12%	16.01%	18.98%	22.31%	28.58%
CDaR at 95%	18.45%	18.90%	19.43%	20.03%	20.82%	21.66%	22.58%	23.59%	24.67%	25.79%	26.96%	28.16%	29.38%	30.59%	31.79%	32.87%	33.97%	35.11%	36.26%	37.45%	38.74%	40.06%	40.74%	41.06%	41.47%	42.52%	45.87%	52.09%	63.28%	94.87%	13.88%	13.72%	13.59%	13.48%	13.26%	13.10%	12.97%	12.90%	12.99%	13.15%	13.33%	13.55%	13.80%	14.10%	14.44%	14.87%	15.39%	15.98%	16.72%	17.61%	18.56%	19.56%	21.29%	23.95%	26.96%	30.50%	35.00%	39.41%	44.13%	60.36%
MAX Drawdown	33.40%	34.61%	35.81%	37.07%	38.61%	40.15%	41.68%	43.22%	44.77%	47.16%	49.82%	52.48%	55.13%	57.77%	60.37%	62.77%	65.20%	67.75%	70.33%	72.96%	75.82%	78.68%	80.90%	82.54%	83.69%	83.30%	82.91%	81.65%	83.54%	132.38%	33.59%	33.62%	33.65%	33.65%	33.58%	33.50%	33.43%	33.36%	33.29%	33.22%	33.15%	33.08%	33.01%	32.95%	32.88%	32.58%	32.33%	32.34%	32.80%	33.42%	34.05%	34.67%	36.26%	38.75%	41.26%	44.24%	50.08%	54.83%	56.35%	76.40%
Average Drawdown	3.75%	3.71%	3.71%	3.75%	3.83%	3.94%	4.07%	4.24%	4.42%	4.61%	4.81%	5.03%	5.25%	5.48%	5.72%	5.95%	6.19%	6.45%	6.72%	7.00%	7.31%	7.63%	8.00%	8.53%	9.32%	10.31%	11.68%	13.88%	17.25%	25.90%	2.86%	2.83%	2.80%	2.79%	2.74%	2.71%	2.69%	2.68%	2.71%	2.77%	2.83%	2.91%	3.00%	3.11%	3.23%	3.40%	3.57%	3.77%	4.00%	4.25%	4.52%	4.79%	5.21%	5.84%	6.73%	8.12%	10.00%	12.28%	14.52%	16.76%
EDaR at 95%	22.41%	23.02%	23.75%	24.58%	25.70%	26.91%	28.18%	29.51%	30.90%	32.34%	33.82%	35.33%	36.87%	38.43%	39.98%	41.38%	42.83%	44.36%	45.92%	47.52%	49.27%	51.04%	52.30%	53.23%	53.91%	53.67%	54.46%	57.41%	67.13%	106.15%	19.59%	19.56%	19.55%	19.52%	19.41%	19.34%	19.28%	19.25%	19.27%	19.31%	19.37%	19.44%	19.53%	19.65%	19.79%	19.91%	20.08%	20.40%	20.99%	21.72%	22.52%	23.36%	24.90%	27.26%	29.99%	33.30%	38.10%	42.81%	48.14%	64.47%
First Lower Partial Moment	0.37%	0.37%	0.37%	0.38%	0.38%	0.39%	0.39%	0.40%	0.41%	0.42%	0.44%	0.45%	0.46%	0.48%	0.50%	0.51%	0.53%	0.55%	0.56%	0.58%	0.60%	0.62%	0.64%	0.67%	0.70%	0.74%	0.81%	0.89%	1.00%	1.16%	0.30%	0.30%	0.31%	0.31%	0.31%	0.32%	0.32%	0.33%	0.34%	0.34%	0.35%	0.36%	0.37%	0.38%	0.39%	0.40%	0.41%	0.42%	0.44%	0.45%	0.46%	0.48%	0.49%	0.51%	0.54%	0.59%	0.66%	0.73%	0.80%	0.85%
Ulcer Index	0.059	0.060	0.061	0.063	0.065	0.067	0.070	0.073	0.076	0.079	0.083	0.086	0.090	0.094	0.097	0.10	0.10	0.11	0.11	0.12	0.12	0.13	0.13	0.13	0.14	0.15	0.17	0.20	0.24	0.36	0.046	0.046	0.046	0.046	0.045	0.044	0.044	0.044	0.044	0.045	0.046	0.047	0.048	0.049	0.051	0.053	0.055	0.058	0.061	0.065	0.069	0.072	0.079	0.088	0.100	0.12	0.14	0.16	0.19	0.24
Gini Mean Difference	1.09%	1.09%	1.10%	1.11%	1.12%	1.14%	1.16%	1.19%	1.22%	1.25%	1.29%	1.33%	1.37%	1.41%	1.46%	1.50%	1.55%	1.60%	1.66%	1.71%	1.76%	1.82%	1.88%	1.96%	2.05%	2.19%	2.40%	2.66%	2.98%	3.49%	0.91%	0.91%	0.92%	0.93%	0.94%	0.95%	0.97%	0.99%	1.01%	1.03%	1.05%	1.08%	1.11%	1.13%	1.17%	1.20%	1.23%	1.26%	1.30%	1.34%	1.38%	1.42%	1.46%	1.52%	1.60%	1.75%	1.93%	2.13%	2.35%	2.54%
Value at Risk at 95%	1.52%	1.53%	1.54%	1.57%	1.59%	1.59%	1.59%	1.63%	1.68%	1.73%	1.78%	1.84%	1.91%	1.97%	2.03%	2.07%	2.12%	2.19%	2.24%	2.32%	2.40%	2.49%	2.58%	2.66%	2.80%	2.97%	3.20%	3.57%	4.02%	4.82%	1.35%	1.34%	1.36%	1.36%	1.35%	1.38%	1.39%	1.42%	1.45%	1.47%	1.48%	1.51%	1.54%	1.58%	1.64%	1.68%	1.73%	1.78%	1.82%	1.88%	1.93%	2.01%	2.06%	2.12%	2.21%	2.43%	2.71%	2.99%	3.29%	3.60%
Drawdown at Risk at 95%	14.10%	14.49%	14.66%	15.01%	15.22%	15.35%	15.64%	16.40%	17.04%	17.80%	18.37%	19.09%	19.83%	20.52%	21.17%	21.67%	22.41%	23.20%	23.84%	24.89%	25.77%	26.62%	27.20%	27.77%	29.62%	32.92%	37.36%	44.20%	55.01%	79.05%	10.58%	10.54%	10.49%	10.33%	10.22%	10.14%	10.06%	9.94%	9.92%	9.90%	9.74%	10.01%	10.23%	10.50%	10.93%	11.54%	12.15%	12.79%	13.57%	14.42%	15.34%	16.36%	17.62%	19.54%	22.15%	25.98%	29.08%	32.80%	37.57%	51.44%
Entropic Risk Measure at 95%	3.00	3.00	3.00	3.00	3.00	3.00	3.00	2.99	2.99	2.99	2.99	2.99	2.99	2.99	2.99	2.99	2.99	2.99	2.99	2.99	2.99	2.99	2.99	2.99	2.99	2.99	2.99	2.99	2.99	2.99	3.00	3.00	3.00	3.00	3.00	3.00	3.00	3.00	3.00	3.00	3.00	3.00	3.00	3.00	3.00	3.00	3.00	2.99	2.99	2.99	2.99	2.99	2.99	2.99	2.99	3.00	3.00	3.00	3.00	3.00
Fourth Central Moment	0.000016%	0.000016%	0.000017%	0.000017%	0.000018%	0.000019%	0.000021%	0.000022%	0.000024%	0.000026%	0.000028%	0.000031%	0.000034%	0.000037%	0.000041%	0.000045%	0.000050%	0.000055%	0.000062%	0.000070%	0.000079%	0.000088%	0.000097%	0.00011%	0.00012%	0.00016%	0.00027%	0.00047%	0.00085%	0.0020%	0.000016%	0.000016%	0.000017%	0.000018%	0.000019%	0.000020%	0.000021%	0.000022%	0.000023%	0.000024%	0.000026%	0.000028%	0.000029%	0.000031%	0.000033%	0.000035%	0.000037%	0.000040%	0.000043%	0.000046%	0.000050%	0.000054%	0.000058%	0.000062%	0.000070%	0.000097%	0.00014%	0.00023%	0.00037%	0.00077%
Fourth Lower Partial Moment	0.000006%	0.000006%	0.000006%	0.000006%	0.000006%	0.000006%	0.000007%	0.000007%	0.000007%	0.000008%	0.000009%	0.000010%	0.000011%	0.000012%	0.000014%	0.000016%	0.000018%	0.000020%	0.000023%	0.000026%	0.000029%	0.000033%	0.000038%	0.000046%	0.000056%	0.000088%	0.00016%	0.00031%	0.00059%	0.0014%	0.000008%	0.000008%	0.000008%	0.000008%	0.000008%	0.000009%	0.000009%	0.000009%	0.000009%	0.000010%	0.000010%	0.000011%	0.000011%	0.000011%	0.000012%	0.000012%	0.000013%	0.000014%	0.000015%	0.000016%	0.000017%	0.000019%	0.000020%	0.000021%	0.000024%	0.000035%	0.000057%	0.000098%	0.00017%	0.00042%
Skew	26.18%	27.63%	28.80%	29.68%	30.29%	30.54%	30.46%	30.07%	29.42%	28.54%	27.50%	26.32%	25.06%	23.76%	22.43%	21.04%	19.81%	18.96%	18.23%	17.47%	16.62%	15.72%	13.34%	9.76%	5.83%	-2.14%	-11.06%	-19.84%	-27.79%	-37.39%	0.11%	1.66%	3.43%	5.49%	8.23%	11.10%	13.99%	16.88%	19.70%	22.40%	24.93%	27.31%	29.48%	31.32%	33.01%	35.06%	36.87%	37.81%	36.87%	35.75%	34.15%	32.59%	32.96%	34.39%	35.10%	34.55%	32.65%	28.74%	21.77%	-2.90%
Kurtosis	1291.65%	1314.19%	1327.57%	1332.93%	1335.68%	1330.10%	1317.27%	1298.53%	1275.27%	1248.90%	1220.76%	1191.72%	1162.62%	1133.92%	1106.01%	1077.58%	1053.12%	1037.16%	1030.03%	1021.73%	1010.78%	999.43%	969.27%	922.64%	869.32%	859.88%	940.30%	1060.15%	1193.52%	1428.95%	2041.35%	2076.87%	2102.02%	2117.46%	2126.55%	2124.76%	2112.60%	2091.31%	2061.98%	2025.82%	1984.31%	1938.84%	1890.89%	1843.01%	1794.00%	1742.88%	1695.44%	1648.82%	1589.85%	1536.43%	1494.34%	1453.67%	1394.40%	1315.91%	1235.13%	1177.90%	1176.14%	1210.04%	1301.37%	1739.55%
Sharpe Ratio	0.059	0.062	0.064	0.066	0.068	0.070	0.071	0.072	0.073	0.073	0.074	0.074	0.074	0.074	0.074	0.074	0.073	0.073	0.072	0.072	0.072	0.071	0.070	0.069	0.067	0.064	0.059	0.054	0.049	0.042	0.058	0.059	0.059	0.060	0.061	0.062	0.063	0.064	0.064	0.065	0.065	0.065	0.066	0.066	0.066	0.066	0.066	0.065	0.065	0.064	0.064	0.063	0.062	0.059	0.055	0.049	0.043	0.038	0.034	0.035
Annualized Sharpe Ratio	0.94	0.98	1.02	1.05	1.08	1.11	1.13	1.14	1.16	1.16	1.17	1.17	1.17	1.17	1.17	1.17	1.16	1.16	1.15	1.14	1.14	1.13	1.12	1.10	1.07	1.02	0.94	0.86	0.78	0.67	0.92	0.93	0.94	0.95	0.97	0.98	1.00	1.01	1.02	1.03	1.03	1.04	1.04	1.04	1.04	1.04	1.04	1.04	1.03	1.02	1.01	1.00	0.98	0.94	0.88	0.78	0.69	0.60	0.53	0.55
Sortino Ratio	0.084	0.088	0.091	0.094	0.097	0.099	0.10	0.10	0.10	0.10	0.10	0.10	0.10	0.10	0.10	0.10	0.10	0.10	0.10	0.10	0.10	0.10	0.100	0.098	0.096	0.091	0.085	0.077	0.070	0.060	0.081	0.082	0.083	0.083	0.085	0.087	0.088	0.089	0.090	0.091	0.091	0.092	0.092	0.093	0.093	0.093	0.093	0.093	0.092	0.091	0.091	0.090	0.088	0.084	0.079	0.070	0.062	0.054	0.048	0.049
Annualized Sortino Ratio	1.34	1.40	1.45	1.50	1.54	1.57	1.60	1.62	1.64	1.65	1.66	1.66	1.66	1.66	1.66	1.65	1.65	1.64	1.63	1.62	1.61	1.60	1.58	1.56	1.52	1.45	1.34	1.23	1.11	0.96	1.28	1.30	1.31	1.32	1.35	1.37	1.40	1.41	1.43	1.44	1.45	1.46	1.47	1.47	1.47	1.48	1.48	1.47	1.46	1.45	1.44	1.42	1.39	1.33	1.26	1.12	0.98	0.86	0.76	0.78
Mean Absolute Deviation Ratio	0.084	0.088	0.091	0.094	0.097	0.099	0.10	0.10	0.10	0.10	0.10	0.10	0.10	0.10	0.10	0.10	0.10	0.10	0.10	0.10	0.099	0.099	0.098	0.096	0.094	0.090	0.084	0.078	0.071	0.062	0.090	0.091	0.092	0.093	0.095	0.096	0.097	0.098	0.099	0.099	0.099	0.099	0.099	0.099	0.098	0.098	0.097	0.096	0.095	0.094	0.093	0.092	0.089	0.085	0.079	0.070	0.062	0.054	0.049	0.053
First Lower Partial Moment Ratio	0.17	0.18	0.18	0.19	0.19	0.20	0.20	0.20	0.21	0.21	0.21	0.21	0.21	0.21	0.21	0.21	0.20	0.20	0.20	0.20	0.20	0.20	0.20	0.19	0.19	0.18	0.17	0.16	0.14	0.12	0.18	0.18	0.18	0.19	0.19	0.19	0.19	0.20	0.20	0.20	0.20	0.20	0.20	0.20	0.20	0.20	0.19	0.19	0.19	0.19	0.19	0.18	0.18	0.17	0.16	0.14	0.12	0.11	0.098	0.11
Value at Risk Ratio at 95%	0.041	0.042	0.044	0.045	0.046	0.048	0.050	0.050	0.051	0.051	0.051	0.051	0.050	0.050	0.050	0.051	0.051	0.051	0.051	0.050	0.050	0.049	0.049	0.048	0.047	0.045	0.043	0.039	0.035	0.030	0.040	0.041	0.041	0.042	0.044	0.044	0.045	0.045	0.046	0.046	0.047	0.047	0.047	0.047	0.047	0.047	0.046	0.046	0.046	0.045	0.045	0.044	0.043	0.041	0.039	0.034	0.030	0.026	0.024	0.025
CVaR Ratio at 95%	0.027	0.028	0.029	0.030	0.031	0.032	0.032	0.033	0.033	0.033	0.034	0.034	0.034	0.034	0.034	0.034	0.033	0.033	0.033	0.033	0.033	0.033	0.032	0.032	0.031	0.029	0.027	0.025	0.022	0.019	0.025	0.025	0.026	0.026	0.027	0.027	0.027	0.028	0.028	0.028	0.029	0.029	0.029	0.029	0.029	0.029	0.029	0.029	0.029	0.029	0.029	0.028	0.028	0.027	0.025	0.023	0.020	0.017	0.015	0.015
Entropic Risk Measure Ratio at 95%	0.00021	0.00022	0.00023	0.00024	0.00025	0.00026	0.00026	0.00027	0.00028	0.00029	0.00030	0.00031	0.00032	0.00033	0.00034	0.00035	0.00036	0.00037	0.00038	0.00039	0.00040	0.00041	0.00042	0.00043	0.00044	0.00045	0.00046	0.00047	0.00048	0.00048	0.00018	0.00018	0.00019	0.00019	0.00020	0.00020	0.00021	0.00021	0.00022	0.00023	0.00023	0.00024	0.00024	0.00025	0.00026	0.00026	0.00027	0.00027	0.00028	0.00028	0.00029	0.00029	0.00029	0.00029	0.00029	0.00028	0.00027	0.00026	0.00026	0.00030
EVaR Ratio at 95%	0.015	0.016	0.016	0.017	0.018	0.018	0.019	0.019	0.020	0.020	0.020	0.020	0.020	0.020	0.020	0.020	0.019	0.019	0.019	0.019	0.019	0.019	0.018	0.018	0.018	0.016	0.013	0.011	0.0096	0.0078	0.012	0.012	0.012	0.012	0.012	0.013	0.013	0.013	0.013	0.014	0.014	0.014	0.014	0.014	0.015	0.015	0.015	0.015	0.015	0.015	0.015	0.015	0.015	0.015	0.015	0.012	0.010	0.0084	0.0071	0.0065
Worst Realization Ratio	0.0074	0.0079	0.0085	0.0090	0.0095	0.010	0.010	0.010	0.0100	0.0099	0.0098	0.0097	0.0096	0.0096	0.0095	0.0094	0.0094	0.0093	0.0092	0.0091	0.0090	0.0089	0.0089	0.0089	0.0092	0.0084	0.0068	0.0056	0.0048	0.0037	0.0063	0.0063	0.0064	0.0064	0.0065	0.0066	0.0067	0.0068	0.0069	0.0070	0.0071	0.0072	0.0073	0.0073	0.0074	0.0075	0.0076	0.0076	0.0076	0.0076	0.0076	0.0075	0.0076	0.0078	0.0082	0.0063	0.0051	0.0041	0.0035	0.0032
Drawdown at Risk Ratio at 95%	0.0044	0.0045	0.0046	0.0047	0.0048	0.0050	0.0051	0.0050	0.0050	0.0049	0.0049	0.0049	0.0049	0.0048	0.0048	0.0048	0.0048	0.0048	0.0048	0.0047	0.0046	0.0046	0.0046	0.0046	0.0044	0.0041	0.0037	0.0032	0.0026	0.0018	0.0051	0.0052	0.0054	0.0056	0.0058	0.0060	0.0062	0.0065	0.0067	0.0069	0.0071	0.0071	0.0071	0.0071	0.0070	0.0068	0.0066	0.0064	0.0061	0.0059	0.0056	0.0054	0.0050	0.0045	0.0039	0.0032	0.0028	0.0024	0.0021	0.0018
CDaR Ratio at 95%	0.0034	0.0034	0.0035	0.0035	0.0035	0.0035	0.0035	0.0035	0.0034	0.0034	0.0034	0.0033	0.0033	0.0032	0.0032	0.0032	0.0032	0.0032	0.0031	0.0031	0.0031	0.0031	0.0031	0.0031	0.0032	0.0031	0.0030	0.0027	0.0022	0.0015	0.0039	0.0040	0.0041	0.0043	0.0045	0.0046	0.0048	0.0050	0.0051	0.0052	0.0052	0.0053	0.0053	0.0053	0.0053	0.0053	0.0052	0.0051	0.0050	0.0048	0.0046	0.0045	0.0041	0.0036	0.0032	0.0027	0.0023	0.0020	0.0018	0.0015
Calmar Ratio	0.0019	0.0019	0.0019	0.0019	0.0019	0.0019	0.0019	0.0019	0.0019	0.0019	0.0018	0.0018	0.0018	0.0017	0.0017	0.0017	0.0017	0.0016	0.0016	0.0016	0.0016	0.0016	0.0015	0.0016	0.0016	0.0016	0.0016	0.0017	0.0017	0.0011	0.0016	0.0016	0.0017	0.0017	0.0018	0.0018	0.0019	0.0019	0.0020	0.0020	0.0021	0.0022	0.0022	0.0023	0.0023	0.0024	0.0025	0.0025	0.0025	0.0025	0.0025	0.0025	0.0024	0.0022	0.0021	0.0019	0.0016	0.0014	0.0014	0.0012
Average Drawdown Ratio	0.017	0.018	0.018	0.019	0.019	0.019	0.019	0.019	0.019	0.019	0.019	0.019	0.018	0.018	0.018	0.018	0.017	0.017	0.017	0.017	0.016	0.016	0.016	0.015	0.014	0.013	0.012	0.010	0.0083	0.0056	0.019	0.020	0.020	0.021	0.022	0.022	0.023	0.024	0.024	0.025	0.025	0.025	0.024	0.024	0.024	0.023	0.022	0.022	0.021	0.020	0.019	0.018	0.017	0.015	0.013	0.010	0.0081	0.0064	0.0053	0.0054
EDaR Ratio at 95%	0.0028	0.0028	0.0029	0.0029	0.0029	0.0028	0.0028	0.0028	0.0028	0.0027	0.0027	0.0026	0.0026	0.0026	0.0026	0.0025	0.0025	0.0025	0.0025	0.0025	0.0024	0.0024	0.0024	0.0024	0.0024	0.0025	0.0025	0.0024	0.0021	0.0014	0.0028	0.0028	0.0029	0.0029	0.0030	0.0031	0.0032	0.0033	0.0034	0.0035	0.0036	0.0037	0.0037	0.0038	0.0039	0.0039	0.0040	0.0040	0.0040	0.0039	0.0038	0.0038	0.0035	0.0032	0.0029	0.0025	0.0021	0.0018	0.0016	0.0014
Ulcer Index Ratio	0.010	0.011	0.011	0.011	0.011	0.011	0.011	0.011	0.011	0.011	0.011	0.011	0.011	0.011	0.010	0.010	0.010	0.010	0.010	0.010	0.0099	0.0098	0.0097	0.0095	0.0092	0.0087	0.0081	0.0070	0.0058	0.0040	0.012	0.012	0.012	0.013	0.013	0.014	0.014	0.015	0.015	0.015	0.015	0.015	0.015	0.015	0.015	0.015	0.015	0.014	0.014	0.013	0.013	0.012	0.011	0.0099	0.0086	0.0072	0.0059	0.0048	0.0041	0.0038
Gini Mean Difference Ratio	0.057	0.060	0.062	0.064	0.066	0.067	0.068	0.069	0.070	0.070	0.070	0.071	0.071	0.070	0.070	0.070	0.070	0.069	0.069	0.068	0.068	0.067	0.067	0.065	0.064	0.061	0.057	0.052	0.048	0.042	0.060	0.060	0.061	0.062	0.063	0.064	0.065	0.065	0.066	0.066	0.066	0.066	0.066	0.066	0.066	0.066	0.065	0.065	0.064	0.063	0.062	0.062	0.060	0.057	0.054	0.048	0.042	0.037	0.033	0.035
Effective Number of Assets	8.200105310174294	8.59167564989511	8.914061803581202	9.20057593224975	9.669076363624884	10.042905188114835	10.296356964249894	10.407690290603215	10.36833537771513	10.182061194354427	9.864764715987107	9.437583988064256	8.93642825430989	8.39936034148487	7.843069929156864	7.305664213683842	6.7881858925195955	6.322661810342193	5.894001701423032	5.474123745523491	5.083026522037526	4.687782011559598	4.402394853192732	4.022087577333231	3.600838708075875	3.2020819241703986	2.5798056188192846	2.0201935849251913	1.5152184941723117	1.000000132533317	8.200105310174294	8.59167564989511	8.914061803581202	9.20057593224975	9.669076363624884	10.042905188114835	10.296356964249894	10.407690290603215	10.36833537771513	10.182061194354427	9.864764715987107	9.437583988064256	8.93642825430989	8.39936034148487	7.843069929156864	7.305664213683842	6.7881858925195955	6.322661810342193	5.894001701423032	5.474123745523491	5.083026522037526	4.687782011559598	4.402394853192732	4.022087577333231	3.600838708075875	3.2020819241703986	2.5798056188192846	2.0201935849251913	1.5152184941723117	1.000000132533317
Assets Number	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20	20

Instead of providing efficient_frontier_size=30, you can also provide an array of lower bounds for the expected returns using min_return. In the below example, we find the 5 portfolios that minimize the variance under a minimum return constraint of 15%, 20%, 25%, 30% and 35% (annualized):

model = MeanRisk(
    risk_measure=RiskMeasure.VARIANCE,
    min_return=np.array([0.15, 0.20, 0.25, 0.30, 0.35]) / 252,
    portfolio_params=dict(name="Variance"),
)

population = model.fit_predict(X_train)

population.plot_measures(
    x=RiskMeasure.ANNUALIZED_STANDARD_DEVIATION,
    y=PerfMeasure.ANNUALIZED_MEAN,
    color_scale=RatioMeasure.ANNUALIZED_SHARPE_RATIO,
    hover_measures=[RiskMeasure.MAX_DRAWDOWN, RatioMeasure.ANNUALIZED_SORTINO_RATIO],
)