Incorporating transaction costs into a portfolio optimization problem is an important aspect, as it reflects the real-world condition where trading incurs costs, such as brokerage fees, bid-ask spread, and market impact. To account for these costs in portfolio optimization, one can extend the classical mean-variance optimization framework by introducing a transaction cost model in the objective function.
Let’s first recall the standard mean-variance optimization problem, which aims to minimize portfolio risk subject to a target expected return. The objective function can be formulated as:
$$\min_{w} \quad \frac{1}{2}\omega^T \Sigma \omega$$
s.t. μTω = rt
$$\sum_{i=1}^N \omega_i = 1$$
ω ≥ 0
Here, ω represents the portfolio weights, Σ corresponds to the covariance matrix for asset returns, μ denotes the expected return vector for assets, rt is the target return level, and N refers to the number of assets in the portfolio. Positive values for ωi represent long positions, and negative values stand for short positions (if short-selling is allowed).
Now, let us introduce transaction costs into this problem. We can consider a linear transaction cost model, where the cost is proportional to the trade size. For simplicity, we assume that the cost function is symmetric for buying and selling. The transaction cost (TC) can be represented as:
$$\text{TC} = k \sum_{i=1}^N |w_i - w_i^0|$$
Here, k is the proportionality constant which reflects transaction costs per unit of traded asset, wi stands for the new portfolio weight, and wi0 denotes the initial portfolio weight.
The portfolio optimization problem, including both the risk objective and transaction costs, can now be formulated as:
$$\min_{w} \quad \frac{1}{2}\omega^T \Sigma \omega + k \sum_{i=1}^N |w_i - w_i^0|$$
s.t. μTω = rt
$$\sum_{i=1}^N \omega_i = 1$$
ω ≥ 0
This new optimization problem captures the trade-off between minimizing risk and minimizing transaction costs. In this context, the investor finds the optimal portfolio that balances the benefits of risk reduction with the costs of trading.
Solving this problem directly can be challenging due to the absolute values present in the transaction cost term (making the problem non-linear and non-convex). One approach to solve it is to use the "epigraph reformulation" method, wherein we can introduce auxiliary variables and convert this problem into a linear or quadratic program. However, this would be beyond the scope of this answer.
In summary, incorporating transaction costs into a portfolio optimization problem can be achieved by adding a cost term reflecting the linear transaction fee model in the objective function. This modification allows the investor to consider both risk minimization and transaction costs in their portfolio allocation decisions.