  We investigate how price variations of a stock are transformed into profits
and losses (P&Ls) of a trend following strategy. In the frame of a Gaussian
model, we derive the probability distribution of P&Ls and analyze its moments
(mean, variance, skewness and kurtosis) and asymptotic behavior (quantiles). We
show that the asymmetry of the distribution (with often small losses and less
frequent but significant profits) is reminiscent to trend following strategies
and less dependent on peculiarities of price variations. At short times, trend
following strategies admit larger losses than one may anticipate from standard
Gaussian estimates, while smaller losses are ensured at longer times. Simple
explicit formulas characterizing the distribution of P&Ls illustrate the basic
mechanisms of momentum trading, while general matrix representations can be
applied to arbitrary Gaussian models. We also compute explicitly annualized
risk adjusted P&L and strategy turnover to account for transaction costs. We
deduce the trend following optimal timescale and its dependence on both
auto-correlation level and transaction costs. Theoretical results are
illustrated on the Dow Jones index.
