How Many Stocks Are Needed for Diversification?

The chart in Figure 1 initially confirms the familiar finding: as the number of holdings increases, the share of variance not explained by the benchmark declines significantly. In the market-cap-weighted Top X portfolio, it falls from around 46 percent for a single stock to just under 18 percent for ten holdings and about 4 percent for 100 stocks. A portfolio that holds enough large companies at approximately their index weights therefore comes to resemble the overall market relatively quickly. This does not automatically mean, however, that the portfolio is well diversified.
 

Benchmark Proximity Is not the Same as Diversification

Figure 1 primarily measures how well portfolio returns can be explained by the market-cap-weighted benchmark. This is an important perspective, but not the only one. A portfolio comprising the index's largest companies quickly comes to resemble the benchmark because these companies themselves account for a substantial share of the index. The growing similarity is therefore partly mechanical: the more of the index's largest constituents are included, the greater the overlap between the portfolio and the benchmark.

The pattern is different for equal-weighted portfolios. Here, too, the unexplained share of variance initially declines and reaches its minimum at around 100 stocks in our analysis. It then rises again. In the equal-weighted full universe, it stands at around 18 percent. This is not a contradiction: adding stocks spreads the portfolio more broadly across individual companies, but the increasing underweight in large caps and overweight in mid and small caps also move it further away from the benchmark, whose weights are determined by market capitalisation.

As their number of holdings increases, the equal-weighted random portfolios converge towards the equal-weighted full universe rather than the market-cap-weighted index. Adding more stocks can therefore reduce dependence on individual companies while increasing the deviation from the benchmark. In this sense, diversification and benchmark proximity are not the same thing.
 

The Concentration Paradox

The shortcomings of an analysis based solely on the number of holdings become particularly apparent when individual portfolio weights are examined. A portfolio may comprise hundreds of stocks and still be highly concentrated. What matters is not only the number of positions, but also the weight of the largest companies.

In a market-cap-weighted portfolio of 100 stocks, the ten largest companies account for around 47 percent of the total weight (Figure 2, top chart). With 250 stocks, the figure is 36 percent. Even in the full universe of 1,268 stocks, the ten largest companies still account for around 26 percent of the portfolio weight. The nominal number of holdings can therefore rise sharply without the invested capital being distributed anything close to evenly.

An equal-weighted portfolio naturally presents a different picture. From ten stocks onwards, the top-10 weight equals ten divided by the number of holdings. Accordingly, in a portfolio of 100 stocks, 10 percent is invested in the ten largest positions; with 1,268 stocks, the figure is less than 1 percent.

The effective number of stocks, calculated as the reciprocal of the sum of squared weights, summarises this concentration in a single metric. A fully equal-weighted portfolio of 100 stocks has an effective number of stocks of 100. As soon as the weights are distributed unevenly, the figure is lower. In the global full universe, for example, 1,268 nominal positions correspond to only around 95 effective stocks (Figure 2, bottom). The concentration of weights is therefore roughly equivalent to that of an equal-weighted portfolio of 95 stocks.
 

Weighting and Risk Are not the Same

An analysis of portfolio weights alone is also incomplete. For investors, what ultimately matters is which positions drive overall risk. A stock with a weight of 5 percent may contribute more or less than 5 percent to the portfolio's total risk. In addition to its weight, the key factors are its own volatility and its correlation with the remaining positions. For the analysis in Figure 3, portfolio variance is decomposed into additive risk contributions. This makes it possible to assign each position a contribution to total risk. We consider the sum of the ten largest contributions (the corresponding companies need not be the same as the ten largest positions by weight).

In a market-cap-weighted portfolio of 100 companies, the ten stocks with the largest risk contributions account for around 61 percent of total portfolio variance. With 250 stocks, the figure is still around 50 percent; in the full universe, it is about 40 percent. In the equal-weighted portfolio, their share declines much more rapidly: from 18 percent with 100 companies to around 7 percent with 250 stocks and less than 2 percent in the full universe.

Risk concentration can therefore be even more pronounced than concentration by weight alone. In the market-cap-weighted full universe, the ten largest positions account for around 26 percent of the weight but contribute around 40 percent of portfolio variance.
 

The Assenagon Equity Framework

The number of stocks therefore answers only part of the underlying question. It is more useful to distinguish between at least three levels of diversification.

Single-stock diversification: How heavily does the portfolio depend on individual companies? Appropriate metrics include the largest position weight, the top-10 weight, the effective number of stocks and the distribution of risk contributions.

Benchmark dependence: To what extent are the portfolio's movements explained by the benchmark index? Tracking error and the share of variance not explained by the benchmark are appropriate metrics. A low value, however, is not evidence of a balanced distribution of risk.

Economic diversification: How broadly is the portfolio spread across economic risk drivers? These include countries, sectors, currencies, business models and, in particular, factors such as value, size, profitability, leverage, momentum and market sensitivity. A large number of different stocks may have similar factor characteristics and therefore respond to the same market movements.

The Assenagon Equity Framework therefore considers not only a portfolio's single-stock, country and sector structure, but also systematically analyses its factor profiles, fundamentals, sensitivities and risk metrics. The aim is to separate intended return drivers from unintended risks. Under this approach, diversification does not mean avoiding all factor exposures, but deliberately implementing desired factor positions while limiting unwanted concentrations.

For these reasons, a portfolio of 250 stocks may be better diversified in economic terms than a much larger portfolio. What matters is not only the number of positions, but which risk drivers are intentional and which arise unintentionally within the portfolio.
 

For Capital Market Investors

A high number of holdings alone does not guarantee effective diversification. As the number of holdings increases, market-cap-weighted portfolios quickly come to resemble the benchmark, yet may remain heavily dependent on a small number of companies and common risk drivers. Equal-weighted portfolios distribute capital and risk more broadly, but in return deviate more from the benchmark. Diversification is therefore not a one-dimensional objective. What matters is not only how many stocks a portfolio holds, but how many mutually independent risk drivers it encompasses and how heavily it depends on them.