Jay's Asset Allocation Blog

A few new papers

2010-01-06T18:58:00.000-08:00

I've not been too busy with this project lately, but I'm back.

I read an interesting paper recently, A Sharper Angle on Optimization, by Max Golts and Gregory Jones. I'm always looking out for new takes on improving correlation matrices, and ones which have a geometric point of view seem like they should have some intuitive feel. I've only read through this once, and not yet tried to implement it, but I like there ideas. I'm sure I'll be blogging a bit more on this after I've had time to read and digest the paper.

On the Black-Litterman front, I've had several folks send me helpful information, the most recent is Allaj Erwindi giving me the pointer that I could save some trees in my paper by referring to the Sherman-Morrison formula (which I work through in an appendix of the paper without knowing the name). Thanks, the next version of the paper will be shorter and will include references to this formula.

Diversification and Entropy

2009-09-15T18:33:00.000-07:00

Haven't posted in a while.

Just read an interesting paper by Attilio Meucci Managing Diversification. He goes through some interesting workings to come up with a utility function which allows the investor to trade off diversification versus return. I did some work a while back on entropy and diversification, getting to the point of a paper 2/3 of the way done, but lost momentum. Anyways, this gets me stoked back up to look at the problem of diversification as it's one important to an individual investor as well as an institutional investor.

Dr Meucci is doing a bit of a roadshow with this new work under the auspices of Bloomber University, though the Boston even which I was hoping to attend was canceled at the last minute. Hopefully, it'll be rescheduled soon.

Interesting papers

2009-07-13T18:49:00.000-07:00

I've been reading two papers lately related to Black-Litterman and one that isn't. Mark Kritzman's paper on asset allocation in turbulent regimes is pretty interesting, I'm working on some code to try it out. On the Black-Litterman front, I've been checking out The Augmented Black-Litterman Model which covers a new way to integrate factor models with Black-Litterman. I've plans to write a paper comparing some results from a factor model for tactical asset allocation using Wing Cheung's methods and also the more straight forward two-step approach.

Finally I also came across A VaR Black-Litterman Model which is an interesting paper as well. The author goes into some depth of the implementation of their optimization algorithms which intrigues me as they cover a pretty complex problem with integer and linear constraints.

Updated Applet

2009-06-12T20:42:00.000-07:00

So I've updated the Applet, took me longer than I thought. It's still a little under development, but the new one is posted and it should be easier to use. Right now it doesn't allow a whole bunch of changes to the problem yet, that will come later.

It has some tooltips and hopefully it will be more intuitive to use than the previous one. It has the same set of charts, but they aren't in the same place anymore.

Updating the Applet

2009-06-08T19:41:00.001-07:00

I moved the paper to SSRN and now the BLApplet comes up top on Google so I'm trying to clean it up. The old one is quite idiosyncratic with odd keystrokes and now I'm trying to move to a more intuitive design. It is a fun challenge and I haven't been doing much Swing lately at work so it's a fun distraction as well. I will post again when I get it done, though I'm hoping it's not more than a few more days before I can get the alpha release of the new version up there!

Robustification of Black-Litterman

2009-05-28T19:34:00.000-07:00

I'm reading a paper, "Comparison and robustification of Bayes and
Black-Litterman models", by Schottle, Werner and Zagst. It is interesting in how they build up a Bayesian framework to get uncertainty in the variance of the distribution. Usually in Black-Litterman, the only uncertainty in the variance comes from the uncertainty in the estimate of the mean, so this is a way to add another degree of freedom to the problem. I'd like to build up the MATLAB code to work on this and see how it goes.

Of course, I've also come across some interesting papers on factor models with Black-Litterman and need to digest those. So many things to do. I'm adding links and quick descriptions of all the papers to my RSS feed, so you can pick the papers up from there if you are interested.

Qian and Gorman

2009-05-23T06:28:00.001-07:00

I've been working on understanding paper for oh so long, and finally figured it out. Not sure if it's a testament to other than my follow through on the task, but it feels nice.

It seems they came up with a posterior variance formula different from the Black-Litterman model's posterior variance, and with some less than optimal characteristics. It doesn't include mixing, and it doesn't generally decrease because of the mixing, it can increase. They describe it as allowing the investor to specify views on covariance, but I am stuck on the math. It is not so obvious how the investors view mixes (at least to me), for example any non-zero investors view on variance will increase the posterior variance.

I need to read a bit more on how it should be used in order to really understand what they've done.

I've added a new section to the paper covering this analysis, but in the end it's just tying up a loose end and not really adding anything new to the puzzle.

I've also come across some more new papers, including a few out of Lehman in 2007 which cover some intriguing ways of handling factor models. Factor models and Black-Litterman is definitely an area that I am interested in.

Paper has moved to SSRN

2009-04-26T19:53:00.000-07:00

I've been tracking downloads of the paper from my own site, but I submitted it to SSRN some time ago and it went out in an electronic journal last week, so I'm pushing all downloads through their site to see how many are real and how many are spiders. Should be interesting to see.

I've been quite busy with some night school the last few months, but once it finishes in 3 weeks I'll be able to spend some more time on the asset allocation topics again. Looking forward to running an example through and documenting it.

Shrinkage Methods

2009-03-23T03:29:00.001-07:00

The akutan open source project includes code that implements a few shrinkage methods. Spherical shrinkage is the latest one. It comes from Attilio Meucci's book. I've also revisited and finally found and fixed the bugs in the Ledoit-Wolf shrinkage code. I haven't updated the applet, but CVS has been updated.

It is interesting to compare the efficient frontier drawn for these two methods, the Ledoit-Wolf shrinkage frontier is very close and just to the right of the "standard" efficient frontier. When you look at the portfolios you see they are more diversified than those on the "standard" efficient frontier. When you consider spherical shrinkage, the efficient frontier can be far to the right in parts and then for the example I was considering, was actually to the left of the "standard" efficient frontier. This is because the spherical shrinkage also shrinks the means, so the efficient frontier can move both in the risk direction and the return direction.

Do we need to use constrained optimization with Idzorek's method

2009-03-12T20:47:00.000-07:00

Another reader has asked me whether they need to use constraints during the process of backing the uncertainty in a view out of a confidence when using Idzorek's method.

The thrust of their argument was that their view at 100% certainty was too extreme, and thus they would not consider holding the portfolio generated at 100% certainty.

Idzorek's method provides a simple and clean way to determine the variance of the view by backing it out from a confidence and the change in the weights based on an individual view. You don't need to use a constrained reverse optimization process, and in fact that would change the problem from a simple linear one to a much more complex one. If your views are too extreme, you can shrink the size of the return (Q vector) for that view, however what is important is the impact of the views at the confidence level you hold in them (what you do with the variance backed out from Idzorek's method) rather than the 100% confidence portfolio.

So even if you will use a constrained optimization with the final posterior returns, you do not need to use any type of constraints when computing the 100% confidence posterior as part of Idzorek's method. In fact if you use the formulas included in my paper, you don't really even see the 100% confidence portfolio.

Currency as an Asset Class and Black-Litterman

2009-03-11T03:20:00.000-07:00

I was recently asked a question regarding how to integrate currency as an Asset Class into the Black-Litterman model. This raises several interesting questions. I am hoping this post answers a few, rather than making it more murky.

I am of the opinion that currency is not an asset class. Here are some ideas in that direction.

I do not believe that institutional investors consider currency as a separate asset class. They do often seem to segregate hedge fund investments as if they are a separate asset class, and may place currency managers there. Check out the asset allocations of major endowments for example, I don't recall seeing one with currency called out as a separate asset class.

Within the initial Black-Litterman papers, they considered a global equilibrium that was partially hedged and thus considered currency in this light. It was not considered a separately investable asset class. You essentially hold currency along with the assets denominated in them.

When using a model like Black-Litterman and the CAPM market model, an asset class market capitalization does not include derivatives, only what we might call the cash market. In F/X, as an asset class this might mean holding a basket of currencies as an asset, but you would want to at least earn a deposit rate on the currency which means rather than a currency investment, we could look at it as a basket of short term bonds.

Finally, the criteria for identifying something as an asset class is that they should be homogenous, mutually exclusive, diversifying, contribute to a the market cap and capable of absorbing liquidity. Currency meets most of those criteria, except for being mutually exclusive. If we desire foreign equities to be an asset class, and currency as well then we need to use the hedged foreign equities as our asset class, but that is really just a long position in foreign equity and a short position in the currency. The CAPM equilibrium model in the original Black-Litterman paper deals with this type of model where currency hedging is included in the model, but currency is not an asset class.

For further information, there are some papers linked to from the site which consider issues of whether commodities are an asset class and how to compute their market cap, and also a similar paper dealing with real estate.

I have not found any similar papers discussing currency as an asset class on the internet as of this time, but I haven't searched exhaustively.

More Consistent Formulas

2009-02-17T03:26:00.000-08:00

Another update to the paper is out, this time I really went through and tried to clean up all the formulas to make them more consistent. This is a general problem in the literature, where different authors use the same symbol for different things sometimes and other times they use different symbols for the same thing.

There are probably still a few holdouts in the paper, but the general rule is I use E(r) for the scalar expected return on a portfolio, and π as the vector of returns from something. I've tried to keep σ and ω straight as well.

Also updated the cookbook to match the paper. This should address the most obvious questions I have been receiving lately regarding my inconsistent formulas.

As always, comments, questions and suggestions are welcome either via the blog comments or via email.

More updates and downside risk measures

2009-02-14T19:15:00.000-08:00

Another update the the paper is coming out. Michele Costola caught me playing a little fast and loose with π and E(r) in the paper. Somebody else had already noticed this in the cookbook webpage, but now I'm also fixing up the paper. Probably most people won't notice much, but the symbols will now be more consistent throughout the paper.

I read an interesting paper, "The Black-Litterman Model for Active Portfolio Management" by Da Silva, Lee and Pornrojnangkool. The thrust of their argument is that it isn't always right to consider the prior portfolio as being the mean-variance efficient portfolio from CAPM. If one is using Black-Litterman for active management then one should be considering information ratio. I'd like to add this paper to the literature survey paper in some upcoming round.

MATLAB Black-Litterman

2009-02-04T03:27:00.000-08:00

Finally have access to MATLAB so I was able to port the He and Litterman paper example from SciLab to MATLAB and put it up on the website. Next step will be to move the Idzorek example over as well.

My next project will be to work an example from the literature which generates the views from a multi-factor model. There are a few of these, and Anisa Salomans thesis provides her SciLab source code and I have been able to find all the data freely on the internet.

Updated Paper (reprise)

2009-02-02T03:25:00.000-08:00

I updated the paper with a few more tweaks and it's up on the site. I've also posted it to ssrn. The next area I plan on working on here is a multi-factor model to generate the views. Anisa Salomons wrote a thesis on Black-Litterman which I've mentioned before in the blog, and it is her example I'm trying to make a bit more transparent and document so others can easily reproduce it. I've finally tracked down all the data I need (and that I think she used), now I just need to get the scripts in place to do the work. One area I look forward to trying out, is whether using a posterior variance improves the results or does not. She states that she tested this and it was worse, but didn't include any evidence in her thesis and it wasn't on topic.

In project news my move to cleanup the javadocs and get everything squared away for a 1.0 release is coming along. Still have a ways to go, but it's feeling a little closer each day. Hoping to have it done in the next 2 weeks.

Updated Paper on the site

2009-01-24T19:28:00.000-08:00

I've finally finished my updating of the paper. This new January 22nd version includes what I think is an improved section on Idzorek's extension and a more thorough treatment of measuring tilt/distance/compatibility of the prior, views and posterior. I need to make a quick update with some small edits, but it's basically all there right now.

Haven't been adding as much to the RSS feed of papers I've been reading, but I did get a second batch in there and will shortly put up some more papers.

I'm also in the process of cleaning up the akutan open source project, trying to document everything that is in there. I'm also trying to separate examples and unit tests from the re-usable part. My goal is to get a release 1.0 jar up on sourceforge in the next few weeks.

Added a Reading List to blacklitterman.org

2008-12-13T19:52:00.001-08:00

I've added a reading list RSS feed at blacklitterman.org. It's just getting started so not much content and a little creaky to use. It should improve over time as I get used to it.

My goal is to add links to papers with relevant content to the Black-Litterman model and Asset Allocation in general.

I've started out with a series of papers on Asset classes, and integrating them with the Black-Litterman model. Several of the papers are authored by Thomas Idzorek from Ibbotson.

Updating the Black-Litterman paper

2008-11-25T20:15:00.000-08:00

Had to stop work on all the other fun things to update the paper on Black-Litterman. Turns out the section on Idzorek's method has some inconsistencies in the example tables (all the A ones don't make much sense), though the formulas and the results are all good. Anyways, that needs to be rewritten and I'm adding a section on Braga and Natale's work on TEV, probably a consistent example across Meucci and Fusai, and Braga and Natale to show how the two methods of measuring distance differ. I'd also like to add a third metric, but not finding one I like that much so far.

Then it's on to updating the website and adding some more SciLab scripts for various Black-Litterman examples.

CVaR

2008-11-07T06:02:00.000-08:00

After switching to use the GPL open source linear optimizer lp_solve I can now optimize an asset allocation with a CVaR objective. It does not look quite as exciting as I had hoped, and it is slow, but it will be an interesting new tool in the toolbox.

Next stop will probably be back to work on some Black-Litterman ideas and maybe providing some richer (but accessible) tools which implement it. I still have in the back of my mind that I'd like to know if using posterior variance makes the results better or worse. I would also like to come up with a standard set of data to apply across all the various methods so that they could be compared apples to apples in one place.

Always lots to do.

Commodities in Black-Litterman

2008-10-29T18:27:00.000-07:00

For those people trying to use the Black Litterman to model asset classes other than equities and bonds, the challenge of commodities is an interesting one. How does one figure out the market capitalization of commodities, and what is the relationship between the ownership of reserves by natural resource companies (thus included in their market capitalization), and how do futures which might exceed the size of the spot market by some multiple fit in.

Tom Idzorek wrote a paper which covers some of these topics. I've seen a few more around the internet and I'll try to start putting them up here and also on the blacklitterman.org website.

Updated blacklitterman.org

2008-10-29T18:15:00.000-07:00

I updated the blacklitterman.org website with a definitely working Scilab script for the He and Litterman paper, and hopefully working Matlab scripts for the same thing. I should get a chance to debug the Matlab scripts in the next day. It is one of the most common queries on my website, looking for Matlab Black-Litterman scripts.

I've been having a lot of good discussions with co-workers about how one might structure a tool for Black-Litterman work. It's helping me continue to get a better handle on the various ideas and how they all interrelate. At this point it seems the math is the easy part.

I'm hoping to try and pull this together and update the paper since it's been almost 6 months since the last update. Not too many new things to put in there, but maybe some more flesh on the bones.

Also trying to clean things up in the website html a bit, finally moving to a dreamweaver template for the site so it's easier to make site wide changes.

Random Matrix Theory

2008-10-28T04:48:00.000-07:00

Between Rosenow, et al and Potter, et al I've figured out a few ways people suggest to 'clean' a 'noisy' correlation matrix. Basically take the largest eigenvalues and either average the rest, or set the rest to 0 depending, reconstitute the matrix and set the diagonal to 1. By setting the noisy eigenvalues to 0, you have a sort of shrinkage towards uncorrelated assets which is interesting. I haven't quite thought through what the implication of that is on a mean-variance optimization.

Asset Allocation Technology

2008-10-19T19:07:00.000-07:00

This blog is really discussing the technology of Asset Allocation, and hopefully some on the topic of the asset allocation process. I will not be discussing potential asset allocations.
Most recently I began to work on building some programs to perform portfolio optimization using CVaR, Conditional Value at Risk as the objective function. This requires a linear optimization algorithm, which I don't currently have in my toolbox. Someplace along the way of building this linear optimizer I ran into the topic of Random Matrix Theory.
Econophysics, it always draws me in. I think it's because of my Physics background, and how appealing their approaches are. Entropy is another Econophysics concept that I've spent some time looking at, but it hasn't added up to much in terms of portfolio optimization yet.
As far as Random Matrix Theory it also seems interesting, and it's not to difficult to figure out which part of the correlation matrix appears to be noisy, but it is somewhat complex to put the updated correlation matrix back together. While there are many papers on the topic, they don't usually provide a lot of information on re-assembling the correlation matrix. I'm hoping to make some progress in this direction at some point myself.
In the meantime I'm back working on my linear least squares optimizer. I was trying to work with the augmented form of the equations, but so far I have some errors in my math as my matrices are singular. Ouch.

CVaR

2008-10-03T20:19:00.000-07:00

I decided to take a look at CVaR as an objective function. It has some desirable features, it can be solved using simulation and a good linear solver for sparse problems. This was all good in theory, but my Java simplex solver just isn't up to the task, so it's required me to go back to square 1 and build myself a new linear interior points solver. Given that I built a reasonable interior points solution for dense quadratic problems, I'm hoping it's not too much of a stretch to build a reasonable one for sparse linear problems.

lp_solve seems like a good tool in the problem space, but it's C so integration is a bit of work and I'd rather have a pure java solution in the short term even if it's not quite so good. I am not trying to solve general purpose problems, just CVaR.

I'm hoping to get this up on the site in the next few weeks, it depends on how much time I have for my hobby.

Asset Allocation Workbench

2008-09-23T18:49:00.001-07:00

So I've started work on this tool. I was stuck on making it an eclipse plugin and once I gave up on that and went for a simple Swing app things started falling into place. Hopefully I'll get the latest checked into the akutan cvs at sourceforge by the weekend.

I'm still noodling with entropy and information theory as some sort of aid in portfolio optimization as I like the elegance of the math, but the point was made to me that for the most part this is just shrinkage in weight space during the optimization vs the more common shrinkage in return space (prior to the optimization). Unfortunately clean math doesn't always get us anything useful in asset allocation.

Given that I've started on the asset allocation workbench project, I'm starting to look at more concrete portfolio optimization techniques I can fold in, at the moment it's CVaR, but next I hope to plumb a bit of higher order moments. I've built up a bit of a wish list from other folks over the years which I'll be adding to the blog as a means to capture requirements so to speak.