1. Pereira (2000) Motivation Motivation Motivation Experimental Design Experimental Design Experimental Design Experimental Design Performance Evaluation Performance Evaluation Performance Evaluation Performance Evaluation Experimental Results Experimental Results Experimental Results Experimental Results Structured Investments Group County Investment Management Level 19, 255 George Street Sydney NSW 2000

2. Motivation The majority of the studies examining trading rules chose both the rules and their parameter values arbitrarily. The majority of the studies examining trading rules chose both the rules and their parameter values arbitrarily. However this approach leaves these studies open to the criticisms of data-snooping and the possibility of a survivorship bias. However this approach leaves these studies open to the criticisms of data-snooping and the possibility of a survivorship bias.data-snoopingsurvivorship biasdata-snoopingsurvivorship bias

3. Methodology By choosing trading rules based on an optimization procedure utilizing in-sample data and testing the performance of these rules out-of-sample, this bias can be avoided or at least reduced. By choosing trading rules based on an optimization procedure utilizing in-sample data and testing the performance of these rules out-of-sample, this bias can be avoided or at least reduced.

4. Purpose In this paper the forecasting ability and economic profitability of some popular technical trading rules applied to the Australian share market are investigated using a standard genetic algorithm optimization procedure. In this paper the forecasting ability and economic profitability of some popular technical trading rules applied to the Australian share market are investigated using a standard genetic algorithm optimization procedure.

5. Purpose The genetic algorithm can be used to search for the optimal parameter values for the GMA and GOS trading rules. The genetic algorithm can be used to search for the optimal parameter values for the GMA and GOS trading rules.

6. Data Snooping Lo A.W., and A.G.MacKinley (1990), ``Data Snooping Biases in Tests of Financial Asset Pricing Models,’’ The Review of Financial Studies, Vol. 3, pp. 431--467. Lo A.W., and A.G.MacKinley (1990), ``Data Snooping Biases in Tests of Financial Asset Pricing Models,’’ The Review of Financial Studies, Vol. 3, pp. 431--467.

7. Survivorship Bias Brown S., W. Goetzmann, and S. Ross (1995), ``Survival,’’ Journal of Finance, Vol. 50, pp. 853--873. Brown S., W. Goetzmann, and S. Ross (1995), ``Survival,’’ Journal of Finance, Vol. 50, pp. 853--873.

8. Experimental Design Encoding a Trading Rule Encoding a Trading Rule Encoding a Trading Rule Encoding a Trading Rule Fitness Function Fitness Function Fitness Function Fitness Function Genetic Operation Genetic Operation Genetic Operation Genetic Operation

9. Encoding a Trading Rule A GMA Rule A GMA Rule A GMA Rule A GMA Rule A GOS Rule A GOS Rule A GOS Rule A GOS Rule

10. Encoding the GMA Rule Potential solutions to the problem of optimization of the parameters of the GMA rule Potential solutions to the problem of optimization of the parameters of the GMA rule can be represented by the vector can be represented by the vector

11. Meaning of and The periodicity of the two MAs have a range defined by The periodicity of the two MAs have a range defined by and and where represents the maximum length of the moving average. where represents the maximum length of the moving average.

12. Meaning of The filter parameter has a range given by The filter parameter has a range given by where represents the maximum filter where represents the maximum filter value. value.

13. Parameters Setting For this study =250 days and =100 basis points. For this study =250 days and =100 basis points. These limiting values are consistent with what is used in practice. These limiting values are consistent with what is used in practice. Also, results from a preliminary investigation, indicated that higher parameter values generally produced losses. Also, results from a preliminary investigation, indicated that higher parameter values generally produced losses.

14. Number of Bits Used In order to satisfy the limiting values given above, the binary representations for and In order to satisfy the limiting values given above, the binary representations for and are each given by a vector consisting of eight elements. are each given by a vector consisting of eight elements. For the filter parameter ( ) a seven bit vector is required. For the filter parameter ( ) a seven bit vector is required.

15. A GMA Rule Therefore, the binary representation for the GMA rule can be defined by a row vector consisting of twenty three elements stated as Therefore, the binary representation for the GMA rule can be defined by a row vector consisting of twenty three elements stated as ( ): 8 bits ( ): 8 bits ( ): 7 bits ( ): 7 bits 23 bits

16. Encoding the GOS Rule For the GOS rule, For the GOS rule, candidates can be represented by the vector

17. Meaning of The parameter on the channel rule represents the number of the most recent historical observations used to calculate either the maximum or minimum price. The parameter on the channel rule represents the number of the most recent historical observations used to calculate either the maximum or minimum price.

18. Parameter Setting The parameter is restricted to the values The parameter is restricted to the values The range for is given by basis points. The range for is given by basis points.

19. A GOS Rule The binary representations for the order statistics based rule can be defined by a row vector consisting of twenty elements stated as The binary representations for the order statistics based rule can be defined by a row vector consisting of twenty elements stated as ( ): 8 bits ( ): 8 bits ( ): 10 bits ( ): 10 bits ( ): 1 bit ( ): 1 bit 20 bits

20. Fitness Function Each candidate's performance can be assessed in terms of this objective function, which can take numerous forms depending upon specific investor preferences. Each candidate's performance can be assessed in terms of this objective function, which can take numerous forms depending upon specific investor preferences. Given that individuals are generally risk averse, performance should be defined in terms of both risk and return. Given that individuals are generally risk averse, performance should be defined in terms of both risk and return.

21. Modified Sharpe Ratio The Sharpe ratio is given by The Sharpe ratio is given by where is the average annualized trading rule return, is the standard deviation of daily trading rule returns, while Y is equal to the number of trading days per year. where is the average annualized trading rule return, is the standard deviation of daily trading rule returns, while Y is equal to the number of trading days per year.

22. Modified Sharpe Ratio This formulation is actually a modified version of the original Sharpe ratio which uses average excess returns, defined as the difference between average market return and the risk-free rate. This formulation is actually a modified version of the original Sharpe ratio which uses average excess returns, defined as the difference between average market return and the risk-free rate.

23. Trading Rule BUYSELL ``In’’ the Market``Out of’’ the Market Earning Market Rate of Return Risk-Free Rate of Return

24. Returns Thus the trading rule return over the entire period of 0 to N can be calculated as Thus the trading rule return over the entire period of 0 to N can be calculated aswhere

25. Transaction Costs An adjustment for transaction costs is given by the last term which consists of the product of the cost per transaction (tc) and the number of transactions (T). An adjustment for transaction costs is given by the last term which consists of the product of the cost per transaction (tc) and the number of transactions (T). Transaction costs of 0.2 percent per trade are considered for the in-sample optimization of the trading rules. Transaction costs of 0.2 percent per trade are considered for the in-sample optimization of the trading rules.

26. Genetic Operation Selection Method: Ranking-Based Procedure Selection Method: Ranking-Based ProcedureRanking-Based ProcedureRanking-Based Procedure Crossover Style: One Point Crossover Crossover Style: One Point Crossover Parameter Settings: Parameter Settings:

27. Ranking-Based Selection In the genetic algorithm developed in this paper a copy of the best candidate replaces the worst candidate. In the genetic algorithm developed in this paper a copy of the best candidate replaces the worst candidate.

28. Performance Evaluation Profitability Profitability Profitability Directional Predictability Directional Predictability Directional Predictability Directional Predictability Bootstrap Bootstrap Bootstrap

29. Measure for Trading Rule Performance Some Difficulties Some Difficulties Some Difficulties Some Difficulties Measure 1 Measure 1 Measure 1 Measure 1 Measure 2 Measure 2 Measure 2 Measure 2

30. Some Difficulties The true profitability of technical trading rules is hard to measure given the difficulties in properly accounting for the risks and costs associated with trading. The true profitability of technical trading rules is hard to measure given the difficulties in properly accounting for the risks and costs associated with trading.costs Benchmark Benchmark Benchmark

31. Costs Trading costs include not only transaction costs and taxes, but also hidden costs involved in the collection and analysis of information. Trading costs include not only transaction costs and taxes, but also hidden costs involved in the collection and analysis of information.  Institutional Traders vs. Individual Traders Institutional Traders Institutional Traders  The Break-Even Cost The Break-Even Cost The Break-Even Cost

32. Institutional Traders’ Costs According to Sweeney (1988), large institutional investors are able to achieve one-way transaction costs in the range of 0.1 to 0.2 percent. According to Sweeney (1988), large institutional investors are able to achieve one-way transaction costs in the range of 0.1 to 0.2 percent.Sweeney (1988),Sweeney (1988),

33. Sweeney (1988) Sweeney R. J. (1988), ``Some New Filter Rule Tests: Methods and Results,’’ Journal of Financial and Quantitative Analysis, Vol. 23, pp. 285--301. Sweeney R. J. (1988), ``Some New Filter Rule Tests: Methods and Results,’’ Journal of Financial and Quantitative Analysis, Vol. 23, pp. 285--301.

34. The Break-Even Transaction Cost This is the level of transaction costs which offsets trading rule revenue with costs, leading to zero trading profits. This is the level of transaction costs which offsets trading rule revenue with costs, leading to zero trading profits. See Bessembinder and Chan (1995). See Bessembinder and Chan (1995).Bessembinder and Chan (1995).Bessembinder and Chan (1995).

35. Bessembinder and Chan (1995) Bessembinder H., and K. Chan (1995), ``The Profitability of Technical Trading Rules in the Asian Stock Markets,’’. Pacific Basin Finance Journal, Vol. 3, pp. 257--284 Bessembinder H., and K. Chan (1995), ``The Profitability of Technical Trading Rules in the Asian Stock Markets,’’. Pacific Basin Finance Journal, Vol. 3, pp. 257--284

36. Benchmark To evaluate trading rule profitability, it is necessary to compare trading rule returns to an appropriate benchmark. To evaluate trading rule profitability, it is necessary to compare trading rule returns to an appropriate benchmark. Since the trading rules considered in this paper restrict short selling, they do not always lead to a position being held in the market and therefore are less risky than a passive buy and hold benchmark strategy, which always holds a long position in the market. Since the trading rules considered in this paper restrict short selling, they do not always lead to a position being held in the market and therefore are less risky than a passive buy and hold benchmark strategy, which always holds a long position in the market.

37. Weighted Average as A Benchmark Therefore, the appropriate benchmark is constructed by taking a weighted average of the return from being long in the market and the return from holding no position in the market and thus earning the risk free rate of return. Therefore, the appropriate benchmark is constructed by taking a weighted average of the return from being long in the market and the return from holding no position in the market and thus earning the risk free rate of return.

38. Risk-Adjusted Buy and Hold Strategy The return on this risk-adjusted buy and hold strategy can be written as The return on this risk-adjusted buy and hold strategy can be written as where is the proportion of trading days that the rule is out of the market. where is the proportion of trading days that the rule is out of the market.

39. A Portfolio of ``In’’ and ``Out’’ This return represents the expected return from investing in both the risk-free asset and the market according to the weights and (1- ) respectively. This return represents the expected return from investing in both the risk-free asset and the market according to the weights and (1- ) respectively. There is also an adjustment for transaction costs incurred due to purchasing the market portfolio on the first trading day and selling it on the last day of trading. There is also an adjustment for transaction costs incurred due to purchasing the market portfolio on the first trading day and selling it on the last day of trading.

40. Measure 1: Excess Returns Therefore, trading rule performance relative to the benchmark can be measured by excess returns Therefore, trading rule performance relative to the benchmark can be measured by excess returns where r represents the total return for a particular trading rule calculated from where r represents the total return for a particular trading rule calculated from

41. Measure 2: Sharpe Ratio Since investors and traders also care about the risk incurred in deriving these returns, a Sharpe ratio based on excess returns can be calculated using Equation Since investors and traders also care about the risk incurred in deriving these returns, a Sharpe ratio based on excess returns can be calculated using Equation

42. Directional Predictability To investigate the statistical significance of the forecasting power of the buy and sell signals, traditional t tests can be employed to examine whether the trading rules issue buy (or sell) signals on days when the return on the market is on average higher (or lower) than the unconditional mean return for the market. To investigate the statistical significance of the forecasting power of the buy and sell signals, traditional t tests can be employed to examine whether the trading rules issue buy (or sell) signals on days when the return on the market is on average higher (or lower) than the unconditional mean return for the market.traditional t teststraditional t tests

43. Test Statistics T-test for Buy (Sell) Signal T-test for Buy (Sell) Signal T-test for Buy (Sell) Signal T-test for Buy (Sell) Signal T-test for the Difference between Buy and Sell Signals T-test for the Difference between Buy and Sell Signals T-test for the Difference between Buy and Sell Signals T-test for the Difference between Buy and Sell Signals Cumby-Modest Test Cumby-Modest Test Cumby-Modest Test Cumby-Modest Test

44. Test Statistic (Buy) The t-statistic used to test the predictive ability of the buy signals is The t-statistic used to test the predictive ability of the buy signals is where represents the average daily return following a buy signal and is the number of days that the trading rule returns a buy signal. where represents the average daily return following a buy signal and is the number of days that the trading rule returns a buy signal.

45. Null and Alternative Hypothesis The null and alternative hypotheses can be stated as The null and alternative hypotheses can be stated as

46. Test Statistic (Buy and Sell) To test whether the difference between the mean return on the market following a buy signal and the mean return on the market following a sell signal is statistically significant, a t-test can be specified as To test whether the difference between the mean return on the market following a buy signal and the mean return on the market following a sell signal is statistically significant, a t-test can be specified as

47. The Null and Alternative Hypothesis The null and alternative hypotheses are The null and alternative hypotheses are

48. Cumby-Modest Test Cumby and Modest (1987) suggested a test based on the following regression Cumby and Modest (1987) suggested a test based on the following regression Cumby and Modest (1987) Cumby and Modest (1987)

49. Cumby and Modest (1987) Cumby R. E., and D. M. Modest (1987), ``Testing for Market Timing Ability: A Framework for Forecast Evaluation,’’ Journal of Financial Economics, Vol. 19, pp. 169--189. Cumby R. E., and D. M. Modest (1987), ``Testing for Market Timing Ability: A Framework for Forecast Evaluation,’’ Journal of Financial Economics, Vol. 19, pp. 169--189.

50. Experimental Results Data Data Data Results Results Results

51. Data Data Description Data Description Data Description Data Description Data Split Data Split Data Split Data Split Basic Statistics Basic Statistics Basic Statistics Basic Statistics

52. Data Description The Daily Closing All Ordinaries Accumulation index The Daily Closing All Ordinaries Accumulation index The Daily 90 day Reserve Bank of Australia Bill Dealer Rate. The Daily 90 day Reserve Bank of Australia Bill Dealer Rate. Period: 4/1/1982 to 31/12/97 (4065 observations) Period: 4/1/1982 to 31/12/97 (4065 observations) Source: the Equinet Pty Ltd data base. Source: the Equinet Pty Ltd data base.

53. Data Split: In-sample optimization period: In-sample optimization period:  from 4/1/1982 to 31/12/89 Out-of-sample test period Out-of-sample test period  from 2/1/1990 to 31/12/97.

54. Basic Statistics Table 2 provides summary statistics for the continuously compounded daily returns ( ) on the All Ordinaries index. Table 2 provides summary statistics for the continuously compounded daily returns ( ) on the All Ordinaries index. Table 2 Table 2 Significant first-order serial correlation in share indices is a well known stylized fact due to the inclusion of thinly-traded small shares in share market indices. Significant first-order serial correlation in share indices is a well known stylized fact due to the inclusion of thinly-traded small shares in share market indices.

55. Basic Statistics This result is not surprising given that the All Ordinaries Accumulation index is comprised of over 300 shares, which includes a significant amount of small shares. This result is not surprising given that the All Ordinaries Accumulation index is comprised of over 300 shares, which includes a significant amount of small shares. There also appears to be significant higher order serial correlation as indicated by the heteroscedasticity-adjusted Box-Pierce Q statistic (ABP). There also appears to be significant higher order serial correlation as indicated by the heteroscedasticity-adjusted Box-Pierce Q statistic (ABP).

57. Results In-Sample Period In-Sample Period In-Sample Period In-Sample Period Out-of-Sample Period Out-of-Sample Period  Economic Profitability Economic ProfitabilityEconomic Profitability  Predictive Ability Predictive AbilityPredictive Ability  Bootstrap Bootstrap Non-Synchronous Trading Bias Non-Synchronous Trading Bias Non-Synchronous Trading Bias Non-Synchronous Trading Bias

58. In-Sample Period The GA-optimal parameter values for the trading rules found during the in-sample period based on transaction costs of 10 basis points are reported in Table 3. The GA-optimal parameter values for the trading rules found during the in-sample period based on transaction costs of 10 basis points are reported in Table 3.Table 3.Table 3. The returns and the Sharpe ratios are high, even compared to the buy and hold return of 18.79 percent per annum and the corresponding Sharpe ratio of 0.86 percent per unit of standard deviation. The returns and the Sharpe ratios are high, even compared to the buy and hold return of 18.79 percent per annum and the corresponding Sharpe ratio of 0.86 percent per unit of standard deviation.

59. In-Sample Period The best GMA rule can be described as a 14 day MA rule with a 64 basis point filter, while the best GOS rule can be described as a 9 day channel rule with a 21 basis point filter. The best GMA rule can be described as a 14 day MA rule with a 64 basis point filter, while the best GOS rule can be described as a 9 day channel rule with a 21 basis point filter.

61. Table Description The Sharpe ratio (SR) is calculated as the ratio of annualised returns ( ) to standard deviation. The Sharpe ratio (SR) is calculated as the ratio of annualised returns ( ) to standard deviation. The number of times the best rule was found in 10 trials (No.) is given in the fifth column. The number of times the best rule was found in 10 trials (No.) is given in the fifth column.

62. Table Description The average number of iterations completed until the best rule was found (Best after) is reported in column 8.. The average number of iterations completed until the best rule was found (Best after) is reported in column 8.. The average number of iterations completed for one trial (No. of iters) is reported in the second last column. The average number of iterations completed for one trial (No. of iters) is reported in the second last column.

63. Economic Profitability The out-of-sample performance statistics are reported in Table 4. The out-of-sample performance statistics are reported in Table 4.Table 4Table 4 In terms of maximum drawdown, both rules are much less risky than the buy and hold, which has a maximum drawdown of -69 percent. In terms of maximum drawdown, both rules are much less risky than the buy and hold, which has a maximum drawdown of -69 percent.

64. Economic Profitability The robustness of the results is investigated across different non-overlapping sub- periods. The robustness of the results is investigated across different non-overlapping sub- periods. A sub-period analysis of the performance results, show that this good performance deteriorates over time. A sub-period analysis of the performance results, show that this good performance deteriorates over time. In the last couple of years neither rule is able to outperform the benchmark. In the last couple of years neither rule is able to outperform the benchmark.

66. Table Description The break-even level of transaction cost is given by. The break-even level of transaction cost is given by. Trading frequency is measured by the average number of trades per year. Trading frequency is measured by the average number of trades per year. represents the proportion of trades that yield positive excess returns. represents the proportion of trades that yield positive excess returns.

67. Table Description The average excess return on winning and losing trades is given by and respectively. The average excess return on winning and losing trades is given by and respectively. The maximum drawdown represents the largest drop in cumulative excess returns. The maximum drawdown represents the largest drop in cumulative excess returns.

68. Predictive Ability The results for the predictive ability of the trading rules are reported in Table 5. The results for the predictive ability of the trading rules are reported in Table 5.Table 5Table 5

70. Cumby-Modest Test Both rules display some evidence of significant predictive ability as indicated by the t-statistics in the second last column of Table 5. Both rules display some evidence of significant predictive ability as indicated by the t-statistics in the second last column of Table 5. This result is confirmed by the final column in the table which reports the t-statistic based on the Cumby-Modest market timing test. This result is confirmed by the final column in the table which reports the t-statistic based on the Cumby-Modest market timing test.

71. T test and Volatility However, individually the buy and sell signals do not seem to have any significant predictive ability. However, individually the buy and sell signals do not seem to have any significant predictive ability. All the rules issue buy (or sell) signals when the excess returns on the market are on average less (or more) volatile as indicated by the volatility of returns following buy ( ) and sell ( ) signals respectively. All the rules issue buy (or sell) signals when the excess returns on the market are on average less (or more) volatile as indicated by the volatility of returns following buy ( ) and sell ( ) signals respectively.

72. Sub-Period Analysis A sub-period analysis of these results indicates that the difference between the average return following buy signals and the average return following sell is no longer significant. A sub-period analysis of these results indicates that the difference between the average return following buy signals and the average return following sell is no longer significant.

73. Bootstrap The bootstrap approach is applied to both the trading rule performance and predictive ability results. The bootstrap approach is applied to both the trading rule performance and predictive ability results. The simulated p-values for the various measures of performance and predictive ability are given in Table 6. The simulated p-values for the various measures of performance and predictive ability are given in Table 6.Table 6Table 6

74. Bootstrap P-Value

75. The results for the entire out-of-sample period provide evidence that the rules have significant better forecasting power and profitability in the original series than in the bootstrap series which the return series are generated by a random walk process. The results for the entire out-of-sample period provide evidence that the rules have significant better forecasting power and profitability in the original series than in the bootstrap series which the return series are generated by a random walk process.

76. Non-Synchronous Trading Bias It is important to evaluate the sensitivity of the results to the significant persistence in returns, which are reported in Table 2. It is important to evaluate the sensitivity of the results to the significant persistence in returns, which are reported in Table 2. Since the existence of thinly traded shares in the index can introduce a non-synchronous trading bias or return measurement error. Since the existence of thinly traded shares in the index can introduce a non-synchronous trading bias or return measurement error. These returns might not be exploitable in practice. These returns might not be exploitable in practice.

77. Trade with A Delay To investigate this issue, the performance of the trading rules is simulated based on trades occurring with a delay of one day. To investigate this issue, the performance of the trading rules is simulated based on trades occurring with a delay of one day. This should remove any first order autocorrelation bias due to non-synchronous trading. This should remove any first order autocorrelation bias due to non-synchronous trading.

78. Results As can be seen from Table 7, the rules are still profitable over the out-of-sample test period, although there has been a substantial reduction in performance. As can be seen from Table 7, the rules are still profitable over the out-of-sample test period, although there has been a substantial reduction in performance.Table 7Table 7 Furthermore, there appears to be weak, if any, evidence of predictive ability. Furthermore, there appears to be weak, if any, evidence of predictive ability.

80. Break-Even Transaction Costs The break-even transaction costs have been reduced to approximately 0.25 percent per trade. The break-even transaction costs have been reduced to approximately 0.25 percent per trade. This probably lower than the costs faced by most financial institutions. This probably lower than the costs faced by most financial institutions. This probably lower than the costs faced by most financial institutions. This probably lower than the costs faced by most financial institutions. Thus, there does not appear to be sufficient evidence to conclude that the trading rules are economically profitable. Thus, there does not appear to be sufficient evidence to conclude that the trading rules are economically profitable.

81. Stamp Duty Costs and Taxes Since stamp duty and taxes are also incurred on all trades, which have been ignored in this evaluation of trading rule performance. Since stamp duty and taxes are also incurred on all trades, which have been ignored in this evaluation of trading rule performance. Stamp duty costs per trade in Australia are currently 0.15 percent. Stamp duty costs per trade in Australia are currently 0.15 percent. Taxation costs vary across institutions and different individuals. Taxation costs vary across institutions and different individuals.

82. Liquidity Costs Also during volatile periods liquidity costs, as reflected by the bid-ask spread, could increase substantially. Also during volatile periods liquidity costs, as reflected by the bid-ask spread, could increase substantially. Even for large shares this increase could be in the order of 0.5 to 1 percent. Even for large shares this increase could be in the order of 0.5 to 1 percent.