Please explain like I am a high school student: Conclusions
A few existing studies use stochastic optimal control theory to solve the problem of optimal trend-following with transaction costs. However, despite its theoretical appropriateness, this approach often renders the problem analytically intractable. Furthermore, finding numerical solutions through this approach tends to involve considerable complexity.

In this paper, we introduce a novel approach that balances theoretical simplicity and practical applicability. Our approach makes the problem of optimal trend-following with transaction costs much more manageable, relying on a heuristic control mechanism confined to a moving average framework. Consequently, we reshape the problem into a partially tractable form that can be tackled numerically using standard and efficient optimization methods.

Our model’s effectiveness becomes evident as we showcase its capability to provide a satisfactory solution, significantly reducing trading costs. By embracing this approach, we bridge the gap between intricate theoretical considerations and pragmatic solutions, offering a feasible path to optimizing trend-following strategies while considering the impact of transaction costs.

Our approach holds significant practical relevance due to its alignment with investment professionals’ familiarity with various moving averages and the corresponding trend-following principles. As a result, the underlying concept is readily understandable, and the optimization procedure is notably straightforward. Importantly, our approach offers theoretical justification for utilizing moving average crossover rules.

The innovations introduced in our paper can be outlined as follows. First, we derive the expression for expected transaction costs and analytically solve for the return weights of the trading indicator that minimizes these costs. Remarkably, we demonstrate that this trading indicator utilizes the sine weighted moving average of returns. The shape of this weighting function resembles a semi-circle. Second, we deduce analytical solutions for the expected return and Sharpe ratio of the trend-following strategy in the presence of transaction costs. The availability of these analytical solutions facilitates the numerical optimization of return weights within a trading indicator, utilizing conventional and efficient optimization methods.

Our model considers financial asset returns that adhere to an 
 process. Empirical observations suggest a reasonable assumption that these autoregressive coefficients exhibit a linearly decreasing trend. Our third significant contribution revolves around conducting a direct numerical optimization process to ascertain the optimal return weights for the trading indicator that maximizes the profitability of the trend-following strategy.

Our numerical computations yield remarkably intuitive outcomes. In scenarios without transaction costs, the shape of the optimal trading indicator’s return-weighting function mirrors that of the AR coefficients. This similarity is employed to enhance the accuracy of trend direction forecasts. Conversely, when transaction costs are substantial, the trading indicator prioritizes minimizing such costs, leading to a return-weighting function resembling a semi-circle. In cases falling between these two extremes – when transaction costs are neither negligibly small nor exceedingly large – the optimal return-weighting function adopts a hump-like shape that bridges the gap between linearly decreasing and semi-circle shapes. This shape can be effectively approximated using return weights from a moving average crossover rule.

Lastly, in our empirical study, we validate our theoretical model’s predictions and showcase the benefits of the optimal trend-following strategy with transaction costs using data from several real-world financial markets. To begin, our study demonstrates that the empirical results align well with the theoretical predictions established by our model. This correspondence between theory and practice supports the reliability of our proposed framework. Furthermore, by drawing on data from various financial markets, we present compelling evidence that highlights the superiority of our optimal trend-following strategy, particularly when transaction costs are factored in. This evidence adds weight to our argument for the practical effectiveness of our approach in real-world investment contexts.