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## Pereira (2000)

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**Pereira (2000)**• Motivation • Experimental Design • Performance Evaluation • Experimental Results Structured Investments Group County Investment Management Level 19, 255 George Street Sydney NSW 2000**Motivation**• 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.**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.**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.**Purpose**• The genetic algorithm can be used to search for the optimal parameter values for the GMA and GOS trading rules.**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.**Survivorship Bias**• Brown S., W. Goetzmann, and S. Ross (1995), ``Survival,’’ Journal of Finance, Vol. 50, pp. 853--873.**Experimental Design**• Encoding a Trading Rule • Fitness Function • Genetic Operation**Encoding a Trading Rule**• A GMA Rule • A GOS Rule**Encoding the GMA Rule**• Potential solutions to the problem of optimization of the parameters of the GMA rule can be represented by the vector**Meaning of and**• The periodicity of the two MAs have a range defined by and where represents the maximum length of the moving average.**Meaning of**• The filter parameter has a range given by where represents the maximum filter value.**Parameters Setting**• For this study =250 days and =100 basis points. • 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.**Number of Bits Used**• In order to satisfy the limiting values given above, the binary representations for and are each given by a vector consisting of eight elements. • For the filter parameter ( ) a seven bit vector is required.**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 • ( ): 8 bits • ( ): 8 bits • ( ): 7 bits 23 bits**Encoding the GOS Rule**• For the GOS rule, candidates can be represented by the vector**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.**Parameter Setting**• The parameter is restricted to the values • The range for is given by basis points.**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 ( ): 8 bits ( ): 10 bits ( ): 1 bit ( ): 1 bit 20 bits**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. • Given that individuals are generally risk averse, performance should be defined in terms of both riskand return.**Modified Sharpe Ratio**• 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.**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.**Trading Rule**BUY SELL ``In’’ the Market ``Out of’’ the Market Market Rate of Return Risk-Free Rate of Return Earning**Returns**• Thus the trading rule return over the entire period of 0 to N can be calculated as where**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). • Transaction costs of 0.2 percent per trade are considered for the in-sample optimization of the trading rules.**Genetic Operation**• Selection Method: Ranking-Based Procedure • Crossover Style: One Point Crossover • Parameter Settings:**Ranking-Based Selection**• In the genetic algorithm developed in this paper a copy of the best candidate replaces the worst candidate.**Performance Evaluation**• Profitability • Directional Predictability • Bootstrap**Measure for Trading Rule Performance**• Some Difficulties • Measure 1 • Measure 2**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. • Benchmark**Costs**• 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 • The Break-Even Cost**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.**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.**The Break-Even Transaction Cost**• This is the level of transaction costs which offsets trading rule revenue with costs, leading to zero trading profits. • See Bessembinder and Chan (1995).**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**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.**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.**Risk-Adjusted Buy and Hold Strategy**• 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.**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. • 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.**Measure 1: 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**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**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.**Test Statistics**• T-test for Buy (Sell) Signal • T-test for the Difference between Buy and Sell Signals • Cumby-Modest Test**Test Statistic (Buy)**• 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.**Null and Alternative Hypothesis**• The null and alternative hypotheses can be stated as**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**The Null and Alternative Hypothesis**• The null and alternative hypotheses are**Cumby-Modest Test**• Cumby and Modest (1987) suggested a test based on the following regression**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.**Experimental Results**• Data • Results