Mathematical formulations¶

cvxrisk solves minimum-risk portfolio problems directly with the Clarabel conic solver. This page documents the exact conic program each risk model builds in its solve_minrisk method, so the implementations can be audited against the mathematics.

All problems share the budget constraint and weight bounds

$$\sum_i w_i = 1, \qquad \ell_w \le w \le u_w,$$

plus any user-supplied linear constraints $\ell \le a^T w \le u$ (equalities when $\ell = u$). A base portfolio $b$ (default $0$) turns the problem into tracking-error minimization over the active position $w - b$.

Sample covariance¶

With $R$ the upper-triangular Cholesky factor of the covariance matrix ($R^T R = \Sigma$), the problem is the second-order cone program

$$\min_{t,\,w} t \quad \text{s.t.} \quad \lVert R (w - b) \rVert_2 \le t$$

over the variables $x = [t, w]$. The optimal $t$ is the portfolio volatility $\sqrt{(w-b)^T \Sigma (w-b)}$.

Factor model¶

With factor exposure $\beta \in \mathbb{R}^{k \times n}$, factor covariance Cholesky factor $R_f$ ($R_f^T R_f = \Sigma_f$), and idiosyncratic volatilities $\sigma$, the problem introduces the factor position $y = \beta w$ as an explicit variable:

$$\min_{t,\,w,\,y} t \quad \text{s.t.} \quad \left\lVert \begin{bmatrix} R_f\,(y - \beta b) \ \operatorname{diag}(\sigma)\,(w - b) \end{bmatrix} \right\rVert_2 \le t, \qquad y = \beta w, \qquad \ell_y \le y \le u_y$$

over $x = [t, w, y]$. The optimal $t$ equals the total tracking error $\sqrt{(w-b)^T (\beta^T \Sigma_f \beta + \operatorname{diag}(\sigma^2)) (w-b)}$, combining systematic and idiosyncratic risk of the active position.

Conditional Value at Risk¶

Given scenario returns $R \in \mathbb{R}^{T \times n}$ and confidence level $\alpha$, the model minimizes the expected loss in the worst $k = \lfloor T (1-\alpha) \rfloor$ scenarios using the Rockafellar–Uryasev linear program over $x = [w, \gamma, u]$:

$$\min_{w,\,\gamma,\,u} \gamma + \frac{1}{k} \sum_{t=1}^{T} u_t \quad \text{s.t.} \quad u \ge -R\,(w - b) - \gamma, \qquad u \ge 0,$$

where $\gamma$ plays the role of the Value at Risk and $u$ the scenario losses beyond it.

Implementation¶

The shared assembly of these programs — stacking constraint blocks, mapping user constraints to cones, and invoking Clarabel — lives in cvx.core.ConeProgramBuilder. Each model contributes only its model-specific blocks (the SOC above for the covariance models, the scenario constraints for CVaR); identity-like blocks are built directly in sparse form so large universes and scenario sets stay memory-efficient.