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The search for appropriate quantitative techniques to construct long-short equity strategies is not a last moment development in the financial markets. Newcomers in this game are constantly joining the traditional players and, currently, the most fervent searchers in quantitative strategies are the hedge funds involved in equity trading. Their operational flexibility and lack of constraints are ideally suited to allow them to benefit from the application of these types of trading strategies.
In this line of research, we are proposing several 'cointegration' trading strategies, such as index tracking and statistical arbitrage. The first strategy aims to replicate a benchmark in terms of returns and volatility, while the other seeks to generate steady returns under all market circumstances. As opposed to other traditional trading strategies, the portfolio optimisation is based on cointegration rather than correlation. When applied to the stocks in DJIA, these trading strategies have produced very steady results. For example, between January 1995 and December 2001 the most successful self-financing statistical arbitrage strategies returned (net of transaction and repo costs) approximately 10% with roughly 2% annual volatility and negligible correlation with the market (Alexander and Dimitriu, 2002). The funded strategies discussed in that paper return far more, of course, but have slightly higher volatility. In most years our strategies had Sharpe ratios close to one.
The applicability of the cointegration technique to asset allocation was pioneered by Lucas (1997) and Alexander (1999). Its key characteristics, i.e. mean reverting tracking error, enhanced weights stability and better use of the information comprised in the stock prices, allow a flexible design of various funded and self-financing trading strategies, from index and enhanced index tracking, to long-short market neutral and alpha transfer techniques.
Why use cointegration
in portfolio management?
However, the correlation analysis is valid only for stationary variables. This requires de-trending the prices and other level financial variables, which are usually found to be integrated of order one or higher. Taking the first difference in log prices is the standard procedure for ensuring stationarity and leads all further inference to be based on returns. This procedure has, however, the disadvantage of loosing valuable information. In particular, de-trending the variables before the analysis removes any possibility to detect common trends in prices. By contrast, the aim of the cointegration analysis is to detect any stochastic trend in the price data and use these common trends for a dynamic analysis of correlation in returns (Alexander, 2001).
The fundamental remark justifying the application of the cointegration concept to, for example, stock prices analysis, is that a system of non-stationary stock prices in level can share common stochastic trends (Stock and Watson, 1991). According to Beveridge and Nelson (1981), a variable has a stochastic trend if its difference has a stationary invertible ARMA(p,q) representation plus a deterministic component. Since ARIMA(p,1,q) models seem to characterise many financial variables, it follows that the growth in these variables can be described by stochastic trends.
When applied to stock prices in a stock market index, cointegration exists when there exists at least one portfolio of stocks that has a stationary tracking error (we use the term tracking error as statisticians do, to denote the difference between the index and the portfolio). In other words, cointegration exists when there is mean reversion in the price spread between the portfolio and the index. This finding does not provide any information for forecasting the individual prices in the system, or the position of the system at some point in the future, but it provides the valuable information that, irrespective to its position, the prices of the portfolio and the index will stay together on a long-run basis.
Outline of the trading
Additionally, being constructed on a rather long history of prices, the portfolio weights tend to ignore short-term movements in stock prices, such as bubbles or just noise, and focus on the long-run behaviour of the prices. As already shown, the fact that the tracking error is, by construction, mean reverting ensures that the tracking portfolio will stay 'tied together' with the index in the long-run, irrespective to short-term movements in individual stock prices. There can, however, be short-term de-correlations between the tracking portfolio and the index. In fact, this is a potential source of 'alpha', i.e. excess return, in the tracking portfolios.
A number of trading strategies can be constructed based on cointegration relationships:
A. Index tracking
The first stage, stock selection, can be the result of proprietary selection models, technical analysis, or just stock picking skills of a portfolio manager. The degree of cointegration and consequently the tracking performance will depend very much on the selection process. However critical, the selection process does not have special features in a cointegration-based tracking technique. It constitutes rather a control variable in identifying the most appropriate tracking strategy.
The second stage of index tracking concerns determining the portfolio holdings in each of the stocks selected in the previous stage. The stocks weights in each portfolio are estimated based on the ordinary least square (OLS) coefficients of the cointegration equation regressing the index log-price on the portfolio stocks log-prices over a given calibration period prior to the portfolio's construction moment. We note that the application of OLS to non-stationary dependent variables such as log(index) is only valid in the special case of a cointegration relationship. Unless the residuals from the cointegration regression are found to be stationary, the OLS coefficients will be inconsistent and further inference based on them will be invalid. Therefore, testing for cointegration is an essential step in constructing cointegration-based tracking portfolios.
Further to estimation, the OLS coefficients are normalised to sum up to one, thus providing the weights of each stock in the tracking portfolio.
As shown in Alexander and Dimitriu (2002), even by using simple stock selection rules such as ranking the stocks by their weight in the index, the tracking portfolios constructed based on this methodology can accurately replicate the market index.
B. Enhanced index tracking
and statistical arbitrage
This type of statistical arbitrage strategy should generate returns according to the 'plus'/'minus' spread (i.e. double alpha) with fairly low volatility and no significant correlation with the market returns. We expect to become more and more difficult to find cointegrated portfolios as the magnitude of the spread between the benchmarks tracked increases. The cointegration relationship between the market index and its component stocks has a solid rationale, but this is not necessarily the case for portfolios tracking artificial benchmarks, which may be chosen to over-perform the market index by 50%, for example. The difficulty in finding an appropriate cointegration relationship leads to an increased instability of the stock weights, higher transaction costs and higher volatility of returns. To avoid this, it is essential to ensure that all the portfolios tracking 'plus' or 'minus' benchmarks pass the cointegration test.
C. Combining index tracking
with statistical arbitrage
This type of strategy, combining enhanced index tracking with statistical arbitrage, should have a high correlation with the market index, while gaining alpha from two sources: first, the excess return from enhanced index tracking and then, double alpha from the statistical arbitrage. When applying alpha transfer strategies to the stocks in DJIA, Alexander and Dimitriu (2002) have obtained significantly better Sharpe ratios then the market, even after accounting for transaction and repo costs.
Alexander, C.O. (2001) Market Models: A Guide to Financial Data Analysis, John Wiley, pp. 347-388
Alexander, C.O. and A. Dimitriu (2002) "The Cointegration Alpha: Enhanced Index Tracking and Long-Short Equity Market Neutral Strategies", Discussion Paper 2002-08, ISMA Centre Discussion Papers in Finance Series
Beveridge, S. and C.R. Nelson (1981) "A New Approach to Decomposition of Economic Time Series into Permanent and Transitory Components with Particular Attention to Measurement of the Business Cycle", Journal of Monetary Economics 7, pp. 151-74
Lucas, A. (1997) "Strategic and Tactical Asset Allocation and the Effect of Long -Run Equilibrium Relations", Research Memorandum 1997-42, Vrije Universiteit Amsterdam
Stock, J.H. and M.W. Watson (1991) "Variable Trends in Economic Time Series", in R.F. Engle and C.W.J. Granger (eds) (1991) Long-run Economic Relationships, Oxford University Press, pp.17-50
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