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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):

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:

- Equally-weighted risk contributions: a new method to build risk balanced diversified portfolios by S. Maillard, T. Roncalli and J. Teiletche (2008)
- On the property of equally-weighted risk contributions portfolios by S. Maillard, T. Roncalli and J. Teiletche (2008)
- Analytical Solution for the Equal-Risk-Contribution Portfolio
- Matlab code for Equal Risk Contribution Portfolio by Farid Moussaoui

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) # http://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.

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