Using Leverage to Fool Investors

How can schemers use statistics to fool investors? In the October 2024 revision of their paper entitled "The Art of Financial Illusion: How to Use Martingale Betting Systems to Fool People", Carlo Zarattini and Andrew Aziz illustrate use of a Martingale betting system to shape the short-term profitability of trading strategies. This system involves increasing the bet (or trade size) after every loss to recover losses and even yield a profit. Specifically, they run 10,000 trials each for three strategies trading Invesco QQQ Trust (QQQ) daily during 2022, all initially capitalized at $1,000:

1. Base - randomly initiate a 100% long or 100% short position at a random time during regular trading hours with 1:1 leverage and a stop-gain and a stop-loss both $0.20 from the entry price. When no stop triggers, close the position at 4:00PM.
2. Martingale - same as base, but double the leverage after each loss and restore it to 1:1 after a win.
3. Martingale + Target Cumulative Profit - same as base but vary the leverage (in terms of number of shares traded) to target a constant cumulative profit of $0.79 per trading day. In other words, the target profit increases by $0.79 every trading day.

They assume a commission rate of $0.0005 per share. For the second and third strategies, they limit leverage to 500:1. Using intraday prices for QQQ from the end of December 2021 through the end of December 2022, they find that: