Pairs Trading and Strategies and the CAPM - Dissertation Example

? PAIRS TRADING AND STRATEGIES AND THE CAPM Banking and finance 23 June Pairs trading is a trading strategy usedto exploit markets that are out of equilibrium, assuming that overtime they will move to rational equilibrium. This paper reviews the idea of whether the pairs-trading market can be systematically beaten. The study will examine the methods of quantitative-motivated trading in pairs. This study will be aimed at testing a pair selection rule for choosing a matching partner for stocks by looking for a stock that can minimise the sum of the squared deviations between the two normalized price series and to ascertain whether this could be a robust way to find a pair and whether different and more efficient rules may be implemented. Besides testing a model, this study will also be testing market efficiency and using use Cointegration as a decision rule for pair selection, try to ascertain whether different and more efficient rules may be implemented. In order to calculate asset returns we need the Capital Asset Pricing Model (CAPM) which gives predictions on how to measure risk and the relationship between risk and return. The relationship of expected return is linear and is necessary to explain differences in returns among securities. Introduction Pairs trading include tested methods used to identify and invest in pairs. This was developed by Morgan Stanley in the 1980’s and is today one of the most commonly used strategies in the finance and trading industry. Using this strategy, an investor looks at two assets, whose prices have moved together in the past. As the price spread widens, the investor takes a short position in the outperforming asset and a long position in the underperforming asset hoping that the spread will move back again, thereby generating profits. If history then repeats itself, prices will congregate and the arbitrageur will earn revenue. For example, if the U.S. equity markets were efficient at all times, risk-adjusted returns from pairs trading would never be positive. The Morgan Stanley group disbanded in 1989 after a couple of bad years of performance, pairs trading has since then become an progressively more well known market-neutral investment strategy used by investors as well as hedge funds. The increased popularity of quantitative based statistical arbitrage strategies has also been affecting the profits. The Capital Asset Pricing Model (CAPM) is a vital area of financial management that has contributed to finance becoming a scientific and fully fledged discipline of study. There abounds criticism that the Capital Asset Pricing Model is somewhat unrealistic due to the assumptions that it is based upon. This includes the assumption that investors would only require returns on the systematic risking of their portfolios, due to the removal of the unsystematic risk which can hence be ignored. The market neutral portfolios are constructed using just two securities, consisting of a long position in one and a short position in the other, in a predetermined ratio. The two versions of pairs trading in the equity markets are statistical arbitrage pairs and risk arbitrage pairs. A Statistical arbitrage pair trading is based on the idea of relative pricing. The underlying premise in relative pricing is that stocks with similar characteristics must be priced more or less the same. The spread in the case may be thought of as a degree of mutual mispricing, so the greater the spread, the higher the magnitude of mispricing and hence a greater scope of profit. The strategy involves assuming a long-short position when the spread is substantially away from the mean. It is expected that the mispricing will be correct. The position is then reversed and profits are made when the spread reverts. The pairs trading strategy might be justified within an equilibrium asset-pricing framework with non-stationary common factors as noted in Fund & Hsieh (1999). Asset returns can be computed by Capital Asset Pricing Model (CAPM). Sharpe (1964) and Lintner (1965) introduced the Capital Asset Pricing Model independently. It is often said that forecasting the market for the short term is a certain way to burn your fingers and so investing for the long haul is the solution. Deep theories from diverse disciplines like statistics, graph theory, probability, physics, geometry etc. have been applied to elucidate diverse aspects of market behaviour. There have been some failures in the CAPM theory, and this dissertation will examine the issues involved in trying and overcome these failures. Going by some studies, ? was proven not to be the only factor to affect asset returns (Marcus W, 2010), and so some other variables will be mentioned in the literature review, and tested to see their effects on security returns. Literature review This section reviews the existing literature on the issue of the possibility of systematically beating the pairs-trading market, and also examines some methods of quantitative-motivated trading in pairs. Some of the available literature on this topic has collectively discussed the main approaches utilised in pairs-trading, including the Cointegration method, the distance method (non-parametric), stochastic spread method and the more extensive stochastic residual spread method. The Cointegration method The Cointegration method, as outlined by Vidyamurthy (2004) is one of the strategies aimed at parameterising pairs-trading through the explicit modelling of the mean reverting characteristics of the spread in the pairs. The Distance method This non-parametric method was adopted by Gatev, Goetzmann & Rouwenhorst (1999) for empirical testing in the pairs-trading market. Within this method, co-movement is measured in a pair by the sum of squared differences or the distance between both normalised price series. Gatev et al. (1999) constructed an index of cumulative total returns for different stocks over a given formation period, choosing a matching partner for the stocks by looking for the security that can minimise the sum of the squared deviations between the two normalized price series (-2 S.D. + 2 S.D.). The distance method exploits the statistical relationship between any 2 instruments or securities at a particular price level. The formation of stock pairs involves extensively matching the normalised day-to-day prices that include any re-invented dividends. Gatev et al. (1999) note that as stocks belong to the same category as stated by S&P, the results by sector and the unrestricted pairs can be utilised in testing for how robust the identified net profits are. The study by Gatev et al (1999) also employed the use of unrestricted pair trade samples, with the trading rules for entering and exiting positions being based on a measure of standard deviation. The basis of this method is a standard deviation metric involving opening and closing of positions when there is a deviation of price by 3 or more historical shifts from the value estimated when the pair is formed, and open trades are closed out during the next price crossing (Gatev et al. 1999). In Nath (2003), a distance approach is utilised in the identification of likely pair trades, although the distance approach fails to identify any mutually exclusive pairs. In Nath (2003), a record is made for the distances between pairs in an empirical distribution format to ensure that a trade can be opened for a given pair when an observed distance moves across the fifteen percentile. In contrast to the study by Gatev et al (1999), the approach adopted by Nath (2003) enables a given security to be traded against various other securities at the same time. Nath’s approach further differs from that used in Gatev et al (1999) in the sense that Gatev et al (1999) did not try to include measures of risk management in the adopted trading approach, whereas in Nath (2003), a stop loss trigger is incorporated with the aim of closing positions in the event of a 5 percentile negative deviation. Using this strategy incorporates a maximum trading period which entails the closing of all open positions if the distance does not revert to equilibrium state within the specified time-frame, and also involves the rule that new trades on any particular pair will be prohibited when there is an early closing of trades prior to mean reversion (Studentmund 2006). The stochastic spread and the stochastic residual spread methods The stochastic spread and the stochastic residual spread methods came into use more recently than the cointegration and the distance method. The stochastic spread approach was proposed by Van der Hoe and Malcolm (2005), while the stochastic residual spread approach was put forward by Do and Hamza (2006). The stochastic spread method, stochastic residual spread method and the Cointegration method are all aimed at parameterising pairs-trading through the explicit modelling of the mean reverting characteristics of the spread in the pairs. As noted in Elliot et al (2005), the stochastic spread method discusses an approach to pairs-trading that explicitly models the mean reverting behaviour of the pairs’ spread in a continued time setting. Vidyamurthy, (2004) states that pairs-trading is used to exploit markets that are out of equilibrium, assuming that over time they will move to rational equilibrium (Vidyamurthy 2004). A pair trade is a portfolio consisting of a long position in one asset and short position in another in a predetermined ratio, and it is a broadly applied investment strategy in the financial industry. Short positions can remove any exposure to market risk. The pairs trading strategy might thus be justified within an equilibrium asset-pricing framework with non-stationary common factors. Asset returns can be computed by Capital Asset Pricing Model (CAPM). Sharpe (1964) and Lintner (1965) introduced the CAPM independently with the purpose of forecasting the relationship between expected risk and return. It has been practically tested that the Capital Asset Pricing Model has its limitations. Firstly, it’s hard to calculate the covariance of returns between assets. Secondly, its mathematical method makes it difficult in practice. Third, portfolio model is assumed to be stable. The model was developed from the portfolio choice model introduced in Fama & French (2004) which states that an investor selects a portfolio at time t-1, and gets his return at time t. CAPM was constructed on an assumption that investors are risk averse, and the mean and variance (risk and return relationship) of their one-period investment are the key points they give attention to (Fama & French 2004). More assumptions to the CAPM indicate that borrowing and lending is at a risk-free rate and is same for all investors, and that investors are assumed to agree with the expected investments. It has been practically tested that the Capital Asset Pricing Model has limitations. Firstly, it’s hard to calculate the covariance of returns between assets. Secondly, mathematical method makes it difficult in practice. Third, portfolio model is assumed to be stable. The model was developed from the portfolio choice model introduced in Fama & French (2004) that states that an investor selects a portfolio at time t-1, and gets his return at time t. CAPM was constructed on an assumption that investors are risk averse, and the mean and variance (risk and return relationship) of their one-period investment are the key points they give attention to. More assumptions to CAPM say that borrowing and lending is at a risk-free rate and is same for all investors, and that investors are assumed to agree with the expected investments. If the long and short components fluctuate with common non-stationary factors, then the prices of the component portfolios would be co-integrated and the pairs trading strategy will be expected to work. Evidence of exposures to common non-stationary factors would support common non-stationary factor pricing framework. Suppose that prices obey a statistical model of the form, then: pit ???ilplt??it ? k? n Where ?it denotes a weakly dependent error, in accordance with Enders (1995). Assuming also that pit is weakly dependent after differencing once. Under these assumptions, the price vectorptis co-integrated of order 1 with co- integrating rank r = n – k, in the sense of Engle and Granger (1987). Thus, there exist r linearly independent vectors {aq}q= 1,...,rsuch that zq = ?q?pt are weakly dependent. In other words, r linear combinations of prices will not be driven by the k common non-stationary components pl. Note that this interpretation does not imply that the market is inefficient, but rather that certain assets are weakly redundant, so that any deviation of their price from a linear combination of the prices of other assets is expected to be temporary and reverting. As pairs may not open and close at various points, the calculation of the excess return on a portfolio of pairs is a significant issue. Pairs that open and diverge during the trading interval will have a positive cash flow. Pairs that open but do not diverge will only have cash flows on the last day of the trading interval when all positions are closed out. The payoffs to pairs trading strategies are a set of positive cash flows that are randomly distributed during the trading period, and a set of cash flows at the end of trading interval that can either be positive or negative. As the trading gains and losses are computed over long and short position, the payoffs have the interpretation of excess returns. The daily returns on the short and long positions are calculated as value weighted returns as under: rP,t = wi,t = wi,t-1(1+ri,t-1) = (1+ri,1) . .(1+ri,t-1) Where w defines weights and r defines returns. This has a simple interpretation of a buy-and-hold strategy. Gatev, Goetzmann, & Rouwenhorst (2006) documented that pairs trading generates consistent arbitrage profits in the U.S. equity markets, which is considered the most efficient and liquid in the world and so it is easy to assume that the success in pairs trading lies in the identification of security pairs. In the study by Gatev et al. (2006), a purely empirical approach to achieving this end was adopted. They methodically chose pairs based entirely on the historical price movement of securities and checked to see how pairs trading would have fared in a double-blind study. The theoretical explanation for the co-movement of security prices stems from arbitrage pricing theory (APT). According to APT, if two securities have exactly the same risk factor exposures, then the expected return of the two securities for a given time frame is the same. The actual return may, however, differ slightly because of different specific returns for the securities. Let the price of securities A and B at time t be pt A and pt B , and at time t + i be pt + I A and pt + I B , respectively. The return in the time period i for the two securities will be given as: log(ptA) – log(pt + iA) and log(ptB) – log(pt + iB). The linear relationship between returns required for business operations or stock market securities investment and the systematic risks involved can be represented by the Capital Asset Pricing Model formula: E(ri) = Rf + ?i(E(rm) - Rf) Where E(ri) is the required return on the financial asset (i) E(rm) is the average returns on the capital market ?i is the beta value of the financial asset Rf is the risk free ra (Elliott, Van Der Hoek & Malcolm 2005) In order to make comparable the returns on different securities, the CAPM assumes a standardised holding period. For instance, a return over 6 months cannot be held in comparison to a return over 12 months and usually, a holding period of about 1 year is mostly utilised. A major disadvantage of applying this model in investment appraisal is that assuming a single-period time horizon does not agree with the multi-period nature of investment appraisal (Lin & Gulati 2006). Although variables can be assumed constant in successive future periods in the Capital Asset Pricing Model, this may not always be so in real practice. The existing literature suggests that the capital asset pricing model (CAPM) is by far the most common approach used in calculating the cost of equity, despite the fact that it is not universally accepted by all practitioners and academics. Some academics are concerned that there is conflicting empirical evidence about the CAPM’s ability to explain historical stock returns and some business people doubt its relevance to the real world. There are some instances where the capital asset pricing model is regarded to perform less effectively, including small companies, companies with extreme book to market rations, companies with relatively high dividend yields and highly leveraged companies. With a historically small share of overall company financing, the importance of debt financing has often been overlooked and generally far less attention has been paid to the cost of debt than the cost of equity when calculating the cost of capital. Only in certain circumstances, such as leveraged buyouts and project finance has debt moved centre stage. However, lower interest rates and a greater appetite from investors for corporate bond issues mean that companies are looking more towards debt, particularly bond financing, as a core component of business capital. As debt becomes a bigger source of capital for companies, so it becomes more important for the practitioner to be able to estimate the cost of debt accurately. Possible role for market neutrality in the pairs trading strategy Do et al (2006) have categorised a set of long and short equity strategies to either belong to pairs-trading or market-neutral sub-categories. As a result, the apparent mutual exclusivity will render the market neutrality constraint irrelevant to the pairs-trading strategy. As Fund and Hsieh (1999) note, “a strategy can be seen as being market-neutral when it generates returns that are not dependent on the relevant market returns (Fund & Hsieh 1999). Market-neutral funds will usually actively seek the avoidance of major risk factors, instead looking to relative price movement. Nath (2003) and Lin et al (2006) described pairs-trading correctly as an investment strategy that is market-neutral, although such market-neutrality is probably derived from previously known interdependencies in the stocks in question, in combination with a portfolio beta that is equal to zero. Method Empirical framework This study will utilise a Johansen test for co-integration that is based on a Vector Error Correction Model (VECM).The empirical framework can be divided into two sections. The first section will be concerned with the process of selecting the pairs. An analysis of 17 stocks will be carried out including a test on the rule of finding a pair and the tests will be aimed at ascertaining an optimal method for pair selection. The second part will consist of model estimation, identifying and overcoming the problem of parameterization in the Vector Error Correction Model. Table 1: financial stocks used for this study ARBUTHNOT BANKING GROUP PLC BANK OF GEORGIA HLDGS PLC BANK OF SCOTLAND BANKERS INVESTMENT TRUST PLC BANNER CHEMICALS PLC BARCLAYS PLC BEESON GREGORY GROUP PLC BI GROUP PLC CADBURY PLC MATRIX INC & GRW 2 VCT PLC MCLEOD RUSSELL HLDGS PLC PERSONAL ASSETS TRUST PLC PERSONAL GROUP HOLDINGS PETER BLACK HOLDINGS PLC ROYAL INSURANCE HLDGS RPC GROUP PLC SCHRODERS PLC In order to calculate asset returns we need the Capital Asset Pricing Model (CAPM) which gives predictions on how to measure risk and the relationship between risk and return. E(ri) = rf + ?im [E(rm) ? rf], where ?im =cov(ri,rm)/?2(rm) The relationship of expected return and ?i is linear. Only ?i is necessary to explain differences in return among securities. The expected return on an asset with a ? of zero is rf. The expected return of an asset with ? of one is the same as the expected return on the market. The relationship of expected return and ?i is linear. Only ?i is necessary to explain differences in return among securities. The expected return on an asset with a ? of zero is rf. The expected return of an asset with ? of one is the same as the expected return on the market. Motivation The general idea of investing in a marketplace from an assessment point of view is to sell overvalued securities and buy undervalued ones. There have been some failures in the CAPM theory and this dissertation will analyse and attempt to overcome these failures. Moreover, according to some studies, ? was proven not to be the only factor to affect asset returns. To calculate asset returns we need Capital Asset Pricing Model (CAPM). CAPM gives predictions of how to measure risk and the relation between risk and return. Besides testing a model, we are also testing market efficiency. E(ri) = rf + ?im [E(rm) ? rf], where ?im =cov(ri,rm)/?2(rm) The relationship of expected return and ?i is linear. Only ?i is necessary to explain differences in return among securities. The expected return on an asset with a ? of zero is rf. The expected return of an asset with ? of one is the same as the expected return on market. The main observation about our motivating models of the CAPM-APT variety is that they are known to apply perfect co-linearity of prices, which is readily rejected by the data. It is a kind of investment strategy used by many hedge funds. If the spread widens short the high stock and buy the low stock. As the spread narrows again to equilibrium value, the outcome is a profit. If the market spread ceases s to be mean reverting, the investor is open to considerable risk. Hence, investors generally chose in advance a stop-loss level, level of loss above which they would want to close the pair trade. Pairs-trading offers a structure for and insights on relating precise analysis to trading pairs in the equity markets. If the two stocks are from the same financial sector example two IT stocks, one may take this ratio to be unity. The portfolio will be called a spread if this ratio is selected in such a way that resulting portfolio is market neutral, a portfolio with zero beta. The pair selection procedure One of the first decisions within the pairs trading strategy would be which stocks to trade. The most important property in this strategy would be the presence of a statistically significant mean reverting relationship between the stocks, so this study will use Cointegration (Gatev et al. 1999; Engle & Granger 1987) as a decision rule for pair selection and also try to ascertain whether different and more efficient rules may be implemented. The approach adopted by Gatev et al. (1999) involves an index of cumulative total returns for different stocks over a given formation period, choosing a matching partner for the stocks by looking for the security that can minimize the sum of the squared differences. For the pair selection procedure in this study, potential pairs within the 17 financial stocks were tested to ascertain if there is a cointegrating relationship between the stocks. This cointegration approach, as outlined in Vidyamurthy (2004) is aimed at parameterising pairs-trading through the analysis and exploration of possible cointegration and involves a statistical relationship in which 2 time series that have been integrated of the same order are linearly combined in an effort to come up with a single time series that is integrated of a different order (Engle & Granger, 1987). The implementation of the Johansen test here means that for the 17 stocks used for this study, there will be a total of (172-17) ?2 = 136 potential trading stock pairs. First of all, a set of economic variables in long-run equilibrium will be considered when, ?1 X1t + ?2 X2t + …. + ?n Xnt = 0 The long run equilibrium equation can also be rewritten in matrix form as follows, t =  Where  = (?1, ?2,…, ?n) and t = (X1t, X2t,….,Xnt). The equilibrium error will be the deviation from the long run equilibrium, and can be represented by et = t One of the main questions in the development of a trading strategy is the question of maximising the profit function. In Vidyamurthy (2004), a trading strategy was developed that was based on the assumed portfolio dynamics involved. The basic idea for trading here was to open long positions when it is sufficiently below its long-run equilibrium (?-?) and on the other hand entering short positions when it is sufficiently above its long-run value (?+?) (Vidyamurthy, 2004). The position will then be exited with profit when there is a reverting of the portfolio mean to the long-run equilibrium value. Data analysis The data that will be used in this study comprises of daily stock prices of 17 stocks listed on the Financial Times Stock Exchange (FTSE) 100 share price index. The stocks were chosen from the banking sector. The chosen data consists of daily stock prices over a period of 3 years. The motivation here is to use this data for searching the trade pairs from stocks within the same industry. Although pairs-trading is a market neutral strategy, there will probably be some market risk until the portfolio is constructed to have a zero beta. Identifying a cointegrating relationship and knowing that the stocks are most likely driven by similar fundamental factors, leads in the direction that the observed equilibrium relationship is probably going to continue into the future. This is an important feature in pairs-trading, as there is the fundamental risk of illiquidity on the short and also the long side of the market in the process of implementing this kind of strategy. Thus, the relationship of cointegration is likely to remain significant. On the other hand, the random identification of a cointegrating relationship between differing stocks would lead to a relationship that is a more statistical phenomenon in contrast to the deriving of a fundamental relationship from the stocks being driven by a unique set of factors. The ideal portfolio would be that it does not consist of any systematic risk so that the returns generated by the positions will arise from the convergence of residual spread. The point is profits shall not be earned from holding long positions in a bull market since this would not be the actual purpose of developing a trading strategy. By restriction the pairs to stocks within the same industry, it is assumed that those stocks should have a similar exposure to systematic risk or beta. This is the reason the portfolio beta should be close to zero. The most appropriate way would be to choose stocks with the same beta so that the portfolio beta would be close to zero. But since only a sample of stocks was chosen, then the constraint will be enough. Traditional methods of selecting pairs do not guarantee mean reversion, which is a very important aspect of profitable pairs-trading. The results of the data show that the strategy employed in this study produces big deviations around the mean, as well as a high rate of zero crossings. This shows the profitability of the strategy, and is thus a reasonably efficient rule for pair selection. Conclusion Pairs-trading is a kind of investment strategy used by many hedge funds that involves shorting the high stock and buying the low stock when the spread widens. This strategy exploits financial markets that are out of equilibrium, and results in a profit as the spread narrows again to equilibrium value. If the market spread ceases to be mean reverting, the investor will be open to a considerable amount of risk. Hence, investors generally choose in advance a stop-loss level, which is the level of loss above which they would want to close their trades. For this research, the Johansen test for cointegration has been employed in the selection of trading pairs in a pairs-trading framework. It can be seen that in pairs trading, the speed of adjustment coefficients must differ significantly from zero with cointegration. As noted in Do et al (2006) the distance approach, being model-free, enjoys the advantage of not being exposed to and wrong estimation and misspecification in the model, although there may be a lack of forecasting ability with the distance approach as regards the expected holding period or convergence time. In a portfolio which consists of a positive position in one asset and a negative position in another asset, pairs-trading is regarded as a unique form of statistical arbitrage and offers a structure for and insights on relating precise analysis to trading pairs in the equity markets. Previous research has shown that the Capital Asset Pricing Model stands up well against criticism and as such, this model remains an important part of financial management. If the two securities and stocks being traded are from the same financial sector (for example two banking stocks) then this ratio may be taken to indicate unity. The portfolio will be called a spread if this ratio is selected in such a way that the resulting portfolio is market neutral with zero beta. Going by the results of the data analysis, it is evident that the motivating models of the Capital Asset Pricing Model-Arbitrage pricing theory variety is that they are known to apply perfect co-linearity of prices that is readily rejected by the gathered data. The results of the data analysis show that different co-integrated stocks are combinable within particular linear combinations in a way that the resulting dynamics of a portfolio will be controlled by a stationary process. Pairs were ascertained in the course of this study by using 17 financial stocks that are listed on the listed on the Financial Times Stock Exchange (FTSE) 100 share price index, and chosen from the banking sector. The results of the data analysis show that it is viable to combine 2 different cointegrated stocks in a certain linear combination in a way that the dynamics of the resulting portfolio will be ruled by a stationary process. Bibliography Do, B., R. Faff and Hamza, K. 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