Introduction
We compared the risk profiles of two types of portfolios, one based on S&P 500 stocks and the other based on exchangetraded funds (ETFs). Since many of the initially available ETFs tracked largecap indexes such as the S&P 500, it was unclear how much total risk reduction could be achieved with portfolios of ETFs above and beyond portfolios of S&P 500 stocks. The creation of more varied ETFs, however, may provide opportunities for superior diversification. That is, the total risk of a portfolio consisting of ETFs may be lower than that of a portfolio based on the S&P 500. To test how newer ETFs have changed portfolio comparisons, we compared data from an early time period when relatively few ETFs were available to a more recent time period with a greater variety of offerings.
Our results, which are based on data from 2001 through 2007, support the idea that portfolios constructed purely of ETFs may be a costeffective substitute for traditional portfolios comprised of stocks and bonds. Specifically, based on the available universe of investable ETFs, we found that portfolios of ETFs require roughly half as many securities as portfolios of stocks to achieve similar levels of diversification. While in practice the exact number of stocks or ETFs needed to diversify a portfolio will depend on a multitude of factors including which ETFs are selected, our results provide a guidepost to the question, “How many ETFs are enough?”
Risk reduction through the construction of equity portfolios has been a mainstay of finance theory and practice. Harry Markowitz’s original study on portfolio selection and diversification distinguished between two types of risk: undiversifiable or systematic risk and diversifiable or unsystematic risk.^{1} Markowitz’s foundational work has provided the basis for numerous studies on the practical question of how many securities are required to effectively remove unsystematic risk, producing a diversified portfolio. One early study employed data for 470 securities in the S&P 500 index from January 1958 to July 1967 and constructed equally weighted, randomly chosen portfolios to estimate the relationship between portfolio size and risk.^{2} The results indicated that the benefits of diversification were drastically diminished once a portfolio reached eight to 10 stocks.
This approach to diversification, sometimes called naïve diversification, was refined and retested by others. A similar study from a later period concluded that eight stocks were sufficient to effectively diversify a portfolio and that adding any additional securities contributed only marginally to diversification. The data showed that 40% of possible risk reduction was obtained by holding two stocks, 80% by holding eight stocks, 90% by 16 stocks, 95% by 32 stocks, and 99% by 128 stocks.^{3} Although a few authors have argued that more stocks were needed, perhaps even 20 to 40,^{4} most followup studies have confirmed and most finance texts now teach that eight to 20 stocks are sufficient to diversify an equity portfolio.^{5}
While diversification of stock portfolios remains an important issue, the introduction of new financial products has opened the possibility of portfolios comprised of other types of assets. For example, a recent study on the optimal number of hedge funds needed to diversify a portfolio concluded that five to 10 funds would provide most of the diversification benefit. Other research has found substantial diversification benefits from adding multiple mutual funds to a portfolio, even when the mutual funds purported to follow similar strategies.^{6} The availability of a variety of ETFs and the ease with which they can be used to construct portfolios at low cost make them a natural tool to use for diversification. Since ETFs are a relatively new financial instrument and have yet to be studied in depth, we decided to test ETF portfolios against portfolios of stocks.
How We Constructed Our Portfolios
To compare ETF portfolios to stock portfolios and examine how the introduction of newer and more diverse types of ETFs has enabled higher levels of portfolio diversification, we collected available data provided by CSI and Bloomberg for two time periods: January 2001 to December 2003 (89 ETFs) and January 2004 to February 2007 (134 ETFs). This difference in number of funds is due to the relatively small number of ETFs available during the earlier period and the growing number of ETFs after 2003.
Table 1 describes the composition of ETF categories as listed by Morningstar in each time period. Domestic style funds focus on specific categories of equities, such as largecap growth or midcap blend. In 2001, 43.8% of ETFs were stylebased and, of these, 69.2% were largecap funds. As the number of ETFs has expanded, domestic style funds still make up the largest proportion of available funds, although that proportion is decreasing. The secondlargest category, representing 29.2% of available ETFs in 2001 was sectororiented funds, which specialize in industry groups such as technology, financials, health care and utilities. International ETFs made up approximately 27.0% of available ETFs, but between 2001 and 2004, more funds focusing on emerging markets and Asia were introduced.
Table 1: Composition Of ETFs In Each Sample Period

Both periods employed data for the 457 stocks included in the S&P 500 index from January 2001 through February 2007. Fewer than 500 stocks were used because the composition of the S&P 500 changes somewhat each year. While the presence of survivorship bias may change return estimates somewhat, they should have only a small impact on standard deviation estimates.^{7}
To create our portfolios, we assumed an equal dollar amount was invested in each security, and that the composition of each portfolio was randomly selected without replacement from the population of securities in their respective markets. While this is not a diversification strategy we would recommend, this naïve strategy is a useful way to find a lower bound for risk reduction, because most investors can diversify at least as well as a naïve strategy. Note that with naïve diversification, the expected returns of any randomly selected portfolios are equal.
Next, we computed weekly returns for each security in each group over the specified time period. To get the data necessary to compute annual expected portfolio variances, we calculated the average annual variances and covariances of the returns. Annual portfolio variance is given by:^{8}
(1)
where ?^{2} = annual portfolio variance, N = the number of equally weighted securities in a given portfolio, = mean annual variance of all the securities in the population, and = mean annual covariance of all the securities in the population. For a single stock portfolio, N equals 1 and the annual expected portfolio variance equals the expected annual variance of the single security portfolio. As N approaches the size of the population, the annual expected portfolio variance approaches the variance of an equally weighted portfolio of all assets in the population.
Ideally, additional assets should be added to a portfolio until the marginal benefits accrued from risk reduction do not compensate for transaction costs. Transaction costs as a percentage of a portfolio are difficult to estimate and surely depend on portfolio size among other factors; however, for comparison purposes we used a risk reduction threshold of 0.25 percentage points, a number consistent with the findings of other formal studies of optimal portfolio diversification.^{9} That is, diversification was deemed to be sufficient when the addition of one additional security to a portfolio decreased risk by 0.25 percentage points or less.
The resulting annual portfolio standard deviations calculated from the expected variances given by equation 1 are listed in Table 2 and plotted in Figures 1 and 2 for each time period. These figures show the textbook pattern of risk reduction through diversification: portfolio standard deviations are high when the number of securities is small and decline as the number of securities is increased.
Table 2: Portfolio Standard Deviations

The relationship between portfolios of ETFs and stocks was similar for the two time periods. For portfolios with a small number of securities, total risk was greater for the S&P 500 portfolios than the ETF portfolios. ETFs effectively eliminated unsystematic risk with fewer securities. For the ETF portfolios, the benefit of adding securities became negligible at a portfolio size of approximately five to six in both time periods. For the S&P 500 portfolios, the number of stocks needed to remove substantially all of the unsystematic risk was approximately 12 during the more volatile 20012003 period and nine in the more recent time period, a finding consistent with previous studies. These results are summarized in Table 3.
Table 3: Number Of Securities Required For Diversification

While the smaller number of ETFs needed for diversification intuitively results from the fact that ETFs themselves are comprised of a number of stocks and hence, are less volatile individually, this did not translate into lower levels of systematic risk for the population of ETFs. Rather, as the number of securities increased, the standard deviations of stock and ETF portfolios roughly converged. In the earlier and higher volatility time period, the minimum attainable risk was slightly greater than 20.0% (i.e., both portfolios asymptotically approached a standard deviation of roughly 20.0%) while in the later time period, it was approximately 11.5%.
One might have expected the significant number of ETFs dedicated to international equities to reduce portfolio risk relative to S&P stocks. While this foreign exposure likely helped, the risk reduction benefit resulting from foreign ETFs (approximately 28.0% of the population), was offset by the presence of higherbeta ETFs focusing on sectors such as technology (approximately 10.0% of the population).
In the later time period, the composition of the ETF population arguably became more diverse with a 4.5% allocation to bonds and the higher weighting on international outside of Europe. While this change in weightings might have been expected to reduce achievable systematic risk, any reduction in portfolio risk as a result of this allocation was not obviously detectable in the data. This suggests that as of 2004, the population of ETFs was still weighted heavily toward largecap stocks.
Figure 1: Portfolio Standard Deviations, January 2001December 2003
Figure 2: Portfolio Standard Deviations, January 2004February 2007
Lessons For Employing Portfolios Of ETFs
Once a substitute for traditional mutual funds, ETFs now span a wider range of asset classes and provide investors with the flexibility to manage portfolios at relatively low cost. We found that investors building diversified portfolios with ETFs can substantially eliminate unsystematic risk with about half the number of securities as stocks. A portfolio of five to six ETFs would provide about the same level of risk as a portfolio of nine to 12 largecap stocks. It should also be noted that any investor interested in achieving the same level of risk as largecap stocks could do so with only a single S&P 500 ETF. Hence, the actual number of ETFs needed for diversification must depend on the investor’s specific goals. We expect that many investors using ETFs to construct portfolios will want to take advantage of their considerable flexibility to target sectors or styles that present the best opportunities. In this regard, our results indicate that such portfolios require no more than a handful of ETFs.
Although in this respect ETFs do more with less, naïve diversification with ETFs has its limits. Because most of the available funds focus on largecap stocks and many concentrate in highbeta sectors such as technology, reducing portfolio volatilities to levels below that of largecap stocks will require a more targeted approach to diversification that takes into account the overall weightings in the population of available ETFs. These findings are critical for investors considering ETFs as an alternative to stocks.
End Notes
^{2}See John Evans & Stephen Archer, “Diversification and the Reduction of Dispersion: An Empirical Analysis,” Journal of Finance, 1968, Vol. 23, pp. 761767.
^{3}See Lawrence Fisher & James Lorie, “Some Studies of Variability of Returns on Investments in Common Stocks,” The Journal of Business, 1970, Vol. 43, pp. 99134.
^{4}For two examples of diversification studies that estimated that a larger number of securities are needed, see: Ron Bird & Mark Tippett, “Naïve Diversification and Portfolio Risk: A Note,” Management Science, 1986, Vol. 32, pp. 24425 and Meir Statman, “How Many Stocks Make a Diversified Portfolio?” Journal of Financial and Quantitative Analysis, 1987, Vol. 22, pp. 353363.
^{5}A summary of what popular textbooks give as recommendations for how many stocks are needed for diversification is provided by Gerald Newbould & Percy Poon, “The Minimum Number of Stocks Needed for Diversification,” Financial Practice and Education, 1993, Vol. 3, pp. 8587.
^{6}For a study on how many hedge funds are needed for diversification, see FrancoisSerge Lhabitant & Michelle Learned, “Hedge Fund Diversification: How Much is Enough?” The Journal of Alternative Investments, 2002, Vol. 5, pp. 2349. Edward O’Neal, “How Many Mutual Funds Constitute a Diversified Mutual Fund Portfolio?” Financial Analysts Journal, March/April, 1997 explores the characteristics of portfolios of mutual funds.
^{7}In simulation studies of hedge funds, which tend to exhibit higher levels of fund dropouts, the impact of survivorship bias on returns variability has been found to be small. For details, see Lhabitant & Learned.
^{8}Equation (1) can be found in Harry Markowitz, Portfolio Selection: Efficient Diversification of Investments, New York: John Wiley & Sons, 1959.
^{9}Two examples of diversification studies that attempted to estimate an optimal threshold for diversification are: Evans & Archer from endnote number 2 and Statman from endnote number 4.