# How to optimize a portfolio under *both* maximum diversity ratio and minimum variance

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?

• Will QCQP with an upper bound on variance and maximize MDP satisfy your needs? – pavy bez Mar 18 '13 at 14:49
• Sometimes heuristics works best. So before building a elaborate objective function to solve this potentially non linear func, I would check if there is good performance by combining the weights. Since min.var usually spits out a subset weights, why not try and apply Max div on those assets with > 0 weights? – user1234440 Nov 4 '13 at 7:36

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

• 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

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 ).

This is a multiobjective problem and can be solved by building a cloud of portfolios with no constraints on either covariance or correlation and constraints on the return and constraints on either the covariance or correlation (whichever you didn't pick as being unconstrained).

Then, find the efficient (Pareto) frontier of this cloud to find the portfolio that is optimal for both correlation and covariance. This is a QCQP (Quadratically constrained quadratic program) since both correlation and covariance optimization are solved using quadratic programming.

• I'm always interested in clever optimization schemes but I don't see how this could work. My concern, as voiced in my answer, is that both portfolios are unique and in general unequal. So, whatever you do, you can never find a portfolio that achieves both objectives. Still interested in more information about your proposed method. – Bob Jansen May 24 '13 at 18:08
• @BobJansen That's correct, there not a single optimal portfolio. What you will get is a surface of non-dominated portfolio allocations in the dimensions return, (co)variance, and correlation – Bo Lu May 30 '13 at 15:01
• In a standard mean variance framework there is only one portfolio with the lowest attainable variance. If you extend the model there might be more but that does not seem to be the context of this question. – Bob Jansen May 30 '13 at 16:42

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.

• 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