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Quantitative trading The moving average is considered to be an effective tool to reveal trends. It ostensibly removes noise and smoothes underlying data by averaging a chosen number of observations. It is employed in various market forecasting models and trading strategies which exploit well-documented low-order serial correlation and nonlinearity of return series. Empirical evidence from a number of developed and emerging equity markets indicates that moving average models can predict reversals in price and yield statistically and economically significant excess returns over a buy-and-hold strategy, and that this cannot as yet be attributed to measurement errors arising from the use of non-synchronous trading data or the omission of dividend yields.1 Surprisingly, genetic algorithms that simultaneously search for optimal trading strategies typically produce models similar to moving averages.2 This study evaluates the performance of ten double crossover moving average strategies with and without a trading filter band on the Toronto Stock Exchange S&P/TSX Composite Index.3While such a study is of obvious relevance to individual investors, it does have value for institutional investors who, regardless of whether they hold broad market-based investments, either benchmark against a market index or allocate capital to individual investments of varying correlation with the market. The strategy is simple. A long position is held when the short-period average price is greater than the long period average price, and a short position is held when the opposite is true. A filter band is a predefined fixed percentage around the long moving average that the short moving average must pass before the investment position is changed. This presumably filters out weak signals. These moving average strategies can be summarized by MA (p, q, r) model, where p is the length of the short period, q is the length of the long period, and r is the size of the filter band. So, for example, MA (5, 150, 0.01) uses five days to calculate the short moving average, 150 days for the long, and imposes a filter band of 1% above and below the long moving average. To avoid selection biases, this study uses the same parameters as in the seminal study on the topic by Brock, Lakonishok and LeBaron (1992).4 Ito (2003) has studied moving average strategies on TSE 300 data from 1977 to 1995. Here, the same sample is employed for a number of analyses of what will be referred to as the insample. Eight more years of data, 1996 to 2003, were collected and used as an out-of-sample period for comparison purposes. The predictive power of the strategies was evaluated by testing the trading signals for randomness because non-random signals have the potential to be exploited for excess returns. The economic performance of the strategies was then studied by comparing both simple returns and total returns, adjusted for risk and transaction costs, with the return on a buy-and-hold position. Finally, a Consistency Check of the strategies was performed to see whether they maintained their performance ranking over time. Did the best remain the best and the worst the worst? The analysis reveals that the strategies generate nonrandom signals. Despite the large qualitative differences between sample periods, the average holding period of a position and the proportion of long to short positions remain constant. All of the strategies yield excess returns over buy-and-hold, and none of them is riskier than the passive approach. The best performing models have negative betas. The results are largely consistent over various comparison criteria, and are robust to twoway transaction costs of 25 basis points. A rank order test of the models’ risk-adjusted performance points to a degree of forecast reliability. There is also a positive and significant relationship between conditional returns and unconditional variance, which implies higher profits in more volatile periods.5 Previous Research Although Brock, Lakonishok and LeBaron do not find differences in trading model performance across sub-samples, replications of their work by others suggest weakening profits in more recent periods. Sullivan, Timmermann, and White (1999), for instance, document that the best model from 1897 to 1986 does not repeat its superior performance in the out-of-sample period 1987 to 1996. In fact, none of the models is able to maintain its profitability. They offer three possible explanations: that the out-of-sample period is not representative; that the trading models themselves are no longer representative; and that markets have become more efficient. They ultimately favour the third, citing low-cost computing power, lower transaction costs, and increased liquidity as possible causes of heightened market efficiency. Ready (2002) reports similar findings and speculates that arbitrage activities on Wall Street have minimized trading models, return opportunities. Employing data from 1962 to 1996 for NYSE and from 1973 to 1996 for NASDAQ, Kwon and Kish (2002) find weakening profits from trading models and suggest that markets are becoming more efficient in disseminating information to a wider range of investors. Moving average strategies are, however, economically effective where trading costs are low. Isakov and Hollistein (1999) find that investors who face two-way transaction costs of less than 0.66% can benefit on the Swiss market (SBC Index, 1969-1997) and conclude that, for a large fraction of participants, the weak form of market efficiency cannot be rejected. Two separate studies employing large samples, 1935 to 1994, for the FT30 [Hudson, Dempsey and Keasey (1996)] and 1926 to 1991 for the DJIA [Bessembinder and Chan (1998)], estimate identical average break-even two-way transaction costs of 0.8% and 0.78%, respectively. In contrast, Ito (1999) reports an average break-even cost of 1.58% for Canada from 1980 to 1996. The breakeven levels estimated in this study are much lower for roughly the same period, whereas the differences resulting from data samples and trading strategies cannot explain the gap. It appears that Ito did not distinguish between buy-sell differences and actual returns in the 1999 study, as he does in a later study. Data and methodology Returns were calculated as differences of the natural logarithms of closing prices, and annual returns, where used, were estimated using continuous compounding for an average of 252 trading days. Long, short and out-of-market positions are initiated following buy, sell and neutral signals, respectively. A long position yields a return equal to the buy-and-hold; a short position yields a return equal to the negative of the buy-andhold return; and an out-of-market position yields the risk-free rate. The cost of borrowing to sell short was implemented conservatively: it included the dividend yield and the risk-free rate. This method is the least biased when compared to other approaches, such as replacing short positions with out-of-market positions [Ready (2002)], which assumes equality between negative risk premia and the risk-free rate, or a double-or-out strategy [Bessembinder and Chan (1998)], which assumes equality between long and short position returns. Recognizing the non-normality of equity returns, a set of distribution-free tests, the Chi-square, Mood’s median and Wilcoxon’s rank-sum test, were used to test hypotheses about the randomness of buy and sell signals. The premise is that the process of generating buy and sell signal return series is equivalent to a sampling procedure with the MA (p, q, r) model as the sample selection method and the buy-and-hold return series as the underlying population. As long as the selection method is random, buy and sell sampling distributions should approximate the population distribution; otherwise, the model possesses some predictive power. The economic performance was then examined by adjusting the moving average returns for risk and transaction costs. The notion of risk-adjusted returns is fundamental for the precise evaluation of trading strategies, because it demands comparison of returns at equal levels of risk. This study implemented it using the standard Sharpe ratio (1966) and Jensen measure (1968). Subtracting 25 basis points for every two-way transaction accounts for the trading costs presumably borne by institutional investors. Oneway transaction costs were applied only for trading models with filter bands when there was a shift from a neutral to a long or short position or vice versa. Transaction costs were subtracted from the conditional returns on the day they were incurred. The analysis also includes an estimation of the break-even two-way transaction cost across all trading models. Finally, I asked whether there is a significant difference across moving average strategies and, most importantly, whether there is a degree of forecasting reliability. To do this, the trading models were first ranked according to their Sharpe ratios. Then a median exact test for differences across k independent samples of the returns was performed. To estimate the exact significance with 99% confidence, a Monte Carlo simulation was conducted on 10,000 sample “tables.”Whenever the hypothesis that the trading models are not equally effective was rejected, the best and the worst strategies were excluded, and the procedure was repeated on the remaining models until all were tested. I refer to this as the Consistency Check. Empirical Results The distribution-free tests for the randomness of buy and sell signals reveal significant differences between the moving average and buy-and-hold distributions, implying some predictive power to the former. This predictive power is partly reflected in the observation that average returns following sell signals are all significantly positive, and this result is largely consistent with Brock, Lakonishok and LeBaron (1992) and Mills (1997), among others. A few of the tests of the two double crossover strategies, however, fail to reject the null hypotheses from 1996 to 2003 (see Table 1, models not marked with S in the first column). Interestingly, these are the models that have the lowest signal-generating frequency and thus the longest holding periods. Clearly these properties affect their conditional return distributions in a way that brings them closer to the buy-and-hold strategy.
All of the moving average strategies outperform buy-and-hold under the Sharpe criterion with average ratios of 1.897% and 3.1029% for the two periods as compared to –0.4134% and 1.1309% for the passive approach. Nine in-sample and all ten out-of-sample alphas are positive, but for only two models, MA (1, 50, 0) and MA (1, 50, 0.01), are they highly significant (see Table 2, models marked with S in the first column). Moving average models outperformed the buy and-hold strategy by an average of 0.0106% or 2.71% annually, and from 1996 to 2003, by 0.0328% or 8.62%. It is worth pointing out that the two models with the highest alphas have significantly negative betas. This is typical of strategies designed to counter the market, such as those employed by hedge funds. Lastly, while buy-and-hold returns are higher when based on total return series, moving average returns are still superior.
After adjusting returns with two-way transaction costs of 25 basis points, only the MA (5, 150, 0) model is unable to outperform the buy-and-hold in the first period. The other trading strategies require an average two way transaction cost of less than 50 basis points in order to be profitable. Interestingly, despite its high trading frequency, the best performing model is again MA (1, 50, 0.01); it yields approximately 6.47% excess return per year and is profitable up to trading costs of 74 basis points. Accounting for dividend yields, the average break even for a single crossover moving average model is 23 basis points. From 1996 to 2003 the break-even levels increase to 89 basis points for a two-way transaction, which suggests higher profits. This is attributable to the higher market volatility and its relationship to conditional returns. Overall, the break-even costs are strikingly similar to those reported on DJIA, FT30 and SBC.
For 1977 to 1995, the null hypothesis of equality among moving average strategies is rejected at 1% level of significance as long as the two best and the two worst performing models are present. There is virtually no significant difference among trading models ranked from three to eight. The results for the second period are similar. Most important is that the two best performing, MA (1, 50, 0,01) and MA (1, 50, 0), and two worst performing, MA (5, 150, 0,01) and MA (5, 150, 0) strategies are the same, in-sample and out-of-sample (see Table 4, models marked with “*” in the last two columns). Conclusion Acknowledgments Endnotes References For a PDF version of this article, click
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The
returns associated with the moving average strategies are less risky.
Long positions are less volatile than short positions, and overall,
none of the trading models is riskier than buy-and-hold, in- or
out-of-sample. Table 2 highlights the risk properties of moving
average strategies using the Sharpe ratio as well as Jensen’s
Alpha as a performance measure and Beta as the risk parameter. Alpha,
as it is used here, is the average incremental risk-adjusted return
due to the predictive power of the trading models, where positive
and significant intercepts are indicative of superior performance;
the beta is a measure of tendency of conditional returns to move
in line with the market risk premia.
Table
3 summarizes the returns adjusted for transaction costs. Columns
ER and ER25 present the annualized continuously compounded excess
price return differences between moving average strategy and buy-and-hold
returns before and after two-way transaction costs of 25 basis points,
respectively. Column B/E shows the break-even or two-way transaction
cost measured in basis points at which a trading model and buy-and-hold
average returns are equal. Before transaction costs, all strategies
yield higher mean daily returns than buy-and-hold, in-sample and
out-of-sample. MA (1, 50, 0.01) generates the highest return of
0.0686% or 18.87% at an annual rate. This is 9.9% greater than buy-and-hold
return. Overall, it appears that trading models with lower smoothing
parameters tend to yield higher returns, and that the introduction
of trading filter bands improves performance in every case. These
findings are valid when total price returns are employed.
One
gauge of forecast reliability is to see whether the best and worst
performing trading models in the in-sample keep their ranks in the
out-of-sample. Assume that moving average strategies are not significantly
different one from another. Hence, the probability of a trading
model changing rank from best to worst or vice versa in a later
period is equal across all models, and so combining them and using
a weighted signal would be as good as any. If, on the other hand,
trading models differ significantly in performance, then an optimal
strategy will involve separating the best from the worst. That,
in essence, is the Consistency Check, as described previously and
presented here in Table 4. The column Sharpe Ranking shows the trading
models ranked according to their Sharpe ratios, where 1-10 includes
all models, 2-9 excludes the best and the worst after the first
simulation, etc.; columns Lower B and Upper B are the lower and
upper band of the 99% confidence interval calculated with Monte
Carlo simulations based on 10,000 sample tables of the corresponding
p-value. Best Model and Worst Model show the strategies that yield
the highest and the lowest mean returns within those groups.