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I am trying to code up the optimization problem for Max Diversification Portfolios.

The main problem I am having is properly translating the objective function in to code and port it in to the optimizer in general.

How would one approach this? Can this be solved with R's quadprog?

The objective function to maximize is the diversification ratio:

d(P) = P'E / sqrt(P'VP)

Where:

  • E is vector of asset volatilities,
  • P is the vector of weights
  • V is the covariance matrix.
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2 Answers

For the record, the formula for maximum diversification portfolio can be found in this paper.

As you can see from the quadprog documentation, it minimizes problems of the following form:

$$ \min - d'b + \tfrac12 b' D b ~ \text{with} ~ A' b \geq b_0 $$

So clearly, it's not good for your formula.

You can consider optim or one of its extensions for your problem.

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You can find the full R source code for that at the site of Systematic Investor.

For example have a look at this post about Maximum Sharpe Portfolios. There you see that he created the helper function portfolio.allocation.helper for the following optimization methods:

EW=equal.weight.portfolio,
RP=risk.parity.portfolio,
MV=min.var.portfolio,
MD=max.div.portfolio,
MC=min.corr.portfolio,
MC2=min.corr2.portfolio,
MCE=min.corr.excel.portfolio,
MS=max.sharpe.portfolio

Now the full source code can be found here.

You'll want to have a look at max.div.portfolio which is based on the method in:

Toward Maximum Diversification by Y. Choueifaty, Y. Coignard, The Journal of Portfolio Management, Fall 2008, Vol. 35, No. 1: pp. 40-51

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