Sharpe's algorithm for portfolio improvement
Brian G. Peterson wrote:
Just to round out this thread on-list, I've attached a script for the problem as described by Prof. Burkett that uses the Munger and Arrow bounded utility function to construct a portfolio via PortfolioAnalytics using both random portfolios and DEoptim as alternate optimization engines.
Thanks again for your ingenious script and illuminating comments. The only contribution I can make to the discussion now is to provide some references on bounded utility functions. Karl Menger (a mathematician and son of the Austrian economist Carl Menger) made a case for bounded utility functions in a 1934 article in German. An English translation was published as "The Role of Uncertainty in Economics", ch. 16 in Essays in Mathematical Economics in Honor of Oskar Morenstern, ed. Martin Shubik (Princeton University Press, 1967). Other arguments for bounded utility functions have been made by Kenneth J. Arrow, Essays in the Theory of Risk-bearing (Markham, 1971) and by John W. Pratt, Howard Raiffa, and Robert Schlaifer, Introduction to Statistical Decision Theory (MIT Press, 1995), pp. 80-81. The particular utility function I have been using is just one example of a bounded utility function, expressed in terms of gross return. To the best of my knowledge, none of the authorities cited above have recommended this particular function. Best regards, John
I've modified the script to use the 'edhec' data series included with PerformanceAnalytics for reproducibility. As Enrico noted in an earlier post, this utility function tends towards creating very concentrated portfolios in only a small number of assets. Regards, - Brian On Mon, 2011-08-01 at 22:57 -0400, John P. Burkett wrote:
Brian G. Peterson wrote:
On Mon, 2011-08-01 at 17:07 -0400, John P. Burkett wrote:
Enrico Schumann wrote:
what kind of "difficulty" did you encounter? If you would give more details on what you tried, and how, then people should be better able to help you.
Thank you, Enrico, for your prompt and helpful response. Focusing on
Sharpe's algorithm, I originally omitted specifics about difficulties I
have encountered with packages based on other algorithms. However, in
case it is of interest, I will now outline my project and difficulties.
I have about 120 monthly observations on gross real returns for
about 30 assets. [Trying to mitigate estimation error, I've shrunk
observed returns toward a common value as proposed by Philippe Jorion,
"Bayes-Stein Estimation for Portfolio Analysis",
Journal of Financial and Quantitative Analysis, v. 21, n. 3 (Sept.,
1986), pp. 279-92.] I initially specified the utility function as U =
R/(0.5 + R) where R is R is the gross return on a portfolio and
specified long-only constraints. The restriction that portfolio shares
sum to 1 is handled by specifying n-1 variables (where n is the nubmer
of assets) and making the share asset n be 1 minus the sum of other
shares. If I apply DEoptim to a sufficiently small subset of the assets,
it converges and selects a plausible portfolio. Yet if I ask DEoptim
to analyze as many as 30 assets, it fails to converge in any number of
iterations that I've tried. Given the same problem, Rgenoud converged
after 44 hours (more than 2 million generation, if memory serves). I
subsequently tried changing the utility function to ln(R) and asking
Rgenoud to maximize that. Thus far it has run for over 1.5 million
generations without converging. The portfolio shares have barely
changed over many recent generations. Perhaps I could just relax the
default convergence criteria and declare the problem solved for
practical purposes. However, that might result in mistaking a local for
a global maximum. These experiences may just indicate that a 30 asset
portfolio is hard to optimize using general purpose algorithms. Kris
Boudt et al. note that "portfolio problems encountered in practice may
require days to optimize on a personal computer" ("Large-scale portfolio
optimization with DEoptim," p. 1). These experiences motivated my
interest in trying an algorithm, such as that of Sharpe, designed
specifically for portfolio optimization.
As one of the authors of the paper in question, I'll note that I was talking about portfolios with hundreds or thousands of assets, and/or with many rebalancing periods or observations. Monthly series of 120 observations each shouldn't take millions of iterations to converge. It is possible that the starting set is not feasible, given the way your full investment constraint is being applied.
Thanks, Brian. I'll double check my code to see if it's given DEoptim an impossible task.
If your objective function is truly a smooth convex function, then optim() or some other linear or conical solver should be sufficient. If the function is not smooth (and real portfolio objectives and constraints rarely are), then a random portfolios or a global optimizer will be required. Perhaps you should start with a couple hundred random portfolio survey of your feasible space? This may be accomplished in R with Pat Burns' excellent Portfolio Probe (commercial non-free), or with PortfolioAnalytics (open source/free, on R-Forge). PortfolioAnalytics will also allow you to use DEoptim after using random portfolios to get close to the global minima. See out R/Finance seminar presentation here: http://www.rinfinance.com/agenda/2010/Carl+Peterson+Boudt_Tutorial.pdf and the code to replicate here: https://r-forge.r-project.org/scm/viewvc.php/*checkout*/pkg/PortfolioAnalytics/sandbox/script.workshop2010.R?root=returnanalytics PortfolioAnalytics does some of the heavy lifting to set good defaults and give you a good set of starting population values to speed convergence. Also, Pat Burns' Portfolio Probe blog is here: http://www.portfolioprobe.com/blog/
These are very helpful leads. I'll pursue them tomorrow morning. Thanks again! Best regards, John
- Brian
I don't know the paper you mentioned, but I know a paper of W. Sharpe in which he suggests to do repeated zero-sum changes to the portfolio, like "increase one asset by x%, and decrease another one by x%". If that is what you mean, this can be done with a local search.
The algorithm you describe sounds very much like that covered in the articles I cited in my previous note. It's probably the same algorithm. (But actually, other
functions like DEoptim should work just as well. The DEoptim package even comes with a vignette that adresses portfolio optimisation.)
Perhaps I should just be more patient with DEoptim or buy a faster computer!
An example for a local search procedure is given in one of the vignettes of the NMOF package (which is available from https://r-forge.r-project.org/R/?group_id=1128 ), even though I am not sure how self-explanatory the vignette is.
Thank you for the NMOF reference. I've printed "Examples and Extensions
for the NMOF package" and tried the command
install.packages("NMOF", repos = "http://R-Forge.R-project.org")
That command elicited the following messages:
Warning in install.packages("NMOF", repos =
"http://R-Forge.R-project.org") :
argument 'lib' is missing: using
'/home/john/R/i486-pc-linux-gnu-library/2.8'
trying URL 'http://R-Forge.R-project.org/src/contrib/NMOF_0.14-2.tar.gz'
Content type 'application/x-gzip' length 1352123 bytes (1.3 Mb)
opened URL
==================================================
downloaded 1.3 Mb
* Installing *source* package 'NMOF' ...
** R
** data
** moving datasets to lazyload DB
** inst
** preparing package for lazy loading
** help
>>> Building/Updating help pages for package 'NMOF'
Formats: text html latex example
DEopt text html latex example
EuropeanCall text html latex example
GAopt text html latex example
LSopt text html latex example
MA text html latex example
NMOF-package text html latex example
NS text html latex example
NSf text html latex example
PSopt text html latex example
TAopt text html latex example
bundData text html latex example
callHestoncf text html latex example
fundData text html latex example
gridSearch text html latex example
qTable text html latex example
ERROR in file 'repairMatrix.Rd': Environment (text enclosed in {}) found
in \title{...}.
The title must be plain text!
ERROR: building help failed for package 'NMOF'
** Removing '/home/john/R/i486-pc-linux-gnu-library/2.8/NMOF'
The downloaded packages are in
/tmp/RtmpNAuDvf/downloaded_packages
Warning message:
In install.packages("NMOF", repos = "http://R-Forge.R-project.org") :
installation of package 'NMOF' had non-zero exit status
***************************************
Any suggestions for how to successfully install NMOF would be greatly
appreciated.
Best regards,
John
Regards, Enrico
-----Urspr?ngliche Nachricht----- Von: r-sig-finance-bounces at r-project.org [mailto:r-sig-finance-bounces at r-project.org] Im Auftrag von John P. Burkett Gesendet: Montag, 1. August 2011 18:49 An: R-SIG-Finance at r-project.org Betreff: [R-SIG-Finance] Sharpe's algorithm for portfolio improvement An attractive sounding algorithm for maximizing the expected utility of of a portfolio was proposed by William F. Sharpe in "An algorithm for portfolio improvement," Advances in Mathematical Programming and Financial Planning, 1987, pp. 155-170 and summarized by the same author in "Expected utility asset allocation," Financial Analysts Journal, vol. 63, no. 5 (Sep.-Oct., 2007), pp. 18-30. Has this algorithm been implemented in R? If not, is there a substitute that is likely to work well for a user-specified concave utility function? I've tried optim, DEoptim, and Rgenoud but encountered difficulty in getting them to converge for a long-only portfolio of around 30 assets. Best regards, John -- John P. Burkett Department of Economics University of Rhode Island Kingston, RI 02881-0808 USA
-- John P. Burkett Department of Economics University of Rhode Island Kingston, RI 02881-0808 USA phone (401) 874-9195
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John P. Burkett Department of Economics University of Rhode Island Kingston, RI 02881-0808 USA phone (401) 874-9195