On Frequency-Based Optimal Portfolio with Transaction Costs

Technical Summary: On Frequency Dependent Log-Optimal Portfolio with Transaction Costs

Problem Formulation
This paper investigates the log-optimal portfolio problem, which seeks to maximize the Expected Logarithmic Growth (ELG) of an investor's wealth, while explicitly incorporating two critical factors often treated separately or ignored in prior literature: rebalancing frequency and proportional transaction costs. The authors consider a discrete-time framework with $N$ stages and $m \ge 2$ assets, including at least one riskless asset. The investor employs a linear policy where the weight vector $K$ lies on the unit simplex (long-only constraint).

The core challenge addressed is that when proportional transaction costs are non-zero, the standard frequency-dependent log-optimal problem may become intractable. Specifically, the presence of costs can lead to "survival issues" where the account value $V(n)$ drops to zero or below with positive probability, rendering the logarithmic growth rate undefined. Furthermore, the exact optimization problem with costs does not always yield a trivial or closed-form solution, necessitating a rigorous mathematical approach to handle the constraints and potential bankruptcy.

Methodology
The paper employs a combination of convex optimization theory, stochastic analysis, and empirical validation using both intraday and daily data.

1. Concave Program Equivalence: The authors first prove that the frequency-dependent ELG optimization problem with proportional transaction costs is equivalent to a concave program. By defining a fee-adjusted return vector $\tilde{X}_n$ where $\tilde{X}_{n,i} = X_{n,i} - c_i$ (with $c_i$ being the transaction cost), the objective function is shown to be concave over the convex compact set of the unit simplex.
2. Dominance Theorem with Costs: A version of the Dominance Theorem is established. It provides a sufficient condition under which a log-optimal investor should invest 100% of available funds into a single asset $j$. This occurs if the expected ratio of the fee-adjusted returns of any other asset $i$ to asset $j$ is less than or equal to one.
3. Approximation via Quadratic Concave Program: To address the bankruptcy issue where $V(n) \le 0$, the paper proposes a Taylor-based quadratic approximation of the ELG function around $K=0$. This transforms the problem into a concave quadratic program. The authors derive necessary and sufficient optimality conditions (KKT conditions) for this approximate problem.
4. Two-Fund Theorem: A version of the Two-Fund Theorem is proven for the approximate problem. It demonstrates that any convex combination of two optimal weight vectors satisfying the derived optimality conditions is itself an optimal solution.
5. Sliding Window for Online Trading: To adapt to dynamic market conditions where true return distributions are unknown, the authors implement a data-driven sliding window approach. This method solves a sequence of static concave programs over a moving window of historical data, avoiding the computational complexity of stochastic dynamic programming.

Key Results and Empirical Findings
The theoretical framework is validated through extensive empirical studies using intraday minute-by-minute data (five assets including ETFs and Bitcoin) and daily data (a 32-asset portfolio including Dow 30 stocks and Bitcoin).

• Approximation Accuracy: The empirical results show that the optimal weights derived from the quadratic approximation ($\hat{K}^*$) are nearly identical to the true log-optimal weights ($K^*$), validating the utility of the approximation for practical implementation.
• Impact of Transaction Costs and Frequency:
  • High-frequency rebalancing (e.g., every minute) combined with transaction costs significantly degrades performance, often resulting in negative Sharpe ratios and cumulative losses.
  • Increasing the rebalancing period (e.g., every 5 minutes or days) mitigates the impact of transaction costs, leading to positive Sharpe ratios and better performance compared to frequent rebalancing.
  • Under high transaction costs, the optimal strategy often shifts to holding the riskless asset (bank account) or a single dominant asset, rather than a diversified portfolio.
• Dominance in Mid-Sized Portfolios: In the 32-asset daily data example, the Dominance Theorem correctly identified a single dominant asset (Chevron) when transaction costs were present, leading the optimal strategy to concentrate all capital in that asset.
• Sliding Window Performance: The online trading scenario using a sliding window approach demonstrated that the log-optimal strategies (both exact and approximate) generally outperformed a buy-and-hold strategy with equal weights, particularly when the window size was sufficiently large (e.g., 30 days) to smooth out estimation noise.

Significance and Claims
The paper claims to fill a gap in the literature by rigorously integrating rebalancing frequency and transaction costs into the log-optimal portfolio framework. Its primary contributions are:

1. Theoretical Rigor: Proving the equivalence of the cost-inclusive problem to a concave program and establishing conditions for bankruptcy avoidance.
2. Practical Solvability: Providing a tractable quadratic approximation with necessary and sufficient optimality conditions, making the problem solvable for real-world portfolios.
3. Frequency-Cost Trade-off: Quantifying the trade-off between rebalancing frequency and transaction costs, demonstrating that frequent rebalancing is detrimental when costs are non-zero.
4. Online Adaptation: Extending the static optimization framework to an online trading environment via a sliding window approach, offering a computationally efficient alternative to dynamic programming.

The authors maintain a modest tone, acknowledging that the quadratic approximation is most accurate for small returns and that future work could explore short-selling, dividend modeling, and distributional robustness to handle estimation errors in return distributions.