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We investigate how price variations of a stock are transformed into profitsand losses (P&Ls) of a trend following strategy. In the frame of a Gaussianmodel, we derive the probability distribution of P&Ls and analyze its moments(mean, variance, skewness and kurtosis) and asymptotic behavior (quantiles). Weshow that the asymmetry of the distribution (with often small losses and lessfrequent but significant profits) is reminiscent to trend following strategiesand less dependent on peculiarities of price variations. At short times, trendfollowing strategies admit larger losses than one may anticipate from standardGaussian estimates, while smaller losses are ensured at longer times. Simpleexplicit formulas characterizing the distribution of P&Ls illustrate the basicmechanisms of momentum trading, while general matrix representations can beapplied to arbitrary Gaussian models. We also compute explicitly annualizedrisk adjusted P&L and strategy turnover to account for transaction costs. Wededuce the trend following optimal timescale and its dependence on bothauto-correlation level and transaction costs. Theoretical results areillustrated on the Dow Jones index.

Following a Trend with an Exponential Moving Average: Analytical Results for a Gaussian Model