In this blog post I will try to demonstrate using math why trend following is harder than other trading methods that use much shorter timeframes . I will start with the basic factthat a trading system is profitable if, and only if, the sum of its winning trades is larger than the sum of its losing trades:

∑W_{i}> ΣL_{k} i = 1,2,…, N_{W}, k = 1,2,…N_{L} (1)

where the symbol Σ is the summation operator (it takes a bunch of numbers and adds them basically), W_{i} is the amount of the ith winning trade, L_{k} is the amount of the kth losing trade, N_{W} the number of winning trades and N_{L} is the number of losing trades.

For any system other than the Holy Grail it is true that ΣL_{k} ≠ 0, meaning that there will be at least one losing trade and (1) is equivalent to:

pf = ( ∑W_{i}/ ΣL_{k}) > 1 (2)

The ratio of the sum of winning trades to the sum of losing trades is called the profit factor, one of the most important performance parameters in trading. Equation (2) tell us that for a trading system, or investments method in general, to be profitable, the profit factor must be greater than 1, a basic fact.

It is also true that

AvgW = ∑W_{i}/N_{W} (3) and AvgL = ΣL_{k}/N_{L} (4)

Equations (3) and (4) define the average winning and average losing trade. With the use of these two equations, equation 2 becomes

pf = N_{W} × AvgW/N_{L} × Avg L (5)

But it is also true that

N_{W} + N_{L} = N or N_{L} = N – N_{W} (6)

where N is the total number of trades.

We also define the average win to average loss ratio r as follows:

r = AvgW/AvgL (7)

After combining equations (5) – (7) we obtain:

pf = N_{W} × r / (N – N_{W}) (8)

We now divide both the numerator and the denominator of equation (8) by N, which is always greater than zero for any system with more than one trade

pf = (N_{W}/N) × R /( 1 – N_{W}/N ) (9)

Next, we define the win fraction w as the ratio of winning trades to total trades

w = N_{W}/N (10)

We introduce equation (10) into equation (9) and after we re-arranging, the result is:

pf = w × r / (1 – w) (11)

Equation (11) is an expression for the profit factor of a trading system, or investment method in general, as a function of the win fraction w and average win to average loss ratio r (payoff ratio). Solving for the win rate w we obtain:

w = pf/(pf+r) (12)

I call equation (12), which is also derived in my book *Profitability and Systematic Trading *(out of print), the *profitability rule.* It is an equation that relates the minimum value of the win fraction w that is required to achieve a profit factor pf in the presence of an average win to average loss ratio r. For example, for r = 2 and pf = 2, the minimum win rate w is 2/(2+2) = 0.5 or 50%. Thus, if a trading system wins double the amount that it loses on the average, for a profit factor equal to 2 the win rate must be at least 50%.

In intraday and short-term trading the value of r can be controlled using stops. If r is a *controlled variable*, then for a given profit factor pf, the minimum win rate w that is required is determined from equation (12).

In longer timeframes, r is a *random variable* because because both the magnitude of future trends and their volatility is not know in advanced and they are unpredictable. This means that a higher win rate may be required in the future to maintain the same profit factor pf of a trend-following system. This is true due from the stochastic nature of equation (12) in the case of longer-term timeframes. Therefore, it is not true in general that trend-following systems can have lower w than short-term or intraday systems. That is based on the assumption that future trends will have at least the same magnitude on the average as past ones and the volatility of the market will remain about the same so that exit methods used in the past will be as effective. In the 2007 – 2009 period there were several failures of trend-following systems although trends were there due to higher market volatility. Many more losing trades during whipsaw periods means cause a lower payoff ratio r and a demand for a higher win fraction Also, during periods when trends have smaller magnitude, the value of r decreases, also causing a demand for higher w. But increasing the win fraction w is not always easy, especially when volatility is high.

One of the conclusions of the above analysis, based on the formula derived, implies that trend-following systems must be designed so that they have a high enough win rate. This may cause an overall reduction of the profit factor because a higher win rate means making sure that the system operates like a short-term timeframes during whipsaw. This balance between the performance during sideways markets and trending markets is what may allow a trend-following system to survive higher volatility and smaller trends in the future. However, most developers of trend-following systems try to maximize trend capturing potential after setting parameters based on historical data, thinking that short-term performance does not matter, a sort of hindsight bias or wishful thinking. That causes win rate to drop. High win rate does matter as the profitability rule, equation (12), tell us. Every trend following system must have a short-term position trading aspect to it that is balanced well for longer-term survival and this means a higher win rate than it is usually thought.

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**Disclosure:** no relevant positions.

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