Two horizons on one synthetic series
Suppose an asset is priced at 100, 110, 121 and 108.9. The first two changes are +10% each and the third is −10%; the three-step total remains +8.9%. Calling this a complete reversal of the growth would be wrong. If another asset rose 12% over the full period, the first had a positive absolute result but lost relatively. The winner depends on horizon and comparison base.
The prices are invented only to demonstrate classification. They do not establish a strategy or probability of continuation. Preserve both absolute and relative outcomes.
Do not mix formation with performance
If a rule selects securities using recent-month returns, those months explain selection and are not post-selection earnings. The next period begins only after all required information is available. Any gap between formation and observation must be stated and fixed, not changed to remove inconvenient losses.
Winner and loser comparisons require equal inclusion rules, comparable corporate adjustments and delisted securities. A long-only half and a complete long-short result are different objects. If shorting is assumed, borrowing cost, availability and execution must be modelled separately.
History has no single effect for every horizon
Poterba and Summers describe return dependence that varies by horizon and discuss limited statistical power. Kim, Nelson and Starz show sensitivity of mean-reversion conclusions to historical subperiods. Daniel and Moskowitz examine rare but substantial momentum-strategy losses. A positive historical average therefore does not eliminate severe adverse sequences or provide a universal signal that avoids them in advance.
Test explanations, not only correlation
A proposed cause may involve information reaction, participant constraints or risk compensation. State distinct observable consequences for each explanation. A mechanism that fits every outcome is difficult to test and does not establish the size of the effect after costs or today’s trade price.
Compare predefined horizons and publish all results. Measure turnover, concentration, worst periods and sensitivity to removing individual securities. Overlapping intervals are dependent; the number of rows is larger than the number of independent repetitions.
A frozen educational selection protocol
Skill is not measured by quick reaction or trader confidence. A testable process records the rule version, errors, failed tests and refusal to change limits after losses. In a fictional monthly experiment, the universe is ordinary shares existing at selection, including later exits. The last 120 daily observations and a trading-halt flag are available; missing data are not filled with future prices. Require at least 100 observed closes, no active halt and median 20-day turnover of at least 1,000,000 arbitrary units. The threshold is illustrative, not a real-market liquidity claim.
Rank by total return over 100 daily intervals ending 20 intervals before selection. Select the top 20% rounded down; ties use a stable identifier. Within the list, “acceleration” means the latest 20 observed intervals exceed the preceding 20 by at least five percentage points. “Breakout after compression” means the close exceeds all previous 20 closes and the range of the previous 10 divided by their mean is below 0.05. These rules use no future data.
Enter no earlier than the next available session. Model execution separately from the signal price. One exit is after 20 sessions; another exits when the security leaves the list at the next monthly review, no later than 40 sessions. Publish them separately. A 5% stop means an attempt to close after crossing the threshold at the next available execution, not a guaranteed 5% loss.
Portfolio implementation limits
In a five-security educational portfolio, assign each weight 0.2, with no borrowing or short positions. Returns of 10%, 5%, −5%, 0% and −10% produce a gross result of zero. With cycle expenses of 0.4% of capital, the net result is −0.4%. The number of profitable securities does not determine the portfolio result. A sector cap of two securities must be fixed before testing, or “diversification” becomes hindsight.
Turnover is the sum of absolute weight changes under an explicit convention about whether buys and sells count as two sides. Also test a common daily shock, sector concentration and order size relative to trading. The purpose is to see what survives executable implementation, not to grant permission to trade momentum.
Sources and limits
[1] Mean Reversion in Stock Prices: Evidence and Implications, NBER working paper 2343.
[2] Mean Reversion in Stock Prices? A Reappraisal of the Empirical Evidence, NBER working paper 2795.
[3] Kent Daniel and Tobias J. Moskowitz, Momentum Crashes, NBER working paper 20439. Historical evidence, not a signal or recommendation.