Portfolio Optimization with Transaction Costs

This section extends the mean-risk optimization framework to include transaction costs by using a quadratic programming approximation.

Problem Formulation

Consider the \(d\)-dimensional portfolio optimization problem:

(1)\[\max_{w \in \mathbb{R}^d} \; w^\top m - c_1 \, r(w) - c_2 \|w - w_0\|_1 \quad \text{s.t.} \quad w^\top \mathbf{e} = 1, \; A w \leq b,\]

where \(w\) is the portfolio weight, \(m \in \mathbb{R}^d\) is the expected return, \(r : \mathbb{R}^d \to \mathbb{R}\) is a convex non-negative risk function, \(w_0\) is the current portfolio weight satisfying the constraints, and \(c_1, c_2 > 0\) are constants.

The three terms in the objective are:

• \(w^\top m\): expected return,

• \(c_1 \, r(w)\): risk penalty,

• \(c_2 \|w - w_0\|_1\): transaction cost (turnover penalty).

When \(r(w) = \rho(w^\top X)\) for a coherent risk measure \(\rho\) and normal mixture returns \(X\), the dimension reduction of Mean-Risk Optimization for Normal Mixture Distributions cannot be applied directly due to the transaction cost term.

Quadratic Approximation

When transaction costs constrain the solution to be close to \(w_0\), we can approximate the convex risk function by its Taylor expansion:

\[r(w) \approx r(w_0) + (w - w_0)^\top \nabla r(w_0) + \frac{1}{2} (w - w_0)^\top H_r(w_0) (w - w_0),\]

where \(\nabla r\) is the gradient and \(H_r\) is the Hessian of \(r\). The approximate optimization problem becomes:

(2)\[\max_{w \in \mathbb{R}^d} \; w^\top (m - c_1 \nabla r(w_0)) - \frac{c_1}{2} (w - w_0)^\top H_r(w_0) (w - w_0) - c_2 \|w - w_0\|_1 \quad \text{s.t.} \quad w^\top \mathbf{e} = 1, \; A w \leq b.\]

Reduction to Quadratic Programming

Problem (2) is convex but non-smooth at \(w_0\). It can be reformulated as a quadratic program by introducing buy and sell variables. Let \(v = (v^+; v^-)\) where \(v^+, v^- \in \mathbb{R}^d\) with \(v^+, v^- \geq 0\) and \(w = w_0 + v^+ - v^-\). Then:

(3)\[\max_{v \geq 0} \; v^\top \tilde{m} - \frac{c_1}{2} v^\top \tilde{H} v \quad \text{s.t.} \quad v^\top \tilde{\mathbf{e}} = 0, \; \tilde{A} v \leq \tilde{b},\]

where

\[\begin{split}\tilde{m} = \begin{pmatrix} m - c_1 \nabla r(w_0) - c_2 \mathbf{e} \\ -m + c_1 \nabla r(w_0) - c_2 \mathbf{e} \end{pmatrix}, \quad \tilde{H} = \begin{pmatrix} H_r(w_0) & -H_r(w_0) \\ -H_r(w_0) & H_r(w_0) \end{pmatrix},\end{split}\]

\[\begin{split}\tilde{\mathbf{e}} = \begin{pmatrix} \mathbf{e} \\ -\mathbf{e} \end{pmatrix}, \quad \tilde{A} = (A \; -A), \quad \tilde{b} = b - A w_0.\end{split}\]

If \(v^*\) is the solution of (3), then the optimal portfolio weight is:

\[w^* = w_0 + (I \; -I) \, v^*.\]

Note

The matrix \(\tilde{H}\) is not full rank, so (3) is not strictly convex. In practice, a small regularization of the zero eigenvalues may be needed for certain QP solvers. One should verify that the objective at \(w^*\) exceeds the objective at \(w_0\); otherwise, holding the current position is preferable.

The gradient \(\nabla r(w_0)\) and Hessian \(H_r(w_0)\) for CVaR risk can be computed using the formulas in CVaR Derivatives for Normal Mixture Distributions. The QP formulation is typically orders of magnitude faster than solving the original non-smooth convex problem.