Uncovering Trend Rules by SSRN

Paul Beekhuizen

Robeco Asset Management

Winfried G. Hallerbach

Robeco Asset Management, Quantitative Strategies

May 11, 2015


Trend rules are widely used to infer whether financial markets show an upward or downward trend. By taking suitable long or short positions, one can profit from a continuation of these trends. Conventionally, trend rules are based on moving averages (MAs) of prices rather than returns, which obscures how much weight is assigned to different historical time periods. In this paper, we show how to uncover the underlying historical weighting schemes of price MAs and combinations of price MAs. This leads to surprising and useful insights about popular trend rules, for example that some trend rules have inverted information decay (i.e., distant returns have more weight than recent ones) or hidden mean-reversion patterns. This opens the possibility for improving the trend rule by analyzing the added value of the mean reversion part. We advocate designing trend rules in terms of returns instead of prices, as they offer more flexibility and allow for adjusting trend rules to autocorrelation patterns in returns.

Uncovering Trend Rules – Introduction

Trend rules are widely used to time financial markets. When historical price patterns persist to some extent into the future they can be exploited to predict the future direction of prices. Trend rules have their origin in technical analysis1 and are based on technical indicators computed from historical prices. Undoubtedly the most popular technical indicators are moving averages (MAs) and combinations of MAs. The simplest trend rule uses one N-period MA and prescribes taking a long (short) position when the current price is above (below) this MA, thus capitalizing on the persistence of the trend. More complex trend rules use combinations of long term and short term MAs, or even multiple hierarchically stacked MAs designed to incorporate the acceleration or deceleration of a trend.

It is general practice to define MAs in terms of price levels.2 This is surprising. Firstly, trend (or time series momentum) implies a degree of persistence in price movements and hence focuses on positive or negative changes in prices, rather than price levels. Indeed, a trend implies some dependence structure in the time series of returns.3

Secondly, using some combination of past price levels (as is done when using MAs of prices) obscures the weighting scheme assigned to separate historical periods because price levels cumulate returns over different historical periods. A return, in contrast, is unambiguously linked to a specific time period. Using trend rules defined in terms of returns therefore allows one to acknowledge the differences in importance or weight given to historical time periods. This is important as there will be some degree of information decay rendering the more distant history less relevant than the more recent past. In addition, it is important to know whether some implied historical weights are positive or negative, thus allowing for the distinction between trend persistence and mean-reversion.

Because of this greater transparency, we favor trend indicators defined in terms of returns. The contribution of our paper is that we show how to uncover the weighting schemes implied by conventional price MAs, both in a theoretical and an empirical fashion.

The analysis of weighting schemes in terms of returns reveals surprising and useful information about trend rules. As a first example, we analyze trend rules that combine a short and a long price MA. Except for the special case where the short MA has length 1 (so the current price is compared to a MA of past prices), such rules have a hump-shaped weighting scheme: the weight of the most recent returns start low and increase up to a maximum, whereafter the weights decline again. Combinations of multiple trend rules may even lead to multiple humps.

Secondly, we analyze trend rules with a skip period. With such rules, the long MA is calculated over the period starting when the short MA ends. We show that such trend rules are simply rescaled versions of trend rules without a skip period, but with a longer window over which the long MA is computed.

Thirdly, the weighting scheme of the popular (but complex) MACD rule turns out to have a hump-shaped information decay as well. Moreover, it has as much negative weight as positive weight. As negative weights imply a mean reversion rule (positive past returns imply negative signals), the MACD rule is in fact just as much a trend rule as it is a mean reversion rule.

Trend Rules

Trend Rules

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