An Empirical Investigation of Portfolios with Little Idiosyncratic Risk

The objective of this study is to answer the following research question: How large is a diversified portfolio? Although previous work is abundant, very little progress has been made in answering this question since the seminal work of Evans and Archer (1968). This study proposes two approaches to address the research question. The first approach is to measure the rate of risk reduction as diversification increases. For the first approach, I identify two kinds of risks: (1) risk that portfolio returns vary across time (Evans and Archer (1968), and Campbell et al. (2001)); and (2) risk that returns vary across portfolios of the same size (Elton and Gruber (1977), and O'Neil (1997)). I show that the times series risk reaches an asymptote as portfolio size increases. Cross sectional risk, on the other hand, does not appears to reach an asymptote as portfolio size increases. The second approach consists of comparing portfolios' performance to a benchmark portfolio that is assumed to be diversified (Statman (1987)). I develop a performance index. The performance index is calculated, for any given test portfolio, as the ratio of the Sharpe-like measure of the test portfolio to the Sharpe-like measure of the benchmark portfolio that is assumed to be diversified. The index is based on the intuition that an increase in portfolio size reduces times series risk and cross sectional risk, and increases transaction costs. Portfolio size is worth increasing as long as the marginal increase in the performance index from a decrease in risk is greater than the marginal decrease of the performance index from an increase in transaction costs. Diversification is attained when the value of the index reaches one. The results of my simulations indicate that the size of a well diversified portfolio is at the very least 30. This number can be substantially higher if, for example, the investment horizon length, the benchmark portfolio, and/or the cost of investing in the benchmark portfolio are changed. The active diversification strategy considered in this study, which consists of optimizing randomly selected portfolios, does not seem to produce smaller diversified portfolios. This result supports the market efficiency hypothesis.