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I have a follow-on question to questions that appeared here and was not sure if the right way was to ask in the comments or post a new question.

My question is: how can I optimize a portfolio to suit both minimum variance as well as max diversification. Essentially the minimum variance portfolio that is most diversified.

I can formulate a quadratic optimization for either MVP (minimum variance) or MDP (max diversification) as per choueifaty et al.

But I don't know how to craft a quadratic program that optimizes for both at the same time. Is it even possible with a quadratic program or do I have to use some other optimization procedure?

The source questions are here:

Reduce correlation in output of Minimum Variance Portfolio Optimization

How do I find the most diversified portfolio, or least correlated subset, of stocks?

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Will QCQP with an upper bound on variance and maximize MDP satisfy your needs? – pavy bez Mar 18 at 14:49

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There is only one MVP and only one MDP portfolio so, unless these are the same, this will not be possible.

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Is it not possible to optimize towards a blend of the two e.g. 50% MVP 50%MDP, or is that simply equivalent to holding an MVP portfolio and an equal amount invested in an MDP? – nxstock-trader Mar 28 '12 at 22:03
My approach would be to minimize variance subject to constraints on correlation to the portfolio. – Patrick Burns Mar 29 '12 at 7:56
@PatrickBurns agreed. Note: Another idea would be to minimize the sum of the functions. This seems like bad approach to me because you can't really trade the different units of utility. – Bob Jansen Mar 29 '12 at 16:40
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Solve the system for constraints:

  • minimize variance
  • maximize returns
  • for diversity : maximize sum of lengths of all the edges of a minimum spanning tree extracted from the distance matrix i.e. correlation matrix

Would appreciate any kind of academic references with similar thoughts.

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Can you explain more clearly how you would solve for all these constraints together. E.g. would you formulate a quadratic program, do somethe monte-carlo optimization, something else? – nxstock-trader Mar 29 '12 at 18:10
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Hmmm ... my knowledge is limited to MPT ( http://en.wikipedia.org/wiki/Modern_portfolio_theory ) and according to it, this isn't really a problem or the problem isn't formulated correctly, because it is mathematically provable that more diversified a portfolio is, lower is the variance (or risk, have a look at this lecture for example http://academicearth.org/lectures/portfolio-diversification).

Another life example, Standard Life pension funds:

  • "Pension 2 Managed Fund" (variance = 0.207932, expected return = 0.054878)
  • "Pension 2 Stock Exchange Fund" (variance = 0.200217, expected return = 0.053171)

are highly correlated ρ=0.996032, so MPT (at the optimal point, i.e. Portfolio return of those two = 0.050132 and lowest possible Portfolio variance = 0.194857 - reduced by the way) suggests:

  • Weight("Pension 2 Managed Fund") = -1.779723
  • Weight("Pension 2 Stock Exchange Fund") = 2.779723

I.e. short "Pension 2 Managed Fund".

It is actually easy to implement with Octave or MathLab:

However, finding the best portfolio is quite of a task ( http://rtybase.blogspot.co.uk/2011/11/search.html?showComment=1331035896847#c2577055848756808847 ).

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