## What Is Polarized Fractal Efficiency?

Polarized Fractal Efficiency (PFE) is a technical indicator that was developed by Hans Hannula to determine price efficiency over a user-defined period. This indicator fluctuates between -100 and +100, with 0 as the centerline. Securities with a PFE greater than zero are deemed to be trending up, while a reading of less than zero indicates the trend is down. Polarized Fractal Efficiency's signature characteristic is its use of fractal geometry in determining how efficiently a security's price is moving.

$\begin{aligned} &P_i=100\ \times\ \frac{\sqrt{(\textit{Price}_i-\textit{Price}_{i-N})^2+N^2}} {\sum^{N-2}_{j=0}\sqrt{(\textit{Price}_{i-j}-\textit{Price}_{i-j-1})^2+1}}\\ &\textit{if Close}_i<\textit{Close}_{i-1}P=-P\\ &PFE_i=EMA(P_i,M)\\ &\textbf{where:}\\ &N=\text{period of indicator}\\ &M=\text{smoothing period} \end{aligned}$

### Key Takeaways

- PFE is a technical indicator that determines price efficiency over a user-defined period.
- PFE uses fractal geometry to determine how efficiently a security's price is moving.
- The indicator helps traders determine the direction and strength of a trend.

## Understanding Polarized Fractal Efficiency (PFE)

The Polarized Fractal Efficiency (PFE) indicator measures the strength of the trend by its position relative to the zero line. As a general rule, the further the PFE value is away from zero, the stronger and more efficient the given trend is. A PFE value that fluctuates around the zero line could indicate that the supply and demand for the security are in balance and the price may trade sideways.

## Polarized Fractal Efficiency Trading Signals

Generally, strategies using PFE as a signal consider a buy sign as a reversal in the direction of the indicator and its movement from its minimum value to zero. A signal to close a position arises as the value of the indicator reaches its peak above zero. An indicator shift from peak to zero presents a sell signal. As a rule of thumb, traders should buy to cover all short positions after the indicator forms a new minimum.

The PFE indicator also helps assess the strength of a market trend. The higher the indicator value, the stronger the trend. This means 100 indicates a strong uptrend, while a value of -100 indicates a strong downtrend.

## Polarized Fractal Efficiency and Fractal Geometry Models

Hannula's work follows in the footsteps of Benoit Mandelbrot, whose work as a mathematician and contemporary polymath culminate in his now infamous book, *The Misbehavior of Markets: A Fractal View of Financial Turbulence*. Mandelbrot's 2006 book has gone on to score legions of followers for its revolutionary reevaluation of the standard tools and models of modern financial theory. Its premise is elegantly captured through fractal geometry for financial market applications. Hannula's and Mandelbrot's research has brought the much-needed study of chaotic systems to financial systems. Increasingly, participants recognize that Chaos Theory and nonlinear relationships are a significant driver of investment behavior.

## Polarized Fractal Efficiency Practical Example

The PFE indicator generated a buy signal in American Tower Corporation on July 30, 2019, when it reversed from its minimum near to zero. Subsequently, the indicator generated a signal to close positions when it reached its peak above zero on Aug. 12, 2019. Those who traded the PFE indicator in this particular example made $13.34 per share, or 6% ($221.02 sell price, $207.68 buy price). As with all technical indicators, traders should use PFE in conjunction with other forms of technical analysis.