The Distribution of Share Prices and Elastic Time and Volume Weighted Moving Averages

In this article we consider “the distribution of prices paid per share” and derive simple approximation formulas for its mean and other statistical characteristica. The formulas we derive are approximations since the data about the distribution of prices paid by each shareholder for each share is not available. After going through the derivation of the formula we will end up with recursive definitions of time series which could be viewed as random coefficient autoregressive (RCA) time series (see [6, 5]). At this stage, we have no other motivation for the derivation of such formulas than the mere hope that they should supply some useful information (e.g. whether the average shareholder paid for his shares less or more than the current value). Indeed, it turns out that the (approximation to) the mean of the distribution of prices paid per share is (as one might have expected) a smoothing of the time series of share prices. In Section 4.1 we will thus re-define it as "moving average" and compare it to the standard moving averages used to smooth the time series of share prices. However, it is not our intention to introduce yet another smoothing or moving average with yet another set of advantages (and of course disadvantages). Instead, we believe that the value of this article lies in the derivation given in Section 2 of the article, i.e. viewing the moving averages defined later as mean of distributions which model (approximate) the distribution of share prices and time value, respectively. For example the well known Exponential Weighted Moving Average (EWMA) will come out as a special case of our approximation for stocks with non-volatile volume. A crucial difference of the method presented to the standard moving averages is the weightening by volume and the way it is done. To emphasize this we will conclude this introduction by giving two motivations. Although we will illustrate the smoothing defined by the volume weighted moving average in Section 4.1 by applying it to stock price time series, it should be noted