# Maximum Diversification
This tutorial uses the [`MaximumDiversification`](https://skfolio.org/generated/skfolio.optimization.MaximumDiversification.html.md#skfolio.optimization.MaximumDiversification)
optimization to find the portfolio that maximizes the diversification ratio, which is
the ratio of the weighted volatilities over the total volatility.
## 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 starting from 1990-01-02 up to 2022-12-28:
```Python
from plotly.io import show
from sklearn.model_selection import train_test_split
from skfolio import Population
from skfolio.datasets import load_sp500_dataset
from skfolio.optimization import EqualWeighted, MaximumDiversification
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 maximum diversification model and then fit it on the training set:
```Python
model = MaximumDiversification()
model.fit(X_train)
model.weights_
```
```none
array([8.33459971e-02, 6.74138299e-02, 2.93952123e-02, 8.57558650e-02,
4.12145074e-02, 8.80360023e-09, 1.53457231e-08, 4.41151909e-02,
1.70046770e-08, 5.11503226e-02, 6.82590399e-02, 3.02728712e-02,
3.79430055e-03, 9.95058777e-02, 1.48755617e-02, 1.10849163e-01,
1.08087391e-01, 9.45176307e-02, 6.51331697e-02, 2.31402810e-03])
```
To compare this model, we use an equal weighted benchmark using
the [`EqualWeighted`](https://skfolio.org/generated/skfolio.optimization.EqualWeighted.html.md#skfolio.optimization.EqualWeighted) estimator:
```Python
bench = EqualWeighted()
bench.fit(X_train)
bench.weights_
```
```none
array([0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05,
0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05, 0.05])
```
## Diversification Analysis
Let’s analyze the diversification ratio of both models on the training set.
As expected, the maximum diversification model has the highest diversification ratio:
```Python
ptf_model_train = model.predict(X_train)
ptf_bench_train = bench.predict(X_train)
print("Diversification Ratio:")
print(f" Maximum Diversification model: {ptf_model_train.diversification:0.2f}")
print(f" Equal Weighted model: {ptf_bench_train.diversification:0.2f}")
```
```none
Diversification Ratio:
Maximum Diversification model: 1.92
Equal Weighted model: 1.82
```
## Prediction
We predict the model and the benchmark on the test set:
```Python
ptf_model_test = model.predict(X_test)
ptf_bench_test = bench.predict(X_test)
```
## Analysis
For improved analysis, it’s possible to load both predicted portfolios into a
[`Population`](https://skfolio.org/generated/skfolio.population.Population.html.md#skfolio.population.Population):
```Python
population = Population([ptf_model_test, ptf_bench_test])
```
Let’s plot each portfolio composition:
```Python
fig = population.plot_composition()
show(fig)
```
[plotly figure stripped from llms output]
Finally we can show a full summary of both strategies evaluated on the test set:
```Python
population.plot_cumulative_returns()
```
[plotly figure stripped from llms output]
Finally, we print a full summary of both strategies evaluated on the test set:
```Python
population.summary()
```
|
MaximumDiversification |
EqualWeighted |
| Mean |
0.072% |
0.069% |
| Annualized Mean |
18.17% |
17.30% |
| Variance |
0.00012 |
0.00012 |
| Annualized Variance |
3.14% |
2.94% |
| Semi-Variance |
0.000062 |
0.000060 |
| Annualized Semi-Variance |
1.55% |
1.52% |
| Standard Deviation |
1.12% |
1.08% |
| Annualized Standard Deviation |
17.71% |
17.15% |
| Semi-Deviation |
0.79% |
0.78% |
| Annualized Semi-Deviation |
12.46% |
12.32% |
| Mean Absolute Deviation |
0.75% |
0.71% |
| CVaR at 95% |
2.51% |
2.51% |
| EVaR at 95% |
4.78% |
5.43% |
| Worst Realization |
9.07% |
10.77% |
| CDaR at 95% |
14.57% |
13.35% |
| MAX Drawdown |
29.52% |
34.70% |
| Average Drawdown |
3.24% |
2.67% |
| EDaR at 95% |
18.59% |
20.49% |
| First Lower Partial Moment |
0.37% |
0.35% |
| Ulcer Index |
0.051 |
0.044 |
| Gini Mean Difference |
1.12% |
1.06% |
| Value at Risk at 95% |
1.59% |
1.54% |
| Drawdown at Risk at 95% |
11.66% |
9.48% |
| Entropic Risk Measure at 95% |
3.00 |
3.00 |
| Fourth Central Moment |
0.000028% |
0.000027% |
| Fourth Lower Partial Moment |
0.000009% |
0.000013% |
| Skew |
45.26% |
-2.79% |
| Kurtosis |
1802.21% |
1961.33% |
| Sharpe Ratio |
0.065 |
0.064 |
| Annualized Sharpe Ratio |
1.03 |
1.01 |
| Sortino Ratio |
0.092 |
0.088 |
| Annualized Sortino Ratio |
1.46 |
1.40 |
| Mean Absolute Deviation Ratio |
0.096 |
0.097 |
| First Lower Partial Moment Ratio |
0.19 |
0.19 |
| Value at Risk Ratio at 95% |
0.045 |
0.045 |
| CVaR Ratio at 95% |
0.029 |
0.027 |
| Entropic Risk Measure Ratio at 95% |
0.00024 |
0.00023 |
| EVaR Ratio at 95% |
0.015 |
0.013 |
| Worst Realization Ratio |
0.0079 |
0.0064 |
| Drawdown at Risk Ratio at 95% |
0.0062 |
0.0072 |
| CDaR Ratio at 95% |
0.0049 |
0.0051 |
| Calmar Ratio |
0.0024 |
0.0020 |
| Average Drawdown Ratio |
0.022 |
0.026 |
| EDaR Ratio at 95% |
0.0039 |
0.0033 |
| Ulcer Index Ratio |
0.014 |
0.015 |
| Gini Mean Difference Ratio |
0.065 |
0.064 |
| Effective Number of Assets |
12.684721598180525 |
19.999999999999993 |
| Assets Number |
20 |
20 |
**Total running time of the script:** (0 minutes 1.004 seconds)