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A recent study, Optimal Trend Following Rules in Two-State Regime-Switching Models, analyzes trend-following investment strategies and provides a theoretical foundation for identifying optimal trading rules. Unlike previous research, which primarily focused on empirical backtesting of various trading rules, this paper derives the optimal trend-following strategy using a regime-switching framework with two market states: bull and bear markets.

Key findings include:

• In a Markov Switching Model (MSM) with fixed transition probabilities, the Exponential Moving Average (EMA) rule is the optimal trend-following strategy.

• In a semi-Markov Switching Model (ESMSM) where state durations influence transition probabilities, the optimal strategy resembles the Moving Average Convergence/Divergence (MACD) rule.

• Empirical analysis confirms that the theoretically derived optimal rules outperform traditional 10-month Simple Moving Average (SMA) and 12-month momentum strategies in real-world data.

Methodology

• Return Modeling: Uses a two-state regime-switching model to capture bull and bear market dynamics.

• Markov vs. Semi-Markov Models:

Markov Model (MSM): Transition probabilities between states are fixed.

Semi-Markov Model (ESMSM): Transition probabilities depend on how long the market has remained in a given state.

• Deriving the Optimal Trend-Following Rules: Theoretically derives the optimal trading strategy for each model using mathematical and numerical analysis.

• Empirical Testing: Applies the optimal trading rules to international stock market data, including the U.S. and 16 other countries, and compares their performance with existing trend-following strategies.

Key Findings

1. Optimal Trend-Following Strategies Depend on the Market Model

Markov Model (MSM):

• The Exponential Moving Average (EMA) rule is optimal.

• EMA assigns exponentially decaying weights to past prices, allowing it to react efficiently to regime changes.

Semi-Markov Model (ESMSM):

• The optimal rule resembles MACD (Moving Average Convergence/Divergence).

• Like MACD, this rule incorporates both short-term momentum and long-term mean reversion, making it well-suited for markets where state duration affects trend strength.

2. Empirical Analysis: U.S. Market Performance

The study analyzes U.S. market data (1875–2020) and evaluates the performance of:

• Theoretically derived optimal rules (EMA & MACD-like rules)

• 10-month SMA

• 12-month momentum strategy

Results: The optimal rules consistently outperform traditional SMA and momentum-based strategies across different market regimes.

3. Global Market Performance

Using out-of-sample testing on 16 international stock markets, the study finds that:

• Trend-following strategies outperform buy-and-hold in most markets.

• The optimal rules derived from theory often outperform traditional SMA and momentum rules.

• However, in some markets, SMA and momentum strategies perform better, suggesting market structure variations influence strategy effectiveness.

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