[GSoC-PortA] mapping function

Sun Jun 30 18:51:46 CEST 2013

-----Original Message-----
From: gsoc-porta-bounces at lists.r-forge.r-project.org
[mailto:gsoc-porta-bounces at lists.r-forge.r-project.org] On Behalf Of Brian
G. Peterson
Sent: Saturday, June 29, 2013 6:45 AM
To: PortfolioAnalytics
Subject: [GSoC-PortA] mapping function

Based on side conversations with Ross and Peter, I thought I should talk a
little bit about next steps related to the mapping function.

Apologies for the long email, I want to be complete, and I hope that some of
this can make its way to the documentation.

The purpose of the mapping function is to transform a weights vector that
does not meet all the constraints into a weights vector that does meet the
constraints, if one exists, hopefully with a minimum of transformation.

In the random portfolios code, we've used a couple of techniques pioneered
by Pat Burns.  The philosophical idea is that your optimum portfolio is most
likely to exist at the edges of the feasible space.

At the first R/Finance conference, Pat used the analogy of a mountain lake,
where the lake represents the feasible space.  With a combination of lots of
different constraints, the shore of the lake will not be smooth or regular.
The lake (the feasible space) may not take up a large percentage of the
terrain.

If we randomly place a rock anywhere in the terrain, some of them will land
in the lake, inside the feasible space, but most will land outside, on the
slopes of the mountains that surround the lake.  The goal should be to nudge
these towards the shores of the lake (our feasible space).

Having exhausted the analogy, let's talk details.

A slightly more rigorous treatment of the problem is given here:

 <http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1680224>
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1680224

[Doug] This is the 2010 paper, which I just read much of.  Very nice paper.
I find Burns papers pretty useless except for pointing out nice statistical
aspect and promoting PortfolioProbe.   For example in the paper you sent, he
does not say clearly what he is doing in generating the random portfolios
and what he means by out-of-sample.  As you once mentioned, I guess you got
most details through direct conversation with him.

Then I found the new Shaw 2011 paper at
http://papers.ssrn.com/sol3/papers.cfm?abstract_id=1856476. 

It has some nice additional material, e.g., Section 4.3 on lower bounds and
Section 4.4 on bounded shorting, among other.  I still need to finish
reading this version.

In general if you accept random portfolios that violate the constraints,
what is a good way to: (a) Accept only those within a certain distance, with
appropriately defined metric,  of the feasible regions (you don't want to
consider all infeasible solutions - I guess this the reason for min.sum,
max.sum, etc.?), and (b) Assuming an appropriate metric, does one want to
take a solution nearest to a vertex?

It is possible that can use this method directly for random portfolios (and
that we could add the ectra constraint types to DEoptim).  If so, much of
the rest of what I'll write here is irrelevant.  I strongly suspect that
there will be some constraint types that will still need to be 'adjusted'
via a mapping method like the one laid out below, since a stochastic solver
will hand us a vector that needs to be transformed at least in part to move
into the feasible space.  It's alsom not entirely clear to me that the
methods presented in the paper can satisfy all our constraint types.

I think our first step should be to test each constraint type, in some sort
of hierarchy, starting with box constraints (almost all solvers support box
constraints, of course), since some of the other transformations will
violate the box constraints, and we'll need to transform back again.

Each constraint can be evaluated as a logical expression against the weights
vector.  You can see code for doing something similar with time series data
in the sigFormula function in quantstrat. It takes advantage of some base R
functionality that can treat an R object (in this case the weights vector)
as an environment or 'frame'. This allows the columns of the data to be
addressed without any major manipulation, simply by column name (asset name
in the weights vector, possibly after adding names back in).

The code looks something like this:

eval(parse(text=formula), data)

So, 'data' is our weights vector, and 'formula' is an expression that can be
evaluated as a formula by R.  Evaluating this formula will give us TRUE or
FALSE to denote whether the weights vector is in compliance or in violation
of that constraint.  Then, we'll need to transform the weight vector, if
possible, to comply with that constraint.

Specific Cases:

I've implemented this transformation for box constraints in the random
portfolios code.  We don't need the evaluation I'll describe next for box
constraints, because each single weight is handled separately.

min_sum and max_sum leverage constraints can be evaluated without using the
formula, since the formula is simple, and can be expressed in simple R code.
The transformation can be accomplished by transforming the entire vector.
There's code to do this in both the random portfolios code and in
constrained_objective.  It is probably preferable to do the transformation
one weight at a time, as I do in the random portfolios code, to end closer
to the edges of the feasible space, while continuing to take the box
constraints into account.

linear (in)equality constraints and group constraints can be evaluated
generically via the formula method I've described above.  Then individual
weights can be transformed taking the value of the constraint

(<,>,=) into account (along with the box constraints and leverage
constraints).

and so on...

Challenges:

- recovering the transformed vector from a optimization solver that doesn't
directly support a mapping function.  I've got some tricks for this using
environments that we can revisit after we get the basic methodology working.

-allowing for progressively relaxing constraints when the constraints are
simply too restrictive.  Perhaps Doug has some documentation on this as he's
done it in the past, or perhaps we can simply deal with it in the penalty
part of constrained_objective()

Hopefully this was helpful.

Regards,

Brian

--

Brian G. Peterson

 <http://braverock.com/brian/> http://braverock.com/brian/

Ph: 773-459-4973

IM: bgpbraverock

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