4.2.1. Utility

One of the main reasons that EMH is rejected in many cases and there are many market anomalies in the market is that different investors could have different types of utilities.

#### 4.2.2. Investors with Different Shapes in Their Utility Functions

Many scholars, for example, Bernoulli (1954), believe that investors are risk averse; that is, their utility is increasing concave. Many financial models are developed based on the foundational assumption that investors are risk averse or their utility is increasing concave. For example, Markowitz (1952a) developed the mean–variance (MV) portfolio optimization theory based on this assumption.

In reality, investors' utility may not be increasingly concave. It could be increasingly convex (that is, investors are risk-seeking) or S-shaped or reverse S-shaped. Tobin (1958), Hammond (1974), Stoyan (1983), Wong and Li (1999), Li and Wong (1999), Wong (2006, 2007), Wong and Ma (2008), Levy (2015), Bai et al. (2015), Guo and Wong (2016), and many others have built up their theories by assuming that investors could be risk averse or risk seeking.

Kahneman and Tversky (1979) suggested investors' utility<sup>1</sup> could be concave for gains and convex for losses, implying that investors have a S-shaped utility function. On the other hand, Thaler and Johnson (1990) observed that investors are more risk-seeking on gains and more risk-averse on losses, inferring that investors have a reverse S-shaped utility function. Other academics, for example, Levy and Wiener (1998), Levy and Levy (2002, 2004), Wong and Chan (2008), and Bai et al. (2011b) developed their theories based on the assumption that investors possess S-shaped or reverse S-shaped utility function.

## 4.2.3. Other Utility Functions

Academics not only use the shape of utility functions to measure the behaviors of different investors, but also use other forms of utility functions to measure their behaviors, for example, regret-aversion (Guo et al. 2015; Egozcue et al. 2015), disappointment-aversion (Guo et al. 2020), and many others. In addition, Guo et al. (2016) developed the exponential utility function with a 2n-order and established an estimation approach to find the smallest possible n to provide a good approximation for any integer n.

Chan et al. (2019a) proposed using polynomial utility functions to measure the behavior of risk-averters and risk-seekers. Wong and Qiao (2019) proposed including both risk-averse and risk-seeking components to measure the behavior of investors who could gamble and buy insurance together, or buy any less risky and more risky assets at the same time. Egozcue and Wong (2010a) propose a utility function for Segregation and Integration.

#### *4.3. Portfolio Selection and Optimization*

Portfolio optimization and portfolio selection are the founding theories of modern finance, and they are one of the major areas in Behavioral Finance. They are related to Behavioral Finance because different investors with different utilities could make different selections and ge<sup>t</sup> different optimizations. The foundational portfolio optimization theory developed by Markowitz (1952a), to find out how investors will choose their portfolios, requests assumption of risk-aversion on investors.

Nevertheless, the MV portfolio optimization theory developed by Markowitz (1952a) has been found to have serious problem in its (plug-in) estimation (Michaud 1989), while Bai et al. (2009a) not only prove that the serious estimation problem is natural and it is overestimation, not underestimation. In addition, they find out the magnitude of the overestimation. Thus, one is not surprised they can apply the asymptotic properties of eigenmatrices for large sample covariance matrices (Bai et al. 2011c)

<sup>1</sup> Kahneman and Tversky (1979) and others call it value function, while we call it utility function.

to find out the estimation (they call it bootstrap-corrected estimation) that is consistent to the true optimal return.

Nonetheless, the problem of the bootstrap-corrected estimation is that it does not have a closed-form. To solve the problem, Leung et al. (2012) extended the theory by developing the estimation with closed-form, and Bai et al. (2009b) extended the theory of portfolio optimization for the problem of self-financing. In addition, Bai et al. (2016) further extended the model by employing the spectral distribution of the sample covariance to develop the spectral-corrected estimation that performs better than both plug-in and bootstrap-corrected estimations.

The problem of all the above estimations developed by Markowitz (1952a), Bai et al. (2009a, 2009b), Leung et al. (2012), and many others is that the estimations are the same for any investor with risk-averse utility. Nonetheless, it is well-known that di fferent investors could choose di fferent optimal portfolios. In addition, Guo et al. (2019b) established some properties on e fficient frontiers and boundaries of portfolios by including background risk in the model and by using several approaches, including MV, mean–VaR, and mean–CVaR approaches.

Many studies have explored how to ge<sup>t</sup> solutions for di fferent investors. For example, Li et al. (2018) applied the Maslow portfolio selection model (MPSM) to develop a model that could take care of the need of investors with low financial sustainability who will first look into their lower-level (safety) need, and thereafter look into their higher-level (self-actualization) need, to obtain their optimal return. They illustrated their model by comparing the out-of-sample performance of the traditional model and their proposed model by using the real American stock data. They observed that their proposed model outperformed the traditional model to ge<sup>t</sup> the best out-of-sample performance.

We note that one can modify the model developed by Li et al. (2018) to find the optimal return for investors with high financial sustainability who prefer to looking into their higher-level need first, and then satisfying their lower-level need. Though both investors with high and low financial sustainability are risk-averse, their choices are di fferent in the portfolio selections.

There are many applications using the theory of portfolio optimization (for example, Abid et al. (2009, 2013, 2014), Hoang et al. (2015a, 2015b, 2018, 2019), Mroua et al. (2017), Bouri et al. (2018), and many others).
