Solving Conditional Value-at-Risk (CVaR) with CVXPY
In this post you will read about:
the concept of Conditional Value-at-Risk (CVaR),
how to implement CVaR with CVXPY.
Let’s start
Value-at-Risk (VaR) is a widely used risk measure in finance, which estimates the maximum potential loss for a given portfolio over a specific time period with a certain confidence level. However, VaR has some limitations, such as not being a coherent risk measure and not providing information about the tail distribution beyond the VaR level. Thus, it can potentially lead to suboptimal risk management decisions.
Conditional Value-at-Risk (CVaR), also known as Expected Shortfall or Average Value-at-Risk, significantly enhances traditional risk management approaches by addressing the limitations of Value-at-Risk (VaR). Unlike VaR, which estimates the maximum potential loss at a certain confidence level without indicating the severity of losses beyond this threshold, CVaR provides a more detailed view by measuring the expected loss in the tail end of the distribution of potential losses. This capability makes CVaR a crucial tool for understanding and managing the risks associated with extreme market movements.
The Coherency of CVaR
CVaR is a coherent risk measure, satisfying the four essential properties of coherency: sub-additivity, indicating that diversifying a portfolio reduces its risk; monotonicity, meaning that adding a riskier asset cannot reduce the portfolio's overall risk; translation invariance, which ensures that adding a risk-free asset to a portfolio does not change its risk; and positive homogeneity, implying that scaling the portfolio also scales its risk proportionally. These properties make CVaR particularly valuable for constructing and managing portfolios in a consistent and risk-aware manner.
Understanding CVaR
Before diving into the optimization process, let's briefly refresh the concept of CVaR. As mentioned earlier, CVaR represents the expected loss given that the loss exceeds the VaR level. Mathematically, for a given confidence level α (e.g., 95%), CVaR is defined as:
Where X is the portfolio loss, and VaR_α(X) is the Value-at-Risk at the confidence level α, which serves as the threshold for calculating CVaR.
For continuous loss distributions, CVaR can also be calculated using the integral representation, which averages the losses in the tail beyond the VaR level:
Where f(x) is the probability density function of the loss distribution.
Unlike VaR, which only provides information about the potential loss at a specific confidence level, CVaR captures the average of the worst-case losses beyond that level. This additional information about the tail distribution makes CVaR a more comprehensive and coherent risk measure.
Objective Function and Constraints Explained
In the context of portfolio optimization, minimizing CVaR involves finding the set of portfolio weights that reduces the expected tail losses. The objective function used in the optimization problem effectively minimizes the average loss beyond the predetermined VaR level by focusing on the positive deviations from this threshold. The non-negativity and budget constraints ensure that the portfolio does not include short positions and remains fully invested, respectively. These constraints are fundamental in maintaining a realistic and practical investment strategy.
Clarifying the Implementation with CVXPy
CVXPY is a Python-embedded modeling language for convex optimization problems. It allows users to express convex optimization problems in a natural, disciplined modeling style, similar to the way mathematical optimization models are described in textbooks. CVXPY then transforms these high-level descriptions into code that can be evaluated by state-of-the-art numerical solvers, allowing for efficient solving of a wide range of convex optimization problems.
CVXPy facilitates the solving of this optimization problem by employing sophisticated convex optimization solvers. These solvers, such as interior-point methods, are adept at efficiently finding global minima in convex landscapes, making them ideal for CVaR optimization problems. The convex nature of the CVaR optimization problem guarantees that the solution found is not only locally optimal but also globally optimal, providing confidence in the robustness of the resulting portfolio strategy.
It is worth adding that solving these optimization problems can be computationally intensive, particularly for large-scale portfolios or when incorporating complex constraints. So, the choice of the optimization solver and its efficiency can significantly impact the computational time and scalability of CVaR optimization. Traditional optimization techniques, such as quadratic programming or linear programming solvers, may struggle with the complexity of CVaR optimization problems, especially when dealing with a large number of scenarios or assets.
Let’s see the code:
import numpy as np
import cvxpy as cp
# take some random return data
R = np.random.randn(2500, 100)
n, m = R.shape
alpha = 0.95
entropy_weights = np.random.rand(n)
entropy_weights /= np.sum(entropy_weights)
gamma = cp.Variable()
w = cp.Variable(m)
constraints = [w >= 0, cp.sum(w) == 1]
obj = cp.Minimize(cp.sum(cp.multiply(entropy_weights, cp.pos(R @ w - gamma))))
problem = cp.Problem(objective=obj, constraints=constraints)
problem.solve()
print(f"CVaR: {gamma.value}")
Okay, let's break down each line to better understand this method: