Modern portfolio theory has demonstrated the importance of diversification to investors. While diversification is almost always beneficial in reducing portfolio risk, a commonly asked question is: "How much diversification is enough?"

Newbould and Poon (1993) survey a number of U.S. investment textbooks and academic studies, and find that the consensus view is that portfolios consisting of eight to 20 stocks are generally considered well diversified. While Newbould and Poon do not provide a specific number of stocks that would constitute a well-diversified portfolio, they suggest the number would be much greater than 20.

As for Canadian investment textbooks, Cleary and Jones (1999), and Bodie, Kane, Marcus, Perrakis and Ryan (1997) refer to the Statman study, while Sharpe, Alexander, Bailey and Fowler (1997) suggest that 30 stocks is the 'magic' number.1

We examine this issue using Canadian data for a more recent time period in order to provide updated general results that can be used as a reference point for Canadian academics, investment professionals and individual investors.

The Overall Risk
The overall risk of an individual stock is made up of two types of risk, systematic risk (commonly measured by beta) and unsystematic risk. Systematic risk is economy-wide and affects all or virtually all companies. Unsystematic risk is firm-specific and affects only one company (or a small number of companies).

One of the basic tenets of portfolio management is that diversification reduces the overall risk of a portfolio. Unfortunately, diversification cannot eliminate risk entirely because systematic risk cannot be eliminated (since an unexpected increase in interest rates, for example, would increase borrowing costs and would have a negative effect on virtually all companies). Thus, a properly diversified portfolio of stocks will only be susceptible to systematic risk, as virtually all of the unsystematic risk will have been diversified away.

Evans and Archer (1968) examined semi-annual observations on 470 of the stocks listed in the Standard & Poor's index from 1958 to 1967. They concluded that a portfolio consisting of 10 different stocks was sufficiently diversified, stating that the results of their study "raise doubts concerning the economic justification of increasing portfolio sizes beyond 10 or so securities."

Fisher and Lorie (1970) examined the frequency distributions and dispersion of wealth ratios of investments in different-sized portfolios of New York Stock Exchange stocks between 1926 to 1965, with equal initial investments made in each stock in a portfolio.2 They found that "the opportunity to reduce dispersion by increasing the number of stocks in the portfolio is rapidly exhausted. Roughly, 40% of achievable reduction is obtained by holding two stocks; 80% by holding eight stocks; 90% by holding 16 stocks; 95% by holding 32 stocks; and 99% by holding 128 stocks."

Solnik (1974) examined the additional portfolio risk reduction that could be achieved by diversifying internationally. He used data on more than 300 stocks from the U.S. and seven major European markets (the U.K., France, Germany, Italy, Belgium, the Netherlands and Switzerland). He found that, whether hedged against exchange rate risk or not, "an internationally diversified portfolio is likely to carry a much smaller risk than a typical domestic portfolio." Another interesting finding was that well-diversified stock portfolios from most of the European markets had higher proportions of systematic risk than did a well-diversified portfolio of U.S. stocks.

Statman (1987) showed that "a well-diversified portfolio of randomly chosen stocks must include at least 30 stocks for a borrowing investor and 40 stocks for a lending investor. This contradicts the widely accepted notion that the benefits of diversification are virtually exhausted when a portfolio contains approximately 10 stocks." Statman suggested that diversification should increase as long as the marginal benefits (as measured by risk reduction) exceed the marginal costs (as measured by transaction costs).

Newbould and Poon (1993) argued that standard recommendations to form a portfolio with between eight and 20 stocks were flawed, and that "it may be desirable to have substantially more than 20 stocks in a portfolio to eliminate diversifiable risk."

Thirteen Years of Data
We use monthly arithmetic mean rates of return, and the monthly standard deviations of these returns, for TSE-listed stocks over the January 1985 to December 1997 time period. We consider monthly returns rather than annual returns used in several previous studies because investors and portfolio managers are likely to perform some rebalancing of their portfolios at monthly rather than annual intervals, and therefore will be concerned with monthly portfolio volatility. We also split our sample into two equal sub-periods--from January 1985 to June 1991, and from July 1991 to December 1997.

For each time period, we choose stocks randomly to simulate equally-weighted portfolios ranging from one stock to more than 200 stocks.3 For each portfolio size, 5,000 simulated portfolios are constructed, and the results are averaged. It is assumed that there are no subsequent reallocations or rebalancing of the portfolios during the entire time period.

We include all TSE stocks that had complete total return information available for the time period being examined. Thus, stocks of companies that merged with other companies or were delisted from the TSE during each time period were not included in the analysis.4

As a result, we include 222 stocks in our sample for the entire sample period (January 1985 to December 1997), 236 stocks for the first sub-period (January 1985 to June 1991) and 415 stocks for the second sub-period (July 1991 to December 1997). For comparison purposes, we have included summary statistics for the TSE 300 Composite Index, and TSE Equally-Weighted (TSE-EW) and Market-Weighted (TSE-MW) Indexes for all stocks trading on the TSE, which are obtained from the Canadian Financial Markets Research Centre (CFMRC) database (formerly the TSE-Western database).

The Results
Figures 1, 2 and 3 depict the information contained in Tables 1, 2 and 3 respectively. The results presented in Table 1 and depicted in Figure 1 for the entire sample period show that approximately two-thirds of the total risk associated with a random stock can be eliminated by combining all 222 stocks in an equally-weighted portfolio, leaving approximately 33% of the risk. This is slightly above the 27% figure documented by Solnik for U.S. stocks, but below the 39% threshold achieved by Statman who used as many as 1,000 U.S. stocks in his portfolio.

The resulting standard deviation of 4.48% is above the figures of 3.96% and 3.95% for the TSE 300 and the TSE-MW. However, it is below the TSE-EW figure of 5.91%. Consistent with the standard trade-off between risk and return, we note that our equally-weighted portfolio provided higher returns than the two lower risk, market-weighted indexes. In contrast, the TSE-EW had the highest return, consistent with its higher level of risk. This general result holds for both sub-periods as well.

Risk is reduced by 46% for a 10-stock portfolio (versus 51% for Statman), 53% for a 20-stock portfolio (versus 56% for Statman) and 56% for a 30-stock portfolio (versus 58% for Statman). The reduction in risk obtained by the 10-stock portfolio represents 68% of the total risk reduction achievable using all 222 stocks, 78% for the 20-stock portfolio and 84% for the 30-stock portfolio. This in itself shows the substantial benefits of diversifying.

However, considerably more of the unsystematic risk can be eliminated by diversifying even further. For example, 90% of the total risk reduction benefits can be achieved using a 50-stock portfolio, 95% for a 90-stock portfolio and 99.6% for a 200-stock portfolio. Finally, we note that a 60-stock portfolio was able to achieve 91.4% of total benefits of diversification over the entire period, on average.

This suggests that the recently introduced S&P/TSE 60 Index is reasonably diversified for practical purposes. In fact, since the S&P/TSE 60 Index includes stocks from all of the major industry groups by construction, it is almost certain to be better diversified than our randomly constructed portfolios.

Sub-Period Analysis
It is evident that, depending on the time period in question, the proportion of systematic risk to unsystematic risk can vary considerably. Results reported in Table 2 and depicted in Figure 2 for the January 1985 to June 1991 sub-period show that approximately 60% of the total risk can be eliminated when all 236 stocks are combined. This is below the 67% reduction achievable for the entire period. It is also well below the 74% reduction obtainable in the second sub-period using 200 stocks.

In all likelihood, we can attribute a substantial portion of this result to the fact that there was a larger number of stocks (415) available for diversification purposes during the later sub-period. We would also note that the 60-stock portfolios are well diversified in both periods (achieving 96% of the attainable diversification benefits during the first sub-period, and 91% during the second).

We now focus our attention on the 30-stock portfolios constructed during each period of the study, since this is a commonly referred to number of stocks to be included in a well-diversified portfolio. During the first sub-period we observe that a 30-stock portfolio eliminates about 56% of the standard deviation of a one-stock portfolio, which is the identical statistic reported for the entire sample. However, during this sub-period, this reduction represents approximately 93% of the total potential benefits achievable through diversification using this universe of stocks, versus 84% for the entire period. Diversification adds very little beyond 30 stocks during this time period, contrary to what we observed for the entire period.

During the second sub-period, a 30-stock portfolio eliminates about 65% of the standard deviation of a one-stock portfolio, well above the reduction that occurred during the entire period and the first sub-period. This reduction represents about 86% of the total benefits achievable through diversification for this period, which is very close to the percentage for the entire period. During this period, 50 stocks are required to achieve 90% of the total benefits achievable through diversification (100 stocks to achieve 95% of the benefits).

Conclusions
It appears that 30 to 50 Canadian stocks are required to capture most of the benefits associated with diversification. However, substantial benefits occur by diversifying across as few as 10 stocks. In all periods, it is safe to say that the S&P/TSE 60 Index is sufficiently diversified for all practical purposes.

These results are fairly consistent with previous U.S. evidence, although it appears that slightly more Canadian stocks are needed for equivalent diversification benefits. This result is intuitive because of the high concentration of Canadian stocks within a few industries, and the high proportion of resource-based companies listed on the TSE.5

We do not state a number of stocks required in a well-diversified portfolio because, as noted by Statman, the benefits of diversification must be weighed against the cost of excessive diversification in the form of transactions costs and monitoring costs that arise from tracking a large number of stocks.

For individual investors, it is often impractical to hold more than 10 (or even five) stocks in a portfolio. The good news for these investors is that holding 10 stocks in a portfolio provides about two-thirds of the potential benefits of diversification.

In contrast to individuals, professional investors have large amounts of funds to invest, face minimal transaction costs and can efficiently monitor a large universe of stocks. Hence they routinely hold portfolios of 50 stocks or more. Our results confirm the benefits associated with holding such large portfolios, contrary to the conclusions of several previous U.S. studies.

Endnotes
1. Cleary and Jones (1999) also refer to the present study to provide Canadian evidence.

2. The wealth ratio is the ratio of the ending value of the investment portfolio to the initial amount invested in the portfolio.

3. Therefore, our results likely understate the realizable benefits of diversification that can be achieved by diversifying in a more systematic way, such as diversifying with regard to size, industry and/or geographic region.

4. The authors acknowledge that this method of analysis introduces survivorship bias in the data. However, it is consistent with the approach of previous studies. More importantly, it does not detract materially from the stated purpose of this study, which is to provide general guidelines for the potential benefits of diversification.

5. For example, the TSE sub-indexes for metals and minerals, gold and precious minerals and oil and gas companies comprised close to 17% of the market capitalization of the TSE 300 composite index as of March 31, 1999, while financial services companies and utilities accounted for 21% and 13% respectively.

References
Bodie, Z., A. Kane, A. Marcus, S. Perrakis, and P. Ryan, Investments, Second Canadian Edition, Toronto, Irwin, 1997.

Cleary, W.S., and C.P. Jones, Investments: Analysis and Management, First Canadian Edition, Toronto, John Wiley & Sons Canada Limited, 1999.

Evans, J.L. and S.H. Archer, "Diversification and the Reduction of Dispersion: An Empirical Analysis," Journal of Finance, 23 (Dec 1968), 761-767.

Fisher, L. and J.H. Lorie, "Some Studies of Variability of Returns on Investments in Common Stocks," The Journal of Business, 43 (Apr 1970), 99-134.

Newbould, G.D. and P.S. Poon, "The Minimum Number of Stocks Needed for Diversification," Financial Practice and Education, 3 (Fall 1993), 85-87.

Sharpe, W. F., G. J. Alexander, J. V.Bailey and D.J. Fowler, Investments, Second Canadian Edition, Scarborough, Ontario, Prentice Hall Canada Incorporated, 1997.

Solnik, B.H., "Why Not Diversify Internationally Rather Than Domestically?" Financial Analysts Journal, (July-August 1974), 48-54.

Statman, M., "How Many Stocks Make a Diversified Portfolio?" Journal of Financial and Quantitative Analysis, 22 (September 1987), 353-363. Sean Cleary is an assistant professor with the Department of Finance & Management Science, Saint Mary's University in Halifax, Nova Scotia.

David Copp is an assistant professor at the Department of Commerce, Mount Allison University in Sackville, New Brunswick.