Investor preferences and portfolio selection: is diversification an appropriate strategy?

Abstract

This paper analyzes the relationship between diversification and several distributional characteristics that have risk implications for stock returns. We develop a flexible three-parameter distribution to model the stock returns. Using data on the current 30 DJIA stocks, we show that an investor's strategy on diversification depends on the measures of risk for particular concerns. For example, investors who desire to increase positive skewness would hold a less diversified portfolio, while those who care more about extreme losses would hold a more diversified portfolio. Experimenting with a more general pool of stocks yields the same conclusions.

Keywords:

• Diversification
• Asymmetric Generalized t distribution
• Skewness

Acknowledgements

We are grateful to the editor, two anonymous referees, and the participants in the 2003 Conference of High-Frequency Financial Data in Taipei for helpful comments. Naturally, all remaining errors are ours. Yau acknowledges the research support from the National Science Council of the Republic of China (NSC93-2415-H-008-007).

Notes

§Arrow (1974) theoretically shows that risk-averse investors with non-increasing risk aversion prefer positively skewed investment positions.

†Other flexible alternatives include the exponential generalized beta of the second kind (EGB2), transformations of normally distributed variables discussed by Johnson (1949), and a family of modified Weibull distributions proposed by Malevergne and Sornette (2004). Wang et al. (2001) apply the EGB2 to GARCH models. The AGT distribution proposed in this paper is at least equally as flexible as the EGB2 distribution. Whereas the EGB2 distribution imposes limited ranges on higher moments, the AGT distribution has no such limits. The modified Weibull distribution is characterized by only two parameters, but it does not nest the Student's t distribution, the most often used statistical distribution to capture the fat-tail behaviors in asset returns.

†The GT distribution does not restrict the level of kurtosis. In addition, it can be shown that the square of the skewness is less than one plus the kurtosis [(SK)²<KU + 1]. Therefore, a wider range for kurtosis also allows a wider range for skewness.

‡Hansen (1994) uses the same technique to develop an asymmetric t-distribution. In a different framework, Theodossiou (1998) also develops a skewed version of the GT distribution, but his distribution has four parameters and a more complicated pdf.

§We use a logistic transformation to set constraints on the parameters. With this transformation, even if the parameter ω is allowed to vary over the entire real line, θ will be constrained to lie in the region [L, U]. Specifically, p and q are restricted to be between 0 and 50 and r is between −1 and 1.

†Note that we are using the 30 stocks that are currently the components of the Dow Jones Index. Some of them were not in the Index in the earlier part of the sample. As of the date this paper is written, the latest change of the components in the DJIA was on 8 April 2004. Table 1 includes a list of the stock symbols.

‡ The industries represented by the DJIA include materials, electronics, food/beverages/tobacco, financial services, aviation/aerospace, heavy equipment, chemicals, petroleum, automobiles, retail, computer hardware/software/services, pharmaceuticals, household supplies, telecommunications, and entertainment.

† There are two approaches to calculate the distributional characteristics of risks for a portfolio's returns. The first method is to apply estimation schemes to the portfolio returns directly. This is the method employed in this paper and in Campbell et al. (2001). The second approach is to compute the distributional characteristics for the portfolio returns from a multivariate model, in which the dependence structure among individual stock returns needs to be identified beforehand. Examples of this are Guidolin and Timmermann (2006) and Malevergne and Sornette (2004).

† This specification has been used in the literature on yield curves. See, for example, Nelson and Siegel (1987).

† We also use weekly data to estimate the 1% and 5% conditional VaRs at the weekly horizon. All qualitative patterns observed based on the daily data are well preserved in the weekly data. Specifically, the average expected maximum 1% and 5% weekly losses are 10.455% and 6.692%, respectively, for a single-stock portfolio. The diversifiable losses are 3.393% and 2.019%, respectively. More than 92% of the diversifiable losses are gone with a five-stock portfolio.

† We select stocks from size-ordered groups (from the smallest to the biggest), because we want to see the diversification effect when bigger-size stocks are added to the portfolio, in an attempt to compare our results with those from studies such as Campbell et al. (2001). A more general approach is to randomly select a size-group, randomly select another size-group from the other 29 groups, and so on. We also experiment with this more general approach and obtain similar results. The same sampling approach is also applied to the liquidity-based strategy experimented upon later. The results from both experiments do not change our conclusions and are available from the authors upon request.

† In a more recent strand of the literature, Barberis and Huang (2005) and Kumar (2005) argue that models with cumulative prospect-theoretic preferences imply that idiosyncratic skewness should be priced as well. In this case, the relationship between diversification and idiosyncratic skewness is also an interesting topic.