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arxiv: 1907.00212 · v1 · pith:DCTMWD3Gnew · submitted 2019-06-29 · 💱 q-fin.ST

Detailed study of a moving average trading rule

Fernando F. Ferreira , A. Christian Silva , Ju-Yi Yen This is my paper

Pith reviewed 2026-05-25 12:38 UTC · model grok-4.3

classification 💱 q-fin.ST
keywords moving average trading ruleSharpe ratioautocorrelationdriftlook-back periodnon-stationary modelstock indexes
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The pith

Moving average trading rule's Sharpe ratio shows distinct dependence on look-back when driven by autocorrelation versus drift, plus long-term oscillations from non-stationarity.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies a trading rule that uses moving averages of past returns to forecast future returns on stock indexes, with the goal of connecting rule performance to the asset's underlying stochastic process. It derives the Sharpe ratio of the rule directly from the probability distribution function of returns and compares the result to real data across different time scales. The central distinction is that the SR curve versus look-back period takes a markedly different shape when performance is driven by return autocorrelation than when it is driven by the asset's average return (drift). Data indicate that autocorrelation dominates for look-backs of only a few months while drift grows in importance for longer horizons, and a previously unreported long-term oscillation in SR is explained by a non-stationary model.

Core claim

The Sharpe ratio of the moving average rule, when expressed as a function of the underlying asset's return PDF, behaves differently depending on whether performance is driven by autocorrelation or by drift. For look-back periods of a few months the investor is more likely to exploit autocorrelation; for longer look-backs the drift becomes progressively more important. Empirical analysis also reveals a new long-term oscillation in the SR that is accounted for by a non-stationary model.

What carries the argument

Sharpe ratio of the moving average rule expressed as a function of the look-back (portfolio formation) period and derived from the probability distribution function of asset returns.

If this is right

  • Autocorrelation dominates performance for look-back periods of a few months.
  • Drift becomes progressively more important for look-backs larger than a few months.
  • A new long-term oscillation of the SR appears in the data.
  • A non-stationary model reproduces the observed long-term oscillations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Traders could deliberately choose short versus long look-backs to isolate autocorrelation versus drift effects.
  • The non-stationary explanation implies that stationary return models will systematically mis-predict performance at multi-year horizons.
  • The same SR-versus-look-back decomposition could be applied to individual equities or futures to test whether the autocorrelation-drift crossover is universal.
  • Microstructure effects omitted from the PDF derivation may alter the reported dominance patterns once they are restored.

Load-bearing premise

The trading rule's realized returns can be accurately expressed solely as a function of the underlying asset's return PDF without material contributions from transaction costs, liquidity effects, or other market microstructure features.

What would settle it

A simulation or out-of-sample test in which the observed SR versus look-back curve deviates substantially from the PDF-derived prediction once realistic transaction costs are included.

Figures

Figures reproduced from arXiv: 1907.00212 by A. Christian Silva, Fernando F. Ferreira, Ju-Yi Yen.

Figure 1
Figure 1. Figure 1: Increasing red line: SR as a function of the look-back lag [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: In the upper panel: Time series of Log-return simulated from an ARMA(2,2) with the following parameters: [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Time-series (left) and corresponding probability distribution functions (PDF) [right, black solid line] for the [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Log cumulative return of weekly DJIA from 1995 to 2013 (circles). The intervals between green and red vertical [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: First lag autocorrelation for rescaled DJIA log-returns within each regime. Notice the clear change in sign [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Average SR vs. N (look-back) over patches determined by BFAST algorithm. Average is performed over regimes before 1975 (squares) and after 1975 (triangles) as well as all 47 regimes/patches (circles). Symbols represent the data and the solid lines are the best fit theoretical SR given by Eqs. Eqs. (5) and (8). 13 [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Average SR vs. N (look-back) over patches determined by BFAST algorithm. Average is performed over [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: SR vs. portfolio formation period N in weeks. The thin solid lines represent the empirical SR per trading day. The highlighted areas point to the range of N reported in the literature (mostly stocks). Our study goes beyond the range normally described in the literature and uncovers a periodic oscillation on the SR. Dashed lines is the result of theoretical model from Eq. 13. The lower red line is generated… view at source ↗
Figure 9
Figure 9. Figure 9: The standard deviation of the strategy applied to all DJIA data as a function to the portfolio formation period [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: SR vs. portfolio formation period N (weeks) for the strategy (Eq. 2) per 20 year sub-periods. The solid red line refers to the strategy applied to 119 years of the DJIA while the solid black line refers to the strategy applied within a 20 year period. 20 [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: SR for strategy (Eq. 2 as a function of N when applied to log-returns that are scaled/normalized by a measure of the local standard deviation (Eq. 12) [top red line] compared to the same strategy applied to log-returns without normalization (bottom blue line). 25 [PITH_FULL_IMAGE:figures/full_fig_p025_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: SR for strategy (Eq. 2) applied to 11 developed world markets from 1994 to 2016 as a function of portfolio [PITH_FULL_IMAGE:figures/full_fig_p026_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Figure C2. SR for strategy (Eq.2) applied to three indices from 1950 to 2016 as a function of N weeks. For [PITH_FULL_IMAGE:figures/full_fig_p027_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: shows that the SR for trading daily, weekly or monthly overlaps. That is, the results of this study do not depend on the trading frequency. From a practical point of view, trading daily is much more complex and costly. However, trading cost effects are ignored at any point in this study, but clearly should be accounted in any real life implementation. 0.00 0.05 0.10 0.15 0.20 0.25 0 100 200 300 400 500 N … view at source ↗
read the original abstract

We present a detailed study of the performance of a trading rule that uses moving average of past returns to predict future returns on stock indexes. Our main goal is to link performance and the stochastic process of the traded asset. Our study reports short, medium and long term effects by looking at the Sharpe ratio (SR). We calculate the Sharpe ratio of our trading rule as a function of the probability distribution function of the underlying traded asset and compare it with data. We show that if the performance is mainly due to presence of autocorrelation in the returns of the traded assets, the SR as a function of the portfolio formation period (look-back) is very different from performance due to the drift (average return). The SR shows that for look-back periods of a few months the investor is more likely to tap into autocorrelation. However, for look-back larger than few months, the drift of the asset becomes progressively more important. Finally, our empirical work reports a new long-term effect, namely oscillation of the SR and propose a non-stationary model to account for such oscillations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The manuscript examines a moving average trading rule applied to stock indexes. It derives the Sharpe ratio (SR) of the rule as a function of the underlying asset's return PDF, compares the result to empirical data, and claims that the shape of SR versus look-back period distinguishes autocorrelation-driven performance (dominant for look-backs of a few months) from drift-driven performance (increasingly important at longer horizons). A new long-term oscillation in SR is reported and accounted for by a proposed non-stationary model.

Significance. If the PDF-to-SR mapping is rigorously derived without omitted microstructure terms and the non-stationary model is validated out-of-sample, the work would clarify the relative contributions of autocorrelation and drift to moving-average rule performance and introduce an empirically motivated long-term effect. The direct comparison of analytic SR shapes to index data is a positive feature.

major comments (3)
  1. [Abstract / derivation section] Abstract and the section deriving SR from the PDF: the claim that SR is obtained directly as a function of the return PDF supplies no derivation steps, error analysis, or treatment of finite-sample bias, so the asserted clean separation between autocorrelation and drift contributions cannot be evaluated.
  2. [Non-stationary model section] Non-stationary model section: the model is introduced specifically to reproduce the observed long-term SR oscillations; without out-of-sample validation or explicit discussion of how parameters are chosen, the account risks circularity by construction.
  3. [Empirical results / short-horizon analysis] Empirical comparison (short-horizon results): the analysis assumes realized rule returns depend only on the return PDF, yet for look-backs of a few months—where autocorrelation is claimed to dominate—per-trade costs, bid-ask bounce, and slippage are largest and would shift both the level and location of the SR peak, undermining the reported distinction between mechanisms.
minor comments (2)
  1. Notation for the look-back period and the PDF should be introduced once and used consistently; occasional switches between symbols obscure the mapping from theory to figures.
  2. Figure captions should state the exact index, sample period, and rebalancing frequency used for each SR curve so that the data comparison can be reproduced.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the thoughtful comments, which have helped us identify areas for improvement in the manuscript. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract / derivation section] Abstract and the section deriving SR from the PDF: the claim that SR is obtained directly as a function of the return PDF supplies no derivation steps, error analysis, or treatment of finite-sample bias, so the asserted clean separation between autocorrelation and drift contributions cannot be evaluated.

    Authors: We agree that additional detail on the derivation is needed to allow full evaluation. The revised manuscript will include the complete step-by-step derivation of the Sharpe ratio from the return PDF, incorporating error analysis and a treatment of finite-sample bias. This will strengthen the presentation of the separation between autocorrelation-driven and drift-driven contributions. revision: yes

  2. Referee: [Non-stationary model section] Non-stationary model section: the model is introduced specifically to reproduce the observed long-term SR oscillations; without out-of-sample validation or explicit discussion of how parameters are chosen, the account risks circularity by construction.

    Authors: The non-stationary model is proposed to explain the empirically observed oscillations in SR at long horizons. In the revision, we will provide explicit details on how the model parameters are selected based on the data characteristics and discuss the in-sample nature of the fit. While out-of-sample validation would be ideal, the long time scales involved limit available data for such tests; we will note this as a limitation. revision: partial

  3. Referee: [Empirical results / short-horizon analysis] Empirical comparison (short-horizon results): the analysis assumes realized rule returns depend only on the return PDF, yet for look-backs of a few months—where autocorrelation is claimed to dominate—per-trade costs, bid-ask bounce, and slippage are largest and would shift both the level and location of the SR peak, undermining the reported distinction between mechanisms.

    Authors: This is a valid concern for short look-back periods. The current analysis focuses on the theoretical mapping from the return PDF to SR, which corresponds to gross returns. In the revised manuscript, we will add a section discussing the potential impact of transaction costs, bid-ask bounce, and slippage on the short-horizon results and how they might affect the observed SR peak, to better contextualize the distinction between autocorrelation and drift mechanisms. revision: yes

Circularity Check

0 steps flagged

Derivation of SR from return PDF is self-contained; no reduction to fitted inputs or self-citations

full rationale

The paper derives the Sharpe ratio of the moving-average rule directly as a function of the underlying asset's return PDF, then compares the resulting functional forms (for autocorrelation-driven vs. drift-driven cases) to index data. No equations or sections in the provided text reduce a claimed prediction to a parameter fitted on the target quantity itself, nor does any load-bearing step rest on a self-citation whose content is unverified. The non-stationary model is introduced to explain an observed long-term oscillation after the PDF-based derivation; absent explicit equations showing that model parameters are fitted to the same oscillation they purport to explain, the derivation chain remains independent of its empirical targets.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Based on abstract alone, the central claim rests on the unstated premise that the trading rule's performance is fully captured by the first two moments or the full PDF of returns under a stationary or piecewise-stationary process; no explicit free parameters, axioms, or invented entities are identifiable from the provided text.

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