Converting true returns into reported returns: A general theory of linear smoothing and anti-smoothing

In this paper, we present a unified theory of linear smoothing, which looks at the problem from a time-series perspective. We use the term ‘conversion’ to refer to generic operations that create a difference between true returns and reported returns. ‘Smoothing’ occurs when that conversion process leads to a reduction in the variance of the reported returns and we establish the conditions which guarantee smoothing. Most importantly, we discuss situations where ‘anti-smoothing’ can occur, i.e. reported returns become more volatile than their true counterparts. Finally, we present empirical evidence of the presence of both smoothing and anti-smoothing in returns data for a number of different classes of asset.