Simple riskengine for cvxpy
Project description
cvxrisk
We provide an abstract Model
class.
The class is designed to be used in conjunction with cvxpy.
Using this class, we can formulate a function computing a standard minimum
risk portfolio as
import cvxpy as cp
from cvx.risk import Model
def minimum_risk(w: cp.Variable, risk_model: Model, **kwargs) -> cp.Problem:
"""Constructs a minimum variance portfolio.
Args:
w: cp.Variable representing the portfolio weights.
risk_model: A risk model.
Returns:
A convex optimization problem.
"""
return cp.Problem(
cp.Minimize(risk_model.estimate(w, **kwargs)),
[cp.sum(w) == 1, w >= 0] + risk_model.constraints(w, **kwargs)
)
The risk model is injected into the function.
The function is not aware of the precise risk model used.
All risk models are required to implement the estimate
method.
Note that factor risk models work with weights for the assets but also with
weights for the factors.
To stay flexible we are applying the **kwargs
pattern to the function above.
A first example
A first example is a risk model based on the sample covariance matrix. We construct the risk model as follows
import numpy as np
import cvxpy as cp
from cvx.risk.sample import SampleCovariance
riskmodel = SampleCovariance(num=2)
w = cp.Variable(2)
problem = minimum_risk(w, riskmodel)
riskmodel.update(cov=np.array([[1.0, 0.5], [0.5, 2.0]]))
problem.solve()
print(w.value)
The risk model and the actual optimization problem are decoupled. This is good practice and keeps the code clean and maintainable.
In a backtest we don't have to reconstruct the problem in every iteration. We can simply update the risk model with the new data and solve the problem again. The implementation of the risk models is flexible enough to deal with changing dimensions of the underlying weight space.
Risk models
Sample covariance
We offer a SampleCovariance
class as seen above.
Factor risk models
Factor risk models use the projection of the weight vector into a lower dimensional subspace, e.g. each asset is the linear combination of $k$ factors.
r_i = \sum_{j=1}^k f_j \beta_{ji} + \epsilon_i
The factor time series are $f_1, \ldots, f_k$. The loadings are the coefficients $\beta_{ji}$. The residual returns $\epsilon_i$ are assumed to be uncorrelated with the f actors.
Any position $w$ in weight space projects to a position $y = \beta^T w$ in factor space. The variance for a position $w$ is the sum of the variance of the systematic returns explained by the factors and the variance of the idiosyncratic returns.
Var(r) = Var(\beta^T w) + Var(\epsilon w)
We assume the residual returns are uncorrelated and hence
Var(r) = y^T \Sigma_f y + \sum_i w_i^2 Var(\epsilon_i)
where $\Sigma_f$ is the covariance matrix of the factors and $Var(\epsilon_i)$ is the variance of the idiosyncratic returns.
Factor risk models are widely used in practice. Usually two scenarios are distinguished. A first route is to rely on estimates for the factor covariance matrix $\Sigma_f$, the loadings $\beta$ and the volatilities of the idiosyncratic returns $\epsilon_i$. Usually those quantities are provided by external parties, e.g. Barra or Axioma.
An alternative would be to start with the estimation of factor time series $f_1, \ldots, f_k$. Usually they are estimated via a principal component analysis (PCA) of the asset returns. It is then a simple linear regression to compute the loadings $\beta$. The volatilities of the idiosyncratic returns $\epsilon_i$ are computed as the standard deviation of the observed residuals. The factor covariance matrix $\Sigma_f$ may even be diagonal in this case as the factors are orthogonal.
We expose a method to compute the first $k$ principal components.
cvar
We currently also support the conditional value at risk (CVaR) as a risk measure.
Poetry
We assume you share already the love for Poetry. Once you have installed poetry you can perform
make install
to replicate the virtual environment we have defined in pyproject.toml and locked in poetry.lock.
Jupyter
We install JupyterLab on fly within the aforementioned virtual environment. Executing
make jupyter
will install and start the jupyter lab.
Project details
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