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Permanent Portfolio

September 18, 2012 2 comments

First, just a quick update: I’m moving the release date of the SIT package a few months down the road, probably in November.

Now back to the post. Recently I came across a series of interesting posts about the Permanent Portfolio at the GestaltU blog. Today I want to show you how to back-test the Permanent Portfolio using the Systematic Investor Toolbox.

The simple version of the Permanent Portfolio consists of equal allocations to stocks(SPY), gold(GLD), treasuries(TLT), and cash(SHY). [25% allocation each] The portfolio is rebalanced once a year if any allocation breaks out from the 15% – 35% range.

###############################################################################
# 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,GLD,SHY')
		
	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)
		
		# extend GLD with Gold.PM - London Gold afternoon fixing prices
		data$GLD = extend.GLD(data$GLD)
	
	bt.prep(data, align='remove.na')

	#*****************************************************************
	# Code Strategies
	#****************************************************************** 		
	prices = data$prices   
		n = ncol(prices)
		nperiods = nrow(prices)

	# annual
	period.ends = endpoints(prices, 'years')
		period.ends = period.ends[period.ends > 0]		
		period.ends.y = c(1, period.ends)

	# quarterly
	period.ends = endpoints(prices, 'quarters')
		period.ends = period.ends[period.ends > 0]		
		period.ends.q = c(1, period.ends)
					
	models = list()
	
	#*****************************************************************
	# Code Strategies
	#****************************************************************** 	
	target.allocation = matrix(rep(1/n,n), nrow=1)
	
	# Buy & Hold	
	data$weight[] = NA	
		data$weight[period.ends.y[1],] = target.allocation
	models$buy.hold = bt.run.share(data, clean.signal=F)
		
	# Equal Weight Annual
	data$weight[] = NA
		data$weight[period.ends.y,] = ntop(prices[period.ends.y,], n)
	models$equal.weight.y = bt.run.share(data, clean.signal=F)

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

To rebalance base on the 10% threshold (i.e. portfolio weights breaking out from the 15% – 35% range) I will use bt.max.deviation.rebalancing() function introduced in the Backtesting Rebalancing methods post.

	#*****************************************************************
	# Rebalance based on threshold
	#****************************************************************** 	
	# Rebalance only when threshold is broken
	models$threshold.y = bt.max.deviation.rebalancing(data, models$buy.hold, target.allocation, 10/100, 0, period.ends = period.ends.y) 

	# Rebalance only when threshold is broken
	models$threshold.q = bt.max.deviation.rebalancing(data, models$buy.hold, target.allocation, 10/100, 0, period.ends = period.ends.q) 

	#*****************************************************************
	# Create Report
	#******************************************************************       
	plotbt.custom.report.part1(models)       
	
	plotbt.strategy.sidebyside(models)
		
	# Plot Portfolio Turnover for each Rebalancing method
	layout(1:2)
	barplot.with.labels(sapply(models, compute.turnover, data), 'Average Annual Portfolio Turnover', F)
	barplot.with.labels(sapply(models, compute.max.deviation, target.allocation), 'Maximum Deviation from Target Mix')

The Quarterly rebalancing with 10% threshold produces an attractive portfolio with top performance and low turnover.

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

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Categories: Asset Allocation, Backtesting, Portfolio Construction, R, Strategy, Trading Strategies

Adaptive Asset Allocation – Sensitivity Analysis

August 21, 2012 2 comments

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.

Categories: Asset Allocation, Backtesting, Portfolio Construction, R, Strategy, Trading Strategies

Adaptive Asset Allocation

August 14, 2012 9 comments

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.

Categories: Asset Allocation, Backtesting, Portfolio Construction, R, Strategy, Trading Strategies

Gini Efficient Frontier

March 24, 2012 Leave a comment

David Varadi have recently wrote two posts about Gini Coefficient: I Dream of Gini, and Mean-Gini Optimization. I want to show how to use Gini risk measure to construct efficient frontier and compare it with alternative risk measures I discussed previously.

I will use Gini mean difference risk measure – the mean of the difference between every possible pair of returns to construct Mean-Gini Efficient Frontier. I will use methods presented in “The Generation of Mean Gini Efficient Sets” by J. Okunev (1991) paper to construct optimal portfolios.

Let x.i, i= 1,…,N be weights of instruments in the portfolio. Let us denote by r.it the return of i-th asset in the time period t for i= 1,…,N and t= 1,…,T. The portfolio’s Gini mean difference (page 5) can be written as:

\Gamma = \frac{1}{T^2}\sum_{j=1}^{T}\sum_{k>j}^{T}\left | Y_{j} - Y_{k} \right |

It can be formulated as a linear programming problem

\left | Y_{j} - Y_{k} \right | = a_{jk}^{+} - a_{jk}^{-} \newline \newline min \frac{1}{T^2}\sum_{j=1}^{T}\sum_{k>j}^{T}\left ( a_{jk}^{+} - a_{jk}^{-} \right ) \newline \sum_{i=1}^{N}x_{i}\left ( r_{ij} - r_{ik} \right ) - a_{jk}^{+} + a_{jk}^{-} = 0 \newline \left ( \begin{matrix} for & j=1 & to & T, & and & k>j\end{matrix} \right ) \newline a_{jk}^{+}, a_{jk}^{-}\geqslant 0

This linear programming problem can be easily implemented

min.gini.portfolio <- function
(
	ia,		# input assumptions
	constraints	# constraints
)
{
	n = ia$n
	nt = nrow(ia$hist.returns)
		
	# objective : Gini mean difference - the mean of the difference between every possible pair of returns
	#  1/(T^2) * [ SUM <over j = 1,...,T , k>j> a.long.jk + a.short.jk ]
	f.obj = c(rep(0, n), (1/(nt^2)) * rep(1, nt*(nt-1)))

	# adjust constraints, add a.long.jk , a.short.jk
	constraints = add.variables(nt*(nt-1), constraints, lb=0)

	# [ SUM <over i> x.i * (r.ij - r.ik) ] - a.long.jk + a.short.jk = 0
	# for each j = 1,...,T , k>j	
	a = matrix(0, n + nt*(nt-1), nt*(nt-1)/2)
		diag(a[(n+1) : (n + nt*(nt-1)/2), ]) = -1
		diag(a[(n+1+nt*(nt-1)/2) : (n + nt*(nt-1)), ]) = 1
	hist.returns = as.matrix(ia$hist.returns)
	
	i.start = 0
	for(t in 1:(nt-1)) {
		index = (i.start+1) : (i.start + nt -t)
		for(i in 1:n) {
			a[i, index] = ( hist.returns[t,i] - hist.returns[,i] ) [ (t+1) : nt ] 
		}
		i.start = i.start + nt -t		
	}
	constraints = add.constraints(a, 0, '=', constraints)
				
	# setup linear programming	
	f.con = constraints$A
	f.dir = c(rep('=', constraints$meq), rep('>=', len(constraints$b) - constraints$meq))
	f.rhs = constraints$b

	# find optimal solution
	x = NA
	sol = try(solve.LP.bounds('min', f.obj, t(f.con), f.dir, f.rhs, 
					lb = constraints$lb, ub = constraints$ub), TRUE)

	if(!inherits(sol, 'try-error')) {
		x = sol$solution[1:n]
	}

	return( x )
}

Let’s examine efficient frontiers computed under Gini and Standard deviation risk measures using sample historical input assumptions.

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

	#--------------------------------------------------------------------------
	# Create Efficient Frontier
	#--------------------------------------------------------------------------
	ia = aa.test.create.ia.rebal()
	n = ia$n		

	# 0 <= x.i <= 1 
	constraints = new.constraints(n, lb = 0, ub = 1)
		
	# SUM x.i = 1
	constraints = add.constraints(rep(1, n), 1, type = '=', constraints)		

	# create efficient frontier(s)
	ef.risk = portopt(ia, constraints, 50, 'Risk')
	ef.gini = portopt(ia, constraints, 50, 'GINI', min.gini.portfolio)
		
	#--------------------------------------------------------------------------
	# Create Plots
	#--------------------------------------------------------------------------
	layout( matrix(1:4, nrow = 2) )
	plot.ef(ia, list(ef.risk, ef.gini), portfolio.risk, F)	
	plot.ef(ia, list(ef.risk, ef.gini), portfolio.gini.coefficient, F)	

	plot.transition.map(ef.risk)
	plot.transition.map(ef.gini)

The Gini efficient frontier is almost identical to Standard deviation efficient frontier, labeled ‘Risk’. This is not a surprise because asset returns that are used in the sample input assumptions are well behaved. The Gini measure of risk would be most appropriate if asset returns contained large outliers.

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

Next I added Gini risk measure to the mix of Asset Allocation strategies that I examined in the Backtesting Asset Allocation portfolios post.

The Gini portfolios and Minimum Variance portfolios show very similar perfromance

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

Categories: Asset Allocation, Backtesting, Portfolio Construction, R, Risk Measures, Strategy, Trading Strategies

Backtesting Asset Allocation portfolios

March 19, 2012 9 comments

In the last post, Portfolio Optimization: Specify constraints with GNU MathProg language, Paolo and MC raised a question: “How would you construct an equal risk contribution portfolio?” Unfortunately, this problem cannot be expressed as a Linear or Quadratic Programming problem.

The outline for this post:

  • I will show how Equal Risk Contribution portfolio can be formulated and solved using a non-linear solver.
  • I will backtest Equal Risk Contribution portfolio and other Asset Allocation portfolios based on various risk measures I described in the Asset Allocation series of post.

Pat Burns wrote an excellent post: Unproxying weight constraints that explains Risk Contribution – partition the variance of a portfolio into pieces attributed to each asset. The Equal Risk Contribution portfolio is a portfolio that splits total portfolio risk equally among its assets. (The concept is similar to 1/N portfolio – a portfolio that splits total portfolio weight equally among its assets.)

Risk Contributions (risk fractions) can be expressed in terms of portfolio weights (w) and covariance matrix (V):
f=\frac{w*Vw}{w'Vw}

Our objective is to find portfolio weights (w) such that Risk Contributions are equal for all assets. This objective function can be easily coded in R:

	risk.contribution = w * (cov %*% w)
	sum( abs(risk.contribution - mean(risk.contribution)) )

I recommend following references for a detailed discussion of Risk Contributions:

I will use a Nonlinear programming solver, Rdonlp2, which is based on donlp2 routine developed and copyright by Prof. Dr. Peter Spellucci to solve for Equal Risk Contribution portfolio. [Please note that following code might not properly execute on your computer because Rdonlp2 package is required and not available on CRAN]

#--------------------------------------------------------------------------
# Equal Risk Contribution portfolio
#--------------------------------------------------------------------------
ia = aa.test.create.ia()
n = ia$n		

# 0 <= x.i <= 1
constraints = new.constraints(n, lb = 0, ub = 1)

# SUM x.i = 1
constraints = add.constraints(rep(1, n), 1, type = '=', constraints)		

# find Equal Risk Contribution portfolio 
w = find.erc.portfolio(ia, constraints)	

# compute Risk Contributions 	
risk.contributions = portfolio.risk.contribution(w, ia)

Next, I want to expand on the Backtesting Minimum Variance portfolios post to include Equal Risk Contribution portfolio and and other Asset Allocation portfolios based on various risk measures I described in the Asset Allocation series of post.

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

	#*****************************************************************
	# Load historical data
	#****************************************************************** 
	load.packages('quantmod,quadprog,corpcor,lpSolve')
	tickers = spl('SPY,QQQ,EEM,IWM,EFA,TLT,IYR,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='remove.na', dates='1990::2011')
 
	#*****************************************************************
	# Code Strategies
	#****************************************************************** 
	prices = data$prices   
	n = ncol(prices)
	
	# find week ends
	period.ends = endpoints(prices, 'weeks')
		period.ends = period.ends[period.ends > 0]

	#*****************************************************************
	# Create Constraints
	#*****************************************************************
	constraints = new.constraints(n, lb = 0, ub = 1)
	
	# SUM x.i = 1
	constraints = add.constraints(rep(1, n), 1, type = '=', constraints)		

	#*****************************************************************
	# Create Portfolios
	#*****************************************************************			
	ret = prices / mlag(prices) - 1
	start.i = which(period.ends >= (63 + 1))[1]
	
	weight = NA * prices[period.ends,]
	weights = list()
		# Equal Weight 1/N Benchmark
		weights$equal.weight = weight
			weights$equal.weight[] = ntop(prices[period.ends,], n)	
			weights$equal.weight[1:start.i,] = NA
			
		weights$min.var = weight
		weights$min.maxloss = weight
		weights$min.mad = weight
		weights$min.cvar = weight
		weights$min.cdar = weight
		weights$min.cor.insteadof.cov = weight
		weights$min.mad.downside = weight
		weights$min.risk.downside = weight
		
		# following optimizations use a non-linear solver
		weights$erc = weight		
		weights$min.avgcor = weight		
		
	risk.contributions = list()	
		risk.contributions$erc = weight		
		
	# construct portfolios
	for( j in start.i:len(period.ends) ) {
		i = period.ends[j]
		
		# one quarter = 63 days
		hist = ret[ (i- 63 +1):i, ]
		
		# create historical input assumptions
		ia = create.historical.ia(hist, 252)
			s0 = apply(coredata(hist),2,sd)		
			ia$correlation = cor(coredata(hist), use='complete.obs',method='pearson')
			ia$cov = ia$correlation * (s0 %*% t(s0))
			
		# construct portfolios based on various risk measures
		weights$min.var[j,] = min.risk.portfolio(ia, constraints)
		weights$min.maxloss[j,] = min.maxloss.portfolio(ia, constraints)
		weights$min.mad[j,] = min.mad.portfolio(ia, constraints)
		weights$min.cvar[j,] = min.cvar.portfolio(ia, constraints)
		weights$min.cdar[j,] = min.cdar.portfolio(ia, constraints)
		weights$min.cor.insteadof.cov[j,] = min.cor.insteadof.cov.portfolio(ia, constraints)
		weights$min.mad.downside[j,] = min.mad.downside.portfolio(ia, constraints)
		weights$min.risk.downside[j,] = min.risk.downside.portfolio(ia, constraints)

		# following optimizations use a non-linear solver		
		constraints$x0 = weights$erc[(j-1),]
		weights$erc[j,] = find.erc.portfolio(ia, constraints)		
		
		constraints$x0 = weights$min.avgcor[(j-1),]
		weights$min.avgcor[j,] = min.avgcor.portfolio(ia, constraints)						
		
		risk.contributions$erc[j,] = portfolio.risk.contribution(weights$erc[j,], ia)
	}

Next let’s backtest these portfolios and create summary statistics:

	#*****************************************************************
	# Create strategies
	#****************************************************************** 		
	models = list()
	for(i in names(weights)) {
		data$weight[] = NA
			data$weight[period.ends,] = weights[[i]]	
		models[[i]] = bt.run.share(data, clean.signal = F)
	}
		
	#*****************************************************************
	# Create Report
	#****************************************************************** 
	models = rev(models)

	# Plot perfromance
	plotbt(models, plotX = T, log = 'y', LeftMargin = 3)	    	
		mtext('Cumulative Performance', side = 2, line = 1)

	# Plot Strategy Statistics  Side by Side
	plotbt.strategy.sidebyside(models)
	
	# Plot transition maps
	layout(1:len(models))
	for(m in names(models)) {
		plotbt.transition.map(models[[m]]$weight, name=m)
			legend('topright', legend = m, bty = 'n')
	}

	# Plot risk contributions
	layout(1:len(risk.contributions))
	for(m in names(risk.contributions)) {
		plotbt.transition.map(risk.contributions[[m]], name=paste('Risk Contributions',m))
			legend('topright', legend = m, bty = 'n')
	}

	# Compute portfolio concentration and turnover stats based on the
	# On the property of equally-weighted risk contributions portfolios by S. Maillard, 
	# T. Roncalli and J. Teiletche (2008), page 22
	# http://www.thierry-roncalli.com/download/erc.pdf
	out = compute.stats( rev(weights),
		list(Gini=function(w) mean(portfolio.concentration.gini.coefficient(w), na.rm=T),
			Herfindahl=function(w) mean(portfolio.concentration.herfindahl.index(w), na.rm=T),
			Turnover=function(w) 52 * mean(portfolio.turnover(w), na.rm=T)
			)
		)
	
	out[] = plota.format(100 * out, 1, '', '%')
	plot.table(t(out))

The minimum variance (min.risk) portfolio performed very well during that period with 10.5% CAGR and 14% maximum drawdown. The Equal Risk Contribution portfolio (find.erc) also fares well with 10.5% CAGR and 19% maximum drawdown. The 1/N portfolio (equal.weight) is the worst strategy with 7.8% CAGR and 45% maximum drawdown.

One interesting way to modify this strategy is to consider different measures of volatility used to construct a covariance matrix. For example TTR package provides functions for the Garman Klass – Yang Zhang and the Yang Zhang volatility estimation methods. For more details, please have a look at the Different Volatility Measures Effect on Daily MR by Quantum Financier post.

Inspired by the I Dream of Gini by David Varadi, I will show how to create Gini efficient frontier in the next post.

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

Categories: Asset Allocation, Backtesting, Portfolio Construction, R, Risk Measures