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

TL;DR

This paper analytically examines the profit and loss distribution of trend following strategies within a Gaussian model, revealing insights into risk, asymmetry, and optimal timescales, with practical illustrations on the Dow Jones index.

Contribution

It provides explicit formulas and matrix representations for P&L distributions in Gaussian models, enhancing understanding of trend following dynamics and optimal timing.

Findings

Distribution asymmetry favors small losses and significant profits

Short-term strategies face larger-than-expected losses

Optimal timescale depends on auto-correlation and transaction costs

Abstract

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.