howard-lynn-ye Trend-following models
A trend backtest fails in two places, and both produce a smooth equity curve and no error. The signal is easy; the alignment is not. This skill is the paper's definitions checked against the paper, plus the size of each misalignment measured on one seeded panel.
Everything marked ✅ Measured comes from scripts/trend_models.py (numpy + pandas, seed
20260908, 10 synthetic futures x 5,220 daily bars, runs in under 2 s). Everything marked
✅ source-verified is quoted in references/mop-2012.md from the paper itself.
1. What Moskowitz, Ooi & Pedersen (2012) actually specify
✅ source-verified — JFE 104 (2012) 228–250, full quotations in references/mop-2012.md.
| Question | The paper's answer | Where |
|---|---|---|
| Lookback | 12 months, k=12, holding period h=1 month |
§4.1, eq. (5) |
| Skip the most recent month? | No. sign(r_{t−12,t}) runs up to the rebalance date |
eq. (5) |
| Volatility estimator | EWMA of squared daily returns, centre of mass 60 days | eq. (1) |
| Annualization | 261, not 252 | eq. (1) |
| Which sigma sizes which return | σ_{t−1} applied to time-t returns |
§2.4 |
| Vol target | 40% annualized, per position | §4.1 |
sigma_t^2 = 261 * sum_{i>=0} (1 - d) d^i (r_{t-1-i} - rbar_t)^2 , d/(1-d) = 60 days (1)
r^TSMOM_{t,t+1} = sign(r_{t-12,t}) * (40% / sigma_t) * r_{t,t+1} (5)
🚨 The skipped month is the single most common mis-implementation, and it is imported from the wrong paper. The only place MOP mention skipping is footnote 10, p. 240, and it is about the cross-sectional benchmark they compare against: "Asness, Moskowitz, and Pedersen (2010) exclude the most recent month when computing 12-month cross-sectional momentum. For consistency, we follow that convention here." Cross-sectional momentum skips a month to dodge short-term reversal in individual stocks. Time-series momentum on futures does not.
✅ Measured — the same panel, monthly rebalance, 228 months:
| Variant | Ann. return | Ann. vol | Sharpe | Max DD |
|---|---|---|---|---|
| 12 months, nothing skipped (eq. 5) | 7.8% | 14.1% | 0.56 | −24.1% |
| 12−1, most recent month skipped (fn. 10) | 4.3% | 15.2% | 0.29 | −46.5% |
⚠️ On this seeded panel the skip roughly halves the Sharpe and doubles the drawdown. MOP say "our results do not depend on whether the most recent month is excluded or not" on their data — this panel is synthetic and trends persistently by construction, so treat the direction, not the magnitude, as the finding: the skip is a choice, it costs something, and it is not in eq. (5). If you skip, say you skipped.
1b. Two readings of eq. (5) that disagree
⚠️ The prose one paragraph above eq. (5) writes the position size as 40%/σ_{t−1}; eq. (5)
itself prints 40%/σ_t. They agree — because eq. (1) already sums over r_{t−1−i}, so
σ_t excludes r_t. 🚨 Re-implement from eq. (5) alone with a sigma that includes the
contemporaneous return and you have built §3's look-ahead while believing you followed the
paper.
2. The estimator is one line of pandas, and it is exactly eq. (1)
# d = com/(1+com) = 60/61; pandas' com IS the paper's centre of mass d/(1-d)
sigma = np.sqrt(returns.ewm(com=60, adjust=True).var(bias=True) * 261).shift(1)
✅ Measured — d = 0.983607, Σ_i (1−d)d^i·i = 60.0000 days, weights sum to 1.000000.
Against eq. (1) written out longhand with explicit weights and an explicitly weighted mean, the
pandas value differs by a relative 3.188e−16 — machine precision. eq1_variance() in the
script is that longhand reference.
🔑 bias=True is required. The default var() is the unbiased (reliability-weighted) form and
is not eq. (1). adjust=True (the default) is what renormalises the truncated tail.
3. 🚨 Trap 1 — sizing r_t with a sigma that contains r_t
The whole bug is a missing .shift(1). Nothing raises, the position series looks sane, and the
equity curve is smoother than the honest one because every large return has been divided by a
denominator that already knows about it.
✅ Measured — daily-rebalanced TSMOM, identical signal, only the volatility lag changes:
| Volatility estimator | Sharpe with σ_{t−1} |
Sharpe with σ_t |
Gain from the leak |
|---|---|---|---|
| EWMA com=60 (MOP eq. 1) | 0.64 | 0.68 | +0.05 |
| EWMA com=20 | 0.60 | 0.73 | +0.13 |
| EWMA com=5 | 0.60 | 1.01 | +0.41 |
| rolling 60-day | 0.60 | 0.64 | +0.05 |
| rolling 20-day | 0.60 | 0.74 | +0.15 |
| rolling 5-day | 0.60 | 1.28 | +0.68 |
🔑 The size of the leak is set by the estimator's window, not by the strategy. At MOP's own 60-day centre of mass one day is ~1.6% of the weight and the theft is worth +0.05 Sharpe — small enough to survive review. At a 5-day window one day is ~20% of the denominator and the same one-character bug doubles the Sharpe. A short vol window is not a modelling choice here; it is an amplifier on a bug you cannot see.
⚠️ Note the honest column barely moves (0.60–0.64). Volatility targeting is not what makes this strategy work — §5 shows what it actually buys.
4. 🚨 Trap 2 — which return the position is actually paid
Three alignments, one signal. Only the third is tradeable.
| What it assumes | |
|---|---|
| no shift | the position computed from close_t earns close_{t−1} → close_t |
| same close | signal from close_{t−1}, filled at close_{t−1} |
| next open | signal from close_{t−1}, filled at open_t, earns open_t → open_{t+1} |
✅ Measured, Sharpe:
| Model | no shift | same close | next open |
|---|---|---|---|
| TSMOM 12m | 2.51 | 0.64 | 0.60 |
| MA 50/200 | 0.74 | 0.65 | 0.63 |
| Donchian 20/10 | 7.04 | 0.61 | 0.68 |
🚨 Donchian goes from 0.61 to 7.04 on a one-bar misalignment. Breakout rules are the most sensitive family there is, because the rule fires precisely on the bar that moved — crediting that bar's own return to the position it triggered is close to reading the answer.
⚠️ same close → next open is the small one here (0.64 → 0.60 for TSMOM, 0.61 → 0.68 for
Donchian; it can go either way on a given path). It is still not free: a close-to-close backtest
assumes a fill at a price nobody could transact at, and the gap it hides is exactly the overnight
move that trend signals are most exposed to. Report both, and cost the difference with
../../../fin-core/skills/backtest-validation/scripts/cost_curve.py.
5. 🚨 Trap 3 — the look-ahead test that does not fire
../../../fin-core/skills/signal-construction/scripts/assert_causal.py is the right tool and
you should use it. Its default perturbation cannot see trap 2.
✅ Measured, perturbing rows ≥ k and checking that positions at rows ≤ k did not move:
| Detector | Finds trap 1 (sizing) | Finds trap 2 (no shift) |
|---|---|---|
assert_causal's ×2.0 scaling |
yes, all 6 estimators | 🚨 no, on all 3 models |
| overwrite the tail: −90% and +900% days | yes | yes, all 3 models |
🔑 np.sign() is locally flat. Doubling one daily return moves a 12-month log-sum by a
fraction of a percent, the sign does not flip, the position does not change, and the test reports
clean. Trap 1 is caught because 40%/σ is continuous in the perturbed data; trap 2 is not,
because the signal is a step function of it.
So: perturb hard enough to cross the threshold. position_is_causal(fn, frame, k, shock=SHOCKS) overwrites the tail with each extreme return in turn. The correctly shifted
position stays causal under both perturbations — the script asserts that, so the detector is
shown to distinguish rather than to always fire.
⚠️ Two-sided, and worth being precise about. A single −90% day only flips a signal that was
positive; an instrument already trending down is unmoved, so a one-sided shock silently passes
on exactly the names it should catch. Hence SHOCKS = (-0.90, 9.0). And the ×2 row is
data-dependent, not a law: on a smaller 4-asset panel the same ×2 probe does catch the
Donchian leak, by luck, because a price-level breakout threshold happened to sit within 1% of
close_k. It never catches the two sign-based models on either panel. The honest statement is
not "×2 always misses" — it is "×2 is not a reliable probe for a thresholded signal, and you
cannot tell from a green result which case you are in."
🔑 And the second silent failure in the same file: the function under test must rebuild everything from the frame. A closure over a signal computed outside makes the check pass for any input. This skill's own first draft did that, and printed "causal: True" for three demonstrably leaking models.
6. What volatility targeting actually buys
✅ Measured — daily rebalance, same signal, sizing on and off:
| Ann. return | Ann. vol | Sharpe | Max DD | |
|---|---|---|---|---|
| 40% vol target per position | 8.1% | 12.7% | 0.64 | −20.1% |
| sign only, no target | 5.7% | 8.7% | 0.66 | −12.6% |
Sharpe is unchanged. What moves is the concentration: the variance share of the highest-vol
asset (40% annualized) falls from 22% to 10% — and 1/10 = 10% is an even split across ten
instruments. Per-asset Sharpes on this panel run −0.20 to 0.58.
🔑 Vol targeting is a risk-allocation decision, not an alpha. It equalises how much each instrument contributes, which is why it matters on a 58-contract futures panel and does almost nothing on one instrument. ✅ source-verified — MOP agree: "The choice of 40% is inconsequential, but it makes it easier to intuitively compare our portfolios to others."
⚠️ The vol target also is a trade. Rebalancing daily to 40%/σ_t turns a monthly strategy
into a daily one; at MOP's monthly rebalance most of that turnover disappears. Cost it before
you claim it.
7. The other two families, and what is not verified
Donchian / turtle breakout — long on a close above the prior entry-day channel high, flat
on a close below the prior exit-day channel low. 🔑 The channel must be .shift(1)ed: the bar
that breaks out must not be inside its own channel. The classic 20/10 parameters are folklore
here, not source-verified — this library has not checked the original Turtle rules against a
primary document, so treat entry=20, exit=10 as a default, not a finding.
Moving-average crossover — sign(SMA_fast − SMA_slow). 50/200 is convention, not a result.
⚠️ It is the least sensitive of the three to trap 2 (0.74 vs 0.65) precisely because a slow
crossover rarely flips on the bar you are misaligning; that makes it the worst family to use as
a canary for alignment bugs.
🚨 Nothing here is a claim about real markets. The panel is synthetic, with AR(1) drift
(phi=0.997) and GARCH(1,1) volatility, built so trends exist. It exists to size errors
against a fixed strategy, not to establish that trend following works. Any Sharpe above is a
property of the generator.
8. Before you believe a trend backtest
assert_causalon the position, not the signal — with a shock, not a scaling (§5).- Confirm the sigma that sizes
r_tis.shift(1)ed (§3). - Confirm the fill is the next open, and report close-to-close alongside it (§4).
- Count the parameters you tried. Lookback, vol window, vol target, entry/exit, fast/slow —
that is a six-dimensional grid, and every cell is a trial. Log them in
../../../fin-core/skills/backtest-validation/scripts/trial_ledger.pyand test the survivor with../../../fin-core/skills/backtest-validation/scripts/spa_test.py. - Turn the result into a cost curve, not a number —
../../../fin-core/skills/backtest-validation/scripts/cost_curve.py. A daily-rebalanced vol target has real turnover. - Measure the fills you actually get:
../../../fin-core/skills/execution-cost-analysis/SKILL.md.
9. Scripts and where this sits
scripts/trend_models.py — the panel, MOP eq. (1) and eq. (5), Donchian, MA crossover, vol
targeting, and every table above. numpy + pandas, seed 20260908, under 2 s.
references/mop-2012.md — the paper's own words for every ✅ source-verified claim above.
- Computing the indicator, warm-up and repainting —
../../../fin-core/skills/signal-construction/SKILL.md, and its../../../fin-core/skills/signal-construction/scripts/assert_causal.pyand../../../fin-core/skills/signal-construction/scripts/warmup_probe.py, which §5 extends rather than repeats. - Cross-sectional ranking, IC, and factor tests —
../../../fin-core/skills/factor-and-timeseries-research/SKILL.md. - Combining this signal with others, and neutralizing it —
../alpha-combination-and-neutralization/SKILL.md. - How large the position should be given the edge —
../position-sizing-kelly/SKILL.md. - Getting the order done —
../execution-algorithms/SKILL.mdand../../../fin-core/skills/execution-cost-analysis/SKILL.md. - Portfolio-level vol targeting and risk decomposition —
../../../fin-core/skills/portfolio-and-risk/SKILL.md. - Trials, deflated Sharpe, and whether the survivor survives —
../../../fin-core/skills/backtest-validation/SKILL.mdand../../../fin-core/skills/research-integrity-guards/SKILL.md. - Running this at scale in a vectorized engine —
../../../fin-libraries/skills/lib-vectorbt/SKILL.md. - Continuous futures series, roll gaps and back-adjustment (a trend signal on a badly rolled
series measures the roll) —
../../../fin-futures-fx/skills/futures-continuous-contracts/SKILL.md.