A simple framework for analysing bull and bear markets

Bull and bear markets are a common way of describing cycles in equity prices. To fully describe such cycles one would need to know the data generating process (DGP) for equity prices. We begin with a definition of bull and bear markets and use an algorithm based on it to sort a given time series of equity prices into periods that can be designated as bull and bear markets. The rule to do this is then studied analytically and it is shown that bull and bear market characteristics depend upon the DGP for capital gains. By simulation methods we examine a number of DGP's that are known to fit the data quite well random walks, GARCH models, and models with duration dependence. We find that a pure random walk provides as good an explanation of bull and bear markets as the more complex statistical models. In the final section of the paper we look at some asset pricing models that appear in the literature from the viewpoint of their success in producing bull and bear markets which resemble those in the data. *We thank many people who commented on earlier versions of this paper, particularly Allan Timmermann, Mardi Dungey and two anonymous referees. The first author is at the Economics Program, Research School of Social Sciences, Australian National University and Nuffield College, University of Oxford. The second author is at the New Economic School, Moscow. E-mails: arpagan@coombs.anu.edu.au and ksosunov@nes.ru. 1.Introduction In the past two decades a great deal of attention has been paid to the documentation of many features of returns in equity markets. It has been shown that equity returns exhibit a range of features such as an equity premium, volatility clustering and fat-tailed densities e.g. see the surveys by Campbell, Lo and MacKinlay (1997) and Pagan (1996) inter alia. But much less attention has been paid to the feature that probably attracts more commentary than anything else, namely that there are extensive periods of time when equity prices rise and fall. Colloquially these periods of time are referred to as bull and bear markets respectively. Because it is less studied, the objective of this paper is to provide a framework for thinking about and analysing such phases of the market. Since the movement from a bull market to a bear market phase (and conversely) involves a turning point in the market, section 2 proposes a definition of such an event. The idea we use is motivated by similar research that has been carried out when detecting turning points in the business cycle – in particular the seminal work of Bry and Boschan (1971) and its application in King and Plosser (1994), Watson (1994) and Harding and Pagan (2001). We develop an algorithm that seems to be quite successful in locating periods in time that have been thought of as bull and bear markets in US equity prices. Once the turning points are established characteristics of the phases can be identified. This exercise is performed on monthly data for the equivalent of the S&P500 for the U.S. over the years 1835/1-1997/5. 1 Section 3 gives a formal statement of the criterion selected to determine a turning point and section 4 canvasses equivalent forms that are sometimes more useful for analysis. It emerges that the nature of bull and bear markets will depend upon the type of data generating process (DGP) for capital gains in the market. For example, if equity prices follow a random walk with normally distributed increments, all of the characteristics of bull and bear markets will depend solely upon the mean and volatility of capital gains. In section 5 we investigate the type of bull and bear markets generated by a number of statistical models that have been proposed as candidates for the DGP of the capital gainsa random walk, GARCH and EGARCH models and a hidden layer Markov chain model. Section 6 looks at the type of markets generated by DGP’s that come from economic rather than purely statistical models. We provide a general discussion about this and then focus in some detail upon models by Gordon and St-Armour (2000) and Campbell and Cochrane (2000). We also look at some work by Donaldson and Kamstra (1996) which explains the bull and bear markets of the 1920s and we consider some aspects of their explanation. Since economic models are generally of the calibrated variety they are capable of being simulated, whereupon the output from them may be passed through the dating algorithm to establish whether the bull and bear markets they imply match up with what has been observed. In general we find that all models have difficulty in producing realistic bull and bear markets and we use our framework to provide a simple explanation of why this is so. Finally, one by-product of the framework is that it facilitates the study of extreme events such as large increases in stock values during bull markets. We look at this use in the context of equity market behavior in the 1920s. Experiments such as this may also be useful for Value at Risk work focusing on event risk. 2. Dating Algorithms In its earliest manifestation the definition of bull and bear (B&B ) markets seems to correspond to that given in Chauvet and Potter (2000, p90,fn 6), “In stock market terminology, bull (bear) market corresponds to periods of generally increasing (decreasing) market prices”, Recent usage in the financial press however seems to have refined this to insist on the rise (fall) of the market being greater (less) than either 20% or 25% in order to qualify for these names. In many ways the more general definition offered in the quotation above would seem to be closer to that used to describe contractions and expansions in the business cycle literature while the second, by emphasizing extreme movements, would be analogous to “booms” and “busts” in the real economy. We will adopt the first definition, although it will become clear that the analysis could equally have been done with the second and we actually do measure this to some extent. Thus our definition implies that the stock market has gone from a bull to a bear state if prices have declined for a substantial period since their previous (local) peak. This definition does not rule out sequences of negative price movements in stock prices during a bull market or positive ones in bear markets, but we will have to provide some extra rules to restrict the extent of these movements. Given this definition we need to be able to describe turning points in the series. In the business cycle literature an algorithm for doing this was developed Bry and Boschan (BB) (1971). It is important to recognize that the BB program is basically a pattern recognition program and it seeks to isolate the patterns using a sequence of rules. Broadly these are of two types. First, a criterion is needed for deciding on the location of potential peaks and troughs. This is done by finding points which are higher or lower than a window of surrounding points. Secondly, durations between these points are measured and a set of censoring rules is then adopted which restricts the minimal lengths of any phase as well as those of complete cycles. Because one is simply seeking patterns in the data, the philosophy underlying the BB program is relevant to any series, but the nature of asset prices is sufficiently different from real quantities as to suggest that some modification may be needed in the precise way that pattern recognition is performed. In particular, while the determination of a set of initial peaks and troughs of the business cycle is done by using (monthly) data that are smoothed, and from which “outliers” have been removed, this is not as attractive with monthly asset price data. In fact, the process of eliminating “outliers” may actually be suppressing some of the most important movements in the series. Considerations such as these lead us to make a number of modifications to the BB procedures. Our first deviation from BB is not to smooth any of the series, while the second relates to the size of window used in making the initial location of turning points. In the BB program this is six. It is not entirely clear how to choose this parameter when dealing with asset prices. But given the lack of smoothing it seemed sensible to make this slightly longer, and we eventually settled on eight months as the appropriate length for asset prices. Our second deviation relates to some rule for deciding on the minimum time one can spend in any phase. In business cycle dating this is six months. To try to determine something that would be appropriate for stock prices we consider some of the earliest formal literature that emphasizes the terms “bull and bear markets”. This literature, Dow Theory, was developed by Charles Dow at the turn of the century and popularized by W.P Hamilton in editorials in the Wall Street Journal. Dow theory saw the stock market as composed of three distinct movements and distinguished between " The daily fluctuation... a briefer movement typified by the reaction in a bull market or the sharp recovery in a bear market which has been oversold... and the main movement. which decides the trend over a period of many months.” (W.P Hamilton ''Stock Market Analysis'', Wall Street Journal, Aug 9 1919, reprinted in Rhea (1932, p. 181-182)) W.P Hamilton also regarded the main or primary trend as “The broad upward and downward movements known as bull and bear markets” (Rhea(1932, p. 12)) while the secondary reaction was “an important decline in a primary bull market or a rally in a primary bear market. These reactions usually last from three weeks to as many months”. Since this paper shares with Dow theorists a fundamental interest in the primary movements, the quotations above point to a minimal length for a stock market phase of three months. We therefore set it at four months. Some minimum length to the complete cycle also needs to be prescribed. Dow Theory is somewhat vaguer about this. Dow defined a primary bull market as one with a broad up