Trend-Following P&L Distributions and Optimal Timescales in a Gaussian Model

Summary

This document analyzes how a trend-following strategy converts stock price changes into profits and losses under Gaussian assumptions. It derives the P&L distribution and examines its mean, variance, skewness, kurtosis, and tail quantiles. The analysis links the strategy’s asymmetric outcomes—frequent small losses and less frequent larger gains—to the mechanics of trend following rather than to unusual price behavior alone.

The work also gives formulas for annualized risk-adjusted P&L and turnover, then uses them to study the timescale that best balances return and trading costs. It reports that losses at short horizons can be larger than standard Gaussian estimates suggest, while longer horizons constrain losses more. The theoretical results are illustrated using the Dow Jones index. Conclusions depend on the Gaussian modeling framework, and the document does not establish that the formulas capture all real market behavior.

Key ideas

• The study derives a probability distribution for trend-following profits and losses under Gaussian price models.
• Trend-following P&L can be positively asymmetric, with smaller frequent losses and less frequent larger gains.
• Short-horizon losses may exceed what standard Gaussian estimates imply.
• The preferred trend timescale depends on return autocorrelation and transaction costs.
• Turnover and annualized risk-adjusted P&L are used to account for trading costs.

Tags

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


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









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.

Shown in full with attribution under the source's licence. Licence: abstract CC0

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