Commissions
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Today, I want to explain the commission’s functionality build in to Systematic Investor Toolbox(SIT) “share” backtest.
At each rebalance 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) # http://systematicinvestor.wordpress.com/systematicinvestortoolbox/ ############################################################################### 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 = '19700101', 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.
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