The Economic Value of Trading Rules

Transcription

1 The Economic Value of Trading Rules Darryl A Ross School of Banking and Finance, University of New South Wales PRELIMINARY DRAFT, NOT TO BE QUOTED 16 October 2007 Abstract While numerous studies claim that technical trading rules have some predictive power for stock prices, few of these have risk adjusted rule returns. I evaluate the economic value of trading rules by estimating the weight of trading rules in the optimal portfolios of investors with power utility preferences. I find that on the basis of gross returns trading rules form a significant weight in optimal portfolios. This appears to be due to the low correlation with index returns and the skewed pattern of rule returns. However, these benefits do no survive the reality check imposed by transaction costs. After allowing for transaction costs, optimal portfolios rarely include trading rules. These findings, both before and after trading costs, generally persist for several years. This suggests that rule returns are consistent with market efficiency. JEL Classification: E11, G2. Keywords: Trading rules, technical analysis, performance evaluation, asset allocation, market efficiency and persistence. z @student.unsw. 1

2 1 Introduction Technical trading rules claim to predict trends and turning points in stock prices, based on patterns in historic prices. This is a challenge to the weak form of the efficient markets hypothesis (EMH), which holds that current stock prices fully incorporate information in past prices. While academics might be sceptical about trading rules, in practice many investors rely on these rules, investment banks publish newsletters on the topic and relevant software is widely available. Recent research also finds trading rules have some predictive power for stock prices ( Neftci (1991), Brock, Lakonishok & LeBaron (1992), Gencay (1998) and Lo, Mamaysky & Wang (2000)). Other researchers find that the predictive power is eliminated after allowing for transaction costs and the effects of measurement errors and datamining (Hudson, Dempsey & Keasey (1996), Ready (1997), Bessembinder & Chan (1998), Allen & Karjalainen (1999), Ito (1999) and Sullivan, Timmermann & White (1999)). Whether the predictive power translates into superior investment performance remains an open question, however, because few studies in either group have risk adjusted returns. While the performance evaluation literature has extensively debated how to detect forecasting ability and separate the return component due to market timing from stock selection 1, trading rule studies have generally eschewed these performance measures. This may be because the properties of these measures are generally not well understood for timing strategies which short the market when it is forecast to decline, as trading rules do. Instead, trading rules have generally been evaluated in one of two ways. The first approach compares rule returns with the return on a buy-and-hold strategy. It is difficult to interpret the significance of these results as the risks of the two strategies differ, because trading rules switch between long and short positions. Research which adopts this approach mostly finds that rule returns do not exceed that of the a buy-and-hold strategy, but trading rules may still be useful if they are less risky than other assets. The second approach measures the difference between market returns on days when rules are long and are short; this approach omits risk entirely. These two approaches to assessing rule performance also ignore the potential diversification gains from including trading rules in wider asset portfolios. Rule returns are generally not highly correlated with returns on the underlying asset for two reasons. For a rule to be perfectly correlated, it would always have to be long, which happens rarely. As well, while returns on a trading rule with predictive power will be positively correlated on days the price of the traded asset rises, this is offset by the negative correlation on days the price of the traded asset declines. This biases the overall correlation towards zero. Thus, rules which appear to sub-perform other assets when 1 For example, Fama (1972), Grant (1977), Merton (1981), Henriksson & Merton (1981), Admati, Bhattacharya, Pfleiderer & Ross (1986), Grinblatt & Titman (1989),Chen & Knez (1996), Ferson & Schadt (1996) and Goetzmann, Ingersol & Ivkovic (2000). 2

3 considered in isolation may still be valuable in a portfolio of other assets. Promoters of hedge funds, many of which use quantitative techniques, often stress the diversification benefits of these funds. I evaluate the economic value of trading rules by examining their weight in investors optimal portfolios and the performance fee investors would pay to include the rules in their portfolios. Where trading rules possess superior information about future states of the market, these rules should form a significant part of investors optimal portfolios and investors should be willing to pay a positive fee to incorporate these rules in their portfolios. On the other hand, rules lacking predictive power are unlikely to feature in optimal portfolios or command significant performance fees. This paper also differs from previous rule research, as it focusses on the expected return (and risk) of trading rules, rather than realised returns. Firstly, I reconsider the theoretical moments of trading rule returns. In contrast with Praetz (1976), I show that the mean and variance of a rule s returns depend on its predictive ability as well as the mean and variance of the asset it trades. Greater predictive power results in a higher expected return and lower variance because rule returns become more tightly clustered. Second, I empirically estimate the weight of trading rules in investors optimal portfolios and calculate the performance fee investors would pay to add trading rules on the Dow and SP500 to portfolios comprising only the market index and a riskless asset. The optimal portfolios are estimated using utility maximisation, one of the standard workhorses of the asset allocation literature, and monte carlo simulation. The paper finds that on the basis of gross returns trading rules form a significant weight in optimal portfolios. This appears to be due to the low correlation with index returns and the skewed pattern of rule returns, not superior returns due to predictive ability. However, these benefits do no survive the reality check imposed by transaction costs. After allowing for transaction costs, optimal portfolios rarely include trading rules. These findings, both before and after trading costs, generally persist for several years. This suggests that trading rule returns are consistent with market efficiency. The remainder of the paper is structured as follows. Section 2 reviews the literature on trading rules. Section 3 reconsiders the theoretical meanvariance properties of trading rules and outlines the methodology for estimating optimal portfolios and performance fees. Section 4 details the trading rules tested and Section 5 contains the empirical results. Section 6 concludes. 2 Literature Technical trading rules have a long history in the literature, commencing with the pioneering analysis of Cowles (1933). Alexander (1961) reported that returns on filter rules on the Dow Jones Industrial Average and the SP500 index generally did not exceed the buy-and-hold return after trans- 3

4 action costs. After adjusting for the effects of dividend payments, Fama & Blume (1966) found that such rules applied to the stocks in the Dow between 1957 and 1962 earned substantially less than a buy-and-hold strategy, even before transaction costs. Levy (1967) demonstrated that returns on some relative strength trading strategies exceed buy-and-hold returns, but Jensen & Benington (1970) argues that these results could be due to data mining, as only a few of the large sample of rules tested had been theoretically justified ex ante. Levy (1971) also reported that a wide range chart patterns applied to individual securities on the New York Stock Exchange did not outperform the market return after transaction costs. Jensen (1972) concludes there is no important evidence from this period against the weak form of the efficient markets hypothesis. More recently, Treynor & Ferguson (1985), Brown & Jennings (1989), Grundy & McNichols (1989) and Blume, Easley & O Hara (1994) have developed theoretical models in which it is optimal for investors to follow trading rules due to hetrogeneous beliefs or prices not fully revealing private information. In the models of Brock (1998), Westerhoff (2006), Farmer & Joshi (2002) and Chiarella, He & Hommes (2006) the presence of such trading strategies explain many of stylised facts about equity markets, such as volatility clustering and fat tails in the distribution of returns. Empirical studies in the last two decades also provide some evidence that trading rules may have predictive power for stock prices. Brock, Lakonishok & LeBaron (1992) documented significant differences between mean market returns before transaction costs on days moving average and breakout rules were long and short the Dow Index. Market returns following buy signals were also found to be less volatile than returns following sell signals. Bootstrap simulations showed these results are unlikely to be generated by four stochastic models of market returns. The differences in conditional returns reported by Brock, Lakonishok & LeBaron (1992) have also been found in foreign markets (for example, Ratner & Leal (1999)). Neftci (1991) and Gencay (1998) report that adding moving average trading signals to autoregressive models of stock prices improved the out-of-sample forecasting performance of these models. Lo, Mamaysky & Wang (2000) use non-parametric kernal regressions to recognise technical patterns, such as head-and-shoulders and double tops, and conclude that that several patterns provide incremental information on stock prices. Nam, Washer & Chu (2005) find that trading rule profits on the SP500 relate to asymmetries in the return dynamics of the index, while Kavajecz & Odders-White (2004) note links between trading rules and the depth of the order book. The economic significance of these results, however, is unclear. Allen & Karjalainen (1999) and Hudson et al. (1996) find that after transaction costs trading rules on the SP500 and the United Kingdom s FT 300 do not consistently generate excess returns over a buy-and-hold strategy. Bessembinder & Chan (1998) and Ito (1999) find that after conditioning for the measurement errors caused by non-synchronous trading, rule returns are not significant. 4

5 Ready (1997) documents that taking account of price slippage the price movement between when trading rules generate buy or sell signals and when trades are executed significantly reduces the reported profits of trading rules. Sullivan et al. (1999) partially attribute the significance of rule returns to data mining. Further, few trading rule studies have risk adjusted rule returns. Brown, Goetzmann & Kumar (1998) find that the Dow Theory generated a significantly positive risk adjusted returns over the period when the main exponent of the theory was the editor of the Wall Street Journal. Only a single trading rule, however, was tested and Jensen s alpha is an inconsistent measure of performance when a portfolio s beta is not stationary, as is the case with dynamic trading strategies. 2 Sweeney (1988) reports that filter returns for a small sample of companies generated significant risk adjusted returns, as measured by the X-statistic, provided the rules do not short stocks and are executed by floor traders, who face the lowest transaction costs. But the X-statistic does not reflect reflect a rule s predictive power, as it assumes the expected return on a long rule position is the same as the return on a buy-and-hold strategy. 3 Mean-Variance Properties Praetz (1976) analytically derives the expected mean and variance of trading rules, assuming that changes in prices follow a random walk and that the return on a rule when it is long is the average return on the asset traded by the rule. Under these assumptions the approximate mean return of a rule is E(r ds ) = m(1 2f) (1) where m is the expected mean return of the asset traded by the rule and f is the fraction of days the rule is short the asset. A rule which is always long (short) would have the same (opposite) expected return as a buy-and-hold strategy, while a rule which is long half the time would be expected to earn a zero rate of return. The expected variance of rule returns in this model is not significantly different from the expected variance of the underlying asset. This approach has several limitations. One, it does not take account of a trading rule s predictive power all rules long (or short) the traded asset for the same fraction of time have the same expected return regardless of their predictive ability. Clearly a trading rule which has the ability to persistently predict up and down price movements is more valuable to an investor than a rule which does not and should be reflected in expected rates of return. A trading rule with more predictive power should also have a smaller variance, as its returns will be more tightly clustered than a rule which has no forecasting power. 2 See Jensen (1972), Grant (1977) and Grinblatt & Titman (1989). 5

6 On a day a trading rule is long the underlying asset, undoubtedly it earns the same realised return as a buy-and-hold strategy on that day. This does not mean, however, that the expected return on the trading rule which is long must equal that of the buy-and-hold strategy. A trading rule buys or sells an asset, because it is implicitly forecasting that the price of the asset will rise or fall respectively. Where a trading rule has a perfect track record of predicting up and down movements, its expected return on days it is long (short) would be that of (opposite of) a buy-and-hold strategy. Conversely, a rule which always wrongly forecasts price movements will have an expected return opposite to the buy-and-hold strategy when the rule is long and the same as the buy-and-hold strategy when it is short. Rules with a 50:50 track record, hereafter described as having zero predictive power, should earn approximately the risk free rate, because forecasts of these rules are likely to be as wrong as right. Where a rule has a track record better or worse than 50:50, which I call positive or negative forecasting power, its return should lie between zero and the maximum and minimum noted above. A model of the moments of trading rule returns which formalises these insights is developed below. This model is deliberately kept simple to highlight the key influences on the moments of rule returns. The model encompasses a broad range of return generating processes commonly used to model market returns, including autocorrrelated, generalised autoregressive conditional hetroscedasticity (GARCH), stochastic volatility and Levy processes. Assume trading rules may switch position at the end of each day and the investor s investment horizon is also one day that is, the rule s decision interval is the same as the investor s evaluation interval. Investors are also assumed to be rational, in that they formulate the expected return of a asset as the probability weighted average of all possible return outcomes for that asset. Under these conditions, the expected return on the buy-and-hold strategy is E(r b ) = r a f(r a )dr a, (2) where f(r a )is the density function of the asset s returns. A trading rule is effectively a specialised portfolio which takes one of two positions based on the predicted direction of asset prices: it is either 100 per cent long when asset prices are forecast to rise, or 100 per cent short when prices are expected to fall. Recall that the realised return on a portfolio comprising a single risky asset and a risk free asset is r p,t = (1 α t )r f + α t r a,t = r f + α t (r a,t r f ), (3) 6

7 where r f is the risk free rate, r a,t is the return on the risky asset and α t is the fraction of the portfolio invested in the risky asset. Accordingly, the return on a trading rule which is long is r a and 2r f r a where it is short. Some trading rules start from a neutral position, with the funds invested in the trading rule placed in the riskless asset until the rule initiates its first long or short position. After this initial period these rules behave in the same way as the rules outlined earlier, switching between only long and short positions, and can be studied with the same model. The forecasting ability of trading rules is summarised by two measures: p +, the probability a rule positions itself long conditional on an upward moves in price; and p, the probability it shorts the market conditional on the price declining. These measures are interpreted in this paper as the ability of a rule to successfully predict up and downwards price movements respectively. The accuracy of forecasts is also assumed to be independent of the expected magnitude of price movements. For possible positive asset returns, the rule s expected return is the weighted sum of payoffs from two states. The rule correctly anticipates the price rise and is long, or the forecast is wrong and the rule is short; the weights are p + and 1 p + respectively. Similarly, for each possible negative asset return the expected rule return is the weighted sum of the return earned where the rule successfully forecasts the market will fall and is short and the return generated from being long in a falling market (with weights p and 1 p ). The unconditional expected return of a dynamic trading strategy can be written as: E(r ds ) = p + r a,t + (1 p + )(2r f r a,t ) f(r a )dr a p (2r f r a,t ) + (1 p )r a,t f(r a )dr a =r f + (2p + 1) + (1 2p ) 0 0 (r a r f ) f(r a )dr a (r a r f ) f(r a )dr a (4) A rule s expected return depends on its predictive power and the density function for returns of the underlying asset. Where a trading rule has 50:50 predictive power (p + = p = 0.5) it would be expected to return the risk free rate. This is because, on average, it would earn the market premium on the underlying asset as often it would forego the premium by being short. Trading rules with perfect predictive power (p + = p = 1) would, of course, 7

8 earn the greatest return, while rules which are always wrong (p + = p = 0) earn the least. The expected return on a rule with some positive or negative predictive power (p + = p > 0.5 or p + = p < 0.5) lies between r f and the maximum and minimum noted above. Where a rule is always long (p + = 1 and p = 0) equation (4) collapses to the expected rate of return on the traded asset. If the expected equilibrium rate of return on the underlying asset is positive, as is commonly required by asset pricing models, the ability to forecast up markets is more valuable than the ability to forecast market declines. 3 To put these theoretical results into economic perspective, the mean monthly return of the Dow Jones Industrial Index in the decade to 2006 was 0.45 per cent, the return on a rule with perfect forecasting power would have averaged about 20 per cent per month, while a rule that was always wrong would have lost 0.5 per cent each month. The expected variance of the buy-and-hold strategy is: σ 2 b = E(r 2 a) E(r a ) 2 = ra 2 f(r a )dr a E(r a ) 2. (5) The expected variance of a rule s returns is: σ 2 ds =E(r 2 ds) E(r ds ) 2 = (p + ra,t 2 + (1 p + )(2r f r a,t ) 2 ) f(r a )dr a (p (2r f r a,t ) 2 + (1 p )r 2 a,t) f(r a )dr a E(r ds ) 2 = ra,t 2 f(r a )dr a + (1 p + ) 4r f (r f r a,t ) f(r a )dr a + p 0 0 4r f (r f r a,t ) f(r a )dr a E(r ds ) 2. (6) This shows that the variance of a trading rule s returns depends on the rule s predictive power, in addition to the variance of the underlying asset traded by the rule. Rules which always incorrectly predict market direction (p + = p = 0) have the highest variance, while rules with a perfect fore- 3 Because 0 r a f(r a )dr a > 0 r a f(r a )dr a. 8

9 casting track record (p + = p = 1) have the lowest variance. Rules with a 50:50 track record would have a variance similar to that of the underlying asset, because the second and third terms in equation (6) would largely offset. Where a rule is always long, its variance is the same as the variance of the underlying asset. Skewness, the third moment, is a measure of the asymmetry of the distribution of returns. The skewness of buy-and-hold returns is κ = (r a µ a ) 3 f(r a )dr a = r 3 a 3µ a r 2 a + 3µ 2 ar a µ 3 a f(r a )dr a, (7) where µ a is the expected return on the asset and depends on the distribution assumed. Symmetrical distributions such as the normal and t distributions have a skew of zero, while the log normal distribution is positively skewed. In comparison, the skew of a rule s returns is κ ds = (r ds µ ds ) 3 f(r a )dr a + 0 (r ds µ ds ) 3 f(r a )dr a 0 = (2p + 1)ra 3 3µ ds ra 2 + 3(2p + 1)µ 2 dsr a µ 3 ds f(r a )dr a (1 2p )r 3 a 3µ ds r 2 a + 3(1 2p )µ 2 dsr a µ 3 ds f(r a )dr a. (8) Returns of trading rules with zero predictive power are symmetric and not skewed, even where the return distribution of the underlying asset is itself skewed, as equation (8) collapses to zero when p + = p = 0.5. Trading rules with positive predictive power (p + = p > 0.5) will be positively skewed, because these rules generate positive returns (by being short) when the asset price declines, attenuating the left hand tail of the return distribution. As predictive power increases, this attenuation increases. The return distribution of rules with perfect predictive power in both up and down markets (p + = p = 1) would be left truncated at zero. Rules with less than a 50:50 track record will be negatively skewed. Trading rules which are always long (or short) have the same skew (opposite skew) as a buy-and-hold strategy. This can be verified by substituting p + = 1 and p = 0 or p + = 0 and p = 1 into equation (8). The correlation between a trading rule s returns and the returns on the 9

10 asset traded is ρ ds,b = σ ds,b σ ds σ b. (9) where σ ds,b =E(r ds r a ) E(r ds )E(r a ) = (p + r a,t + (1 p + )(2r f r a,t ))r a,t f(r a )dr a (p (2r f r a,t ) + (1 p )r a,t )r a f(r a )dr a E(r ds )E(r a ) = (2p + 1)ra 2 + (1 p + )2r f r a f(r a )dr a (1 2p )r 2 a + p 2r f r a f(r a )dr a E(r ds )E(r a ) (10) A rule s returns are perfectly correlated (negatively correlated) with that of the underlying asset only when the rule is always long (short). Trading rules with a 50:50 record will have a zero correlation with the buy-andhold strategy. Where a rule has positive or negative forecasting power its correlation with the underlying asset will be biased towards zero, as the first two integrals in the numerator in equation (9) will tend to offset each other because they would have different signs. This model extends to the case where the investment horizon is longer than the decision interval of the trading rule: for example, the trading rule take daily positions, but the investor is interested in monthly returns. 3.1 Optimal Portfolios Assume investors maximise the utility of expected wealth at the end of the investment period by allocating their portfolios at the start of the period between a riskless asset and a single risky asset, a market index, with expected returns of r f and r m,t respectively. Investors may not short sell the market index or borrow to invest in it. The portfolio return over the period, t, is: r p,t = (1 α t )r f + α t r m,t = r f + α t (r m,t r f ) (11) where 0 α t 1 is the fraction of the portfolio invested in the market index. 10

11 Investors are assumed to have power utility preferences. The expected utility of wealth is: E[U(W t+1 )] = E(W )1 γ t γ (12) where γ represents the degree of relative risk aversion. As γ approaches 1 the limit of this equation is logrithmic utility, E[U(W t+1 ]) = log(w t+1 ). Power utility has the attractive property that the level of absolute risk aversion declines as wealth increases. What seems to be a large gamble for someone with little wealth would probably appear less significant for a wealthy investor (Campbell & Viceira (2002)). Arbitrarily setting an investor s starting wealth to 1 and substituting equation (11) into equation (12) gives: E[U(W t+1 )] = (1 + r f + α t (r m,t r f )) 1 γ 1 1 γ (13) Where an investor allocates their portfolio across a risk free asset, a market index and a trading rule on the index, the approach is similar. Investors may neither short either risky asset, or borrow to invest in them. This means the sum of long positions in the market index and the trading rule can not exceed 100 per cent of the value of the portfolio. The return on the two risky assets is represented by a 2 1 vector R t, α t is an 2 1 vector containing the weights for the market index and the trading rule, Y is a conforming vector of 1s and the portfolio return is: The utility function is now: r p,t = (1 α ty )r f + α tr t = r f + α t(r t r f Y ) (14) E[U(W t+1 )] = (1 + r f + α t(r t r f Y )) 1 γ 1 1 γ (15) The utility functions in (13) and (15)would ususally be approximated with mean-variance analysis, or estimated by assuming a specific density function and analytically evaluating the resulting integral. Neither approach is suitable for trading rules. Mean-variance is a good approximation only where returns are not too non-normal. As noted in Section 3, the returns of rules with predictive power are skewed. Multivariate skewed density functions are complex to evaluate analytically. Instead, this paper uses monte carlo simulation to estimate the utility 11

12 functions. This is done as follows. For each annual investment period in the sample, a GARCH model of daily market returns is estimated from the 3 years of returns immediately preceeding the investment period. This model is used to simulate 750 daily asset returns series of 410 trading days each 250 days for the annual investment period itself, plus an earlier 160 trading days to provide the necessary back data for the trading rules. The simulations combine the GARCH model s coefficients with its re-scrambled standardised residuals to generate the daily index returns. Each trading rule is then run over each of the simulated series of index returns. This results in 750 simulated market and rules returns, approximating their joint distribution. A numerical algorithm is then used to estimate the optimal portfolio by finding the weight which maximises average utility across the 750 simulated return paths. Optimal portfolios and performance fees are estimated for γ = 2, the level of risk aversion which corresponds with average index holdings of 50 per cent in the optimal portfolio. 3.2 Performance Fees The economic benefit of including trading rules in investors optimal portfolios is estimated with a utility based performance fee methodology similar to West, Edison & Cho (1993) and Fleming, Kirby & Ostdiek (2001). The annualised performance fee is the charge which equates the utility of the optimal portfolio which includes the technical trading rule with the utility of the optimal portfolio which contains only the risk free asset and market index over the investment horizon. It represents the maximum amount an investor with power preferences would pay to include the rule in their portfolio relative to the alternative of holding a portfolio comprising only the riskless asset and the market index. Performance fees are always zero or greater as, by construction, the optimal portfolio including trading rules would never have a lower expected utility than the optimal portfolio comprising only the risk free rate and the index. 4 Trading Rules and Data This paper examines 4 types of trading rules - moving averages, filter rules, trading range breakouts and the relative strength indicator. 4.1 Moving Averages This rule buys (sells) stock when the first moving average based on a short rolling window of observations is above (below) a second moving average based on a longer window of observations by more than a specified amount, the filter percentage f. This trading rule has been tested extensively and Brock, Lakonishok & LeBaron (1992), Bessembinder and Chan (1997) and 12

13 Ito (1998) document its predictive power. The investor s position is updated as follows: S t = { 1 if MA s,t 1 > MA l,t 1 (1 + f), 1 if MA l,t 1 < MA s,t 1 (1 f). (16) where MA s,t and MA l,t are moving averages based on s and l observations at time t. The moving averages are calculated as follows. MA k,i = k i=1 P t+1 k k (17) where P t is the closing market price on day t, and k is the length of the moving average window. Several combinations are tested, along with different filters. Increasing the filter size reduces the risk of trading whiplash price movements, where a price move is immediately reversed. 4.2 Filter Rules Filter rules are based on the belief that when a stock s price moves significantly up or down, it likely to continue moving in that direction. The rule starts holding the risk free asset. If the closing price on a day is a least f per cent higher (lower) than the previous day s closing price, the investor buys (sells) stock and establishes a long (short) position. This position is held until the stock price moves down (up) by f per cent from the subsequent high (low) price. That is: 1 if S t 1 = 0 and P t 1 > P t 1 (1 + f), 1 if S t 1 = 0 and P t 1 < P t 1 (1 f), S t = 1 if S t 1 = 1 and P t 1 < min(p t 2, P t 3,..., P t k )(1 f), 1 if S t 1 = 1 and P t 1 > max(p t 2, P t 3,..., P t k )(1 + f), S t 1 otherwise. (18) where k is the number of days since the original position was established. Filters of 0.5, 1, 1.5, 2 and 5 per cent are tested. Rules with smaller filter sizes are usually triggered by smaller price movements, which means they change position more frequently but are subject to greater transaction costs. Previous studies of filter rules include Alexander (1964), Fama & Blume (1966) and Brock, Lakonishok & LeBaron (1992). 13

14 4.3 Trading Range Breakouts Trading range breakout rules buy (sell) stock when a a resistance (support) level has been penetrated. The rationale for this rule is that once a stock s price break key levels, known as resistance or support, it is expected to continue rising or falling as more buyers or sellers are subsequently drawn into the market. One reason this may occur in practice is that share traders often set stop loss orders above perceived resistance levels or below supposed support levels, to protect themselves against losses should prices break out of these perceived ranges. For example, a share trader who has sold a stock and wants to limit losses from a price rise could place a stop loss order to buy back the stock just above a perceived resistance level. If the stock price rises above this level, the stop loss order would be executed and the stock bought back. While this would cap the trader s losses, it adds to buying pressure in the market, possibly pushing the stock price higher. Practioners usually determine support and resistance rules from local lows and highs in price. If the closing price is above (below) the highest (lowest) closing price over the previous k days by more than f percentage, the filter amount, this is interpreted as penetration of a resistance (support) level. Accordingly, investors following this rule would buy (sell) stock and establish a long (short) position. 1 if P t 1 > max(p t 2, P t 3,..., P t k )(1 + f), S t = 1 if P t 1 < min(p t 2, P t 3,..., P t k )(1 f), otherwise. S t 1 (19) Periods of 5, 10, 25 and 50 days are used establish local highs and lows and filter amounts of 0.5, 1, 5 and 10 per cent are also tested. 4.4 Relative Strength Indicator The Relative Strength Indicator (RSI) is the smoothed ratio of average absolute recent upward and downward price movements, normalised so the result lies between 0 and 100 (Wilder (1978)). If daily price movements over the window used to calculate the indicator are all upwards, the RSI approaches 100; when the movements are all downwards it approaches 0. RSI theory holds that when a stock s price moves significantly up or down in a short time, the market is overextended in that direction and the price is likely to subsequently retrace in the opposite direction. Thus, when the RSI is approaches 0 or 100, this is interpreted as a signal that investors should buy or sell stock in anticipation that its price will rise or decline. The arithmetic of the RSI is: 14

15 RSI k,t = Ut D t (20) k max( P t, 0) U 0 = k (21) D 0 = i=1 k i=1 min( P t, 0) k U t = U t 1k + max( P t, 0) k D t = D t 1k + min( P t, 0) k (22) (23) (24) where P t is the change in price on day t and k is the number of observations in the window used to estimate the indicator. The rule changes position as follows: 1 if RSI t 1 < RSI buy S t = 1 if RSI t 1 > RSI sell, otherwise. S t 1 (25) where RSI buy and RSI sell are the levels at which the asset are considered overbought and oversold respectively. RSIs with windows of 10, 20 and 35 days are tested; the longer the window of observations the less sensitive the RSI is to price movements. Because technicians differ as to the levels which are considered overbought and oversold the combinations 70-30, and are tested. A wider range require larger price movements to trigger buy or sell signals. 4.5 Sample Data These technical trading rules are applied to two sets of data. Dow Index from 1 January 1897 to 31 December This is the longest index series available for US equity markets and has been used in other major studies (Brock, Lakonishok & LeBaron (1992), Allen & Karjalainen (1999) and Lo, Mamaysky & Wang (2000). However, the Dow is a price index which reflects only capital gains and losses from holding the underlying stocks; it does not include the return from dividends. SP500 total return index from 1 January 1988, the first date from which this index was available on a daily basis, to 31 December It includes dividends. 15

16 After starting from a neutral position, rule buy or sell the index in accordance with trading signal generated by the rule. On a buy signal, 100 per cent of the value of the rule s wealth is invested in the index. On a sell signal, an amount of the index equivalent to the rule s current wealth is sold and the proceeds invested in the risk free asset. Trading rules are assumed to change position at closing price which generated the trading signal - that is, there is no price slippage. The return of the index each day is calculated as the simple percentage change in price. where where P t it is the closing index level on t. Risk Free Rate R index,t = (P t P t 1 ) P t 1 1 (26) The risk free rate between 1967 and 2006 is proxied with the US 3 month Treasury Bill rate obtained from the Federal Reserve Economic Database (FRED). Earlier risk free rates are based on commercial paper rates from Homer & Sylla (2005). Transaction Costs Dynamic trading strategies face two main transaction costs: the commission payable to brokers for executing buy and sell orders; and the bid-offer spread. The rules tested are assumed to be implemented by institutional investors. A one-way cost of 0.23 per cent is used for transactions after 1 May 1975 (Berkowitz, Logue & Noser (1988)), when fixed commissions were abolished. This is consistent with the sum of estimates by Chan & Lakonishok (1993) and Knez & Ready (1996) for commission costs and bid-offer spread for institutional investors. Prior to the deregulation of commissions, a one-way cost of 1.3 per cent based on Stoll & Whaley (1983) is used. Other studies have found that effective transaction costs may be higher (Bhardwaj & Brooks (1992) and Lesmond, Ogden & Trzcinka (1999)). Transaction costs for floor traders are naturally lower than for institutional traders, while costs for retail investors would be greater (Sweeney (1988)). 5 Empirical Results 5.1 Preliminary Analysis Summary statistics for the daily returns on the Dow Jones Industrial Average and the SP500 Total Return Index are shown in Table 1. 16

17 Table 1: Summary Statistics for Market Indices Dow Jones SP500 Total Index Return Index 1 January April December April 2007 Observations 27,550 4,968 Mean Variance Maximum Minimum Skew Kurtosis ρ(1) ρ(2) ρ(3) ρ(4) ρ(5) Rule Returns Moving average rules Table 2 shows the results for moving average rules applied to the Dow. There are several notable features about these results. Moving average rules are generally better at positioning correctly for up rather than down movements in the Dow. The fraction of days on which rules are long when the index increases averages 56 per cent across all moving average rules. In contrast, the expected fraction of days rules are short conditional on the Dow falling is 43 per cent, significantly below the 50 per cent benchmark if a rule randomly selected to be long or short. While rules had expected positive returns before transaction costs, on average, across the 103 annual investment periods examined, few rules had expected returns which exceed the annual expected return on the Dow. The average variance of these returns for each rule was also greater than the expected volatility in annual Dow returns over the sample. Nonetheless, each rule s weight in the optimal portfolio was positive and the standard T test indicates these weights are significantly different from zero. This seems to be driven by the diversification benefits provided by these investment strategies and the skewed return profile, not superior returns to the index. As predicted by the model developed in Section 3, the correlation between a rule s expected returns and the expected return on the Down was, ranging from 0.x to 0.y. Expected rule returns also consistently positively skewed. 17

18 Once account is taken of transaction costs incurred by actively switching between long and short positions, however, the results change markedly. As shown in the second and third last columns in Table 2, the average expected return of all moving average rules is negative after allowing for these costs, and the optimal weight of each rules declines to zero. This is despite the correlation between rule returns and the Dow remaining low. Filter Rules Filter rules with small filter sizes have slightly positive predictive power in up markets, but this predictive power generally declines as the size of the filter amount increases. As with moving average rules, expected filter returns are positively skewed and exhibit low correlation with expected Dow returns. The imposition of transaction costs also significantly reduces rule returns and largely eliminates these rules from investors optimal portfolios. Breakout Rules The pattern of predictive power and pre and post-cost returns is similar to moving averages and filter rules. Relative Strength Rules Unlike the other trading rules tested, RSI rules have better ability to position correctly in falling markets than rising markets. Whereas moving average, filter and breakout rules are short about only 40 per cent of days on which the index declines, RSI rules are short about 60 per cent of days. As with other rules, the gross return gernerated by RSI rules rarely exceeds the expected return on the Dow and the volatility of rule returns is higher. Notwithstanding this, RSI rules form a non-trivial part of optimal portfolios, likely because of the low correlation between rule and index returns. Transaction costs eliminate these rule advantages and post-cost returns are low or negative. Not surprising, RSI rules post costs do not figure in optimal portfolios. 18

19 Table 2: Results for Moving Average Rules on Dow Index The results for each rule in the table (except for T statistics) are the average of the expected outcomes in each of the 103 annual investment periods ( , ). The expected outcomes for a rule in each annual investment period was obtained by monte carlo, averaging the results from 750 simulated asset return series. Asset returns in each investment period were simulated using the parameters of a GARCH model estimated from 3 years of asset returns immediately preceeding the investment period, combined with the rescrambled standardised residuals from the model. Expected rule results in each period were estimated by averaging the results from applying the trading rule to the 750 simulated assetprice series implied by the simulated return series. Rule parameters are shown in the first column. The first parameter is the days in the shorter moving average, the second parameter is the days in the second moving average and the third is the filter amount. Frac Up and Long is the fraction of days on which the asset price is expected to rise that the trading rule is long. Frac Down and Short is the fraction of days on which the asset price is expected to decline that the rule is short. Return is the annual expected percentage return. Variance is the variance of a rule s annual expected returns. Skew is a measure of asymmetry in the expected returns on a rule. Correl is the correlation between a rule s expected returns and the expected return of the asset traded by the rule. Weight is the weight (in percent) of a rule in the optimal portfolio of an investor with power utility preferences (γ = 2). T statistics are shown in parentheses. Before transaction costs After transaction costs Parameters Frac up Frac down Return Variance Skew Correl Weight Return Weight and long and short 1, 5, (8.25) (-9.45) (19.57) (6.68) (4.79) (6.61) (0.53) (NaN) 1, 5, (7.83) (-9.76) (19.95) (7.45) (6.41) (6.53) (0.54) (NaN) 1, 5, (4.70) (-10.36) (19.60) (6.07) (5.53) (7.12) (0.53) (1.34) 3, 15, (7.30) (-8.49) (16.38) (5.50) (3.86) (7.51) (0.55) (1.34) 3, 15, (7.14) (-8.80) (16.74) (6.12) (4.18) (7.19) (0.55) (1.62) 3, 15, (6.50) (-9.47) (17.07) (6.03) (4.26) (7.20) (0.55) (2.05) 5, 30, (6.83) (-8.06) (13.33) (4.61) (3.87) (6.84) (0.56) (2.50) 5, 30, (6.70) (-8.22) (13.57) (4.66) (4.10) (6.92) (0.56) (2.80) 5, 30, (6.36) (-8.66) (13.70) (4.70) (4.40) (6.92) (0.56) (2.64) 15, 150, (5.87) (-6.98) (10.26) (4.67) (4.74) (6.58) (0.60) (3.35) 15, 150, (5.80) (-6.99) (10.26) (4.66) (4.81) (6.47) (0.60) (3.59) 15, 150, (5.68) (-7.11) (10.31) (4.69) (4.85) (6.56) (0.59) (3.86) 19

20 Table 3: Results for Filter Rules on Dow Index The results for each rule in the table (except for T statistics) are the average of the expected outcomes in each of the 103 annual investment periods ( , ). The expected outcomes for a rule in each annual investment period was obtained by monte carlo, averaging the results from 750 simulated asset return series. Asset returns in each investment period were simulated using the parameters of a GARCH model estimated from 3 years of asset returns immediately preceeding the investment period, combined with the rescrambled standardised residuals from the model. Expected rule results in each period were estimated by averaging the results from applying the trading rule to the 750 simulated assetprice series implied by the simulated return series. The rule parameter is the filter amount (in percent). Frac Up and Long is the fraction of days on which the asset price is expected to rise that the trading rule is long. Frac Down and Short is the fraction of days on which the asset price is expected to decline that the rule is short. Return is the annual expected percentage return. Variance is the variance of a rule s annual expected returns. Skew is a measure of asymmetry in the expected returns on a rule. Correl is the correlation between a rule s expected returns and the expected return of the asset traded by the rule. Weight is the weight (in percent) of a rule in the optimal portfolio of an investor with power utility preferences (γ = 2). T statistics are shown in parentheses. Before transaction costs After transaction costs Parameters Frac up Frac down Return Variance Skew Correl Weight Return Weight and long and short (9.15) (-11.19) (19.30) (5.15) (9.42) (5.64) (0.54) (NaN) (7.37) (-12.40) (19.46) (5.55) (9.48) (6.31) (0.53) (NaN) (-1.07) (-13.77) (20.03) (5.21) (9.36) (6.39) (0.49) (NaN) (-7.69) (-15.66) (19.60) (6.91) (8.83) (6.30) (0.41) (1.00) (-12.46) (-18.43) (18.34) (7.96) (8.77) (5.40) (0.32) (1.00) (-17.69) (-22.75) (17.97) (3.29) (8.35) (5.20) (0.24) (1.14) NaN NaN (-43.57) (-44.91) (15.92) (3.35) (4.00) (5.50) (0.08) (3.09) 20

21 Table 4: Results for Breakout Rules on Dow Index The results for each rule in the table (except for T statistics) are the average of the expected outcomes in each of the 103 annual investment periods ( , ). The expected outcomes for a rule in each annual investment period was obtained by monte carlo, averaging the results from 750 simulated asset return series. Asset returns in each investment period were simulated using the parameters of a GARCH model estimated from 3 years of asset returns immediately preceeding the investment period, combined with the rescrambled standardised residuals from the model. Expected rule results in each period were estimated by averaging the results from applying the trading rule to the 750 simulated assetprice series implied by the simulated return series. Rule parameters are shown in the first column. The first parameter is the days in the window used to estimate local price highs and lows. Frac Up and Long is the fraction of days on which the asset price is expected to rise that the trading rule is long. Frac Down and Short is the fraction of days on which the asset price is expected to decline that the rule is short. Return is the annual expected percentage return. Variance is the variance of a rule s annual expected returns. Skew is a measure of asymmetry in the expected returns on a rule. Correl is the correlation between a rule s expected returns and the expected return of the asset traded by the rule. Weight is the weight (in percent) of a rule in the optimal portfolio of an investor with power utility preferences (γ = 2). T statistics are shown in parentheses. Before transaction costs After transaction costs Parameters Frac up Frac down Return Variance Skew Correl Weight Return Weight and long and short 10, (6.32) (-8.93) (15.38) (4.52) (3.08) (7.83) (0.54) (1.78) 10, (5.17) (-9.51) (14.84) (4.68) (3.40) (7.39) (0.54) (2.11) 10, (-2.12) (-10.13) (13.83) (4.87) (2.40) (6.74) (0.48) (2.56) 10, (-19.02) (-12.52) (10.32) (5.06) (-2.25) (5.53) (0.24) (3.49) 20, (5.49) (-9.02) (13.82) (5.00) (3.63) (7.01) (0.55) (2.87) 20, (4.22) (-9.63) (14.02) (4.90) (3.89) (7.28) (0.54) (2.79) 20, (-2.69) (-10.86) (13.74) (4.97) (2.98) (6.58) (0.47) (2.95) 20, (-22.04) (-14.74) (9.92) (5.00) (-1.52) (5.76) (0.22) (3.96) 50, (4.07) (-9.47) (12.59) (4.86) (4.21) (7.27) (0.55) (3.46) 50, (2.71) (-10.30) (12.58) (4.76) (4.25) (7.09) (0.53) (3.42) 50, (-3.82) (-12.31) (11.96) (4.87) (3.51) (6.88) (0.45) (3.68) 50, (-26.71) (-18.59) (9.04) (5.49) (-0.35) (5.86) (0.18) (4.64) 21

22 Table 5: Results for Relative Strength Rules on Dow Index The results for each rule in the table (except for T statistics) are the average of the expected outcomes in each of the 103 annual investment periods ( , ). The expected outcomes for a rule in each annual investment period was obtained by monte carlo, averaging the results from 750 simulated asset return series. Asset returns in each investment period were simulated using the parameters of a GARCH model estimated from 3 years of asset returns immediately preceeding the investment period, combined with the rescrambled standardised residuals from the model. Expected rule results in each period were estimated by averaging the results from applying the trading rule to the 750 simulated assetprice series implied by the simulated return series. Rule parameters are shown in the first column. The first and second parameters are the oversold and overbought levels at which the rule buys and sells the index and the third parameter is the days in the period. amount. Frac Up and Long is the fraction of days on which the asset price is expected to rise that the trading rule is long. Frac Down and Short is the fraction of days on which the asset price is expected to decline that the rule is short. Return is the annual expected percentage return. Variance is the variance of a rule s annual expected returns. Skew is a measure of asymmetry in the expected returns on a rule. Correl is the correlation between a rule s expected returns and the expected return of the asset traded by the rule. Weight is the weight (in percent) of a rule in the optimal portfolio of an investor with power utility preferences (γ = 2). T statistics are shown in parentheses. Before transaction costs After transaction costs Parameters Frac up Frac down Return Variance Skew Correl Weight Return Weight and long and short 20, 80, (-8.41) (9.92) (4.22) (3.33) (-7.06) (3.92) (0.40) (2.29) 20, 80, (-8.74) (9.83) (8.94) (4.27) (-10.64) (5.22) (0.44) (2.87) 20, 80, (-8.50) (9.37) (10.21) (4.84) (-11.35) (5.93) (0.45) (3.22) 25, 75, (-7.29) (8.63) (5.21) (6.16) (-3.22) (3.51) (0.44) (NaN) 25, 75, (-7.68) (9.16) (3.78) (3.82) (-5.26) (3.86) (0.41) (2.12) 25, 75, (-8.14) (9.51) (5.55) (3.89) (-7.91) (4.78) (0.41) (2.93) 25, 75, (-8.35) (9.48) (9.09) (5.53) (-9.84) (5.23) (0.44) (3.08) 30, 70, (-7.03) (8.29) (5.75) (4.69) (-2.32) (3.88) (0.45) (NaN) 30, 70, (-6.95) (8.34) (4.73) (3.61) (-4.50) (2.94) (0.43) (1.51) 30, 70, (-7.48) (8.98) (3.13) (2.42) (-6.01) (3.96) (0.41) (2.52) 30, 70, (-7.90) (9.29) (5.80) (6.61) (-7.75) (5.16) (0.42) (3.08) 22