Investors with these goals could be particularly

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Investors with these goals could be particularly concerned about shortfall risk, the possibility that ending wealth will be less than the target amount. This attitude is consistent with the Safety First principle introduced by Roy (1952). According to Safety First, individuals seek to minimize the chance of some disaster. For a portfolio of assets, the disaster is a gross return less than some arbitrary quantity d . When d is equal to the gross return on a risk-free investment, Levy and Sarnat (1972) show that Safety First and the mean-variance criterion select the same optimal portfolio. In recent decades there has been increased attention to the Safety First philosophy through the related concepts of downside risk and shortfall constraints (Leibowitz and Henriksson, 1989; Leibowitz, Bader and Kogelman, 1996; Sortino and Forsey, 1996). Perhaps the greatest practical impact of these ideas is the widespread adoption of Value at Risk (see Jorion, 2000) for risk management. Our study takes a shortfall approach to determine the appropriate number of stocks in a portfolio of individual securities. Numerous studies attempt to answer the question of how many stocks are enough. Many finance textbooks and popu- lar publications claim that the full benefits of diversification can be obtained with just eight to 20 stocks (Newbould and Poon, 1993, 1996). The claim is based on studies comparing the variance of a single asset to the variance of portfolios with increasingly more securities, an approach pioneered by Evans and Archer (1968). Lai and Seiler (2001) use a similar approach to study diversification within industry groups. Several studies recommend a much larger number of stocks. Elton and Gruber (1977) present a total risk measure that includes the conventional variance of portfolio returns plus the risk that the portfolio’s mean return will be different from the expected market return. By this risk measure, a 15-stock portfolio has 32% more risk and a
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D. L. Domian et al./The Financial Review 42 (2007) 557 570 559 60-security portfolio has 20% more risk than a 100-stock portfolio. Statman (1987) uses a security market line approach to conclude that investors should hold at least 30 securities, or at least 40 if the investor uses no personal leverage. Newbould and Poon (1993, 1996) use the S&P 500 stocks to form 1,000 portfolios of various sizes. For each size portfolio, Newbould and Poon (1993, 1996) calculate the average return and the average variance of return, and show that an investor who wants to be within 5% of the average return and within 20% of the average risk would need more than 100 stocks. Campbell, Lettau, Malkiel and Xu (2001) determine that the number of stocks needed to achieve a given level of diversification has increased in recent years. Domian, Louton and Racine (2003) demonstrate that investors need more than 60 stocks to avoid a significant shortfall risk, defined as the difference between the fifth percentile of a portfolio s distribution and the return from a reference 100-stock portfolio. Milevsky (2003) considers the tradeoff between the number of stocks and the length of the holding period. Statman (2004) finds that the current
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