TrendFollowingSystems (trendfollowing)

Closed-form expected return, Sharpe ratio, and turnover of trend-following systems under white noise, AR(1), and ARFIMA processes — with three complete system implementations (European, American, Time Series Momentum), Monte Carlo verification, and an 84-contract futures dataset spanning 1959–2026.

Paper: Sepp, A. and Lucic, V., The Science and Practice of Trend-Following Systems. Read and download the paper on SSRN: ssrn.com/abstract=3167787 (doi:10.2139/ssrn.3167787). See Citation for the BibTeX entry. The replication material for every figure and table is in papers/tf_systems/.

trendfollowing implements the paper's central result: an exact decomposition of the European trend-following system's P&L into an autocorrelation channel and a squared-drift channel,

$$\bar F_{1y} = h \sum_{m=1}^{\infty} \nu^{m}\rho(m) + \frac{l \sigma_{\mathrm{target}}}{\sqrt{a}} \mu^{2}, \qquad h = l \sigma_{\mathrm{target}} \sqrt{a} \frac{1-\nu}{\nu}$$

where $\rho(m)$ is the autocorrelation function of volatility-normalized returns, $\mu$ their annualized drift, and $\nu$ the filter smoothing parameter of the span. The annualized Sharpe ratio follows in closed form for any causal linear process, with the excess kurtosis of the innovations entering through a single loading. On 84 liquid futures contracts, the closed form applied to sample moments reproduces the realized Sharpe ratios of the European system with a pooled correlation of 0.99 and a regression slope of 0.96.

The package is useful when three things matter:

• You want to select the filter span analytically rather than by grid search: the AR-1 break-even cost is nearly span-invariant ($c^{*}_{\infty} = \sqrt{\pi/2a} \phi/(1-\phi)$, 37–41bp at $\phi = 0.05$), while ARFIMA long memory creates an interior cost-optimal span — two regimes the closed forms separate cleanly.
• You want to predict a contract's trend-following Sharpe ratio from its autocorrelation function and drift before running a backtest, and to attribute realized performance to trend, mean reversion, and drift.
• You want three reference system implementations — continuous EWMA-filter weights, binary crossover positions with ATR stops, and sign-based time series momentum — that run out of the box on the packaged dataset, net of volume-based costs, with portfolio volatility targeting.

The analytics layer is pure numpy/scipy: every formula is a function you can read. The backtest layer builds on qis.

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Installation

git clone https://github.com/ArturSepp/TrendFollowingSystems.git
cd TrendFollowingSystems
pip install -e ".[dev]"

Python >= 3.10 and qis >= 5.0.6. The analytical layer and the Monte Carlo verification run without any data. The empirical layer runs from the dataset packaged in trendfollowing/resources (84 futures contracts, benchmarks, volume-based costs; 1959–2026).

Quickstart

The closed forms re-export at the package top level:

import trendfollowing as tf

# closed-form Sharpe ratio of the European system under an AR-1 process
sr = tf.sharpe_ar1(phi=0.05, long_span=21)                      # 0.336

# the generic formula: any population autocorrelation function
rho = tf.population_acf(n_lags=2000, phi=-0.05, d=0.1)          # ARFIMA(1,d,0)
sr = tf.compute_annualised_sharpe(rho=rho, long_span=250, short_span=20)

# the canonical realized-Sharpe estimator shared by all estimation layers
sr_hat = tf.compute_realized_sharpe(returns=daily_returns, af=260.0)

A portfolio backtest of the paper's LS(250,20) filter on the packaged universe:

from trendfollowing.universe import load_data
from trendfollowing.systems.european import run_european_tf_system

prices, volume_costs, benchmark_prices, descriptive_df, group_order = load_data()
outputs = run_european_tf_system(prices=prices,
                                 long_span=250,
                                 short_span=20,
                                 vol_span=33,                  # volatility estimator span, days
                                 portfolio_covar_span=63,      # portfolio-level volatility targeting
                                 portfolio_target_vol=0.15,
                                 volume_costs=volume_costs,
                                 warmup_period=250)
nav = outputs.portfolio_pnl_net                                # compounded nav, net of costs

Net of volume-based costs and gross of fees, this configuration delivers a Sharpe ratio of 1.10 at a 15.2% realized volatility over 1960–2026 (examples/backtest_european_system.py).

The three systems

European (systems/european.py): continuous weights from a variance-preserving EWMA filter, single or long-short, applied to volatility-normalized returns, with volatility-targeted position sizing. The system of the closed forms.

American (systems/american.py): binary positions from the crossover of two price EWMA filters with an ATR entry buffer and ATR trailing stop-losses, in the tradition of the turtle systems. Position size is fixed at trade inception.

TSMOM (systems/tsmom.py): the normalized sum of signs of volatility-normalized period returns, generalizing Moskowitz–Ooi–Pedersen time series momentum to a period length L and lookback of M periods.

At matched lookbacks the three systems correlate at 80% on average with the SG Trend Index and deliver statistically indistinguishable Sharpe ratios by the Ledoit–Wolf test: 0.47, 0.50, and 0.55 against 0.47 for the SG Trend Index, on monthly returns net of costs and 2/20 fees. The European closed form therefore ranks the performance of all three designs.

Closed-form results

For volatility-normalized returns with autocorrelation function $\rho(m)$ and annualized drift $\mu$, the annualized Sharpe ratio of the European system is

$$SR = \frac{\sqrt{a} A_{\nu} + \mu^{2}/\sqrt{a}} {\sqrt{B_{\nu} + A_{\nu}^{2} + \kappa K_{\nu} + (\mu^{2}/a)(1 + B_{\nu} + 2A_{\nu})}}$$

closed-form under any causal linear process, with the excess kurtosis $\kappa$ of the innovations entering through the single loading $K_{\nu}$. Under trading costs per unit of volatility-normalized turnover, the net Sharpe ratio follows at leading order from an independence-based signal-turnover proxy, and the ARFIMA autocorrelation generating function is the Gauss hypergeometric function $F(d, 1, 1-d; \nu)$. trendfollowing.analytics implements all of the above:

• sharpe.compute_annualised_sharpe(rho, long_span, short_span, sr_underlying) — the generic formula
• sharpe.compute_realized_sharpe(returns, af, ddof) — the canonical estimator $\sqrt{a} \hat E[f_t]/\sqrt{\widehat{\mathrm{Var}}[f_t]}$, equal to qis.compute_sharpe_arithmetic (guarded in the tests)
• sharpe.sharpe_ar1, sharpe.compute_kurtosis_loading, sharpe.compute_signal_moments — per-process forms and loadings
• autocorrelation.population_acf(n_lags, phi, d) — white noise, AR(1), ARFIMA(0,d,0), ARFIMA(1,d,0) (Sowell 1992)
• expected_return.expected_pnl_*, expected_return.expected_turnover — expected return and turnover per process

The closed forms are exact rather than fitted, and Monte Carlo confirms them process by process. The figure below is Figure 6.3 of the paper: the expected annual return, the gross Sharpe ratio, and the net Sharpe ratio of the European system under the ARFIMA process with long memory $d = 0.02$ and AR-1 feature $\phi \in {-0.05, 0, 0.05}$, with the analytic values as lines and the Monte Carlo estimates as markers.

Analytic and Monte Carlo values agree at every span. The net Sharpe ratio in panel (C) attains an interior cost-optimal span, which long memory creates and the AR-1 process does not, because there the cost-optimal span diverges at the break-even cost.

Skewness of aggregated returns

Trend-following returns acquire positive skewness under time aggregation with no drift and no predictability. The daily return multiplies the lagged signal by the current return, so the $T$-day cumulative return loads on the realized autocovariance of the volatility-normalized returns, which makes it a convex payoff on the realized trend. Under white noise the skewness is available in closed form,

$$\varsigma(T) = \frac{6\nu \left( T(1-\nu^{2}) - 1 + \nu^{2T} \right)} {(1-\nu^{2})^{3/2} T^{3/2}}$$

which is zero at one day, positive at every horizon beyond one day, and peaks near half the filter span.

Figure 7.5 of the paper: panel (A) is the closed form across filter spans with Monte Carlo markers, panel (B) is Monte Carlo under white noise, AR(1), and ARFIMA at the span of 100 days, and panel (C) is the empirical profile across the 84 futures contracts, whose median attains 2.33 at the horizon of 55 days against the closed-form 2.35 and whose interquartile range stays positive at every horizon. The right tail of trend-following returns is structural: it requires no forecasting skill, because it holds exactly where the expected return is zero. analytics.skewness.skewness_white_noise(horizon, span) implements the formula.

Empirical illustration

The figure below is Figure 7.3 of the paper: the Sharpe ratio of the European system predicted from each contract's sample autocorrelation function and drift, against the realized backtest Sharpe ratio, across 84 futures contracts and the paper's span grid.

The pooled correlation is 0.99 and the regression slope 0.96 for the European system, 0.89 and 0.73 for TSMOM, and 0.92 and 0.61 for the American system at spans above one month. The practical content: two sample moments of a contract's volatility-normalized returns — its autocorrelation function and its drift — carry nearly all the information a trend-following backtest on that contract produces. Span selection, contract screening, and performance attribution can run on the closed form directly, and the same formula prices the trade-off that costs impose: at realistic futures costs of 40–60bp per unit of volatility-normalized turnover, a short-memory AR-1 alpha at $\phi = 0.05$ sits below its 37–41bp break-even at every span, while long-memory alpha survives at the one-to-three-month cost-optimal spans.

You can reproduce the per-contract exercise in three lines (examples/predict_sharpe_from_acf.py): ES1 predicts 0.227 against a realized 0.206, and Corn predicts 0.625 against 0.620.

The three systems also run out of the box on the packaged dataset. The figure below is Figure 7.2 of the paper: the European, American, and TSMOM systems net of volume-based costs against the SG Trend Index, with the cumulative performance, the running drawdown, and the one-year EWMA correlations.

Examples

Self-contained usage cases in examples/, each runnable directly:

• analytic_sharpe_vs_span.py — the closed-form gross and net Sharpe ratios across spans: the AR-1 knife edge (the cost decides the sign at every span) and the ARFIMA interior optimum. Runs without data.
• backtest_european_system.py — the LS(250,20) portfolio backtest on the packaged 84-contract universe with volume-based costs and portfolio volatility targeting.
• predict_sharpe_from_acf.py — the attribution exercise in miniature: predict the per-contract Sharpe ratio from the sample autocorrelation function and drift, and compare with the realized backtest on the same sample.

Reproducing the paper exhibits

One entry point reproduces every figure, driven by the PaperFigure enum:

python -m papers.tf_systems.replication.reproduce_all_figures

Simulation figures are seed-exact (seed 8) and need no data. The Monte Carlo aggregates behind the process figures and the verification table are cached in papers/tf_systems/replication/results/, so those figures re-render in seconds without re-simulation. See papers/tf_systems/README.md for the figure-by-figure map and the verification catalogue.

Repository layout

trendfollowing/                     the installable library
    analytics/                          closed-form results of the paper
    systems/                            european.py, american.py, tsmom.py
    processes/                          simulation of return-generating processes
    universe.py                         futures universe data layer
    resources/                          packaged dataset: 84 futures series (1959-2026),
                                        benchmarks, volume-based costs, metadata
    backtests.py                        portfolio-level backtests of the three systems (qis)
examples/                           self-contained usage cases
papers/
    tf_systems/                         'The Science and Practice of Trend-Following Systems'
        paper/                              LaTeX source, siamonline class, compiled PDF, figures
        replication/                        exhibit generators, verification scripts, MC caches
tests/                              pytest suite

Data

The dataset in trendfollowing/resources contains the daily prices and USD returns of the 84 futures contracts used in the paper (July 1959 to July 2026), the benchmark series, the volume-based cost schedule, and the instrument metadata. The universe covers the most liquid contracts across global equity, bond, short-rate, currency, and commodity markets. The continuous series are constructed so that their relative returns carry no roll-related jumps and equal the excess returns of the held contract. trendfollowing.universe.load_data() serves all empirical scripts from these files; set TF_RESOURCE_PATH to override with a local folder.

Sharpe convention

All Sharpe ratios of the theory, the attribution, and the report exhibits are annualized arithmetic means over annualized volatility of periodic simple excess returns, $SR = \sqrt{a}\cdot\text{mean}/\text{std}$ — the convention of equation (5.1) of the paper, computed by the shared estimator trendfollowing.compute_realized_sharpe. The regime-conditional Sharpe ratios route through the qis SharpeConvention.ARITHMETIC switch at the manuscript's one-sigma 16/84 quantiles, where the bear, normal, and bull contributions sum to the total Sharpe exactly. See qis/docs/sharpe_conventions.md for the decision record.

Verification

papers/tf_systems/replication/ carries the verification scripts behind the manuscript's claims: the boundary term of the sample-path identity, the Appendix C asymptotics, the GARCH pipeline and ARFIMA truncation checks, and a Monte Carlo regression test of the long-short normalization and the turnover closed form.

cd papers/tf_systems/replication && PYTHONPATH=../../.. python verify_ls_normalization.py

Tests

pytest tests/

Ecosystem

This package is part of an open-source Python stack for quantitative finance — full catalogue at github.com/ArturSepp:

Package	Purpose
qis	Performance analytics, factsheets, and visualisation
optimalportfolios	Portfolio construction and backtesting
factorlasso	Sparse factor models and factor covariance estimation
bbg-fetch	Bloomberg data fetching
trendfollowing (this package)	Trend-following systems: closed-form theory and replication
goal-based-allocation	Dynamic MV allocation under regime-switching jump-diffusions
stochvolmodels	Stochastic volatility pricing analytics
vanilla-option-pricers	Vectorised vanilla option pricers and implied volatility fitters

Dependency links within the stack: optimalportfolios builds on qis and factorlasso; trendfollowing builds on qis.

Citation

If you use trendfollowing in academic work, please cite the paper and the software (see also CITATION.cff):

@article{SeppLucic2026trendfollowing,
  author        = {Sepp, Artur and Lucic, Vladimir},
  title         = {The Science and Practice of Trend-Following Systems},
  year          = {2026},
  eprint        = {2607.19497},
  archivePrefix = {arXiv},
  primaryClass  = {q-fin.ST},
  note          = {SSRN: \url{https://ssrn.com/abstract=3167787}},
  doi           = {10.2139/ssrn.3167787}
}

The paper states the results; this package is what produced them, so a replication should also cite the version it ran:

@software{sepp2026trendfollowing,
  author  = {Sepp, Artur and Lucic, Vladimir},
  title   = {trendfollowing: replication package for The Science and Practice of Trend-Following Systems},
  year    = {2026},
  version = {1.0.5},
  url     = {https://github.com/ArturSepp/TrendFollowingSystems}
}

License

GPL-3.0-or-later — see LICENSE.