Goodness of Portfolio Diversification

Do you think portfolio diversification is ‘too mainstream’? Maybe you are missing a great opportunity. Diversification is the most important tool to obtain the risk-adjusted returns you are looking for in a portfolio. In this article we derive a formula that allows you to judge how well diversified your portfolio is.

First, we have to formalize what can be understood by portfolio risk. Risk is a fuzzy concept in the investment world, but we can cover its measure pretty well with the standard deviation of portfolio assets returns:

In the sense of the above formula (1), we are not selling short through the investment weights, hence all of the weights are chosen to be positive. The standard deviations are positive by definition.

We wrote formula (1) as above, to show that the partial goal of portfolio diversification to reduce risk is achieved through picking assets with low correlations (the other goal being to achieve high returns). That is made clear by looking at the second element of the sum operator inside the square root (i.e. the sum of sums); the only way to take away risk in the formula is to have negative elements in the sums, and the only potentially negative elements are the correlation coefficients. Therefore, a very important part of the investment management job is to find uncorrelated assets, which is not an easy task. It is not enough to look into equities and bonds due to their high correlations (even across sectors, capitalization and countries). It is highly recommended to extend the search to asset classes, active strategies and risk factors.

To further develop some intuition, it is worth to visit the following reductio ad absurdum example. It is especially directed to those that think that there is a thing as “too much diversification”. An extreme case, for example, would be picking two assets with a correlation equal to -1 (for simplicity assume equal weight to both assets and equal volatilities that can be achieved through scaling). In this case, the portfolio risk would be equal to zero, yielding a useless investment. In reality, the correlations are dynamic through time and the only way to achieve a perfectly negative constant correlation would be to open a long and a short* position in the same asset, implying no position at all. It is already hard to correctly diversify, and looking at the above example, don’t be afraid to always look for more diversification.

 Now, to derive a formula to score your portfolio diversification, we can compare the fully diversified portfolio risk against the average risk of each single asset

Through the Cauchy-Schwarz Inequality for covariance, we know that

Then, we can construct the main result of this piece:

The above formula will indicate very good diversification when approaching 1 and very bad when approaching 0.

Now you can test how well diversified your portfolio is, and if the score is not good enough, keep looking for more uncorrelated assets.

 

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*I know we said no short selling, but in the example above you can think of going long some derivative that tracks the inverse of the asset, i.e. having a short position but with a positive weight in the formula (1).