# Trading Mathematics and Trend Following

Came across this blog post which showed the mathematics of trading statistics, and had some interesting conclusions for trend following.

Some quick points, to be making money, Profit Factor must be greater than 1.

• Profit Factor (PF)
• = Gross Gains / Gross Losses
• = (Average win * number of wins) / (Average loss * number of losses)
• = R * w / (1-w)
• where R = Average win / Average loss
• w = win rate, i.e. % number of winners compared to total number of trades

Re-arranging, we have

• w = PF / (PF + R)
• R = PF * (1 – w) / w

Sample numbers showing the minimum R required to break-even (i.e. PF = 1, assuming no transaction costs) for varying win rates.

• w = 90% >> R = 0.11
• w = 80% >> R = 0.25
• w = 70% >> R = 0.43
• w = 60% >> R = 0.67
• w = 50% >> R = 1
• w = 40% >> R = 1.5
• w = 30% >> R = 2.33
• w = 20% >> R = 4
• w = 10% >> R = 9

The style of trading strongly influences the win rate and R (average winner / average loser). For example,

• Styles with high win rates and low R
• Playing options for income
• Styles with lower win rates and high R
• Trend following. For example William Eckhardt shared that his trend following system has a win rate of 35%.

On average, professional traders probably have win rates ranging from 40% to 55%.

For trend following, the blog article makes the point that R is determined by the future volatility of the market, which affects the length of future trends. To survive during periods where trends are shorter (e.g. sideways market), a trend following system needs to be able to operate in short-term time frames.

This speaks well for intraday trend following systems, which should work regardless of the length of the daily trends, so long as volatility is present to create decent sized daily ranges. Nonetheless, the increasing participation of high-frequency trading would likely make trends more erratic, resulting in greater losses from shakeouts, fakes, or very short-term fighting. This would likely increase the difficulty of manual discretionary style intraday trading.

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