*4.7. Diversification*

Diversification is one of the important areas in Behavioral Finance. It is related to Behavioral Finance because di fferent investors will have di fferent behaviors in diversification. In the theory of portfolio selection developed by Markowitz (1952a), investors are assumed to be risk-averse, and there is only one best optimal portfolio that investors should choose. Li et al. (2018) showed that even investors are risk-averse, as they can choose di fferent optimal portfolios if one looks into their safety need first, and thereafter, look into his/her self-actualization need while another one looks into his/her self-actualization need first, and then look into his/her safety need.

In addition, Li and Wong (1999) have shown that if all assets are independently and identically distributed, then it is true that risk-averters will choose the same optimal portfolio, which is the completely diversified portfolio. However, they found that investing in a single asset is the best choice for risk-seekers. Wong (2007) extended the theory for the diversification behaviors of both risk-averters

and risk-seekers to the gain, as well as to the loss, while Guo and Wong (2016) extended the theory further, in order to study the diversification behaviors of both risk-averters and risk-seekers in the multivariate settings.

To date, Li and Wong (1999), Wong (2007), and Guo and Wong (2016) only developed the diversification theory to study the diversification behaviors of both risk-averters and risk-seekers, to compare any pair of asset/portfolio(s) among a single asset, partially diversified portfolios, and completely diversified portfolio, but they have not developed any result in the comparison between any two portfolios of partial diversification. To circumvent the limitation, Egozcue and Wong (2010b) established a diversification theory to compare any pair of partially diversified portfolios, including completely diversified portfolios and individual asset.

Using the out-of-sample performance tool, DeMiguel et al. (2009) showed that the naive 1/N portfolio outperforms the "optimal" portfolio from the 14 models in terms of several commonly used measures in their study, and thus, they drew conclusion that the "optimal" portfolio is not optimal. Hoang et al. (2015b) found that risk-averters agree with Markowtiz to select the portfolios from the e fficient frontier, while risk-seekers agree with De Miguel, Garlappi, and Uppal to select the equal-weighted portfolio. On the other hand, Bouri et al. (2018) found that risk-averters prefer the portfolios from the e fficient frontier for low-risk with-wine portfolios but are indi fferent between the portfolios from the e fficient frontier and the naïve portfolio for any high-risk with-wine portfolios.

Statman (2004) showed that investors do not follow Markowtiz's suggestion to invest in the completely diversified portfolio and do not buy only one stock but buy a few. This is the well-known diversification puzzle. To provide a possible solution to the puzzle, Lozza et al. (2018) showed that investors' choices on optimal assets are similar if their utility are not too di fferent.

Egozcue et al. (2011a) bridged the gap in the literature to develop some properties for the diversification behaviors for investors with reverse S-shaped utility functions that have never studied before. They found that the diversification preference for investors with reverse S-shaped utility functions are complicated and depend on the sensitivities toward losses and gains.
