Econometrics Exploiting Patterns in the S&P 500 and DJI Indices: How to Beat the Market Econometrics Figure 1. S&P index showing local peaks (star) and local troughs (box) SP time series from Aug, 2007 to July, 2009 1600 Maximum model Dependent variable: L_MAX 1500 1400 1300 1200 1100 1000 by: Marco Folpmers 900 800 Whether stock prices follow a random walk has been the central question of the finance discipline during the last decades. The question is relevant for a couple of reasons. First the random walk assumption is inherent in many valuation formulas (especially for derivatives) and secondly an unpredictable path is impossible to exploit for smart traders. Recently, the consensus between supporters and opponents of the random walk seems to be that the walk is not random, some patterns seem to be present, but it is very hard to exploit these regularities. In one of his famous articles about market anomalies professor Richard H. Thaler concludes about anomalies (Thaler, 1987, 200): ‘A natural question to ask is whether these anomalies imply profitable trading strategies. The question turns out to be difficult to answer. […] None of the anomalies seem to offer enormous opportunities for private investors (with normal transaction costs).’ More recently, in 2002, Thaler has again indicated that it is hard to take advantage of mispricings because it might take too long for prices to return to a more sensible level.1 Introduction In this article, we will show how it is possible to identify local peaks and local troughs in the Standard & Poor’s 500 index, how to predict these peaks and troughs and how to exploit them with the help of a fairly straightforward algorithm. We will compare the performance of the algorithm with a buy-and-hold strategy and demonstrate that the algorithm outperforms the buy-and-hold strategy dramatically. The great debate: do stock prices follow a random walk? Proponents of the Efficient Market Hypothesis claim that stock prices follow a random walk and that it should be impossible to predict future movements based on publicly available information. The idea that stock prices are unpredictable and follow a random walk (Geometric Marco Folpmers Dr. Marco Folpmers FRM works for Capgemini Consulting and leads the Financial Risk Management service line of Capgemini Consulting NL. He can be reached at marco. [email protected] Brownian Motion) around their intrinsic values is a fundamental element of the Black-Scholes formula for call and put option valuation and a series of formulas (collectively referred to as Black’s formula) for other types of derivatives such as interest rate derivatives. Whereas some evidence has been found that stock prices may depart from the random walk (e.g. the ‘January effect’, January stock prices tend to exceed the prices in the other months, see Thaler, 1987), the departures found are difficult to explain. On top of that, they are often dismissed as accidental patterns that can easily be identified with the help of abundant data – without any meaning whatsoever. Proponents of the Efficient Market Hypothesis (especially University of Chicago professor Eugene Fama) have not tired of explaining away apparent departures from the unpredictable random walk. In 1994, Merton Miller ascribes apparent mean reversion in the Standard & Poor’s 500 index to ‘a statistical illusion’ (Miller, 1994). On the other hand, followers of the Behavioral Finance school highlight certain inefficiencies and among these inefficiencies are overreactions to information after which adjustment takes place.2 The phenomenon that new information leads to an extreme reaction that is adjusted later on is consistent with a short-term mean reverting behavior (De Bondt & Thaler, 1987). However, as cited above, Richard Thaler, one of the founders of the Behavioral Finance school, concedes that, even though Richard Thaler and Burton Malkiel debate in 2002 at Wharton, see http://knowledge.wharton.upenn.edu/article. cfm?articleid=651. 1 4 AENORM vol. 17 (66) December 2009 Table 1. Maximum model 700 600 Predictor Beta T P (6.43) 37.94 1.07 0.06 Std Error 1.24 18.92 0.30 0.04 Constant GR UP DST_LST_ MIN DST_LST_ MAX (5.18) 2.01 3.61 1.48 0.00 0.04 0.00 0.14 0.08 0.05 1.54 0.12 Model performance 0 100 200 300 Day 400 500 600 inefficiencies may be pointed out, it is nearly impossible to profit from them. In this article we will test whether mean reversion is apparent in stock index data and if so, how it can be exploited. Our analysis departs from previous research since we are not interested in predicting the level of the index stock price, but only in predicting the binary attribute whether or not the price is at a local extreme. The algorithm: the minimum and maximum model The underlying idea behind the algorithm is that the index time series can be modeled as an oscillation with unpredictable amplitude but with predictable frequency. Our aim is to identify local peaks and troughs and not the level of these local extremes. The algorithm is applied to index data since index series are less influenced by idiosyncratic factors. First we define a local peak (L_MAXt) in the daily opening prices of Standard & Poor’s 500 Index (It) as the observation that is the maximum of the d preceding and the d following observations, so: L_MAXt = 1, if It = max(It-d, It-d+1,..., It+d) L_MAXt = 0, otherwise The local trough is defined analogously. We have initialized d to a default value of 6 and applied the definitions of the local peak and trough to a 2-year time series of the S&P, running from August 1, 2007 to July 31, 2009. The result is shown in Figure 1, where local local peaks are shown with the help of a star Pairs Concordant Discordant Nr 2,947.00 267.00 % 91.5% 8.3% Ties Total 6.00 3,220.00 0.2% 100.0% and local troughs with the help of a box. Within the period shown, the index reached its (global) maximum value of 1564.98 on October 10, 2007, and its (global) minimum value of 679.28 on March 10, 2009, a decline of 57% in 15 months. We have also split the sample into a development set (the first 250 observations) and a test set (the last 250 observations). The split is depicted with the help of a vertical line. In order to predict a local extreme, we estimate two models, a maximum model and a minimum model, with the help of the development set. We define the explanatory variables for the maximum model as follows: • GRt: growth rate of the index determined as GRt = It/ It-d-1; • UPt: number of successive upward movements at time t; • DST_LST_MINt: distance of observation t to the most recent minimum before time t; • DST_LST_MAXt: distance of observation t to the most recent maximum before time t. Note that we use only data that is contained in the series itself. For the minimum model we use the same explanatory variable with one exception: DOWNt is used (number of For the information overreaction hypothesis tested as mean reversion, see also De Bondt & Thaler, 1989. De Bondt & Thaler test the mean reversion in the long run (3-7 years; see also Cutler, 1991, for long-term mean reversion). See also Balvers e.a. (2000) where it is concluded that there is strong evidence of mean reversion in stock index prices of 18 countries (16 OECD countries plus Hong Kong and Singapore) over several years. Our purpose is to describe an algorithm that exploits mean reversion within days. Short-term mean reversion for individual stocks has mainly be tested after an extreme performance. 2 AENORM vol. 17 (66) December 2009 5 Econometrics Econometrics Figure 2. performance of algorithm for S&P 500 versus buy-and-hold for test set; left-panel: value development, right-panel: number of index stocks in portfolio Figure 3. 101 test sets 0.4 and-hold strategy outperforms the algo. In all applications we have not quantified the transaction costs of the trading activity of the algorithm. However, we believe that the outperformance is dramatic and that the transaction costs have no significant impact on the results. 0.3 Discussion Split between development set and test set 0.6 Value algo and buy & hold (dotted line) Nr shares algo and Buy & Hold (dotted line) 13000 35 11000 25 10000 20 Nr of shares 30 Value 12000 9000 8000 100 Day 200 300 p_maxt ) = X_max ⋅ b_max 1 − p_maxt v In this equation, p_max is the estimated probability that an observation is a maximum according to the maximum model, X_max is the matrix containing the explanatory variables for the maximum model, preceded by a column containing a one-vector, and b_max is the coefficients of the maximum model reported in Table 1. The estimated probabilities are: exp( X_max ⋅ b_maxt ) 1 + exp( X_max ⋅ b_maxt ) With the help of a cut-off value equal to 0.1, we identify the observations that are flagged as a local peak (p_max above 0.1). Of course, there is a trade-off in determining the cut-off value. If it is too high, the model will more often fail to identify a local peak, while, on the other AENORM 10 0 logit ( p_maxt ) = log( 6 0 6000 successive downward movements at time t) instead of UPt. We now estimate a logit maximum model with L_ MAXt as the dependent variable and the explanatory variables mentioned above as independent variables. The estimation is performed on the development set. The results of the estimation are reported in Table 1. The independent variables GRt and UPt are both significant at the 5% level. The model concordance is high, 91.5%. We can illustrate this concordance also as follows: with the help of the estimated betas we calculate the logit scores as: p_maxt = 0.1 5 0 vol. 17 (66) December 2009 In this paper we have referred to the claim that, although underlying patterns may be present in stock price development, it is impossible to profit from these patterns. We have shown with the help of a straightforward algorithm that this claim is untenable. 0.2 15 7000 5000 Outperformance 0.5 -5 -0.1 250 0 100 Day 200 300 hand, if it is too low, it will generate many ‘false alarms’, observations wrongly flagged as local peak. A minimum model has been estimated using the development set in an analogous way (only using DOWNt instead of UPt). The trading strategy The trading strategy works as follows: • The initial liquidity balance equals € 10,000. For each trading day, a liquidity balance is maintained as well as the number of index shares in the portfolio and their value using current prices. • When a local minimum has been identified, the algorithm buys stocks at the current prices for a monetary amount of 10% of the initial liquidity balance, i.e. € 1,000. The liquidity balance decreases with € 1,000 and the stocks bought are added to the portfolio. • When a local maximum has been identified, the algorithm sells stocks at the current prices for a monetary amount of 10% of the initial liquidity balance, i.e. € 1,000. The liquidity balance increases with € 1,000 and the stocks sold are subtracted from the portfolio. • The entire portfolio is liquidated at the end of the period contained in the test set. The performance is assessed in terms of one-year outperformance of a buy-and-hold strategy. We have first calibrated the parameters for application of the algorithm within the development set. Thus, we arrived at d = 6 (as stated above) and the conversion rate of 260 270 280 290 300 310 320 Observation used for split 330 340 350 10% of the initial balance at suspected peaks (conversion from stocks to liquidity) and troughs (conversion from liquidity to stocks). Generally, a higher conversion rate leads to a more volatile performance of the algorithm. Lower values of d lead to a more active algorithm. i.e. more suspected peaks and troughs and, hence, more trading (conversion of liquidity to stocks or vice versa). Whether a profit can be made from the algorithm can only be illustrated by applying the algorithm, i.e. the minimum and maximum models estimated with the help of the development set and the parameters settings for d, the cut-off (0.1) and the conversion rate, to the subsequent test set. In Figure 2 we show the relative performance of the algorithm when applied to the test set. From the figure we conclude that the algorithm starts at a loss, but the value is almost always above the value of the buy-and-hold portfolio. The one-year return of the buy-and-hold strategy is -20.2%; the one-year return of the algorithm is 21.0%. The outperformance equals 51.7%. In order to prove robustness, we have applied the same model also to the Dow Jones Industrial Average index (DJI) for the same period (again split into a development set and a test set): for the DJI, the minimum and maximum models have been estimated for the same period used for the S&P 500 index. Subsequently, the outcomes of the models have been applied to the same test period. The situation is very similar to the results shown for the S&P 500 index. The outperformance of the algorithm applied to the test set of the DJI equals 59.3%. An objection that could be made, is that we only tested the algo with the help of one test set. In order to counter this objection, we have performed additional tests: we have split the development and test set not only at the 250th observation, but at all observations on the domain [ 250, 350]. We have plotted the outperformance for all these 101 test sets in Figure 3. We conclude that the algo consistently outperforms the buy-and-hold strategy. The average outperformance equals 29.8% and there are only 4 cases in which the buy- References Balvers, R., Y. Wu, E. Gilliland. “Mean reversion across national stock markets and parametric contrarian investment strategies.” Journal of Finance, 55.2 (2000) Bondt, W.F.M. de, R. H. Thaler. “Further evidence on investor overreaction and stock market seasonality.” Journal of Finance, 42.3 (1987):557-581 Bondt, W.F.M. de, R. H. Thaler. “Anomalies: a meanreverting walk down Wall Street.” Journal of Economic Perspectives, 3.1 (1989):189-202 Cutler, D.M., J.M. Poterba, L.H. Summers. “Speculative dynamics.” Review of Economic Studies, 58 (1991):529-546 Fama, E.. “Random Walks In Stock Market Prices.” Financial Analysts Journal, 21.5 (1965):55-59 Folpmers, M.. “Making money in a downturn economy: using the overshooting mechanism of stock prices for an investment strategy.” Journal of Asset Management, 10.1 (2009):1-8 Miller, M.H., J. Muthuswamy, R.E. Whaley. “Mean reversion of Standard & Poor’s 500 Index basis changes: arbitrage-induced or statistical illusion?” Journal of Finance, 49.2 (1994):479-513 Poterba, J.M., L.H. Summers. “Mean reversion in stock prices: evidence and implications.” NBER Working Paper Series, w2343 (1989). Available at SSRN: http://ssrn.com/abstract=227278 Thaler, R.H.. “Anomalies: the January effect.” Journal of Economic Perspectives, 1.1 (1987):197-201 Zeira, J.. “Informational overshooting, booms and crashes.” Journal of Monetary Economics, 43.1 (1999):237-257 AENORM vol. 17 (66) December 2009 7

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