Is Your Portfolio Really Diversified?

Remember how you learned as a kid “Don’t put all your eggs in one basket”? That was probably your first lesson in diversification, the concept of spreading risks so that they don’t all hurt you at once. Your portfolio should be diversified for the exact same reason. But how diversified is it, really?

Even if your portfolio holds several securities from different economic categories, and you think it ought to be diversified because all the categories were recommended by your broker or financial advisor, have you ever verified the true degree of diversification? It’s a great question, especially because now most people unknowingly have highly undiversified portfolios. This in turn means their portfolios are riskier than they think.

In “Modern Portfolio Theory (MPT)”, for which its author Harry Markowitz received a Nobel Memorial Prize in Economics in 1990, the idea of diversification against specific risks in each security was deeply analyzed for the first time. Dr. Harry Markowitz showed that by choosing uncorrelated assets, the overall portfolio could have a damped variation of returns even as the individual securities varied individually. In effect, return deviations of securities canceled each other out. The key to investing, in this framework, is to hold a diversified selection of securities. By choosing a set of uncorrelated securities you are supposed to form a portfolio by combining these securities in varying proportions of your investable assets. (“Correlated” simply means the degree to which two securities’ returns go up and down at the same time. This can be quantified by a statistical process called correlation computation.)

The notion of uncorrelated, and thus diversified, assets has become embedded in investor mentality. But how do you pick the uncorrelated assets from which you construct your portfolio? The correct approach requires statistical analysis, which takes work and time, so the financial industry has chosen an approximate method instead. The approximation may have worked in the past but seems to be failing now in our much more connected capital markets.

To diversify correctly, we need to select a pool of securities and compute their statistical cross correlations. From this computation, we select the uncorrelated securities or add more to the pool for further analysis.

The approximate solution, chosen for simplicity, is to choose securities from substantially different financial, geographic and industry categories. For example, designations such as “small capitalization stocks”, “international large capitalization stocks”, and “government bonds” are all categories that have previously indicated that securities drawn from those categories would be uncorrelated. The problem is that now globally, assets and the markets have become quite correlated, and the sought after diversification is lost with this approximate approach to selection.

For example, a major brokerage recommends the asset classes below for an investor with moderate risk tolerance. We have chosen to construct two portfolios exemplifying the recommendations. The first example uses Exchange Traded Funds (ETFs) to study the resulting true diversification, without trying to optimize any return characteristics. The second portfolio uses three different assets and three of the same assets as the first portfolio.

Table 1: Two Portfolios Satisfying Recommended Diversification

Recommended Asset Class Example
Portfolio 1
Symbol Example
Portfolio 2
Small Cap Equity iShares Russell 2000 Index IWM iShares Russell 2000 Index IWM
International Equity Lazard International Small Cap LZSMX Vanguard European Stock ETF Index VGK
Fixed Income PIMCO Total Return Class D Fund PTTDX Schwab Total Bond Market Fund SWLBX
Cash or Equivalent Schwab Yield Plus Short Term Money Fund SWYSX Schwab Yield Plus Short Term Money Fund SWYSX
Other iShares COMEX Gold Trust IAU iShares COMEX Gold Trust IAU
Large Cap Equity iShares S&P 500 Index IVV Rydex Russell Top 50 XLG


Now let’s examine the actual correlations. Technically, the goal is to select securities whose cross correlations of asset returns are close to zero. So, for the above securities, let us get the daily price data for the 16 months ending 30 APR 07, compute the daily asset returns, and build the matrix of all the pair-wise correlations of those asset returns. The result is shown here for Portfolio 1:

Table 2: Correlation Matrix for Portfolio 1


IWM 1.00 0.48 (0.02) 0.03 0.27 0.89 IWM
LZSMX 1.00 0.12 0.09 0.43 0.51 LZSMX
PTTDX 1.00 0.45 0.08 0.01 PTTDX
SWYSX 1.00 0.09 0.04 SWYSX
IAU 1.00 0.23 IAU
IVV 1.00 IVV


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In general, correlation lies between -1 and +1. Correlations of +1 mean the two assets move exactly in sync, and correlation of -1 means the assets move exactly opposite. Correlation of zero means the assets move with no relationship to each other, the condition we seek for diversification.

The matrix shows the value of the correlation of daily asset returns for the 16 months. Since the correlation between IWM and IAU is identical with IAU and IWM, only half of all the different correlations is shown. Of course, the correlation of an asset with itself is always 1, as seen on the diagonal.

In Table 2, each entry is the correlation between the two assets at the row/column intersection. The entries in black are all between -.05 and +.05, which we take as a close approximation to being uncorrelated.

Notice how only 4 of the possible 15 different pairs of correlations are close to being uncorrelated. This means that 4/15, or 27%, of the possible cross-correlations in this portfolio are close to being uncorrelated, when the ideal would be 15/15.

The Portfolio Diversification X-Ray (sm) (PDX, is simply the ratio of the number of approximately uncorrelated assets to all the possible cross-correlations. The PDX varies from 0 to 1, and bigger is better.

Now we’ll see if we can increase the diversification of Portfolio 1 by changing three assets as shown in Portfolio 2. The result is given in Table 3.

Table 3: Correlation Matrix for Portfolio 2


IWM 1.00 0.71 (0.05) 0.03 0.27 0.81 IWM
VGK 1.00 0.06 0.04 0.44 0.77 VGK
SWLBX 1.00 0.54 0.04 (0.01) SWLBX
SWYSX 1.00 0.09 0.01 SWYSX
IAU 1.00 0.20 IAU
XLG 1.00 XLG


Since we now have five roughly zero correlated assets, we have increased the PDX to 33%.

According to the PDX, this Portfolio 2 is an improvement over Portfolio 1.

The message here is that it is not enough to simply invest in brokers’ recommended asset classes. You need to choose securities that really are uncorrelated to ensure diversification. Of course, you also need to choose assets that have expected returns that are sufficiently large as well, so the problem is a bit trickier! However, once you have a set of diverse asset classes, you can choose among those for best returns.

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