Adaptive Asset Allocation – Sensitivity Analysis

Today I want to continue with Adaptive Asset Allocation theme and examine how the strategy results are sensitive to look-back parameters used for momentum and volatility computations. I will follow the sample steps that were outlined by David Varadi on the robustness of parameters of the Adaptive Asset Allocation algorithm post. Please see my prior post for more infromation.

Let’s start by loading historical prices for 10 ETFs 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,EFA,EWJ,EEM,IYR,RWX,IEF,TLT,DBC,GLD')

	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='keep.all', dates='2004:12::')

    #*****************************************************************
    # Code Strategies
    #******************************************************************
    prices = data$prices  
    n = ncol(prices)
   
    models = list()
   
    # find period ends
    period.ends = endpoints(prices, 'months')
        period.ends = period.ends[period.ends > 0]

Next I wrapped the Combo (Momentum and Volatility weighted) strategy and Adaptive Asset Allocation (AAA) strategy into bt.aaa.combo and bt.aaa.minrisk functions respectively. Following is an example how you can use them:

	#*****************************************************************
	# Test
	#****************************************************************** 
	models = list()
	
	models$combo = bt.aaa.combo(data, period.ends, n.top = 5,
					n.mom = 180, n.vol = 20)
										
	models$aaa = bt.aaa.minrisk(data, period.ends, n.top = 5,
					n.mom = 180, n.vol = 20)
					
	plotbt.custom.report.part1(models) 

Now let’s evaluate all possible combinations of momentum and volatility look back parameters ranging from 1 to 12 months using Combo strategy:

	#*****************************************************************
	# Sensitivity Analysis: bt.aaa.combo / bt.aaa.minrisk
	#****************************************************************** 
	# length of momentum look back
	mom.lens = ( 1 : 12 ) * 20
	# length of volatility look back
	vol.lens = ( 1 : 12 ) * 20

	
	models = list()
	
	# evaluate strategies
	for(n.mom in mom.lens) {
		cat('MOM =', n.mom, '\n')
		
		for(n.vol in vol.lens) {
			cat('\tVOL =', n.vol, '\n')

			models[[ paste('M', n.mom, 'V', n.vol) ]] = 
				bt.aaa.combo(data, period.ends, n.top = 5,
					n.mom = n.mom, n.vol = n.vol)
		}
	}
	
	out = plotbt.strategy.sidebyside(models, return.table=T, make.plot = F)

Finally let’s plot the Sharpe, Cagr, DVR, MaxDD statistics for the each strategy:

	#*****************************************************************
	# Create Report
	#****************************************************************** 
	# allocate matrix to store backtest results
	dummy = matrix('', len(vol.lens), len(mom.lens))
		colnames(dummy) = paste('M', mom.lens)
		rownames(dummy) = paste('V', vol.lens)
		
	names = spl('Sharpe,Cagr,DVR,MaxDD')

	layout(matrix(1:4,nrow=2))	
	for(i in names) {
		dummy[] = ''
		
		for(n.mom in mom.lens)
			for(n.vol in vol.lens)
				dummy[paste('V', n.vol), paste('M', n.mom)] =
					out[i, paste('M', n.mom, 'V', n.vol) ]
					
		plot.table(dummy, smain = i, highlight = T, colorbar = F)

	}

I have also repeated the last two steps for the AAA strategy (bt.aaa.minrisk function):

The results for AAA and Combo strategies are very similar. The shorter term momentum and shorter term volatility produce the best results, but likely at the cost of higher turnover.

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

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Adaptive Asset Allocation

Today I want to highlight a whitepaper about Adaptive Asset Allocation by Butler, Philbrick and Gordillo and the discussion by David Varadi on the robustness of parameters of the Adaptive Asset Allocation algorithm.

In this post I will follow the steps of the Adaptive Asset Allocation paper, and in the next post I will show how to test the sensitivity of parameters of the of the Adaptive Asset Allocation algorithm.

I will use the 10 ETFs that invest into the same asset classes as presented in the paper:

• U.S. Stocks (SPY)
• European Stocks (EFA)
• Japanese Stocks (EWJ)
• Emerging Market Stocks (EEM)
• U.S. REITs (IYR)
• International REITs (RWX)
• U.S. Mid-term Treasuries (IEF)
• U.S. Long-term Treasuries (TLT)
• Commodities (DBC)
• Gold (GLD)

Unfortunately, most of these 10 ETFs only began trading in the end of 2004, so I will only be able to replicate the recent Adaptive Asset Allocation strategy performance.

Let’s start by loading historical prices of 10 ETFs 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,EFA,EWJ,EEM,IYR,RWX,IEF,TLT,DBC,GLD')

	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='keep.all', dates='2004:12::')
    
    #*****************************************************************
    # Code Strategies
    #******************************************************************
    prices = data$prices  
    n = ncol(prices)
   
    models = list()
   
    # find period ends
    period.ends = endpoints(prices, 'months')
        period.ends = period.ends[period.ends > 0]

	# Adaptive Asset Allocation parameters
	n.top = 5		# number of momentum positions
	n.mom = 6*22	# length of momentum look back
	n.vol = 1*22 	# length of volatility look back   

Next, let’s create portfolios as outlined in the whitepaper:

    #*****************************************************************
    # Equal Weight
    #******************************************************************
    data$weight[] = NA
        data$weight[period.ends,] = ntop(prices[period.ends,], n)   
    models$equal.weight = bt.run.share(data, clean.signal=F)

    #*****************************************************************
    # Volatliliy Position Sizing
    #******************************************************************
    ret.log = bt.apply.matrix(prices, ROC, type='continuous')
    hist.vol = bt.apply.matrix(ret.log, runSD, n = n.vol)
   
    adj.vol = 1/hist.vol[period.ends,]
           
    data$weight[] = NA
        data$weight[period.ends,] = adj.vol / rowSums(adj.vol, na.rm=T)    
    models$volatility.weighted = bt.run.share(data, clean.signal=F)
   
    #*****************************************************************
    # Momentum Portfolio
    #*****************************************************************
    momentum = prices / mlag(prices, n.mom)
   
    data$weight[] = NA
        data$weight[period.ends,] = ntop(momentum[period.ends,], n.top)   
    models$momentum = bt.run.share(data, clean.signal=F)
       
    #*****************************************************************
    # Combo: weight positions in the Momentum Portfolio according to Volatliliy
    #*****************************************************************
    weight = ntop(momentum[period.ends,], n.top) * adj.vol
   
    data$weight[] = NA
        data$weight[period.ends,] = weight / rowSums(weight, na.rm=T)   
    models$combo = bt.run.share(data, clean.signal=F,trade.summary = TRUE)

Finally let’s create the Adaptive Asset Allocation portfolio:

    #*****************************************************************   
    # Adaptive Asset Allocation (AAA)
    # weight positions in the Momentum Portfolio according to 
    # the minimum variance algorithm
    #*****************************************************************   
    weight = NA * prices
        weight[period.ends,] = ntop(momentum[period.ends,], n.top)
       
    for( i in period.ends[period.ends >= n.mom] ) {
    	hist = ret.log[ (i - n.vol + 1):i, ]
    	
		# require all assets to have full price history
		include.index = count(hist)== n.vol      

		# also only consider assets in the Momentum Portfolio
        index = ( weight[i,] > 0 ) & include.index
        n = sum(index)
        
		if(n > 0) {					
			hist = hist[ , index]
        
	        # create historical input assumptions
	        ia = create.historical.ia(hist, 252)
	            s0 = apply(coredata(hist),2,sd)       
	            ia$cov = cor(coredata(hist), use='complete.obs',method='pearson') * (s0 %*% t(s0))
	       
			# create constraints: 0<=x<=1, sum(x) = 1
			constraints = new.constraints(n, lb = 0, ub = 1)
			constraints = add.constraints(rep(1, n), 1, type = '=', constraints)       
			
			# compute minimum variance weights				            
	        weight[i,] = 0        
	        weight[i,index] = min.risk.portfolio(ia, constraints)
        }
    }

    # Adaptive Asset Allocation (AAA)
    data$weight[] = NA
        data$weight[period.ends,] = weight[period.ends,]   
    models$aaa = bt.run.share(data, clean.signal=F,trade.summary = TRUE)

The last step is create reports for all models:

    #*****************************************************************
    # Create Report
    #******************************************************************    
    models = rev(models)
   
    plotbt.custom.report.part1(models)       
    plotbt.custom.report.part2(models)       
    plotbt.custom.report.part3(models$combo, trade.summary = TRUE)       
    plotbt.custom.report.part3(models$aaa, trade.summary = TRUE)       

The AAA portfolio performs very well, producing the highest Sharpe ratio and smallest draw-down across all strategies. In the next post I will look at the sensitivity of AAA parameters.

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

Updates

I will be traveling to Freeport, Bahamas during the last week of August. If you want to meet for a coffee drop me a line.

I also plan to release the first version of SIT package at the end of September.

The New 60/40

I want to share a brilliant idea and a great example from the You’re Looking at the Wrong Number post at the GestaltU blog. Today, I will focus on the section of this post that outlines simple steps to improve a typical 60/40 stock/bond portfolio by using risk allocation instead of dollar allocation, and targeting constant portfolio volatility.

I will use SPY (S&P 500) as a proxy for a stock allocation and TLT (20 Year Treasuries) as a proxy for a bond allocation. Let’s start by loading historical prices for SPY and a few fixed income ETF’s 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('SHY,IEF,TLT,SPY')

	data.all <- new.env()
	getSymbols(tickers, src = 'yahoo', from = '1990-01-01', env = data.all, auto.assign = T)	
	for(i in ls(data.all)) data.all[[i]] = adjustOHLC(data.all[[i]], use.Adjusted=T)		
	bt.prep(data.all, align='remove.na')
	
	prices = data.all$prices
		n = ncol(prices)
		nperiods = nrow(prices)

	# normalize all prices and plot them
	prices = prices/ matrix(first(prices), nr=nperiods, nc=n, byrow=T)
	plota.matplot(prices)

The fixed income ETFs are not as safe as you might have been expected. The SHY (Barclays 1-3 Year Treasury Bond Fund) is consistently up with out large draw-downs. However, both IEF (Barclays 7-10 Year Treasury Bond Fund) and TLT (Barclays 20 Year Treasury Bond Fund) do go through periods of loosing money.

Next let’s build a traditional dollar weighted 60/40 stock/bond portfolio and risk weighted version:

	#*****************************************************************
	# Load historical data
	#****************************************************************** 		
	data <- new.env()
		data$stock = data.all$SPY
		data$bond = data.all$TLT	
	bt.prep(data, align='remove.na')

	#*****************************************************************
	# Code Strategies
	#****************************************************************** 
	# all bonds began trading at 2002-07-31
	prices = data$prices
		n = ncol(prices)
		nperiods = nrow(prices)
	
	models = list()
	
	period.ends = endpoints(prices, 'months')
		period.ends = period.ends[period.ends > 0]
	
	#*****************************************************************
	# Traditional, Dollar Weighted 40% Bonds & 60% Stock
	#****************************************************************** 			
	weight.dollar = matrix(c(0.4, 0.6), nr=nperiods, nc=n, byrow=T)
	
	data$weight[] = NA
		data$weight[period.ends,] = weight.dollar[period.ends,]
	models$dollar.w.60.40 = bt.run.share(data, clean.signal=F)

	#*****************************************************************
	# Risk Weighted 40% Bonds & 60% Stock
	#****************************************************************** 				
	ret.log = bt.apply.matrix(prices, ROC, type='continuous')
	hist.vol = sqrt(252) * bt.apply.matrix(ret.log, runSD, n = 21)	
	weight.risk = weight.dollar / hist.vol
		weight.risk = weight.risk / rowSums(weight.risk)
		
	data$weight[] = NA
		data$weight[period.ends,] = weight.risk[period.ends,]
	models$risk.w.60.40 = bt.run.share(data, clean.signal=F)

Next let’s create a risk weighted portfolio with 6% volatility. I will adjust portfolio leverage up/down based on the one month historical volatility to target 6% annual volatility.

	#*****************************************************************
	# Helper function to adjust portfolio leverage to target given volatility
	#****************************************************************** 				
	target.vol.strategy <- function(model, weight, 
		target = 10/100, 
		lookback.len = 21,
		max.portfolio.leverage = 100/100) 
	{	
		ret.log.model = ROC(model$equity, type='continuous')
		hist.vol.model = sqrt(252) * runSD(ret.log.model, n = lookback.len)	
			hist.vol.model = as.vector(hist.vol.model)
		
		weight.target = weight * (target / hist.vol.model)
	
		# limit total leverage		
		rs = rowSums(abs(weight.target))
		weight.target = weight.target / iif(rs > max.portfolio.leverage, rs/max.portfolio.leverage, 1)		
		
		return(weight.target)	
	}
				
	#*****************************************************************
	# Scale Risk Weighted 40% Bonds & 60% Stock strategy to have 6% volatility
	#****************************************************************** 				
	data$weight[] = NA
		data$weight[period.ends,] = target.vol.strategy(models$risk.w.60.40,
						weight.risk, 6/100, 21, 100/100)[period.ends,]
	models$risk.w.60.40.target6 = bt.run.share(data, clean.signal=T)

In the last step, I want to invest portfolio cash position into short-term treasuries (SHY) to improve overall performance and create summary reports:

	#*****************************************************************
	# Same, plus invest cash into SHY
	#****************************************************************** 					
	weight = target.vol.strategy(models$risk.w.60.40, weight.risk, 6/100, 21, 100/100)
	data.all$weight[] = NA
		data.all$weight$SPY[period.ends,] = weight$stock[period.ends,]
		data.all$weight$TLT[period.ends,] = weight$bond[period.ends,]
		
		cash = 1-rowSums(weight)
		data.all$weight$SHY[period.ends,] = cash[period.ends]
	models$risk.w.60.40.target6.cash = bt.run.share(data.all, clean.signal=T)

	#*****************************************************************
	# Create Report
	#****************************************************************** 	
	plotbt.strategy.sidebyside(models)
	
	plotbt.custom.report.part1(models)
		
	plotbt.custom.report.part2(models$risk.w.60.40.target6)		
	plotbt.custom.report.part2(models$risk.w.60.40.target6.cash)		

I’m very happy with results: the returns went down a bit, but portfolio risk adjusted performance is up and draw-downs went down from 30% to 12%. The consistent returns are especially important for a retirement portfolio because it allows you to better plan how much money you can withdraw and sustain your fund for a longer term.

To view the complete source code for this example, please have a look at the bt.new.60.40.test() function in bt.test.r at github.