In this paper, we study the question of optimal portfolio diversification for Canadian institutional investors between 1975 and 2011. We determine the number of stocks in a portfolio required to minimize diversifiable risk. Using daily data, we calculate various measures of risk, including some that account for black swan events. Previous research has analyzed the optimal portfolio sizes for an average investor (Solnik (1974) suggests holding 10-15 stocks, Copp and Cleary (1999) – 30-50 stocks and Kryzanowski and Singh (2010) – 20-25 stocks). We build on our predecessors’ contributions by estimating confidence bands around the average number of stocks in portfolios that provide this level of diversification 90% of the time as opposed to achieving this diversification level on average. The portfolio size based on this 90% confidence band will insure that the desired level of diversification is achieved in 90% of the cases, instead of 50% of the time. Holding 19 to 25 stocks will provide investors with an adequate level of diversification only on average. However, the same size recommendation may not provide this level of diversification for half of the investors. Comparing the year-by-year dynamic of portfolio holdings, we find that through time, the recommended number of stocks is affected by the state of the markets – distressed or quiescent, as well as by average correlations among stocks in the Canadian market.
Most academic literature argues that if diversifiable risk can be eliminated with a relatively small number of stocks, equity funds cannot justify holding an excessive number of stocks, which result in over-diversification and exaggerated fund fees. The same literature favours smaller portfolio sizes (about 10 to 25 stocks) to achieve a well-diversified portfolio. However, the portfolio sizes of most institutional investors are in excess of these recommendations. Holding too many stocks is costly both in terms of transaction costs as well as the opportunity cost of monitoring large diversified portfolios where associated fees dampen overall performance. Holding too few stocks exposes the fund to avoidable firm-specific risk. Previous academic research relies on averages and does not take into account the chance of a particular fund or institutional investor falling below the average, i.e. the risk of not achieving a specific risk target. Most institutional investors would feel more comfortable if they could hold a portfolio that would be well-diversified 90% of the time rather than 50% of the time. We find that for the purpose of diversification, portfolios held by Canadian institutional investors should be larger than those suggested in previous academic literature. In the U.S., firm specific risk has grown over the past thirty years relative to the overall variability of the stock market, while correlations between stocks have correspondingly decreased (see Campbell, Lettau, Malkiel, and Xu, 2001). This reinforces the advisability of larger portfolios. We check whether this result holds for Canada too.
We simulate random portfolios based on actual daily Canadian equity returns over the period 1975 to 2011. We construct equally-weighted random portfolios, each of different size, ranging from portfolios consisting only of one security to a broad market portfolio including all actively traded securities at the time. For each of these different- sized portfolios and each year we calculate time series standard deviation (SD), 1% expected shortfall (ES) and terminal wealth standard deviation (TWSD). We focus on SD as our benchmark to be able to compare our results to the previous literature. The ES is a downside risk measure that accounts for black swan events and is associated with the lowest 1% of the return distribution. The TWSD is a standard benchmark measure for buy-and-hold no-rebalancing portfolios, typically suitable for pension funds. We trace the dynamics of diversification benefits over the past 37 years in the Canadian market.
We find that for institutional investors, especially pension funds (relying on TWSD), who seek to avoid large losses in extreme market events, recommended portfolio sizes are typically larger than those recommended to investors concerned with general deviation from the trend (based on SD as a risk measure). The 90% confidence bands around our average risk measures give us an upper limit to the number of stocks required in a portfolio that assures 90% of the time a 90% risk reduction. When measuring extreme losses with ES1%, we find that, over the period 1975-2011, portfolios of 41 stocks would provide sufficient diversification, but when TWSD is used as a risk measure 61 stocks are required (see Table 1). However, to achieve the same level of risk reduction but only on average (instead of 90% of the time), we find that portfolios are typically smaller (25 stocks relying on SD as a risk measure and 19 stocks based on ES1%). In Section 2, we discuss our approach and the data. In Section 3, we present our results. We conclude in Section 4. Download the full paper.