Package matiming. September 8, 2017

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1 Type Package Title Market Timing with Moving Averages Version 1.0 Author Valeriy Zakamulin Package matiming September 8, 2017 Maintainer Valeriy Zakamulin This package contains functions to compute moving averages, simulate moving average trading strategies, and to perform back-tests and forward tests of these strategies. License GPL-3 LazyData TRUE Depends R (>= 2.10) Imports Rcpp (>= ), zoo, np, e1071 LinkingTo Rcpp RoxygenNote NeedsCompilation yes Archs i386, x64 R topics documented: back.test best.is.strategies descriptive.statistics djia.daily djia.monthly DMA EMA Excret forward.test HMA is.outperformance.plot LMA oos.outperformance.plot outperformance.test Sharpe sim.cdir.strategy

2 2 back.test sim.mac.strategy sim.macd.strategy sim.mae.strategy sim.mom.strategy sim.results.merge SMA Sortino sp500.daily sp500.monthly ZMA Index 28 back.test Finds the optimal trading strategies in a back-test (in-sample test) This function finds the optimal trading strategies in a back-test back.test(results, start.date, end.date, n.first=7, include.bh=false, FUN=Sharpe) results start.date end.date n.first include.bh FUN the results of simulation of a trading strategy or multiple trading strategies. Specifically, this is the result returned by function sim.mom.strategy, sim.mac.strategy, sim.cdir.strategy, sim.macd.strategy, or sim.results.merge the start of the in-sample period the end of the in-sample period the (maximum) number of the best performing strategies in the returned object a logical value indicating whether to consider the Buy-and-Hold (bh) strategy in addition to the moving average strategies the function that computes the performance of a trading strategy. An object to be used by other functions Note The argument results contains the dates vector that defines the historical period over which the trading strategies are simulated. The function FUN is either Excret, Sharpe, or Sortino. Other strategy testing functions: forward.test, outperformance.test

3 best.is.strategies 3 Examples ## Not run: library(matiming) library(xtable) library(zoo) # access the data data <- sp500.monthly dates <- index(data) data <- coredata(data) capret <- data[,"cap"] totret <- data[,"tot"] rfret <- data[,"rf"] # simulate the trading strategies tc < fast <- 1:5 slow <- 2:15 shorts <- F results <- sim.mac.strategy(totret=totret, rfret=rfret, dates=dates, capret=capret, tc=tc, shorts=shorts, fast=fast, slow=slow, FUN=SMA) # define the historical period start.date <- as.date(" ", format="%y-%m-%d") end.date <- as.date(" ", format="%y-%m-%d") n.first <- 5 include.bh <- TRUE res <- back.test(results, start.date=start.date, end.date=end.date, n.first=n.first, include.bh=include.bh, FUN=Sortino) # print out the ranking of trading strategies df <- best.is.strategies(res, to.annual=sqrt(12), plot.results=true) xtab <- xtable(df, digits=3) print(xtab, include.rownames=false) ## End(Not run) best.is.strategies Reports the best trading strategies in a back-test (in-sample test) This function reports the best trading strategies in a back-test best.is.strategies(results, to.annual=sqrt(12), plot.results=false) results to.annual plot.results the results returned by function back.test a coefficient used to annualize the performance measure. For example, when the performance is measured using either Sharpe or Sortino, use sqrt(252) and sqrt(12) when the returns are at the daily and monthly frequency respectively. If the performance is measred by Excret, use 12 a logical value indicating whether to plot the cumulative returns to the best trading strategies and to the buy-and-hold strategy

4 4 descriptive.statistics An object of class data.frame containing three columns: Rank, Rule, and Performance. Other strategy summary functions: descriptive.statistics Examples ## Not run: library(matiming) library(xtable) library(zoo) data <- sp500.monthly dates <- index(data) data <- coredata(data) capret <- data[,"cap"] totret <- data[,"tot"] rfret <- data[,"rf"] tc < fast <- 1:9 slow <- 2:18 res.mac <- sim.mac.strategy(totret=totret, rfret=rfret, dates=dates, capret=capret, tc=tc, fast=fast, slow=slow, FUN=SMA) res.mom <- sim.mom.strategy(totret=totret, rfret=rfret, dates=dates, capret=capret, tc=tc, winsize=slow) results <- sim.results.merge(res.mac, res.mom) start.date <- as.date(" ", format="%y-%m-%d") end.date <- as.date(" ", format="%y-%m-%d") n.first <- 10 include.bh <- TRUE res <- back.test(results, start.date=start.date, end.date=end.date, n.first=n.first, include.bh=include.bh, FUN=Sharpe) df <- best.is.strategies(res, plot.results=true) xtab <- xtable(df, digits=3) print(xtab, include.rownames=false) ## End(Not run) descriptive.statistics Computes the summary statistics of the moving average trading strategy and the corresponding buy-and-hold strategy

5 descriptive.statistics 5 This function computes the summary statistics of the moving average trading strategy and the corresponding buy-and-hold strategy descriptive.statistics(results, nobs.a.year = 12, to.annual=sqrt(12), type=c("ordinary", "block", "stationary"), blen=5, R=1000, automatic=false, plot.results=false) results nobs.a.year to.annual type blen R automatic plot.results the results returned by function forward.test or back.test a scalar that defines the number of observations a year. For example, for monthly data the number of observations a year equals 12 a coefficient used to annualize the performance measure. For example, when the performance is measured using either Sharpe or Sortino, use sqrt(252) and sqrt(12) when the returns are at the daily and monthly frequency respectively. If the performance is measred by Excret, use 12 determines the type of the bootstrap method. The default type is "ordinary" (the method of Efron, 1979). The two other types are "block" (the method of Kunsch, 1989) and "stationary" (the method of Politis and Romano, 1994) a scalar that specifies the block length in the block-bootstrap method a scalar that specifies the number of bootstrap replicates a logical variable that specifies whether to compute the optimal block length for the block-bootstrap and stationary bootstrap a logical value indicating whether to plot the cumulative returns and drawdowns to the moving average strategy and the buy-and-hold strategy An object of class data.frame containing two columns: BH and MA. The first column contains the summary statistics for the Buy-and-Hold strategy, the second column contains the summary statistics for the Moving Average strategy. Each column contains the following rows: Mean returns annualized mean returns Std. deviation annualized standard deviation Minimum return minimum return Maximum return maximum return Skewness skewness of return distribution Kurtosis kurtosis of return distribution Average drawdown Average drawdown Average max drawdown Average of the 10 largest drawdowns Maximum drawdown Maximum drawdown Performance Performance measure P-value P-value of the test of outperformance Rolling 5-year Win Probability that the moving average strategy outperforms its passive counterpart over a 5-year horizon Rolling 10-year Win Probability that the moving average strategy outperforms its passive counterpart over a 10-year horizon

6 6 descriptive.statistics Note The length of the tested period should be longer than 10 years (this is needed to compute the Rolling 10-year Win). P-value is the p-value of the outperformance test. This p-value is computed in the same manner as in function outperformance.test. If results are produced by back.test, then this function computes the descriptive statistics and the p-value of the best strategy in the back test. In this case the p-value is not adjusted for the data-mining bias; hence, this p-value is unreliable. Otherwise, if results are produced by forward.test, then this function computes the descriptive statistics and the p-value of the out-of-sample strategy. Other strategy summary functions: best.is.strategies Examples ## Not run: library(matiming) library(xtable) library(zoo) data <- sp500.monthly dates <- index(data) data <- coredata(data) capret <- data[,"cap"] totret <- data[,"tot"] rfret <- data[,"rf"] tc < fast <- 1:9 slow <- 2:18 res.mac <- sim.mac.strategy(totret=totret, rfret=rfret, dates=dates, capret=capret, tc=tc, fast=fast, slow=slow, FUN=SMA) res.mom <- sim.mom.strategy(totret=totret, rfret=rfret, dates=dates, capret=capret, tc=tc, winsize=slow) results <- sim.results.merge(res.mac, res.mom) start.date <- as.date(" ", format="%y-%m-%d") end.is.date <- as.date(" ", format="%y-%m-%d") end.oos.date <- as.date(" ", format="%y-%m-%d") refit.every <- 1 rolling <- FALSE res <- forward.test(results, start.date, end.is.date, end.oos.date, refit.every=refit.every, rolling=rolling, FUN=Sharpe) df <- descriptive.statistics(res, plot.results=true) xtab <- xtable(df, digits=2) print(xtab) ## End(Not run)

7 djia.daily 7 djia.daily Daily returns of the Dow Jones Industrial Average index Format The daily returns of the DJIA index and the risk-free rate of return over the period from July 1926 to December 2015 djia.daily An object of class "zoo" containing the following variables: TOT total return (capital gain plus dividend yield) CAP capital gain return RF risk-free rate of return Other data sets: djia.monthly, sp500.daily, sp500.monthly djia.monthly Monthly returns of the Dow Jones Industrial Average index Format The monthly returns of the DJIA index and the risk-free rate of return over the period from July 1926 to December 2015 djia.monthly An object of class "zoo" containing the following variables: TOT total return (capital gain plus dividend yield) CAP capital gain return RF risk-free rate of return Other data sets: djia.daily, sp500.daily, sp500.monthly

8 8 EMA DMA Computes the Double Exponential Moving Average (DMA) This function computes the DMA of prices DMA(price, n) price n the vector of prices the length of the averaging window (a scalar) the DMA of prices (a vector) References Mulloy P. G. (1994). Smoothing Data With Faster Moving Averages, Technical Analysis of Stocks and Commodities, 12 (1), Other moving average functions: EMA, HMA, LMA, SMA, ZMA EMA Computes the Exponential Moving Average (EMA) This function computes the infinite EMA of prices EMA(price, n) price n the vector of prices the length of the averaging window (a scalar. the EMA of prices (a vector)

9 Excret 9 Note By definition, n is the length of the averaging window in SMA that has the same average lag time as EMA Other moving average functions: DMA, HMA, LMA, SMA, ZMA Excret Computes the mean excess return This function computes the mean excess return. Excess return is the return in excess of the risk-free rate of return Excret(x) x the vector of excess returns the mean excess return (a scalar) Other performance measures: Sharpe, Sortino forward.test Simulates the returns to the out-of-sample trading strategy This function simulates the returns to the out-of-sample trading strategy forward.test(results, start.date, end.is.date, end.oos.date, refit.every=1, rolling=false, include.bh=false, FUN=Sharpe)

10 10 forward.test results start.date end.is.date end.oos.date refit.every rolling include.bh FUN the results of simulation of a trading strategy or multiple trading strategies. Specifically, this is the result returned by function sim.mom.strategy, sim.mac.strategy, sim.cdir.strategy, sim.macd.strategy, or sim.results.merge the start of the in-sample period the end of the initial in-sample period the end of the out-of-sample period this parameter determines how often the best trading strategy in a back test is determined a logical value indicating whether to use the expanding (if FALSE) or rolling (if TRUE) in-sample window a logical value indicating whether to consider the Buy-and-Hold (bh) strategy in addition to the moving average strategies the function that computes the performance of a trading strategy An object to be used by other functions Note The argument results contains the dates vector that defines the historical period over which the trading strategies are simulated. The function FUN is either Excret, Sharpe, or Sortino. This function only simulates the out-of-sample strategy; to test the outperformance hypothesis, use either function outperformance.test or function descriptive.statistics. Other strategy testing functions: back.test, outperformance.test Examples ## Not run: library(matiming) library(xtable) library(zoo) # access the data data <- sp500.monthly dates <- index(data) data <- coredata(data) capret <- data[,"cap"] totret <- data[,"tot"] rfret <- data[,"rf"] # simulate the trading strategies tc < fast <- 1:5 slow <- 2:15 shorts <- F results <- sim.mac.strategy(totret=totret, rfret=rfret, dates=dates, capret=capret, tc=tc, shorts=shorts, fast=fast, slow=slow, FUN=SMA)

11 HMA 11 # perform the out-of-sample test start.date <- as.date(" ", format="%y-%m-%d") end.is.date <- as.date(" ", format="%y-%m-%d") end.oos.date <- as.date(" ", format="%y-%m-%d") refit.every <- 5 include.bh <- TRUE rolling <- TRUE res <- forward.test(results, start.date=start.date, end.is.date=end.is.date, end.oos.date=end.oos.date, refit.every=refit.every, rolling=rolling, include.bh=include.bh, FUN=Sortino) # report the descriptive statistics and the p-value of the test df <- descriptive.statistics(res) xtab <- xtable(df, digits=2) print(xtab) ## End(Not run) HMA Computes the Hull Moving Average (HMA) This function computes the HMA of prices HMA(price, n) price n the vector of prices the length of the averaging window (a scalar) the HMA of prices (a vector) References Hull, A. (2005). How to Reduce Lag in a Moving Average, Other moving average functions: DMA, EMA, LMA, SMA, ZMA

12 12 is.outperformance.plot is.outperformance.plot Plots the trading strategies rolling outperformance This function plots the trading strategies rolling outperformance is.outperformance.plot(results, start.date, split.date, end.date, FUN=Sharpe) results start.date split.date end.date FUN the results of the back-test or simulation of a trading strategy or multiple trading strategies. The former is the result returned by function back.test. The latter is the result returned by function sim.mom.strategy, sim.mac.strategy, sim.cdir.strategy, sim.macd.strategy, or sim.results.merge the start of the first rolling period the end of the first rolling period the end of the total period the function that computes the performance of a trading strategy. The function FUN is either Excret, Sharpe, or Sortino. Note None The argument results contains the dates vector that defines the historical period over which the trading strategies are simulated. The function FUN is only used if results is the result of simulation. The outperformance is defined as the difference between the performance of a trading strategy and the performance of the corresponding passive strategy. The number of plotted strategies is limited by 5. Other plotting functions: oos.outperformance.plot Examples ## Not run: library(matiming) library(zoo) # access the data data <- sp500.monthly dates <- index(data) data <- coredata(data) capret <- data[,"cap"] totret <- data[,"tot"]

13 LMA 13 rfret <- data[,"rf"] # simulate the SMA(10) strategy tc < fast <- 1 slow <- 10 shorts <- F results <- sim.mac.strategy(totret=totret, rfret=rfret, dates=dates, capret=capret, tc=tc, shorts=shorts, fast=fast, slow=slow, FUN=SMA) # plot the rolling outperformance start.date <- as.date(" ", format="%y-%m-%d") split.date <- as.date(" ", format="%y-%m-%d") end.date <- as.date(" ", format="%y-%m-%d") is.outperformance.plot(results, start.date=start.date, split.date=split.date, end.date=end.date, FUN=Sharpe) ## End(Not run) LMA Computes the Linear Moving Average (LMA) This function computes the LMA of prices LMA(price, n) price n the vector of prices the length of the averaging window (a scalar) the LMA of prices (a vector) Other moving average functions: DMA, EMA, HMA, SMA, ZMA

14 14 oos.outperformance.plot oos.outperformance.plot Plots the rolling outperformance of the OOS strategy This function plots the rolling outperformance of the out-of-sample trading strategy oos.outperformance.plot(results, start.date, split.date, end.date) results start.date split.date end.date the results returned by function forward.test the start of the first rolling period the end of the first rolling period the end of the total period Note None The argument results contains, among other variables, the dates vector that defines the historical period over which the trading strategies are simulated. The outperformance is defined as the difference between the performance of a trading strategy and the performance of the corresponding passive strategy. Other plotting functions: is.outperformance.plot Examples ## Not run: library(matiming) library(zoo) data <- sp500.monthly dates <- index(data) data <- coredata(data) capret <- data[,"cap"] totret <- data[,"tot"] rfret <- data[,"rf"] tc < fast <- 1:9 slow <- 2:18 res.mac <- sim.mac.strategy(totret=totret, rfret=rfret, dates=dates,

15 outperformance.test 15 capret=capret, tc=tc, fast=fast, slow=slow, FUN=SMA) res.mom <- sim.mom.strategy(totret=totret, rfret=rfret, dates=dates, capret=capret, tc=tc, winsize=slow) results <- sim.results.merge(res.mac, res.mom) start.date <- as.date(" ", format="%y-%m-%d") end.is.date <- as.date(" ", format="%y-%m-%d") end.oos.date <- as.date(" ", format="%y-%m-%d") res <- forward.test(results, start.date=start.date, end.is.date=end.is.date, end.oos.date=end.oos.date) # plot the rolling outperformance start.date <- as.date(" ", format="%y-%m-%d") split.date <- as.date(" ", format="%y-%m-%d") end.date <- as.date(" ", format="%y-%m-%d") oos.outperformance.plot(res, start.date=start.date, split.date=split.date, end.date=end.date) ## End(Not run) outperformance.test Tests whether the moving average strategy outperforms its passive counterpart This function tests whether the moving average strategy outperforms its passive counterpart. All tests are based on using a bootstrap method. outperformance.test(results, digits=3, to.annual=sqrt(12), type=c("ordinary", "block","stationary"), blen=5, R=1000, automatic=false) results digits to.annual type blen the results returned by function forward.test or back.test defines the number of digits after the decimal delimiter to use in reporting the performance of the moving average and buy-and-hold strategies a coefficient used to annualize the performance measure. For example, when the performance is measured using either Sharpe or Sortino, use sqrt(252) and sqrt(12) when the returns are at the daily and monthly frequency respectively. If the performance is measred by Excret, use 12 determines the type of the bootstrap method. The default type is "ordinary" (the method of Efron, 1979). The two other types are "block" (the method of Kunsch, 1989) and "stationary" (the method of Politis and Romano, 1994) a scalar that specifies the block length in the block-bootstrap method

16 16 outperformance.test R automatic a scalar that specifies the number of bootstrap replicates a logical variable that specifies whether to compute the optimal block length for the block-bootstrap and stationary bootstrap Note The p-value of the outperformance test. If results are produced by back.test, then this function tests whether the best strategy in the back test outperforms the buy-and-hold strategy. Note that in this case the p-value is not ajdusted for the data-mining bias. Otherwise, if results are produced by forward.test, then this function tests whether the out-of-sample strategy outperforms the buy-and-hold strategy. Regardless of the performance measure used (denoted by P M), this function performs the tests of the following null hypothesis: H 0 : P M MA P M BH, where P M MA is the performance of the moving average strategy and P M BH is the performance of the buy-and-hold strategy. In words, the null hypothesis is that the performance of the moving average strategy is not better than the performance of the buy-and-hold strategy. All tests are based on non-parametric bootstrap methods. The three bootstrap methods that are implemented are: (1) the ordinary bootstrap method of Efron (1979), (2) the block-bootstrap method of Kunsch (1989), and (3) the stationary block-bootstrap method of Politis and Romano (1994). In a block-bootstrap method one needs to either specify manually the block length or to allow the function to automatically determine the optimal block length. The optimal block length is computed using the method described in Patton, Politis and White (2009). To determine the optimal block length, function b.star from package np is used (Hayfield and Racine (2008)). References Efron, B., Bootstrap Methods: Another Look at the Jackknife. Annals of Statistics, 1979, 7, Kunsch, H.R., The Jacknife and the Bootstrap for General Stationary Observations. Annals of Statistics, 1989, 17, Politis, D. and Romano, J., The Stationary Bootstrap. Journal of the American Statistical Association, 1994, 89, Politis, D.N. and H. White (2004), Automatic block-length selection for the dependent bootstrap, Econometric Reviews 23 (1), Patton, A. and D.N. Politis and H. White (2009), CORRECTION TO: Automatic block-length selection for the dependent bootstrap by D. Politis and H. White, Econometric Reviews 28 (4), Hayfield, T. and J. Racine (2008). Nonparametric Econometrics: The np Package. Journal of Statistical Software, 27 (5). Other strategy testing functions: back.test, forward.test Examples # Suppose that "results" are the results of simulations of different trading strategies ## Not run: start.date <- as.date(" ", format="%y-%m-%d")

17 Sharpe 17 end.is.date <- as.date(" ", format="%y-%m-%d") end.oos.date <- as.date(" ", format="%y-%m-%d") refit.every <- 5 include.bh <- TRUE res <- forward.test(results, start.date=start.date, end.is.date=end.is.date, end.oos.date=end.oos.date, refit.every=refit.every, include.bh=include.bh, FUN=Sortino) outperformance.test(res, type="stationary", automatic=true) ## End(Not run) Sharpe Computes the Sharpe ratio This function computes the Sharpe ratio Sharpe(x) x the vector of excess returns the Sharpe ratio (a scalar) References Sharpe, W.F. (1994). The Sharpe Ratio. Journal of Portfolio Management, 21 (1), Other performance measures: Excret, Sortino sim.cdir.strategy Simulates the returns to the Moving Average Change of Direction (CDIR) strategy This function simulates the returns to the CDIR strategy sim.cdir.strategy(totret, rfret, dates, capret=null, tc=0, shorts = FALSE, winsize=200, FUN=SMA)

18 18 sim.cdir.strategy totret rfret dates capret tc shorts winsize FUN the vector of total returns the vector of risk-free returns the vector of dates the vector of capital gain returns the amount of one-way transaction costs (a scalar) a logical value indicating the type of the strategy when a Sell signal is generated. If shorts=false, money is just invested in cash; otherwise, the asset is sold short a vector or a scalar that determines the size of the window used to compute a moving average a function that determines the type of a moving average An object to be used by other functions. This object contains the returns to the simulated strategies and the description of each strategy (the type of a moving average and the sizes of the windows) Note The function FUN is either SMA, LMA, EMA, DMA, HMA or ZMA. The trading signal is computed using the vector of capital gain returns capret. If capret=null, then the trading signal is computed using the vector of total returns totret. If winsize is a vector, then the function simulates a number of trading strategies for all window sizes such that winsize > 1 Other strategy simulation functions: sim.mac.strategy, sim.macd.strategy, sim.mae.strategy, sim.mom.strategy Examples ## Not run: data <- djia.monthly # use the monthly data for DJIA index dates <- index(data) data <- coredata(data) capret <- data[,"cap"] totret <- data[,"tot"] rfret <- data[,"rf"] tc < # this defines 0.25% one-way transaction costs winsize <- 2:15 shorts <- FALSE results <- sim.cdir.strategy(totret=totret, rfret=rfret, dates=dates, capret=capret, tc=tc, shorts=shorts, winsize=winsize, FUN=LMA) ## End(Not run)

19 sim.mac.strategy 19 sim.mac.strategy Simulates the returns to the Moving Average Crossover (MAC) strategy This function simulates the returns to the MAC strategy sim.mac.strategy(totret, rfret, dates, capret=null, tc=0, shorts = FALSE, fast=50, slow=200, FUN=SMA) totret rfret dates capret tc shorts fast slow FUN the vector of total returns the vector of risk-free returns the vector of dates the vector of capital gain returns the amount of one-way transaction costs (a scalar) a logical value indicating the type of the strategy when a Sell signal is generated. If shorts=false, money is just invested in cash; otherwise, the asset is sold short a vector or a scalar that determines the size of the shorter averaging window a vector or a scalar that determines the size of the longer averaging window a function that determines the type of a moving average An object to be used by other functions. This object contains the returns to the simulated strategies and the description of each strategy (the type of a moving average, the sizes of the shorter and the longer windows) Note The function FUN is either SMA, LMA, EMA, DMA, HMA or ZMA. The trading signal is computed using the vector of capital gain returns capret. If capret=null, then the trading signal is computed using the vector of total returns totret. If one of the arguments fast or slow is a vector, then the function simulates a number of trading strategies for all combinations of fast and slow such that fast < slow Other strategy simulation functions: sim.cdir.strategy, sim.macd.strategy, sim.mae.strategy, sim.mom.strategy

20 20 sim.macd.strategy Examples ## Not run: data <- djia.monthly # use the monthly data for DJIA index dates <- index(data) data <- coredata(data) capret <- data[,"cap"] totret <- data[,"tot"] rfret <- data[,"rf"] tc < # this defines 0.25% one-way transaction costs fast <- 1:8 # define the sizes of the fast (shorter) averaging window slow <- 2:15 # define the sizes of the slow (longer) averaging window shorts <- FALSE results <- sim.mac.strategy(totret=totret, rfret=rfret, dates=dates, capret=capret, tc=tc, shorts=shorts, fast=fast, slow=slow, FUN=EMA) ## End(Not run) sim.macd.strategy Simulates the returns to the Moving Average Convergence Divergence (MACD) strategy This function simulates the returns to the MACD strategy sim.macd.strategy(totret, rfret, dates, capret=null, tc=0, shorts = FALSE, fast=12, slow=26, final=9, FUN=EMA) totret rfret dates capret tc shorts fast slow final FUN the vector of total returns the vector of risk-free returns the vector of dates the vector of capital gain returns the amount of one-way transaction costs (a scalar) a logical value indicating the type of the strategy when a Sell signal is generated. If shorts=false, money is just invested in cash; otherwise, the asset is sold short a vector or a scalar that determines the size of the shorter averaging window a vector or a scalar that determines the size of the longer averaging window a vector or a scalar that determines the size of the window for final smoothing a function that determines the type of a moving average An object to be used by other functions. This object contains the returns to the simulated strategies and the description of each strategy (the type of a moving average, the sizes of the shorter and the longer windows, the size of the window for final smoothing)

21 sim.mae.strategy 21 Note The function FUN is either SMA, LMA, EMA, DMA, HMA or ZMA. The trading signal is computed using the vector of capital gain returns capret. If capret=null, then the trading signal is computed using the vector of total returns totret. If one of the arguments fast, slow or final is a vector, then the function simulates a number of trading strategies for all combinations of fast, slow and final such that fast < slow and final > 1. Other strategy simulation functions: sim.cdir.strategy, sim.mac.strategy, sim.mae.strategy, sim.mom.strategy Examples ## Not run: data <- djia.daily # use the daily data for DJIA index dates <- index(data) data <- coredata(data) capret <- data[,"cap"] totret <- data[,"tot"] rfret <- data[,"rf"] tc < # this defines 0.25% one-way transaction costs fast <- 1:15 # define the sizes of the fast (shorter) averaging window slow <- 2:45 # define the sizes of the slow (longer) averaging window final <- 2:20 # defines the sizes of the window for final smoothing shorts <- FALSE results <- sim.macd.strategy(totret=totret, rfret=rfret, dates=dates, capret=capret, tc=tc, shorts=shorts, fast=fast, slow=slow, final=final, FUN=EMA) ## End(Not run) sim.mae.strategy Simulates the returns to the Moving Average Envelope (MAE) strategy This function simulates the returns to the MAE strategy sim.mae.strategy(totret, rfret, dates, capret=null, tc=0, shorts = FALSE, winsize=200, percentage=1, FUN=SMA) totret rfret dates capret tc the vector of total returns the vector of risk-free returns the vector of dates the vector of capital gain returns the amount of one-way transaction costs (a scalar)

22 22 sim.mom.strategy shorts winsize percentage FUN a logical value indicating the type of the strategy when a Sell signal is generated. If shorts=false, money is just invested in cash; otherwise, the asset is sold short a vector or a scalar that determines the size of the averaging window a vector or a scalar that determines the envelope percentage a function that determines the type of a moving average An object to be used by other functions. This object contains the returns to the simulated strategies and the description of each strategy (the type of a moving average, the sizes of the averaging windows, and the envelope percentages) Note The function FUN is either SMA, LMA, EMA, DMA, HMA or ZMA. The trading signal is computed using the vector of capital gain returns capret. If capret=null, then the trading signal is computed using the vector of total returns totret. Other strategy simulation functions: sim.cdir.strategy, sim.mac.strategy, sim.macd.strategy, sim.mom.strategy Examples ## Not run: data <- djia.monthly # use the monthly data for DJIA index dates <- index(data) data <- coredata(data) capret <- data[,"cap"] totret <- data[,"tot"] rfret <- data[,"rf"] tc < # this defines 0.25% one-way transaction costs winsize <- 2:20 percentage <- 0:10 shorts <- FALSE results <- sim.mae.strategy(totret=totret, rfret=rfret, dates=dates, capret=capret, tc=tc, winsize=winsize, percentage=percentage, slow=slow, FUN=EMA) ## End(Not run) sim.mom.strategy Simulates the returns to the Momentum (MOM) strategy This function simulates the returns to the MOM strategy

23 sim.mom.strategy 23 sim.mom.strategy(totret, rfret, dates, capret=null, tc=0, shorts = FALSE, winsize=200) totret rfret dates capret tc shorts winsize the vector of total returns the vector of risk-free returns the vector of dates the vector of capital gain returns the amount of one-way transaction costs (a scalar) a logical value indicating the type of the strategy when a Sell signal is generated. If shorts=false, money is just invested in cash; otherwise, the asset is sold short a vector or a scalar that determines the size of the window used to compute the MOM An object to be used by other functions. This object contains the returns to the simulated strategies and the description of each strategy Note The trading signal is computed using the vector of capital gain returns capret. If capret=null, then the trading signal is computed using the vector of total returns totret. If winsize is a vector, then the function simulates a number of trading strategies for all window sizes such that winsize > 1 Other strategy simulation functions: sim.cdir.strategy, sim.mac.strategy, sim.macd.strategy, sim.mae.strategy Examples ## Not run: data <- djia.monthly # use the monthly data for DJIA index dates <- index(data) data <- coredata(data) capret <- data[,"cap"] totret <- data[,"tot"] rfret <- data[,"rf"] tc < # this defines 0.25% one-way transaction costs winsize <- 2:15 shorts <- FALSE results <- sim.mom.strategy(totret=totret, rfret=rfret, dates=dates, capret=capret, tc=tc, shorts=shorts, winsize=winsize) ## End(Not run)

24 24 SMA sim.results.merge Merges the results of simulating the returns to different trading strategies This function merges the results of simulating the returns to different trading strategies sim.results.merge(x, y,...) x y the result of simulation of a trading strategy. Specifically, this is the result returned by function sim.mom.strategy, sim.mac.strategy, sim.cdir.strategy, or sim.macd.strategy the result of simulation of another trading strategy. Again, this is the result returned by function sim.mom.strategy, sim.mac.strategy, sim.cdir.strategy, or sim.macd.strategy... other possible results of simulations if more than two results must be merged An object to be used by other functions. This object is a result of merging two or multiple results of simulating the returns to different trading strategies Examples # Suppose that "res1", "res2", and "res3" are the results of simulations of # three different trading strategies. The following command merges the three results ## Not run: results <- sim.results.merge(res1, res2, res3) ## End(Not run) SMA Computes the Simple Moving Average (SMA) This function computes the SMA of prices SMA(price, n) price n the vector of prices the length of the averaging window (a scalar)

25 Sortino 25 the SMA of prices (a vector) Other moving average functions: DMA, EMA, HMA, LMA, ZMA Sortino Computes the Sortino ratio This function computes the Sortino ratio Sortino(x) x the vector of excess returns the Sortino ratio (a scalar) References Sortino, F.A. and Price, L.N. (1994). Performance measurement in a downside risk framework. Journal of Investing, 3 (3), Other performance measures: Excret, Sharpe sp500.daily Daily returns of the Standard and Poor s Composite index The daily returns of the S&P Composite index and the risk-free rate of return over the period from July 1926 to December 2015 sp500.daily

26 26 ZMA Format An object of class "zoo" containing the following variables: TOT total return (capital gain plus dividend yield) CAP capital gain return RF risk-free rate of return Other data sets: djia.daily, djia.monthly, sp500.monthly sp500.monthly Monthly returns of the Standard and Poor s Composite index Format The monthly returns of the S&P Composite index and the risk-free rate of return over the period from January 1857 to December 2015 sp500.monthly An object of class "zoo" containing the following variables: TOT total return (capital gain plus dividend yield) CAP capital gain return RF risk-free rate of return Other data sets: djia.daily, djia.monthly, sp500.daily ZMA Computes the Zero Lag Exponential Moving Average (ZMA) This function computes the ZMA of prices ZMA(price, n) price n the vector of prices the length of the averaging window (a scalar)

27 ZMA 27 the ZMA of prices (a vector) References Ehlers J. F. and Way R. (2010). Zero Lag (Well, Almost), Technical Analysis of Stocks and Commodities, 28 (12), Other moving average functions: DMA, EMA, HMA, LMA, SMA

28 Index back.test, 2, 3, 5, 6, 10, 12, 15, 16 best.is.strategies, 3, 6 descriptive.statistics, 4, 4, 10 djia.daily, 7, 7, 26 djia.monthly, 7, 7, 26 DMA, 8, 9, 11, 13, 18, 19, 21, 22, 25, 27 EMA, 8, 8, 11, 13, 18, 19, 21, 22, 25, 27 Excret, 2, 3, 5, 9, 10, 12, 15, 17, 25 forward.test, 2, 5, 6, 9, HMA, 8, 9, 11, 13, 18, 19, 21, 22, 25, 27 is.outperformance.plot, 12, 14 LMA, 8, 9, 11, 13, 18, 19, 21, 22, 25, 27 oos.outperformance.plot, 12, 14 outperformance.test, 2, 6, 10, 15 Sharpe, 2, 3, 5, 9, 10, 12, 15, 17, 25 sim.cdir.strategy, 2, 10, 12, 17, 19, sim.mac.strategy, 2, 10, 12, 18, 19, sim.macd.strategy, 2, 10, 12, 18, 19, 20, sim.mae.strategy, 18, 19, 21, 21, 23 sim.mom.strategy, 2, 10, 12, 18, 19, 21, 22, 22, 24 sim.results.merge, 2, 10, 12, 24 SMA, 8, 9, 11, 13, 18, 19, 21, 22, 24, 27 Sortino, 2, 3, 5, 9, 10, 12, 15, 17, 25 sp500.daily, 7, 25, 26 sp500.monthly, 7, 26, 26 ZMA, 8, 9, 11, 13, 18, 19, 21, 22, 25, 26 28