# Minimum CVaR
This tutorial uses the [`MeanRisk`](https://skfolio.org/generated/skfolio.optimization.MeanRisk.html.md#skfolio.optimization.MeanRisk) optimization to find the
minimum CVaR (Conditional Value at Risk) portfolio.
## 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.
Prices are transformed into linear returns (see [data preparation](https://skfolio.org/user_guide/data_preparation.html.md#data-preparation)) and split into a training set and a test set without shuffling to
avoid [data leakage](https://skfolio.org/user_guide/model_selection.html.md#data-leakage).
```Python
import numpy as np
from plotly.io import show
from sklearn.model_selection import train_test_split
from skfolio import Population, RiskMeasure
from skfolio.datasets import load_sp500_dataset
from skfolio.optimization import EqualWeighted, MeanRisk, ObjectiveFunction
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)
print(X_train.head())
```
```none
AAPL AMD BAC ... UNH WMT XOM
Date ...
1990-01-03 0.007576 -0.030303 0.008045 ... -0.019355 0.000000 -0.010079
1990-01-04 0.003759 -0.015500 -0.021355 ... -0.009868 -0.005201 -0.009933
1990-01-05 0.003745 -0.031996 -0.021821 ... -0.043189 -0.010732 -0.005267
1990-01-08 0.003731 0.000000 0.005633 ... -0.020833 0.013630 0.015381
1990-01-09 -0.007435 0.016527 0.000000 ... -0.024823 -0.026619 -0.020114
[5 rows x 20 columns]
```
## Model
We create a Minimum CVaR model and then fit it on the training set.
`portfolio_params` are parameters passed to the [`Portfolio`](https://skfolio.org/generated/skfolio.portfolio.Portfolio.html.md#skfolio.portfolio.Portfolio)
returned by the `predict` method. It can be
omitted, here we use it to give a name to our minimum CVaR portfolio:
```Python
model = MeanRisk(
risk_measure=RiskMeasure.CVAR,
objective_function=ObjectiveFunction.MINIMIZE_RISK,
portfolio_params=dict(name="Min CVaR"),
)
model.fit(X_train)
model.weights_
```
```none
array([2.16562597e-02, 1.41765615e-12, 3.64939426e-13, 1.47358244e-02,
1.35927719e-01, 1.91067310e-12, 3.58410853e-12, 2.10319534e-01,
4.89865658e-13, 8.14734697e-02, 1.92817428e-02, 4.13677567e-12,
7.79977277e-12, 1.26155640e-01, 4.04653978e-12, 1.52708774e-01,
1.20106197e-02, 6.41727678e-03, 1.01024052e-01, 1.18289088e-01])
```
To compare this model, we use an equal-weighted benchmark using
[`EqualWeighted`](https://skfolio.org/generated/skfolio.optimization.EqualWeighted.html.md#skfolio.optimization.EqualWeighted):
```Python
benchmark = EqualWeighted(portfolio_params=dict(name="Equal Weighted"))
# Even if `X` has no impact (as it is equal weighted), we still need to call `fit` for
# API consistency.
benchmark.fit(X_train)
benchmark.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])
```
## Prediction
We predict the model and the benchmark on the test set:
```Python
pred_model = model.predict(X_test)
pred_bench = benchmark.predict(X_test)
```
The `predict` method returns a [`Portfolio`](https://skfolio.org/generated/skfolio.portfolio.Portfolio.html.md#skfolio.portfolio.Portfolio) object.
[`Portfolio`](https://skfolio.org/generated/skfolio.portfolio.Portfolio.html.md#skfolio.portfolio.Portfolio) is an array-container making it compatible
with `scikit-learn` tools: calling `np.asarray(pred_model)` gives the portfolio
returns (same as `pred_model.returns`):
```Python
np.asarray(pred_model)
```
```none
array([ 0.00354605, 0.00342423, -0.00105085, ..., 0.01069377,
0.00542224, -0.01217676], shape=(2743,))
```
The [`Portfolio`](https://skfolio.org/generated/skfolio.portfolio.Portfolio.html.md#skfolio.portfolio.Portfolio) class contains a vast number of properties
and methods used for analysis.
For example:
* pred_model.plot_cumulative_returns()
* pred_model.plot_composition()
* pred_model.summary()
```Python
print(pred_model.cvar)
print(pred_bench.cvar)
```
```none
0.02174155209025761
0.025061083134673378
```
## Analysis
For improved analysis, we load both predicted portfolios into a
[`Population`](https://skfolio.org/generated/skfolio.population.Population.html.md#skfolio.population.Population):
```Python
population = Population([pred_model, pred_bench])
```
The [`Population`](https://skfolio.org/generated/skfolio.population.Population.html.md#skfolio.population.Population) class also contains a
vast number of properties and methods used for analysis.
Let’s plot each portfolio composition:
```Python
population.plot_composition()
```
[plotly figure stripped from llms output]
#### NOTE
Every `plot` methods in `skfolio` returns a `plotly` figure.
To display a plotly figure, you may need to call `show()` and change the
default renderer: [https://plotly.com/python/renderers/](https://plotly.com/python/renderers/)
Let’s plot each portfolio cumulative returns:
```Python
fig = population.plot_cumulative_returns()
# show(fig) is only used for the documentation sticker.
show(fig)
```
[plotly figure stripped from llms output]
Finally, let’s display the full summary of both strategies evaluated on the test
set:
```Python
population.summary()
```
|
Min CVaR |
Equal Weighted |
| Mean |
0.052% |
0.069% |
| Annualized Mean |
13.07% |
17.30% |
| Variance |
0.000090 |
0.00012 |
| Annualized Variance |
2.26% |
2.94% |
| Semi-Variance |
0.000046 |
0.000060 |
| Annualized Semi-Variance |
1.16% |
1.52% |
| Standard Deviation |
0.95% |
1.08% |
| Annualized Standard Deviation |
15.02% |
17.15% |
| Semi-Deviation |
0.68% |
0.78% |
| Annualized Semi-Deviation |
10.76% |
12.32% |
| Mean Absolute Deviation |
0.61% |
0.71% |
| CVaR at 95% |
2.17% |
2.51% |
| EVaR at 95% |
4.62% |
5.43% |
| Worst Realization |
8.53% |
10.77% |
| CDaR at 95% |
14.81% |
13.35% |
| MAX Drawdown |
34.70% |
34.70% |
| Average Drawdown |
3.12% |
2.67% |
| EDaR at 95% |
20.48% |
20.49% |
| First Lower Partial Moment |
0.31% |
0.35% |
| Ulcer Index |
0.050 |
0.044 |
| Gini Mean Difference |
0.92% |
1.06% |
| Value at Risk at 95% |
1.33% |
1.54% |
| Drawdown at Risk at 95% |
11.23% |
9.48% |
| Entropic Risk Measure at 95% |
3.00 |
3.00 |
| Fourth Central Moment |
0.000016% |
0.000027% |
| Fourth Lower Partial Moment |
0.000008% |
0.000013% |
| Skew |
-0.13% |
-2.79% |
| Kurtosis |
1977.71% |
1961.33% |
| Sharpe Ratio |
0.055 |
0.064 |
| Annualized Sharpe Ratio |
0.87 |
1.01 |
| Sortino Ratio |
0.076 |
0.088 |
| Annualized Sortino Ratio |
1.21 |
1.40 |
| Mean Absolute Deviation Ratio |
0.085 |
0.097 |
| First Lower Partial Moment Ratio |
0.17 |
0.19 |
| Value at Risk Ratio at 95% |
0.039 |
0.045 |
| CVaR Ratio at 95% |
0.024 |
0.027 |
| Entropic Risk Measure Ratio at 95% |
0.00017 |
0.00023 |
| EVaR Ratio at 95% |
0.011 |
0.013 |
| Worst Realization Ratio |
0.0061 |
0.0064 |
| Drawdown at Risk Ratio at 95% |
0.0046 |
0.0072 |
| CDaR Ratio at 95% |
0.0035 |
0.0051 |
| Calmar Ratio |
0.0015 |
0.0020 |
| Average Drawdown Ratio |
0.017 |
0.026 |
| EDaR Ratio at 95% |
0.0025 |
0.0033 |
| Ulcer Index Ratio |
0.010 |
0.015 |
| Gini Mean Difference Ratio |
0.056 |
0.064 |
| Effective Number of Assets |
7.46127671221519 |
19.999999999999993 |
| Assets Number |
20 |
20 |
## Conclusion
From the analysis on the test set, we see that the Minimum CVaR portfolio outperforms
the equal-weighted benchmark for all deviation and shortfall risk measures, except for
the drawdown measures, and underperforms for the mean and ratio measures.
#### SEE ALSO
This was a toy example, for more advanced concepts check the
[user guide](https://skfolio.org/user_guide/index.html.md#user-guide) or the [other examples](https://skfolio.org/auto_examples/index.html.md#general-examples).
**Total running time of the script:** (0 minutes 3.183 seconds)