**1. Introduction**

There are many studies that link Big Data, Computational Science, Economics, Finance, Marketing, Management, and Psychology. The analysis of Big Data, medium-sized data and small data will be argued to be an important aspect of Computational Science.

There are many papers of a multidisciplinary nature that have been published in different areas. As such, there is substantial interesting and important research that has been undertaken in risk and financial managemen<sup>t</sup> that is related to Big Data, Computational Science, Economics, Finance, Marketing, Management, Psychology, and cognate areas. As this paper discusses research that is closely related to the interests of the authors, it is focused primarily on the disciplines associated with Big Data, Computational Science, Economics, Finance, Marketing, Management, and Psychology.

Given the general level of confusion for academics and practitioners as to what constitutes big data, the paper provides a definition of big data, and distinguishes between important issues that are associated with big data, small data, and large data sets that are not necessarily satisfy the definition of big data.

Therefore, the paper discusses recent research in the areas of risk and financial managemen<sup>t</sup> as they relate to Big Data, Computational Science, Economics, Finance, Marketing, Management, and Psychology, and cognate disciplines. The intention is to disseminate ideas to researchers who may consider working in the areas of risk and financial managemen<sup>t</sup> in connection with Big Data, Computational Science, Economics, Finance, Marketing, Management, and Psychology.

As many, if not all, theorems for small data do not hold for big data, and, thus, analysis of big data becomes a separate topic, different from that of small data. In addition, Computational Science for both big and small data can be applied to many cognate areas, including Science, Engineering, Medical Science, Experimental Science, Psychology, Social Science, Economics, Finance, Management, and Business.

In this paper, we will discuss different types of utility functions, stochastic dominance (SD), mean-risk (MR) models, portfolio optimization (PO), and other financial, economic, marketing and managemen<sup>t</sup> models as these topics are popular in Big Data, Computational Science, Economics, Finance, and Management. Academics could develop theory, and thereafter develop econometric and statistical models to estimate the associated parameters to analyze some interesting issues in Big Data, Computational Science, Economics, Finance, Marketing, Management, and Psychology.

Academics could then conduct simulations to examine whether the estimators calculated for estimation and hypothesis testing have good size and high power. Thereafter, academics and practitioners could apply their theories to analyze some interesting issues in the seven disciplines and other cognate areas.

In many situations, academics and practitioners have been applying existing theories and empirical methods to big data without initially verifying their appropriateness or suitability. We sugges<sup>t</sup> that academics and practitioners should develop or seek appropriate models for big data before applying the techniques that might be available. The properties of estimators and tests should be verified for applications to big data.

In this paper, we review an extensive literature in economics, finance, marketing, management, psychology and computational science as we are familiar with these areas as part of our research programs on risk and financial management. There are strong theoretical links among these areas. For example, we provide a discussion in Section 2.1 that Li et al. (2018) is related to finance, economics, management, psychology, decision-making, marketing, big data, and computation science. We note that this paper is not the only research that is related to several different disciplines. Most of the research discussed in this paper is related to several different disciplines, as is highlighted in the title of the paper.

The plan of the remainder of the paper is as follows. In Section 2, a number of comprehensive theoretical models of risk and portfolio optimization are discussed. Alternative statistical and econometric models of risk and portfolio optimization are analyzed in Section 3. Alternative procedures for conducting simulations are examined in Section 4. A brief discussion of empirical models in several cognate disciplines is presented in Section 5. Some concluding remarks are given in Section 6.

#### **2. Theoretical Models**

It is important to commence any rigorous research in computational sciences for big data as well as small data in the areas of Economics, Finance, and Management by developing appropriate theoretical models. The authors have been developing some theories to extend those that have been discussed in a number of existing literature reviews. We discuss some of our research in the following subsections.

#### *2.1. Portfolio Optimization*

The mean–variance (MV) portfolio optimization procedure is the milestone of modern finance theory for asset allocation, investment diversification, and optimal portfolio construction (Markowitz 1952b). In the procedure, investors select portfolios that maximize profit subject to achieving a specified level of calculated risk or, equivalently, minimize variance subject to obtaining a predetermined level of expected gain. However, the estimates have been demonstrated to depart seriously from their theoretic optimal returns. Michaud (1989) and others have found the MV-optimized portfolios do more harm than good. Bai et al. (2009a) have proved that this phenomenon is natural.

We note that the estimates have been demonstrated to depart seriously from their theoretical optimal returns. The MV-optimized portfolios can do more harm than good for big data, especially as the number of parameters being estimated increases with the increasing dimension. For small data or big data, where the number of parameters to be estimated is fixed, the estimates do not depart significantly from their theoretical optimal returns.

Recently, Li et al. (2018) extended Maslow (1943) need hierarchy theory and the two-level optimization approach by developing the framework of the Malsow portfolio selection model (MPSM). The authors were able to do this by solving the two optimization problems to meet the need of individuals with low financial sustainability, who prefer to satisfy their lower-level (safety) need before seeking a higher-level (self-actualization) need to maximize the optimal returns. They also find a solution for some investors with high financial sustainability.

In this paper, we review an extensive selection of the literature in economics, finance, marketing, management, psychology computational science. For example, Li et al. (2018) analyse an investment issue, so that it is related to finance. The paper discusses decision-making for investors with low and high financial sustainability, so the paper is related to management, psychology, and decision-making. The paper is also related to marketing funds to investors with low and high financial sustainability, and hence is related to marketing. Moreover, the paper requires analysis of big data, not so big data, and small data, so it is also related to big data and computational science. In addition, investment changes under different economic conditions, so that the paper is also related to economics.

#### *2.2. Cost of Capital*

Gordon and Shapiro (1956) develop the dividend yield plus growth model for individual firms while Thompson (1985) improves the theory by combining the model with analysis of past dividends to estimate the cost of capital and its 'reliability'. Thompson and Wong (1991) estimate the cost of capital using discounted cash flow (DCF) methods that require forecasting dividends.

Thompson and Wong (1996) extend the theory by proving the existence and uniqueness of a solution for the cost of equity capital, and the cost of equity function is continuously differentiable. Wong and Chan (2004) have extended their theory by proving the existence and uniqueness of reliability.

#### *2.3. Behavorial Models*

Barberis et al. (1998) and others use Bayesian models to explain investors' behavioral biases by using the conservatism heuristics and representativeness heuristics in making decisions. Lam et al. (2010) extend the theory by developing a model of weight assignments using a pseudo-Bayesian approach that reflects investors' behavioral biases.

They use the model to explain several financial anomalies, including excess, volatility, short-run underreaction, long-run overreaction, and magnitude effects. Lam et al. (2012) extend their work by developing additional properties for the pseudo-Bayesian approach that reflects investors' behavioral biases, and explain the linkage between these market anomalies and investors' behavioral biases.

Fung et al. (2011) extend their work by incorporating the pseudo-Bayesian model with the impact of a financial crisis. They derive properties of stock returns during the financial crisis and recovery from the crisis.

Guo et al. (2017b) extend the model by assuming that the earnings shock of an asset follows a random walk model, with and without drift, to incorporate the impact of financial crises. They assume the earning shock follows an exponential family distribution to accommodate symmetric as well as asymmetric information. By using this model setting, they develop some properties on the expected earnings shock and its volatility, and establish properties of investor behavior on the stock price and its volatility during financial crises and subsequent recovery.

Thereafter, they develop properties to explain excess volatility, short-term underreaction, long-term overreaction, and their magnitude effects during financial crises and subsequent recovery. Egozcue and Wong (2010a) extend prospect theory, mental accounting, and the hedonic editing model by developing an analytical theory to explain the behavior of investors with extended value functions in segregating or integrating multiple outcomes when evaluating mental accounting.

Whether to keep products segregated (that is, unbundled) or integrate some or all of them (that is, bundle) has been a problem of profound interest in areas such as portfolio theory in finance, risk capital allocations in insurance, and marketing of consumer products. Such decisions are inherently complex and depend on factors such as the underlying product values and consumer preferences, the latter being frequently described using value functions, also known as utility functions in economics.

Egozcue et al. (2012a) develop decision rules for multiple products, which we generally call 'exposure units' to naturally cover manifold scenarios spanning well beyond 'products'. The findings show, for example, that the celebrated Thaler's principles of mental accounting hold as originally postulated when the values of all exposure units are positive (that is, all are gains) or all negative (that is, all are losses).

In the case of exposure unit mixed-sign values, decision rules are much more complex and rely on cataloging the Bell-number of cases that grow very fast, depending on the number of exposure units. Consequently, in this paper, we provide detailed rules for the integration and segregation decisions in the case up to three exposure units, and partial rules for the arbitrary number of units

We note that the theory of decision maker's behavior developed by Egozcue and Wong (2010a) and Egozcue et al. (2012a) is for marketing, and they develop a theory for consumer behavior.

#### *2.4. Modelling Different Types of Investors*

We have been developing some theories, estimation, and testing to examine different utility functions and the preferences of different types of investors. We summarize some of the results here. Readers may refer to Sriboonchita et al. (2009) and Bai et al. (2018) for further information.

#### 2.4.1. Different Types of Utility Functions

Lien (2008) compares the exponential utility function with its second-order approximation under the normality assumption in the optimal production and hedging decision framework. Guo et al. (2016a) extend the theory by comparing the exponential utility function with a 2n-order approximation for any integer n. In addition, they propose an approach with an illustration to determine the smallest n that provides a good approximation.

#### 2.4.2. Stochastic Dominance

We have been developing several theories in stochastic dominance, and discuss some here.

#### Behavior of Risk Averters and Risk Seekers

Wong and Li (1999) develop some properties for the convex stochastic dominance to compare the preferences of different combinations of several assets for both risk-averse and risk-seeking investors. In addition, they compare the preferences between a convex combinations of several continuous distributions and a single continuous distribution. In addition, Li and Wong (1999) develop some SD theorems for the location-and-scale family and linear combinations of random variables for risk seekers and risk averters.

Wong (2007) extends their work by introducing the first three orders of both ascending SD (ASD) and descending SD (DSD) to decisions in business planning and investment for risk-averse and risk-seeking decision makers so that they can compare both return and loss. The author provides tools to identify the first-order SD prospects and discern arbitrage opportunities that could increase

their expected utility and expected wealth. Wong (2007) also introduces the mean–variance (MV) rule to decisions in business planning or investment on both return and loss for both risk-averse and risk-seeking decision makers, and show that the rule is equivalent to the SD rule under some conditions.

Chan et al. (2016) analyse properties of SD for both risk-averse and risk-seeking SD (RSD) for risk-seeking investors, which, in turn, enables an examination of their behavior. They first discuss the basic properties of SD and RSD that link SD and RSD to expected-utility maximization. Thereafter, they prove that a hierarchy exists in both SD and RSD relationships and that the higher orders of SD and RSD can be inferred by the lower orders of SD and RSD, but not vice-versa. Furthermore, they study the conditions in which third-order SD preferences are 'the opposite of' or 'the same as' their counterpart third-order RSD preferences.

In addition, they establish the relationship between the orders of the variances and that of the integrals for two assets, which enables us to establish certain relationships between the dominance of the variances and the second- and third-order SD and RSD for two assets under the condition of equal means. The theory developed in the paper provides a set of tools that enables investors to identify prospects for first-, second-, and third-order SD and RSD, and so enables investors to improve their investment decisions.

Another contribution in the paper is that the authors recommend checking the dominance of the means of the distributions to draw inferences for the preferences for two different assets for third-order risk averters and risk seekers. They illustrate this idea by comparing the investment behavior of both third-order risk averters and risk seekers in bonds and stocks.

Guo and Wong (2016) extend some univariate SD results to multivariate SD (MSD) for both risk averters and risk seekers, respectively, to n order for any n > 0 when the attributes are assumed to be independent and the utility is assumed to be additively separable. Under these assumptions, they develop some properties for MSD for both risk averters and risk seekers. For example, they prove that MSD are equivalent to the expected-utility maximization for both risk averters and risk seekers, respectively.

They show that the hierarchical relationship exists for MSD, and establish some dual relationships between the MSD for risk averters and risk seekers. They develop some properties for non-negative combinations and convex combinations random variables of MSD, and develop the theory of MSD for the preferences of both risk averters and risk seekers on diversification. At last, they discuss some MSD relationships when attributes are dependent, and discuss the importance and the use of the results developed in their paper.

#### Behavior of Investors with S-Shaped and Reverse S-Shaped Utility Functions

Wong and Chan (2008) extend the work on Prospect SD (PSD) and Markowitz SD (MSD) to the first three orders, and link the corresponding S-shaped and reverse S-shaped utility functions to the first three orders. They provide experiments to illustrate each case of the MSD and PSD to the first three orders, and demonstrate that the higher order MSD and PSD cannot be replaced by the lower order MSD and PSD. Furthermore, they show that a hierarchy exists in both PSD and MSD relationships, arbitrage opportunities exist in the first orders of both PSD and MSD, and for any two prospects under certain conditions, their third order MSD preference will be 'the opposite of' or 'the same as' their counterpart third order PSD preferences.

#### 2.4.3. Almost Stochastic Dominance

Guo et al. (2013) provide further information on both the expected-utility maximization and the hierarchy property. For almost SD (ASD), Leshno and Levy (2002) propose a definition, and Tzeng et al. (2013) modify it to provide another definition. Guo et al. (2013) show that the former has the hierarchy property but not expected-utility maximization, whereas the latter has the expected-utility maximization but not the hierarchy property.

Guo et al. (2014) establish necessary conditions for ASD criteria of various orders. These conditions take the form of restrictions on algebraic combinations of moments of the probability distributions in question. The relevant set of conditions depends on the relevant order of ASD but not on the critical value for the admissible violation area. These conditions can help to reduce the information requirement and computational burden in practical applications. A numerical example and an empirical application for historical stock market data illustrate the moment conditions. The first four moment conditions, in particular, seem appealing for many applications.

Guo et al. (2016b) extend ASD theory for risk averters to include ASD for risk-seeking investors. Thereafter, they study the relationship between ASD for risk seekers and ASD for risk averters. Tsetlin et al. (2015) develop the theory of generalized ASD (GASD). Guo et al. (2016b) discuss the advantages and disadvantages of ASD and GASD.

#### *2.5. Indifference Curves*

Meyer (1987) extends MV theory to include comparisons among distributions that differ only by location and scale parameters, and include general utility functions with only convexity or concavity restrictions. Wong (2006) extends both Meyer (1987) and Tobin (1958) by showing that the indifference curve is convex upwards for risk averters, concave downwards for risk lovers, and horizontal for risk neutral investors, in order to include the general conditions stated by Meyer (1987). In addition, Wong (2006) develops some properties among the first- and second-order SD efficient sets and the mean–variance efficient set.

Wong and Ma (2008) extend the work on the location-scale (LS) family with general n random seed sources in a multivariate setting. In addition, they develop some properties for general non-expected utility functions defined over the LS family, and characterize the shapes of the indifference curves induced by the location-scale expected utility functions and non-expected utility functions. Thereafter, they develop properties for well-defined partial orders and dominance relations defined over the LS family, including the first- and second-order stochastic dominances, the mean–variance rule, and location-scale dominance.

Broll et al. (2010) discuss prospect theory and establish general results concerning certain covariances from which they can, in turn, infer properties of indifference curves and hedging decisions within prospect theory.
