## What Is Polarized Fractal Efficiency (PFE)?

The term polarized fractal efficiency (PFE) refers to a technical indicator that is used to determine the price efficiency for an investment over a user-defined period. It fluctuates between -100 and 100 where the centerline is at zero. Traders can use the PFE to see where a security they're interested in is headed. The price shows an upward trend if the PFE goes above zero and a downward trend if it drops below zero. The indicator was developed by author, engineer, programmer, and trader Hans Hannula.

### Key Takeaways

- Polarized fractal efficiency 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.
- It also helps traders determine and assess the strength of a market trend.
- PFE helps traders determine the direction and strength of a trend.
- The indicator, which was developed by author and trader Hans Hannula, was first mentioned in a 1994 issue of
*Technical Analysis of Stocks & Commodities*magazine.

## How the Polarized Fractal Efficiency (PFE) Works

The polarized fractal efficiency indicator was developed by author, engineer, programmer, and trader Hans Hannula. The idea was first mentioned in 1994 in the January issue of *Technical Analysis of Stocks & Commodities *magazine. It fluctuates between -100 and +100, with zero as the centerline. Securities with a PFE greater than zero trend up, while a reading of less than zero indicates a downward trend.

The PFE indicator measures the strength of a 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.

Strategies that generally use the 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. As such, 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.

The signature characteristic of the PFE is its use of fractal geometry in determining how efficiently a security's price is moving.

## Formula for Calculating Polarized Fractal Efficiency (PFE)

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

The polarized fractal efficiency is not a predictor. Rather, it reflects how the price has performed in the past.

## Special Considerations

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 significant drivers of investment behavior.

## Polarized Fractal Efficiency (PFE) Example

The PFE indicator generated a buy signal in American Tower (AMT) 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.