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Probabilistic Momentum

February 17, 2014 13 comments

David Varadi has recently discussed an interesting strategy in the
Are Simple Momentum Strategies Too Dumb? Introducing Probabilistic Momentum post. David also provided the Probabilistic Momentum Spreadsheet if you are interested in doing computations in Excel. Today I want to show how you can test such strategy using the Systematic Investor Toolbox:

###############################################################################
# Load Systematic Investor Toolbox (SIT)
# https://systematicinvestor.wordpress.com/systematic-investor-toolbox/
###############################################################################
setInternet2(TRUE)
con = gzcon(url('http://www.systematicportfolio.com/sit.gz', 'rb'))
    source(con)
close(con)
	#*****************************************************************
	# Load historical data
	#****************************************************************** 
	load.packages('quantmod')
		
	tickers = spl('SPY,TLT')
		
	data <- new.env()
	getSymbols(tickers, src = 'yahoo', from = '1980-01-01', env = data, auto.assign = T)
		for(i in ls(data)) data[[i]] = adjustOHLC(data[[i]], use.Adjusted=T)
	bt.prep(data, align='remove.na', dates='2005::')
 
	
	#*****************************************************************
	# Setup
	#****************************************************************** 
	lookback.len = 60
	
	prices = data$prices
	
	models = list()
	
	#*****************************************************************
	# Simple Momentum
	#****************************************************************** 
	momentum = prices / mlag(prices, lookback.len)
	data$weight[] = NA
		data$weight$SPY[] = momentum$SPY > momentum$TLT
		data$weight$TLT[] = momentum$SPY <= momentum$TLT
	models$Simple  = bt.run.share(data, clean.signal=T) 	

The Simple Momentum strategy invests into SPY if SPY’s momentum if greater than TLT’s momentum, and invests into TLT otherwise.

	#*****************************************************************
	# Probabilistic Momentum
	#****************************************************************** 
	confidence.level = 60/100
	ret = prices / mlag(prices) - 1 

	ir = sqrt(lookback.len) * runMean(ret$SPY - ret$TLT, lookback.len) / runSD(ret$SPY - ret$TLT, lookback.len)
	momentum.p = pt(ir, lookback.len - 1)
		
	data$weight[] = NA
		data$weight$SPY[] = iif(cross.up(momentum.p, confidence.level), 1, iif(cross.dn(momentum.p, (1 - confidence.level)), 0,NA))
		data$weight$TLT[] = iif(cross.dn(momentum.p, (1 - confidence.level)), 1, iif(cross.up(momentum.p, confidence.level), 0,NA))
	models$Probabilistic  = bt.run.share(data, clean.signal=T) 	

The Probabilistic Momentum strategy is using Probabilistic Momentum measure and Confidence Level to decide on allocation. Strategy invests into SPY if SPY vs TLT Probabilistic Momentum is above Confidence Level and invests into TLT is SPY vs TLT Probabilistic Momentum is below 1 – Confidence Level.

To make Strategy a bit more attractive, I added a version that can leverage SPY allocation by 50%

	#*****************************************************************
	# Probabilistic Momentum + SPY Leverage 
	#****************************************************************** 
	data$weight[] = NA
		data$weight$SPY[] = iif(cross.up(momentum.p, confidence.level), 1, iif(cross.up(momentum.p, (1 - confidence.level)), 0,NA))
		data$weight$TLT[] = iif(cross.dn(momentum.p, (1 - confidence.level)), 1, iif(cross.up(momentum.p, confidence.level), 0,NA))
	models$Probabilistic.Leverage = bt.run.share(data, clean.signal=T) 	

	#*****************************************************************
	# Create Report
	#******************************************************************    
	strategy.performance.snapshoot(models, T)

plot1

The back-test results look very similar to the ones reported in the Are Simple Momentum Strategies Too Dumb? Introducing Probabilistic Momentum post.

However, I was not able to exactly reproduce the transition plots. Looks like my interpretation is producing more whipsaw when desired.

	#*****************************************************************
	# Visualize Signal
	#******************************************************************        
	cols = spl('steelblue1,steelblue')
	prices = scale.one(data$prices)

	layout(1:3)
	
	plota(prices$SPY, type='l', ylim=range(prices), plotX=F, col=cols[1], lwd=2)
	plota.lines(prices$TLT, type='l', plotX=F, col=cols[2], lwd=2)
		plota.legend('SPY,TLT',cols,as.list(prices))

	highlight = models$Probabilistic$weight$SPY > 0
		plota.control$col.x.highlight = iif(highlight, cols[1], cols[2])
	plota(models$Probabilistic$equity, type='l', plotX=F, x.highlight = highlight | T)
		plota.legend('Probabilistic,SPY,TLT',c('black',cols))
				
	highlight = models$Simple$weight$SPY > 0
		plota.control$col.x.highlight = iif(highlight, cols[1], cols[2])
	plota(models$Simple$equity, type='l', plotX=T, x.highlight = highlight | T)
		plota.legend('Simple,SPY,TLT',c('black',cols))	

plot2

David thank you very much for sharing your great ideas. I would encourage readers to play with this strategy and report back.

To view the complete source code for this example, please have a look at the bt.probabilistic.momentum.test() function in bt.test.r at github.

Weekend Reading: F-Squared

December 7, 2013 7 comments

Mebane Faber posted another interesting blog post: Building a Simple Sector Rotation on Momentum and Trend that caught my interest. Today I want to show how you can test such strategy using the Systematic Investor Toolbox:

###############################################################################
# Load Systematic Investor Toolbox (SIT)
# https://systematicinvestor.wordpress.com/systematic-investor-toolbox/
###############################################################################
setInternet2(TRUE)
con = gzcon(url('http://www.systematicportfolio.com/sit.gz', 'rb'))
    source(con)
close(con)

	#*****************************************************************
	# Load historical data
	#******************************************************************    
	load.packages('quantmod')		
	
	data = new.env()
	# load historical market returns
	temp = get.fama.french.data('F-F_Research_Data_Factors', periodicity = '',download = T, clean = T)
		ret = cbind(temp[[1]]$Mkt.RF + temp[[1]]$RF, temp[[1]]$RF)
		price = bt.apply.matrix(ret / 100, function(x) cumprod(1 + x))
	data$SPY = make.stock.xts( price$Mkt.RF )
	data$SHY = make.stock.xts( price$RF )
	
	# load historical sector returns
	temp = get.fama.french.data('10_Industry_Portfolios', periodicity = '',download = T, clean = T)		
		ret = temp[[1]]
		price = bt.apply.matrix(ret[,1:9] / 100, function(x) cumprod(1 + x))
	for(n in names(price)) data[[n]] = make.stock.xts( price[,n] )
	
	# align dates
	data$symbolnames = c(names(price), 'SHY', 'SPY')
	bt.prep(data, align='remove.na', dates='2000::')

	# back-test dates
	bt.dates = '2001:04::'

	#*****************************************************************
	# Setup
	#****************************************************************** 	
	prices = data$prices  
	n = ncol(data$prices)
		
	models = list()
	
	#*****************************************************************
	# Benchmark Strategies
	#****************************************************************** 			
	data$weight[] = NA
		data$weight$SPY[1] = 1
	models$SPY = bt.run.share(data, clean.signal=F, dates=bt.dates)
			
	weight = prices
		weight$SPY = NA
		weight$SHY = NA
	
	data$weight[] = NA
		data$weight[] = ntop(weight[], n)
	models$EW = bt.run.share(data, clean.signal=F, dates=bt.dates)
	
	#*****************************************************************
	# Code Strategies
	# http://www.mebanefaber.com/2013/12/04/square-root-of-f-squared/
	#****************************************************************** 			
	sma = bt.apply.matrix(prices, SMA, 10)
	
	# create position score
	position.score = sma
	position.score[ prices < sma ] = NA
		position.score$SHY = NA	
		position.score$SPY = NA	
	
	# equal weight allocation
	weight = ntop(position.score[], n)	
	
	# number of invested funds
	n.selected = rowSums(weight != 0)
	
	# cash logic
	weight$SHY[n.selected == 0,] = 1
	
	weight[n.selected == 1,] = 0.25 * weight[n.selected == 1,]
	weight$SHY[n.selected == 1,] = 0.75
	
	weight[n.selected == 2,] = 0.5 * weight[n.selected == 2,]
	weight$SHY[n.selected == 2,] = 0.5
	
	weight[n.selected == 3,] = 0.75 * weight[n.selected == 3,]
	weight$SHY[n.selected == 3,] = 0.25
	
	# cbind(round(100*weight,0), n.selected)	
	
	data$weight[] = NA
		data$weight[] = weight
	models$strategy1 = bt.run.share(data, clean.signal=F, dates=bt.dates)	
	
	#*****************************************************************
	# Create Report
	#******************************************************************       	
	strategy.performance.snapshoot(models, one.page = T)

plot1

Mebane thank you very much for sharing your great ideas. I would encourage readers to play with this strategy and report back.

Please note that I back-tested the strategy using the monthly observations. The strategy’s draw-down is around 17% using monthly data. If we switch to the daily data, the strategy’s draw-down goes to around 22%. There was one really bad month in 2002.

To view the complete source code for this example, please have a look at the bt.mebanefaber.f.squared.test() function in bt.test.r at github.

Averaged Input Assumptions and Momentum

December 5, 2013 5 comments

Today I want to share another interesting idea contributed by Pierre Chretien. Pierre suggested using Averaged Input Assumptions and Momentum to create reasonably quiet strategy. The averaging techniques are used to avoid over-fitting any particular frequency.

To create Averaged Input Assumptions we combine returns over different look-back periods, giving more weight to the recent returns, to form overall Input Assumptions.

create.ia.averaged <- function(lookbacks, n.lag) {
	lookbacks = lookbacks
	n.lag = n.lag

	function(hist.returns, index=1:ncol(hist.returns), hist.all)
	{	
		nperiods = nrow(hist.returns)
		
		temp = c()
		for (n.lookback in lookbacks) 
			temp = rbind(temp, hist.returns[(nperiods - n.lookback - n.lag + 1):(nperiods - n.lag), ])
		create.ia(temp, index, hist.all)
	}	
}

To create Averaged Momentum we take a look-back weighted avaerage of momentums computed over different look-back periods.

momentum.averaged <- function(prices, 
	lookbacks = c(20,60,120,250) ,	# length of momentum look back
	n.lag = 3
) {
	momentum = 0 * prices
	for (n.lookback in lookbacks) {
		part.mom = mlag(prices, n.lag) / mlag(prices, n.lookback + n.lag) - 1
		momentum = momentum + 252 / n.lookback * part.mom
	}
	momentum / len(lookbacks)
}

Next let’s compare using historical Input Assumptions vs Averaged Input Assumptions and Momentum vs Averaged Momentum. I will consider Absolute Momentum (not cross sectional), for more information about relative and absolute momentum, please see

###############################################################################
# Load Systematic Investor Toolbox (SIT)
# https://systematicinvestor.wordpress.com/systematic-investor-toolbox/
###############################################################################
setInternet2(TRUE)
con = gzcon(url('http://www.systematicportfolio.com/sit.gz', 'rb'))
    source(con)
close(con)
 
	#*****************************************************************
	# Load historical data
	#****************************************************************** 
	load.packages('quantmod')
		
	# 10 funds
	tickers = spl('Us.Eq = VTI + VTSMX,
	Eurpoe.Eq = IEV + FIEUX,
	Japan.Eq = EWJ + FJPNX,
	Emer.Eq = EEM + VEIEX,
	Re = RWX + VNQ + VGSIX,		
	Com = DBC + QRAAX,
	Gold = GLD + SCGDX,
	Long.Tr = TLT + VUSTX,
	Mid.Tr = IEF + VFITX,
	Short.Tr = SHY + VFISX') 
	
	start.date = 1998
	
	dates = paste(start.date,'::',sep='') 
	
	data <- new.env()
	getSymbols.extra(tickers, src = 'yahoo', from = '1980-01-01', env = data, set.symbolnames = T, auto.assign = T)
		for(i in data$symbolnames) data[[i]] = adjustOHLC(data[[i]], use.Adjusted=T)
	bt.prep(data, align='keep.all', dates=paste(start.date-2,':12::',sep=''), fill.gaps = T)

	#*****************************************************************
	# Setup
	#****************************************************************** 		
	prices = data$prices   
		n = ncol(prices)
		nperiods = nrow(prices)
		
	periodicity = 'months'
	period.ends = endpoints(prices, periodicity)
		period.ends = period.ends[period.ends > 0]
		
	max.product.exposure = 0.6	
	
	#*****************************************************************
	# Input Assumptions
	#****************************************************************** 	
	lookback.len = 40
	create.ia.fn = create.ia
	
	# input assumptions are averaged on 20, 40, 60 days using 1 day lag
	ia.array = c(20,40,60)
	avg.create.ia.fn = create.ia.averaged(ia.array, 1)

	#*****************************************************************
	# Momentum
	#****************************************************************** 	
	universe = prices > 0
	
	mom.lookback.len = 120	
	momentum = prices / mlag(prices, mom.lookback.len) - 1
	mom.universe = ifna(momentum > 0, F)
	
	# momentum is averaged on 20,60,120,250 days using 3 day lag
	mom.array = c(20,60,120,250)	
	avg.momentum = momentum.averaged(prices, mom.array, 3)
	avgmom.universe = ifna(avg.momentum > 0, F)

	#*****************************************************************
	# Algos
	#****************************************************************** 	
	min.risk.fns = list(
		EW = equal.weight.portfolio,
		MV = min.var.portfolio,
		MCE = min.corr.excel.portfolio,
				
		MV.RSO = rso.portfolio(min.var.portfolio, 3, 100, const.ub = max.product.exposure),
		MCE.RSO = rso.portfolio(min.corr.excel.portfolio, 3, 100, const.ub = max.product.exposure)
	)

	#*****************************************************************
	# Code Strategies
	#****************************************************************** 	
make.strategy.custom <- function(name, create.ia.fn, lookback.len, universe, env) {
	obj = portfolio.allocation.helper(data$prices, 
		periodicity = periodicity,
		universe = universe,
		lookback.len = lookback.len,
		create.ia.fn = create.ia.fn,
		const.ub = max.product.exposure,
		min.risk.fns = min.risk.fns,
		adjust2positive.definite = F
	)
	env[[name]] = create.strategies(obj, data, prefix=paste(name,'.',sep=''))$models
}


	models <- new.env()	
	make.strategy.custom('ia.none'        , create.ia.fn    , lookback.len, universe       , models)
	make.strategy.custom('ia.mom'         , create.ia.fn    , lookback.len, mom.universe   , models)
	make.strategy.custom('ia.avg_mom'     , create.ia.fn    , lookback.len, avgmom.universe, models)
	make.strategy.custom('avg_ia.none'    , avg.create.ia.fn, 252         , universe       , models)
	make.strategy.custom('avg_ia.mom'     , avg.create.ia.fn, 252         , mom.universe   , models)
	make.strategy.custom('avg_ia.avg_mom' , avg.create.ia.fn, 252         , avgmom.universe, models)
	
	#*****************************************************************
	# Create Report
	#*****************************************************************		
strategy.snapshot.custom <- function(models, n = 0, title = NULL) {
	if (n > 0)
		models = models[ as.vector(matrix(1:len(models),ncol=n, byrow=T)) ]	

	layout(1:3)	
	plotbt(models, plotX = T, log = 'y', LeftMargin = 3, main = title)
		mtext('Cumulative Performance', side = 2, line = 1)
	plotbt.strategy.sidebyside(models)
	barplot.with.labels(sapply(models, compute.turnover, data), 'Average Annual Portfolio Turnover', T)	
}

	# basic vs basic + momentum => momentum filter has better results
	models.final = c(models$ia.none, models$ia.mom)
	strategy.snapshot.custom(models.final, len(min.risk.fns), 'Momentum Filter')

	# basic vs basic + avg ia => averaged ia reduce turnover
	models.final = c(models$ia.none, models$avg_ia.none)
	strategy.snapshot.custom(models.final, len(min.risk.fns), 'Averaged Input Assumptions')

	# basic + momentum vs basic + avg.momentum => mixed results for averaged momentum
	models.final = c(models$ia.mom, models$ia.avg_mom)
	strategy.snapshot.custom(models.final, len(min.risk.fns), 'Averaged Momentum')

	# basic + momentum vs avg ia + avg.momentum
	models.final = c(models$ia.mom, models$avg_ia.avg_mom)
	strategy.snapshot.custom(models.final, len(min.risk.fns), 'Averaged vs Base')	

Above, I compared results for the following 4 cases:
1. Adding Momentum filter: all algos perfrom better
plot3

2. Input Assumptions vs Averaged Input Assumptions: returns are very similar, but using Averaged Input Assumptions helps reduce portfolio turnover.
plot2

3. Momentum vs Averaged Momentum: returns are very similar, but using Averaged Momentum increases portfolio turnover.
plot1

4. historical Input Assumptions + Momentum vs Averaged Input Assumptions + Averaged Momentum: results are mixed, no consistent advantage of using Averaged methods
plot4

Overall, the Averaged methods is a very interesting idea and I hope you will experiemtn with it and share your findings, like Pierre. Pierre, again thank you very much for sharing.

The full source code and example for the bt.averaged.test() function is available in bt.test.r at github.

Running Back-tests in parallel

November 11, 2013 2 comments

Once you start experimenting with many different asset allocation algorithms, the computation time of running the back-tests can be substantial. One simple way to solve the computation time problem is to run the back-tests in parallel. I.e. if the asset allocation algorithm does not use the prior period holdings to make decision about current allocation, we can run many periods in parallel.

In the Update for Backtesting Asset Allocation Portfolios post, I show cased the portfolio.allocation.helper() function in strategy.r at github. The portfolio.allocation.helper() function is a user-friendly interface to evaluate multiple asset allocation algorithms over given asset universe in a sequential fashion.

Following is a sample code from the Update for Backtesting Asset Allocation Portfolios post:

    #*****************************************************************
    # Code Strategies
    #******************************************************************                    
    obj = portfolio.allocation.helper(data$prices,
        periodicity = 'months', lookback.len = 60,
        min.risk.fns = list(
            EW=equal.weight.portfolio,
            RP=risk.parity.portfolio,
            MD=max.div.portfolio,                      
             
            MV=min.var.portfolio,
            MVE=min.var.excel.portfolio,
            MV2=min.var2.portfolio,
             
            MC=min.corr.portfolio,
            MCE=min.corr.excel.portfolio,
            MC2=min.corr2.portfolio,
             
            MS=max.sharpe.portfolio(),
            ERC = equal.risk.contribution.portfolio,
 
            # target retunr / risk
            TRET.12 = target.return.portfolio(12/100),                             
            TRISK.10 = target.risk.portfolio(10/100),
         
            # rso
            RSO.RP.5 = rso.portfolio(risk.parity.portfolio, 5, 500),
             
            # others
            MMaxLoss = min.maxloss.portfolio,
            MMad = min.mad.portfolio,
            MCVaR = min.cvar.portfolio,
            MCDaR = min.cdar.portfolio,
            MMadDown = min.mad.downside.portfolio,
            MRiskDown = min.risk.downside.portfolio,
            MCorCov = min.cor.insteadof.cov.portfolio
        )
    )

To run the same strategies in parallel, I created the portfolio.allocation.helper.parallel() function in strategy.r at github. There is one extra input that you need to specify: cores – number of CPU processors used for computations.

For example, the code below will use 2 CPU processors to run back-test computations. It will run faster than the portfolio.allocation.helper() function.

    #*****************************************************************
    # Code Strategies
    #******************************************************************                    
    obj = portfolio.allocation.helper.parallel(cores = 2, data$prices,
        periodicity = 'months', lookback.len = 60,
        min.risk.fns = list(
            EW=equal.weight.portfolio,
            RP=risk.parity.portfolio,
            MD=max.div.portfolio,                      
             
            MV=min.var.portfolio,
            MVE=min.var.excel.portfolio,
            MV2=min.var2.portfolio,
             
            MC=min.corr.portfolio,
            MCE=min.corr.excel.portfolio,
            MC2=min.corr2.portfolio,
             
            MS=max.sharpe.portfolio(),
            ERC = equal.risk.contribution.portfolio,
 
            # target retunr / risk
            TRET.12 = target.return.portfolio(12/100),                             
            TRISK.10 = target.risk.portfolio(10/100),
         
            # rso
            RSO.RP.5 = rso.portfolio(risk.parity.portfolio, 5, 500),
             
            # others
            MMaxLoss = min.maxloss.portfolio,
            MMad = min.mad.portfolio,
            MCVaR = min.cvar.portfolio,
            MCDaR = min.cdar.portfolio,
            MMadDown = min.mad.downside.portfolio,
            MRiskDown = min.risk.downside.portfolio,
            MCorCov = min.cor.insteadof.cov.portfolio
        )
    )

Hopefully, I did not ruin your prolong lunch plans 🙂

Commissions

November 5, 2013 1 comment

Today, I want to explain the commission’s functionality build in to Systematic Investor Toolbox(SIT) “share” back-test.

At each re-balance time the capital is allocated given the weight such that

	share = weight * capital / price
	cash  = capital - share * price

For example, if weight is 100% (i.e. fully invested) and capital = $100 and price = $10 then

	share = 10 shares
	cash = $0

The period return is equal to

	return = [share * price + cash] / [share * price.yesterday + cash] - 1

The total return is equal to

	total.return = [1 + return.0] * [1 + return.1] * ... * [1 + return.n] - 1

The period returns constructed this way let me construct portfolio returns without using loops
I.e. if share, price, cash are matrices, then

	portfolio.return = rowSums((share * price + cash) / (share * mlag(price) + cash) - 1)

and

	equity = cumprod(1 + portfolio.return)

To introduce commissions into above framework, I had to make following assumptions.
Let’ assume that P0 and P1 are stock prices and com is commission that is very small relative to stock price then

	[P0 - com] / P0 is close (equal) to P0 / [P0 + com] and
	[P0 - com] * P1 is close (equal) to P0 * [P1 - com]

Given commissions, the period return formula used in SIT is equal to

	return = [share * price + cash - commission] / [share * price.yesterday + cash] - 1

Now let’s look at the example of trade with commissions:

	Let's say we are fully invested (i.e. cash = 0 and capital = share * P0)

	opening trade cost = share * P0 + com
	closing  trade cost = share * P1 - com

	return = [closing  trade cost] / [opening trade cost] - 1
	= [share * P1 - com] / [share * P0 + com] - 1

In SIT, these computations are equivalent to

	([capital - com] / [capital]) * ([share * P1 + cash - com] / [share * P0 + cash]) - 1

	< given that cash = 0 and capital = share * P0 >
	= ([share * P0 - com] / [share * P0]) * ([share * P1 - com] / [share * P0]) - 1

	< given [P0 - com] / P0 ~ P0 / [P0 + com] >
	= ([share * P0] / [share * P0 + com] / ) * ([share * P1 - com] / [share * P0]) - 1

	= [share * P1 - com] / [share * P0 + com] - 1

Hence, as long as commissions are small relative to whole trade the returns produced with SIT will be very close to the true returns.

SIT currently supports following 3 types of commissions

  • cents / share commission
    	commission = abs(share - mlag(share)) * commission$cps
    
  • fixed commission
    	commission = sign(abs(share - mlag(share))) * commission$fixed
    
  • percentage commission
    	commission = price * abs(share - mlag(share)) * commission$percentage
    

You can mix and match these commission methods in any way,

	the total.commission = cps.commission + fixed.commission + percentage.commission

and period return is equal to

	return = (share * price + cash - total.commission) / (share * mlag(price) + cash) - 1

Next let’s see the impact of different type of commissions. There two ways to specify the commissions.

  • commission = 0.1

    will be translated in

    commission = list(cps = 0.1, fixed = 0.0, percentage = 0.0)
  • commission = list(cps = 0.0, fixed = 0.0, percentage = 0.0)
###############################################################################
# Load Systematic Investor Toolbox (SIT)
# https://systematicinvestor.wordpress.com/systematic-investor-toolbox/
###############################################################################
setInternet2(TRUE)
con = gzcon(url('http://www.systematicportfolio.com/sit.gz', 'rb'))
    source(con)
close(con)

	#*****************************************************************
	# Load historical data
	#****************************************************************** 
	load.packages('quantmod')	
	tickers = spl('EEM')

	data <- new.env()
	getSymbols(tickers, src = 'yahoo', from = '1970-01-01', env = data, auto.assign = T)
		for(i in ls(data)) data[[i]] = adjustOHLC(data[[i]], use.Adjusted=T)			
	bt.prep(data, align='keep.all', dates='2013:08::2013:09')	
		
	#*****************************************************************
	# Code Strategies
	#****************************************************************** 		
	buy.date = '2013:08:14'	
	sell.date = '2013:08:15'
	day.after.sell.date = '2013:08:16'		
	
	capital = 100000
	prices = data$prices
	share = as.double(capital / prices[buy.date])
	
	# helper function to compute trade return
	comp.ret <- function(sell.trade.cost, buy.trade.cost) { round(100 * (as.double(sell.trade.cost) / as.double(buy.trade.cost) - 1), 2) }
	
	#*****************************************************************
	# Zero commission
	#****************************************************************** 
	data$weight[] = NA
		data$weight[buy.date] = 1
		data$weight[sell.date] = 0
		commission = 0.0
	model = bt.run.share(data, commission = commission, capital = capital, silent = T)
		
	comp.ret( share * prices[sell.date], share * prices[buy.date] )		
	comp.ret( model$equity[day.after.sell.date], model$equity[buy.date] )		
			
	#*****************************************************************
	# 10c cps commission
	# cents / share commission
   	#   trade cost = abs(share - mlag(share)) * commission$cps	
	#****************************************************************** 
	data$weight[] = NA
		data$weight[buy.date] = 1
		data$weight[sell.date] = 0
		commission = 0.1
	model = bt.run.share(data, commission = commission, capital = capital, silent = T)

	comp.ret( share * (prices[sell.date] - commission), share * (prices[buy.date] + commission) )
	comp.ret( model$equity[day.after.sell.date], model$equity[buy.date] )		
	
	#*****************************************************************
	# $5 fixed commission
	# fixed commission per trade to more effectively to penalize for turnover
   	#   trade cost = sign(abs(share - mlag(share))) * commission$fixed	
	#****************************************************************** 
	data$weight[] = NA
		data$weight[buy.date] = 1
		data$weight[sell.date] = 0
		commission = list(cps = 0.0, fixed = 5.0, percentage = 0.0)	
	model = bt.run.share(data, commission = commission, capital = capital, silent = T)

	comp.ret( share * prices[sell.date] - commission$fixed, share * prices[buy.date] + commission$fixed )
	comp.ret( model$equity[day.after.sell.date], model$equity[buy.date] )		
	
	#*****************************************************************
	# % commission
	# percentage commission
	#   trade cost = price * abs(share - mlag(share)) * commission$percentage	
	#****************************************************************** 
	data$weight[] = NA
		data$weight[buy.date] = 1
		data$weight[sell.date] = 0
		commission = list(cps = 0.0, fixed = 0.0, percentage = 1/100)	
	model = bt.run.share(data, commission = commission, capital = capital, silent = T)

	comp.ret( share * prices[sell.date] * (1 - commission$percentage), share * prices[buy.date] * (1 + commission$percentage) )
	comp.ret( model$equity[day.after.sell.date], model$equity[buy.date] )	

To view the complete source code for this example, please have a look at the bt.commission.test() function in bt.test.r at github.