Home > Asset Allocation, Portfolio Construction, R, Risk Measures > The Most Diversified or The Least Correlated Efficient Frontier

The Most Diversified or The Least Correlated Efficient Frontier

The “Minimum Correlation Algorithm” is a term I stumbled at the CSS Analytics blog. This is an Interesting Risk Measure that in my interpretation means: minimizing Average Portfolio Correlation with each Asset Class for a given level of return.

One might try to use Correlation instead of Covariance matrix in mean-variance optimization, but this approach, as I will show below, will not produce the least correlated portfolios.

The Average Portfolio Correlation with each Asset Class:

P = \sum_{i=1}^{n}w_{i}X_{i} = w^{T} \ast X  \newline\newline  \sigma_{P} = w^{T} \ast COV \ast w  \newline\newline  COV\left ( P,X_{i} \right ) = COV\left ( \sum_{i=1}^{n}w_{i}X_{i},X_{i} \right )  =w_{1}COV\left ( X_{1},X_{i} \right )+...+w_{n}COV\left ( X_{n},X_{i} \right )  \newline\newline  \rho_{P,X_{i}} = \frac{COV\left ( P,X_{i} \right )}{\sigma_{P}\ast\sigma_{X_{i}}}  \newline\newline  Average Portfolio Correlation = \frac{1}{n}\sum_{i=1}^{n}\rho_{P,X_{i}}

This formula can be easily coded in R:

portfolio.sigma = sqrt( t(weight) %*% assets.cov %*% weight )
mean( ( weight %*% assets.cov ) / ( assets.sigma * portfolio.sigma ) )

# Alternatively
portfolio.returns = weight %*% t(assets.hist.returns)	
mean(cor(assets.hist.returns, portfolio.returns)) 

I’m not aware of the method to transform this formula in to the linear programming, so I will use a Nonlinear programming solver, Rdonlp2, which is based on donlp2 routine developed and copyright by Prof. Dr. Peter Spellucci. Following code might not properly execute on your computer because Rdonlp2 is only available for R version 2.9 or below.

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

	# 0 <= x.i <= 0.8 
	constraints = new.constraints(n, lb = 0, ub = 0.8)
	
	# 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.cor.insteadof.cov = portopt(ia, constraints, 50, 'Cor instead of Cov', min.cor.insteadof.cov.portfolio)
	ef.avgcor = portopt(ia, constraints, 50, 'AvgCor', min.avgcor.portfolio)

	# Plot multiple Efficient Frontiers	
	layout(1:2)
	plot.ef(ia, list(ef.risk, ef.avgcor, ef.cor.insteadof.cov), portfolio.risk, F)	
	plot.ef(ia, list(ef.risk, ef.avgcor, ef.cor.insteadof.cov), portfolio.avgcor, F)	

	# Plot multiple Transition Maps	
	layout( matrix(1:4, nrow = 2) )
	plot.transition.map(ef.risk)
	plot.transition.map(ef.avgcor)
	plot.transition.map(ef.cor.insteadof.cov)

	# visualize input assumptions
	plot.ia(ia)

Using Correlation instead of Covariance matrix in mean-variance optimization is a very bad idea to produce the least correlated portfolios. The ‘Cor instead of Cov’ efficient frontier actually increases average portfolio correlation compared to the standard ‘Risk’ efficient frontier.

The portfolio composition of the Average Correlation efficient frontier is split between gold (GLD) and bonds (TLT) at the lower risk levels. This is not surprising because both gold and bonds have positive expected returns and low correlation to the other assets.

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

Following is the complete source code for minimizing Average Portfolio Correlation with each Asset Class function:

min.avgcor.portfolio <- function
(
	ia,			# input assumptions
	constraints		# constraints
)
{
	require(Rdonlp2)

	cov = ia$cov[1:ia$n, 1:ia$n]
	s = sqrt(diag(cov))
	
	# avgcor
	fn <- function(x){
		sd_x = sqrt( t(x) %*% cov %*% x )
		mean( ( x %*% cov ) / ( s * sd_x ) )
	}
	
	# control structure
	cntl <- donlp2.control(silent = T, iterma =10000, nstep = 100, epsx = 1e-10)	
	
	# lower/upper bounds
	par.l = constraints$lb
	par.u = constraints$ub
	
	# intial guess
	p = rep(1,n)
	if(!is.null(constraints$x0)) p = constraints$x0
		
	# linear constraints
	A = t(constraints$A)
	lin.l = constraints$b
	lin.u = constraints$b
	lin.u[ -c(1:constraints$meq) ] = +Inf

	# find solution
	sol = donlp2(p, fn, 
			par.lower=par.l, par.upper=par.u, 
			A=A, lin.u=lin.u, lin.l=lin.l, 
			control=cntl)
	x = sol$par

	return( x )
}
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  1. david varadi
    October 27, 2011 at 2:02 pm

    nice post– however some important comments: minimizing correlations optimally implies equivalent volatility for the portfolio assets—furthermore non-linear solutions are useful but not necessary to solve this problem. the weights must be constrained to be non-negative. a linear solution will suffice subject to inequality constraints.
    best
    dv (from css)

    • October 27, 2011 at 3:11 pm

      David, thank you. Can you please share, how do you formulate the Average Portfolio Correlation with each Asset Class as a linear programming problem?

      • October 29, 2011 at 4:45 pm

        hi, here is a link:THE MINIMUM VARIANCE PROPERTY
        That is, it must possess the minimum variance property. We will show that under
        the conditions set out in the classical linear model the OLS estimators are BLUE …

        homepages.uel.ac.uk/…/FE2007%20The%20Minimum%20Variance%20Property%20notes

        best
        david

  2. October 29, 2011 at 9:36 am

    How I would approach this in Portfolio Probe would be to have the correlation matrix in the utility and constrain the portfolio variance to some value.

    • October 29, 2011 at 1:29 pm

      Pat, I tried that in ‘Cor instead of Cov’ efficient frontier. To create ‘Cor instead of Cov’ efficient frontier, I used Correlation instead of Covariance matrix in mean-variance optimization.

      Unfortunately, this approach, does not produce the least correlated portfolios : the portfolios that minimize Average Portfolio Correlation with each Asset Class. Actually the ‘Cor instead of Cov’ efficient frontier, colored with green in the chart above, is very inefficient in the Average Portfolio Correlation framework.

      • October 29, 2011 at 6:00 pm

        Okay, I see. Your function is linear in the weights rather than quadratic (not counting the computation of the portfolio variance).

  3. October 31, 2011 at 11:40 am

    Thinking about this, I’ve decided that I quite like it.

  4. ron
    November 16, 2011 at 3:46 pm

    Would one not be interested in finding the portfolio that has the least weighted sum of the correlation with it’s underlying assets, rather than the portfolio that has the least mean correlation with the underlying assets (subject to constraints)? In a long only case it’s possible that the optimal allocation could be a corner solution, ie. entirely allocate in one asset. Depending on how you quantify portfolio correlation, a portfolio with only one asset could be considered to have a correlation of 1. The diversification of such a portfolio would be minimal, which generally speaking is regarded as undesirable.

    Also you said in referring to gold and bonds splitting the portoflio allocation for low levels of risk “This is not surprising because both gold and bonds have positive expected returns and low correlation to the other assets”.

    The fact the the expected (historical mean) returns are possitive has nothing to do with the allocation process.

    Perhaps one may want to create a ratio of expected returns over the portfolio correlation (or the exponential of the portfolio correlation) and try to maximize that. Of course how one defines portfolio correlation is important, and then i guess one would need to justify such a ratio.

    All and all, interesting stuff.

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