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Academic research on trend-following investing has almost exclusively focused on testing various trading rules' profitability. However, all existing trend-following rules are essentially ad hoc, lacking a solid theoretical justification for their optimality. This paper aims to address this gap in the literature. Specifically, we examine the optimal trend-following when the returns follow a two-state process, randomly switching between bull and bear markets. We show that if a Markov model governs the return process, it is optimal to follow the trend using the Exponential Moving Average rule. However, the Markov model is unrealistic because it does not represent the bull and bear market duration times correctly. It is more sensible to model the return process by a semi-Markov model where the state termination probability increases with age. Under this framework, the optimal trend-following rule resembles the Moving Average Convergence/Divergence rule. We confirm the validity of the semi-Markov model with an empirical study demonstrating that the theoretically optimal trading rule outperforms the popular 10-month Simple Moving Average and 12-month Momentum rules across a universe of international markets. [ABSTRACT FROM AUTHOR]

There is a big controversy among both investment professionals and academics regarding how the termination probability of a market state depends on its age. Using more than two centuries of data on the broad US stock market index, we revisit the duration dependence in bull and bear markets. Our results suggest that the duration dependence for both bull and bear markets is a nonlinear function of the state age. It appears that the duration dependence in bear markets is strictly positive. For 93% of the bull markets, the duration dependence is also positive. Only about 7% of the bull markets, those with the longest durations, do not exhibit positive duration dependence. We also compare a few selected theoretical distributions on their ability to describe the duration dependence in bull and bear markets. Our results advocate that the gamma distribution most often provides the best fit for both the survivor and hazard functions of bull and bear markets. However, our results reveal that none of the selected distributions accurately describes the right tail of the hazard functions. [ABSTRACT FROM AUTHOR]

The recent literature on stock return predictability suggests that it varies substantially across economic states, being strongest during bad economic times. In line with this evidence, we document that stock volatility predictability is also state dependent. In particular, in this paper, we use a large data set of high-frequency data on individual stocks and a few popular time-series volatility models to comprehensively examine how volatility forecastability varies across bull and bear states of the stock market. We find that the volatility forecast horizon is substantially longer when the market is in a bear state than when it is in a bull state. In addition, over all but the shortest horizons, the volatility forecast accuracy is higher when the market is in a bear state. This difference increases as the forecast horizon lengthens. Our study concludes that stock volatility predictability is strongest during bad economic times, proxied by bear market states. [ABSTRACT FROM AUTHOR]