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.

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.