Ever since the stock market decline of 2008 resulting from the bursting of the U.S. housing bubble and the ensuing Global Financial Crisis (otherwise known as the "Great Recession"), investors have become obsessed with limiting their exposure to risk. Many different ratios exist to measure this risk - for example, the Sharpe ratio is a good way of quantifying risk adjusted returns. However, these ratios are used to gauge returns only. I propose that an investor should start using a new ratio - the DEG Diversification ratio - to help assess his or her risk when evaluating whether or not to invest in a mutual fund or similar portfolio of stocks. This ratio can even be used to assess risk within one's own portfolio. The DEG Diversification ratio can be a useful tool in determining if the number of holdings within a portfolio is adequate enough for the risk taken on in that portfolio.
Every seasoned investor is familiar with the proverb "Don't put all your eggs in one basket". A similar situation occurs when an investor has only one stock in his or her portfolio. Such a portfolio is undiversified and is very risky. A single stock can have a very high level of risk associated with it. A portfolio consisting of many stocks will have less risk than a portfolio of a single stock. Generally speaking, a portfolio's risk decreases as more stocks are added to the portfolio. This "diversifiable" risk can be reduced by diversifying among stocks. In 1977, Elton and Gruber worked out an empirical example of the gains from diversification (see Works Referenced section for citation). Their results are summarized in the following table. This example shows that the effects from diversification begin to plateau once the size of a portfolio reaches 30 stocks.
Thus, one can say that the minimum number of stocks required to adequately minimize diversifiable risk is 30.
Derivation of the DEG Diversification Ratio
The DEG Diversification Ratio is a measure of the risk adjusted diversification of a portfolio of stocks or mutual fund. It can be used to determine if a portfolio's diversification, or number of stocks, accounting for it's risk, is greater than or less than the risk adjusted diversification of the average portfolio.
The DEG Diversification Ratio requires the following inputs:
- N - the number of stocks in the portfolio
- the standard deviation of the portfolio
The ratio starts out by taking the number of stocks in the portfolio, N, and dividing it by 30, the minimum number of stocks required to be adequately diversified. The reason for doing this is that a portfolio with exactly 30 stocks will have a ratio of 1.00, which indicates that the portfolio is 100% diversified. Any number of stocks less than 30 will indicate that the portfolio is less than 100% diversified (or not diversified enough), and any number of stocks over 30 will indicated that the portfolio is more than 100% diversified (or sufficiently diversified).
Continuing, one then takes this ratio and divides it by 0.32, or two times the average daily annualized standard deviation of the S&P 500 Index (this number, 0.18, is taken from Investopedia, see Works Referenced section for citation). The reason for doing this is that with 95% probability, the standard deviation of a portfolio of stocks will fall within two standard deviations of the mean of the average daily annualized standard deviation of the S&P 500. It is highly improbable that a sufficiently diversified portfolio will have an annual standard deviation that exceeds 0.32. By doing this, the maximum amount of risk is taken into consideration.
In the base case for the equation thus far derived, having just one stock in the portfolio leads to a result of about 0.10417. However, when encountering this situation, one should realize that it corresponds with no diversification. A portfolio of one asset won't be diversified because that one asset has perfect correlation with itself. Thus, one must normalize the equation by subtracting this base case result (highlighted in yellow in the first row below) from all results. See table below for a visual representation thus far.
One will notice that in the case of exactly 30 stocks, the normalized result is about 3.02. If this process is replicated for any N number of stocks and any standard deviation, the resulting number must be greater 3.02 to ensure adequate diversification.
The equation that I have thus far described can be written as follows:
Further simplifying this equation yields the following:
In order for this equation to be meaningful, values are plugged into the equation, with N=100 (this assumes that the average portfolio or mutual fund consists of 100 stocks) and standard deviation equal to 0.18 (the average daily annualized standard deviation of the S&P 500). The result of doing so is 5.13. Thus, it can be said that the DEG Diversification Ratio of the average mutual fund with a standard deviation of 0.18 is 5.13.
Finally, in order to have something to compare against while actually using this ratio, the whole thing is divided by 5.13.
This equation can then be used in a meaningful way.
The DEG Diversification Ratio can be used to compare two or more mutual funds. I will use it to compare two large cap growth funds. The first is the Laudus Growth Investors US Large Cap Growth (MUTF:LGILX) fund, with an overall 5-star Morningstar rating (very impressive!). The Laudus fund has 44 stocks and a standard deviation of 0.1814. The second fund is the T. Rowe Price Growth Stock Fund (MUTF:PRGFX), with an overall 4-star Morningstar rating. This fund has 108 stocks and a standard deviation of 0.1908. The Laudus fund has a DEG Diversification Ratio of 0.33, which is 67% less diversified than the average diversified mutual fund when adjusted for risk. The T. Rowe Price fund has a DEG Diversification Ratio of 1.02, which is 2% more diversified than the average diversified mutual fund when adjusted for risk. While the Laudus fund certainly beats T. Rowe Price's in terms of performance, the T. Rowe Price fund comes out a winner in terms of risk adjusted diversification.
While evaluating mutual funds and other portfolios, investors should use more than just return related ratios, such as the Sharpe ratio, to arrive at any conclusions. The DEG Diversification Ratio can be a very useful tool when trying to assess the diversification of a portfolio of stocks.
E. J. Elton and M. J. Gruber, "Risk Reduction and Portfolio Size: An Analytic Solution," Journal of Business 50 (October 1977), pp. 415-37
Disclosure: I have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours.