The moving averages demystified

Abstract

A common method in technical analysis is the construction of moving averages along time series of stock prices. We show that they present a practical interest for physicists, and raise new questions on fundamental ground. Indeed, self-affine signals characterized by a defined roughness exponent H can be investigated through moving averages. The density ρ of crossing points between two moving averages is shown to be a measure of long-range power-law correlations in a signal. Finally, we present a specific transform with which various structures in a signal, e.g. trends, cycles, noise, etc. can be investigated in a systematic way.

Introduction

In Physics, theories are mainly motivated by observations. Conversely, experiments are set up in order to either confirm or infirm a theory. In Finance, the situation is quite different: there is a huge gap between econometry and empirical finance. Indeed, the hypothesis of a pure random stock market is almost taken for granted in econometry while it is definetely not considered as such in empirical finance. Econophysicists are trying to fill the above gap as Stanley said in his contribution “Can Physics contribute to Finance?” [1].

The question of the present contribution is quite the opposite: “Can Finance contribute to Physics?”. The answer of this question is undoubtedly YES. We will illustrate this answer in the particular case of a technical tool, the so-called moving averages.

Moving averages are common tools in Technical Analysis [2], [3]. By definition, a moving average $\text{y}\text{̄}$ at time t of a signal y is

$$\text{y}\text{̄}\text{=}\text{1}\text{T}\text{∑}\text{i=0}\text{T−1}\text{y(t−i)}\text{,}$$

where T is the time interval over which the average is calculated. It is easy to show that if the trend of y(t) is positive, the moving average $\text{y}\text{̄}$ will be below y, while $\text{y}\text{̄}\text{>y}$ when the trend is negative.

Consider two different moving averages $\text{y}\text{̄}{}_1$ and $\text{y}\text{̄}{}_2$ characterized, respectively, by T₁ and T₂ intervals such that T₂>T₁. These moving averages are illustrated in Fig. 1 for the specific case of a typical financial time series, i.e. the evolution of Apple stock price from January 1st 1987 till December 31th 1996, and for the parameter values T₁=50 and T₂=200. The crossings of $\text{y}\text{̄}{}_1$ and $\text{y}\text{̄}{}_2$ coincide with drastic changes of the trend of y(t). If y(t) increases for a long period before decreasing rapidly, $\text{y}\text{̄}{}_1$ will cross $\text{y}\text{̄}{}_2$ from above. This event is called a “death cross” in empirical finance [2]. On the contrary, if $\text{y}{}_1\text{̄}$ crosses $\text{y}\text{̄}{}_2$ from below, the crossing point coincides with an upsurge of the signal y(t). This event is called a “gold cross”. Chartists often try to “extrapolate” the evolution of $\text{y}\text{̄}{}_1$ and $\text{y}\text{̄}{}_2$ expecting “gold” or “death” crosses. Most computers on trading places are equiped for performing this kind of analysis and forecasting [3]. Such curves are automatically displayed on stock charts on most trading softwares. Obviously, the positions of the crossing points are determined by the past history of the data and not to the future such that the forecasting in empirical finance as based on moving average “recipes” are far from a short-/long-range quantitative forecasting.

As physicists, we do not endorse these charting methods. Even though moving averages seem to be poor statistical measures, we will however see in this paper that they present some very practical interest for physicists and raise new questions in statistical physics.