Tag Archives: Modern Portfolio Theory

MPT and retirement planning

In a previous post, I questioned how diversified a portfolio of just three assets could really be.  This wasn’t really fair, since the whole point was to avoid such an analysis.  To provide a fairer test, I will use 15 assets for this analysis, all of which are choices taken from an actual retirement plan. The 15 securities listed here consist of 14 mutual funds and a stock.  Choices are predominately focused on US equities, though is a REIT fund (FARCX), two international funds (RERGX, VTRIX), a bond fund (PPTRX), and a money market account.  Given this expanded universe of choices, what will theory tell us is best?

I used Matlab and it’s financial toolbox to retrieve four years of monthly prices for each of the funds.  Since one of the funds is a target date fund, data was not available for it before then.  The data was provided free of charge by Yahoo Finance.  These prices where used to calculate monthly returns, which were in turn used to get means and covariance of the assets.  Once this was done, Matlab’s financial toolbox was used to find the optimal weights.  If you really want to get in to the details of such a calculation, this book explains it well, but be warned:  linear algebra and calculus is required, so in practice, this problem is best left for software.  Below is a screen capture of the results, as well as a graph of the efficient frontier.


The results of the analysis show that fewer assets are better, but raise a few questions as well.  Efficient portfolios generally consist of 3 assets.  This seems to support the idea that a few assets will do.  Yet while these portfolios are efficient with respect the risk reward trade off, they take no account for where the return comes from.  In this case, the funds are for the most part positively correlated over the time period, so funds that would provide diversification are ignored for the higher yielding choices. This points to the period being too short, or more importantly, the number of choices is too small.  The needed number of assets to consider may be close to 100! This seems to indicate that for retirement planning, MPT may be a nice theory, but it’s real value is as a lesson about the value of diversification.

But what’s this say for our choices? There is no substitute for true diversification.  Getting exposure to assets that are uncorrelated is key.  Considering the 15-asset universe, almost every portfolio on the efficient frontier consisted of 3 assets, but a truly diversified portfolio consisting of the choices offered may be better off with a couple more funds.  In this case, examining the top holdings of the funds would provide as much insight as this analysis did!

In search of diversification

In this past weekends Wall Street Journal, I came across an article entitled Funds Investing:  Make More Money and worry less.   At first glance this article seemed to be stating the obvious.  However, after reading it, I began to wonder.  Can one really get diversification from as little as 3 assets?  After some rough analysis, it seems people may need to worry a little bit more then the article suggests.

The article is actually a summary of ways “lazy” people save for retirement.  Lazy, here does not indicate sloth, but rather to retirement ideas named “the margarita”, “the coffeehouse” and “the no brainer”.  The idea of these “lazy” portfolios is that they don’t require a lot of attention or financial know how to set up.  Given the state of most people’s retirements, lets compare the ideas of these retirement plans, requiring little more then your contributions, to conventional wisdom about what it takes to have a solid retirement.

But suppose we have some time, as well as a computer with Matlab or Excel installed on it.  The question I want to ask is:  given these small quantity of assets that make up these lazy portfolios, what do other philosophies about portfolio management say about theses savings ideas?

What follows is based Modern Portfolio Theory, as told in Burton Makail’s “A Random Walk down Wall Street”.   The mathematical analysis, carried out in Matlab, can be done by hand using a general optimization technique called the method of Lagrangian Multipliers.

Modern Portfolio Theory says that the best portfolios lie along the efficient frontier.  This is a line representing the portfolios that invest all the available funds lie on a hyperbola.  The top half of this hyperbola represents what is called the efficient frontier.  These are the portfolios that invest all money, and receive a higher return then the one that lies on the bottom half.  The portfolio located at the “point” is called the Minimum variance portfolio.  The efficient frontier for a portfolio consisting of the 3 mutual funds is shown below.


Taken from Matlab 2012

So consider the portfolio mentioned specifically in the article.  It is a portfolio of 3 vanguard funds, (40% VTSMX (Total Stock Market), 40% VGTSX (Intl. Stock Index), and 20% VBMFX, a bond fund). Using Matlab to calculate things like mean, variance, and covariance of the three assets, using data from Morningstar, we can optimize the weighting of the assets to get the optimal combination.  The weights of the portfolios that lie on the efficient frontier are listed below.


Using the Vanguard funds listed in the article, as well as data on historical returns from Morningstar, I computed the efficient frontier for the 3 assets given in the article.  This analysis shows that while the portfolio may provide exposure “…to every major equity offering in the world”, it is not exactly efficient.  The second asset (the international stock fund) seems to have very little place in this portfolio despite the unique exposure it brings.  Clearly blindly quantitatively optimizing your portfolio may not leave you diversified, but being lazy may not get you there either.