Combining Minimum Variance and Maximum Diversification in Portfolio Optimization
Summary
The document considers how to balance minimum-variance portfolio construction with maximum diversification. It presents several responses: the minimum-variance and maximum-diversification portfolios are each unique and generally differ; treating both objectives together is therefore a multiobjective problem. One proposed approach is to generate portfolios under return and risk or correlation constraints, then identify the Pareto-efficient frontier. The resulting formulation may be a quadratically constrained quadratic program.
Other responses point to mean-variance theory, an example involving two highly correlated funds and short selling, and a suggestion to measure diversification through the total edge length of a minimum spanning tree built from asset correlations. These are discussion contributions rather than a developed, validated procedure. The document does not specify a single objective, constraints, or estimation choices, and its claims about diversification and variance are not reconciled across the answers. The proposed approaches need careful formulation before practical use.
Key ideas
- Minimum-variance and maximum-diversification portfolios generally represent different objectives.
- Balancing variance and diversification can be treated as a multiobjective optimization problem.
- A Pareto frontier can show portfolios that trade off covariance risk and correlation-based diversification.
- A quadratically constrained quadratic program is suggested for combining quadratic objectives or constraints.
- A minimum spanning tree measure is proposed as one alternative way to express diversification.
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Full text
# How to optimize a portfolio under *both* maximum diversity ratio and minimum variance
# 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?
## Answer by Bob Jansen (score 2)
https://quant.stackexchange.com/a/3136
There is only one MVP and only one MDP portfolio so, unless these are the same, this will not be possible.
## Answer by rtybase (score 1)
https://quant.stackexchange.com/a/3143
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:
- http://www.calculatinginvestor.com/2011/06/07/efficient-frontier-1/
- http://www.calculatinginvestor.com/2011/06/14/efficient-frontier-part-2/
However, finding the best portfolio is quite of a task ( http://rtybase.blogspot.co.uk/2011/11/search.html?showComment=1331035896847#c2577055848756808847 ).
## Answer by Bo Lu (score 1)
https://quant.stackexchange.com/a/8048
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.
## Answer by Abhi (score -2)
https://quant.stackexchange.com/a/3140
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.