[R-SIG-Finance] mean-(scalar) portfolio optimization

[Patrick Burns]

If you are serious about portfolio optimization, then
you need to confront integer constraints such as a
maximum number of assets to trade and a maximum
number of assets in the portfolio.  The "nice" optimization
algorithms are going to fail in this case.


Patrick Burns
patrick at burns-stat.com
+44 (0)20 8525 0696
http://www.burns-stat.com
(home of S Poetry and "A Guide for the Unwilling S User")

Brian G. Peterson wrote:

>>Brian G. Peterson wrote:
>>    
>>
>>>The R function solve.QP is used by several authors to solve classic
>>>Markowitz mean-variance optimization using solve.QP and a covariance
>>>matrix.
>>>
>>>Many other classes of portfolio optimization solve for the weighting
>>>vector w using a scalar measure of risk, such as VaR, Sortino, Omega,
>>>etc.
>>>
>>>Basically, this class of problems could be expressed as:
>>>
>>>let w' be the desired portfolio weights
>>>let R be a set of returns for various instruments
>>>
>>>solve for a weighting vector w such that risk is minimized
>>>
>>>w' = min(risk(R))
>>>
>>>solve for a weighting vector w such that return is maximized over risk
>>>budget y
>>>
>>>w'=max(mean(R)) such that risk(R)<.05
>>>
>>>and other similar formulations.
>>>
>>>solve.QP does not appear to be appropriate for these kinds of
>>>optimization.  The functions 'optim' and 'optimize' seem to return
>>>scalar values, solving only for a single minima or maxima, and not
>>>for the vector (although I may be misunderstanding them).
>>>
>>>Does anyone have any pointers on how you might go about solving these
>>>kinds of optimization problems in R?  I apologize if this is a simple
>>>problem that I haven't been able to find a reference for online. I
>>>will happily post the optimizer code once it's working.
>>>      
>>>
>
>On Wednesday 23 August 2006 12:01, Patrick Burns wrote:
>  
>
>>You are misunderstanding 'optim' -- it optimizes
>>a function over one argument but that argument can be
>>a vector.
>>    
>>
>
>Patrick, 
>
>Thanks for correcting me on 'optim'.  I'll take a closer look.  I am aware 
>of the problem of finding *a* minima, but not necessarily the best 
>minima, in a series that could have more than one.
>
>  
>
>>However the utilities that you mention are hard to
>>optimize.  See 'A Data-driven optimization heuristic
>>for downside risk minimization' by Gilli et al.
>>    
>>
>
>This paper basically advocates a constrained brute-force estimation.
>
>I'd like to avoid that if possible, for reasons of computational 
>complexity.  That's clearly not the only approach being advocated in the 
>current literature.
>
>
>This one:
>http://citeseer.ist.psu.edu/lemus99portfolio.html 
>Portfolio Optimization w/ Quantile-based Risk Measures
>Gerardo Jose Lemus Rodriguez, MIT, 1999
>
>does a pretty good comparison of quantile-based, gradient methods, and 
>non-gradient methods, and looks like it has some good prototypes that 
>could be implemented.  From this paper, it looks like a non-parametric 
>gradient estimator should give acceptable results with minimal 
>computational effort.
>
>This paper:
>http://www.edhec-risk.com/site_edhecrisk/public/features/RISKReview.2005-12-19.1651
>Investing in Hedge Funds: Adding Value through Active Style Allocation 
>Decisions
>Martellini, Vaissié, and Ziemann 2005
>
>uses w' = min(VaR(95%)) with constraints on weight 
>to good effect to establish strategic weighting, but does not provide the 
>math for solving directly for the weighting vector, only expected return 
>under a a four-moment CAPM model, or a four-moment Taylor expansion that 
>could be transformed and solved for w'.
>
>This paper:
>http://www.banque-france.fr/gb/publications/ner/1-108.htm
>Optimal Portfolio Allocation Under Higher Moments
>
>solves for a differentiable series of nonlinear equations into a four 
>moment CAPM model.  This is another relatively intensive approach that 
>I'd like to avoid.
>
>I could reference other papers, but I think that these are representative.  
>I was hoping to spark some discussion of optimization around scalar 
>measures of risk such as VaR, Omega, or Expected Shortfall.  
>
>I'm hoping that others on this list have done something similar and would 
>be willing to point me more directly towards implementation in R, as 
>estimation and optimization function in R are still pretty foreign to me.
>
>Regards,
>
>  - Brian
>
>
>  
>