Pereira (2000)

MotivationMotivation Experimental DesignExperimental Design Performance EvaluationPerformance Evaluation Experimental ResultsExperimental Results

Structured Investments GroupCounty Investment ManagementLevel 19, 255 George StreetSydney NSW 2000

Motivation

The majority of the studies examining The majority of the studies examining trading rules chose both the rules and their trading rules chose both the rules and their parameter values parameter values arbitrarilyarbitrarily..

However this approach leaves these studies However this approach leaves these studies open to the criticisms of open to the criticisms of data-snoopingdata-snooping and and the possibility of a the possibility of a survivorship biassurvivorship bias. .

Methodology

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

Purpose

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

Purpose

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

Data Snooping

Lo A.W., and A.G.MacKinley (1990), Lo A.W., and A.G.MacKinley (1990), ``Data Snooping Biases in Tests of ``Data Snooping Biases in Tests of Financial Asset Pricing ModelsFinancial Asset Pricing Models,’’ ,’’ The The Review of Financial StudiesReview of Financial Studies, , Vol. 3, pp. Vol. 3, pp. 431--467.431--467.

Survivorship Bias

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

Experimental Design

Encoding a Trading RuleEncoding a Trading Rule Fitness FunctionFitness Function Genetic OperationGenetic Operation

Encoding a Trading Rule

A GMA RuleA GMA Rule A GOS RuleA GOS Rule

Encoding the GMA Rule

Potential solutions to the problem of Potential solutions to the problem of optimization of the parameters of the GMA optimization of the parameters of the GMA rule rule

can be represented by the vectorcan be represented by the vector

Meaning of and

The periodicity of the two MAs have a The periodicity of the two MAs have a range defined byrange defined by

andand

where represents the maximum length of where represents the maximum length of the moving average.the moving average.

Meaning of

The filter parameter has a range given byThe filter parameter has a range given by

where represents the maximum filterwhere represents the maximum filter

value. value.

Parameters Setting

For this study =250 days and =100 For this study =250 days and =100 basis points.basis points.

These limiting values are consistent with These limiting values are consistent with what is used in practice.what is used in practice.

Also, results from a preliminary Also, results from a preliminary investigation, indicated that higher investigation, indicated that higher parameter values generally produced losses.parameter values generally produced losses.

Number of Bits Used

In order to satisfy the limiting values given In order to satisfy the limiting values given above, the binary representations for andabove, the binary representations for and

are each given by a vector consisting of are each given by a vector consisting of eighteight elements. elements.

For the filter parameter ( ) a seven bit For the filter parameter ( ) a seven bit vector is required.vector is required.

A GMA Rule

Therefore, the binary representation for the Therefore, the binary representation for the GMA rule can be defined by a row vector GMA rule can be defined by a row vector consisting of twenty three elements stated asconsisting of twenty three elements stated as

( ): 8 ( ): 8 bitsbits ( ): 8 ( ): 8 bitsbits ( ): 7 ( ): 7 bits bits 23 bits

Encoding the GOS Rule

For the GOS rule,For the GOS rule,

candidates can be represented by the vectorcandidates can be represented by the vector

Meaning of

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

Parameter Setting

The parameter is restricted to the valuesThe parameter is restricted to the values

The range for is given by The range for is given by basis points.basis points.

A GOS Rule

The binary representations for the order The binary representations for the order statistics based rule can be defined by a row statistics based rule can be defined by a row vector consisting of twenty elements stated asvector consisting of twenty elements stated as

( ): 8 bits( ): 8 bits

( ): 10 ( ): 10 bitsbits

( ): 1 ( ): 1 bitbit

( ): 1 ( ): 1 bitbit20 bits

Fitness Function

Each candidate's performance can be Each candidate's performance can be assessed in terms of this objective function, assessed in terms of this objective function, which can take numerous forms depending which can take numerous forms depending upon specific investor preferences.upon specific investor preferences.

Given that individuals are generally risk Given that individuals are generally risk averse, performance should be defined in averse, performance should be defined in terms of both terms of both riskrisk and and returnreturn. .

Modified Sharpe Ratio

The Sharpe ratio is given byThe Sharpe ratio is given by

where is the average annualized trading where is the average annualized trading rule return, is the standard deviation of rule return, is the standard deviation of daily trading rule returns, while Y is equal daily trading rule returns, while Y is equal to the number of trading days per year. to the number of trading days per year.

Modified Sharpe Ratio

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

Trading Rule

BUY SELL

``In’’ the Market ``Out of’’ the Market

EarningMarket Rate of Return

Risk-Free Rate of Return

Returns

Thus the trading rule return over the entire Thus the trading rule return over the entire period of 0 to N can be calculated asperiod of 0 to N can be calculated as

wherewhere

Transaction Costs

An adjustment for transaction costs is given An adjustment for transaction costs is given by the last term which consists of the by the last term which consists of the product of the cost per transaction (tc) and product of the cost per transaction (tc) and the number of transactions (T).the number of transactions (T).

Transaction costs of Transaction costs of 0.2 percent0.2 percent per trade per trade are considered for the in-sample are considered for the in-sample optimization of the trading rules.optimization of the trading rules.

Genetic Operation

Selection Method: Selection Method: Ranking-Based ProcedureRanking-Based Procedure

Crossover Style: One Point CrossoverCrossover Style: One Point Crossover Parameter Settings:Parameter Settings:

Ranking-Based Selection

In the genetic algorithm developed in this In the genetic algorithm developed in this paper a copy of the best candidate replaces paper a copy of the best candidate replaces the worst candidate.the worst candidate.

Performance Evaluation

ProfitabilityProfitability Directional PredictabilityDirectional Predictability Bootstrap Bootstrap

Measure for Trading Rule Performance Some DifficultiesSome Difficulties Measure 1Measure 1 Measure 2Measure 2

Some Difficulties

The The true profitability of technical trading true profitability of technical trading rules is hard to measurerules is hard to measure given the given the difficulties in properly accounting for the difficulties in properly accounting for the risksrisks and and costscosts associated with trading. associated with trading.

BenchmarkBenchmark

Costs

Trading costs include not only Trading costs include not only transaction transaction costscosts and taxes, but also and taxes, but also hidden costshidden costs involved in the collection and analysis of involved in the collection and analysis of information.information. Institutional TradersInstitutional Traders vs. Individual vs. Individual

TradersTraders The Break-Even CostThe Break-Even Cost

Institutional Traders’ Costs

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

Sweeney (1988)

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

The Break-Even Transaction Cost This is the level of transaction costs which This is the level of transaction costs which

offsets trading rule revenue with costs, offsets trading rule revenue with costs, leading to zero trading profits.leading to zero trading profits.

See See BessembinderBessembinder and Chan (1995). and Chan (1995).

Bessembinder and Chan (1995)

Bessembinder H., and K. Chan (1995), Bessembinder H., and K. Chan (1995), ````The Profitability of Technical Trading The Profitability of Technical Trading Rules in the Asian Stock Markets,Rules in the Asian Stock Markets,’’. ’’. Pacific Basin Finance JournalPacific Basin Finance Journal, Vol. 3 , pp. , Vol. 3 , pp. 257--284257--284

Benchmark To evaluate trading rule profitability, it is To evaluate trading rule profitability, it is

necessary to compare trading rule returns to necessary to compare trading rule returns to an appropriate benchmark.an appropriate benchmark.

Since the trading rules considered in this Since the trading rules considered in this paper paper restrictrestrict short selling, they do not short selling, they do not always lead to a position being held in the always lead to a position being held in the market and therefore are less risky than a market and therefore are less risky than a passive passive buy and hold benchmark buy and hold benchmark strategystrategy, which always holds a long , which always holds a long position in the market.position in the market.

Weighted Average as A Benchmark Therefore, the appropriate benchmark is Therefore, the appropriate benchmark is

constructed by taking a weighted average of constructed by taking a weighted average of the return from being long in the market the return from being long in the market and the return from holding no position in and the return from holding no position in the market and thus earning the risk free the market and thus earning the risk free rate of return.rate of return.

Risk-Adjusted Buy and Hold Strategy The return on this risk-adjusted buy and The return on this risk-adjusted buy and

hold strategy can be written ashold strategy can be written as

where is the proportion of trading days where is the proportion of trading days that the rule is out of the market.that the rule is out of the market.

A Portfolio of ``In’’ and ``Out’’

This return represents the expected return This return represents the expected return from investing in both the risk-free asset from investing in both the risk-free asset and the market according to the weights and the market according to the weights and (1- ) respectively.and (1- ) respectively.

There is also an adjustment for transaction There is also an adjustment for transaction costs incurred due to purchasing the market costs incurred due to purchasing the market portfolio on the first trading day and selling portfolio on the first trading day and selling it on the last day of trading.it on the last day of trading.

Measure 1: Excess Returns

Therefore, trading rule performance relative Therefore, trading rule performance relative to the benchmark can be measured by to the benchmark can be measured by excess returnsexcess returns

where r represents the total return for a where r represents the total return for a particular trading rule calculated fromparticular trading rule calculated from

Measure 2: Sharpe Ratio

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

Directional Predictability

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

Test Statistics

T-test for Buy (Sell) SignalT-test for Buy (Sell) Signal T-test for the Difference between Buy and ST-test for the Difference between Buy and S

ell Signalsell Signals CumbyCumby-Modest Test-Modest Test

Test Statistic (Buy)

The t-statistic used to test the predictive ability The t-statistic used to test the predictive ability of the buy signals isof the buy signals is

where represents the average daily return where represents the average daily return following a buy signal and is the number of following a buy signal and is the number of days that the trading rule returns a buy signal.days that the trading rule returns a buy signal.

Null and Alternative Hypothesis

The null and alternative hypotheses can be The null and alternative hypotheses can be stated asstated as

Test Statistic (Buy and Sell)

To test whether the difference between the To test whether the difference between the mean return on the market following a buy mean return on the market following a buy signal and the mean return on the market signal and the mean return on the market following a sell signal is statistically following a sell signal is statistically significant, a t-test can be specified assignificant, a t-test can be specified as

The Null and Alternative Hypothesis The null and alternative hypotheses areThe null and alternative hypotheses are

Cumby-Modest Test

CumbyCumby and Modest (1987) and Modest (1987) suggested a test suggested a test based on the following regressionbased on the following regression

Cumby and Modest (1987)

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

Experimental Results

DataData ResultsResults

Data

Data DescriptionData Description Data SplitData Split Basic StatisticsBasic Statistics

Data Description

The Daily Closing All Ordinaries The Daily Closing All Ordinaries Accumulation index Accumulation index

The Daily 90 day Reserve Bank of The Daily 90 day Reserve Bank of Australia Bill Dealer Rate.Australia Bill Dealer Rate.

Period: 4/1/1982 to 31/12/97 (4065 Period: 4/1/1982 to 31/12/97 (4065 observations)observations)

Source: the Equinet Pty Ltd data base.Source: the Equinet Pty Ltd data base.

Data Split:

In-sample optimization period: In-sample optimization period: from 4/1/1982 to 31/12/89 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.from 2/1/1990 to 31/12/97.

Basic Statistics

Table 2Table 2 provides summary statistics for the provides summary statistics for the continuously compounded daily returns continuously compounded daily returns ( ) on the All Ordinaries index.( ) on the All Ordinaries index.

Significant first-order serial correlation in Significant first-order serial correlation in share indices is a well known stylized fact share indices is a well known stylized fact due to the inclusion of thinly-traded small due to the inclusion of thinly-traded small shares in share market indices.shares in share market indices.

Basic Statistics

This result is not surprising given that the All This result is not surprising given that the All Ordinaries Accumulation index is comprised of Ordinaries Accumulation index is comprised of over 300 shares, which includes a significant over 300 shares, which includes a significant amount of small shares.amount of small shares.

There also appears to be significant higher There also appears to be significant higher order serial correlation as indicated by the order serial correlation as indicated by the heteroscedasticity-adjusted Box-Pierce Q heteroscedasticity-adjusted Box-Pierce Q statistic (ABP).statistic (ABP).

Results

In-Sample PeriodIn-Sample Period Out-of-Sample PeriodOut-of-Sample Period

Economic ProfitabilityEconomic Profitability Predictive AbilityPredictive Ability BootstrapBootstrap

Non-Synchronous Trading BiasNon-Synchronous Trading Bias

In-Sample Period

The GA-optimal parameter values for the trading The GA-optimal parameter values for the trading rules found during the rules found during the in-sample periodin-sample period based on based on transaction costs of 10 basis points are reported in transaction costs of 10 basis points are reported in Table 3.Table 3.

The returns and the Sharpe ratios are high, even The returns and the Sharpe ratios are high, even compared to the buy and hold return of compared to the buy and hold return of 18.7918.79 percent per annum and the corresponding Sharpe percent per annum and the corresponding Sharpe ratio of ratio of 0.860.86 percent per unit of standard deviation. percent per unit of standard deviation.

In-Sample Period

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

Table Description

The Sharpe ratio (SR) is calculated as the The Sharpe ratio (SR) is calculated as the ratio of annualised returns ( ) to standard ratio of annualised returns ( ) to standard deviation.deviation.

The number of times the best rule was The number of times the best rule was found in 10 trials (No.) is given in the fifth found in 10 trials (No.) is given in the fifth column.column.

Table Description

The average number of iterations completed The average number of iterations completed until the best rule was found (Best after) is until the best rule was found (Best after) is reported in column 8..reported in column 8..

The average number of iterations completed The average number of iterations completed for one trial (No. of iters) is reported in the for one trial (No. of iters) is reported in the second last column.second last column.

Economic Profitability

The out-of-sample performance statistics The out-of-sample performance statistics are reported in are reported in Table 4Table 4..

In terms of maximum drawdown, both rules In terms of maximum drawdown, both rules are much less risky than the buy and hold, are much less risky than the buy and hold, which has a maximum drawdown of -69 which has a maximum drawdown of -69 percent.percent.

Economic Profitability

The robustness of the results is investigated The robustness of the results is investigated across different non-overlapping sub-across different non-overlapping sub-periods.periods.

A sub-period analysis of the performance A sub-period analysis of the performance results, show that this good performance results, show that this good performance deteriorates over time. deteriorates over time.

In the last couple of years neither rule is In the last couple of years neither rule is able to outperform the benchmark.able to outperform the benchmark.

Table Description

The break-even level of transaction cost is The break-even level of transaction cost is given by .given by .

Trading frequency is measured by the Trading frequency is measured by the average number of trades average number of trades per yearper year..

represents the proportion of trades represents the proportion of trades that yield positive excess returns. that yield positive excess returns.

Table Description

The average excess return on winning and The average excess return on winning and losing trades is given by and losing trades is given by and respectively.respectively.

The maximum drawdown The maximum drawdown represents the largest drop in cumulative represents the largest drop in cumulative excess returns.excess returns.

Predictive Ability

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

Cumby-Modest Test

Both rules display some evidence of Both rules display some evidence of significant predictive ability as indicated by significant predictive ability as indicated by the t-statistics in the second last column of the t-statistics in the second last column of Table 5.Table 5.

This result is confirmed by the final column This result is confirmed by the final column in the table which reports the t-statistic in the table which reports the t-statistic based on the Cumby-Modest market timing based on the Cumby-Modest market timing test.test.

T test and Volatility

However, individually the buy and sell However, individually the buy and sell signals do not seem to have any significant signals do not seem to have any significant predictive ability.predictive ability.

All the rules issue buy (or sell) signals when All the rules issue buy (or sell) signals when the excess returns on the market are on the excess returns on the market are on average less (or more) volatile as indicated average less (or more) volatile as indicated by the volatility of returns following buy by the volatility of returns following buy ( ) and sell ( ) signals respectively.( ) and sell ( ) signals respectively.

Sub-Period Analysis

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

Bootstrap

The bootstrap approach is applied to both The bootstrap approach is applied to both the trading rule performance and predictive the trading rule performance and predictive ability results.ability results.

The The simulated p-valuessimulated p-values for the various for the various measures of performance and predictive measures of performance and predictive ability are given in Table 6.ability are given in Table 6.

Bootstrap P-Value

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

Non-Synchronous Trading Bias

It is important to evaluate the sensitivity of the It is important to evaluate the sensitivity of the results to the significant results to the significant persistence in returnspersistence in returns, , which are reported in Table 2.which are reported in Table 2.

Since the existence of thinly traded shares in the Since the existence of thinly traded shares in the index can introduce index can introduce a non-synchronous trading a non-synchronous trading biasbias or return measurement error. or return measurement error.

These returns might not be exploitable in These returns might not be exploitable in practice. practice.

Trade with A Delay

To investigate this issue, the performance of To investigate this issue, the performance of the trading rules is simulated based on the trading rules is simulated based on trades occurring with a delay of one day.trades occurring with a delay of one day.

This should remove any first order This should remove any first order autocorrelation bias due to non-autocorrelation bias due to non-synchronous trading.synchronous trading.

Results

As can be seen from Table 7, the rules are As can be seen from Table 7, the rules are still profitable over the out-of-sample test still profitable over the out-of-sample test period, although there has been a substantial period, although there has been a substantial reduction in performance.reduction in performance.

Furthermore, there appears to be weak, if Furthermore, there appears to be weak, if any, evidence of predictive ability.any, evidence of predictive ability.

Break-Even Transaction Costs

The break-even transaction costs have been The break-even transaction costs have been reduced to approximately 0.25 percent per reduced to approximately 0.25 percent per trade.trade.

This probably lower than the costs faced by This probably lower than the costs faced by most financial institutions.most financial institutions.

Thus, there does not appear to be sufficient Thus, there does not appear to be sufficient evidence to conclude that the trading rules evidence to conclude that the trading rules are economically profitable.are economically profitable.

Stamp Duty Costs and Taxes

Since stamp duty and taxes are also Since stamp duty and taxes are also incurred on all trades, which have been incurred on all trades, which have been ignored in this evaluation of trading rule ignored in this evaluation of trading rule performance.performance.

Stamp duty costs per trade in Australia are Stamp duty costs per trade in Australia are currently 0.15 percent. currently 0.15 percent.

Taxation costs vary across institutions and Taxation costs vary across institutions and different individuals.different individuals.

Liquidity Costs

Also during volatile periods Also during volatile periods liquidity costsliquidity costs, , as reflected by the as reflected by the bid-ask spreadbid-ask spread, could , could increase substantially.increase substantially.

Even for large shares this increase could be Even for large shares this increase could be in the order of 0.5 to 1 percent.in the order of 0.5 to 1 percent.