*2.6. Diversification*

Wong and Li (1999) extend the theory of convex SD (Fishburn 1974) by including any distribution function, developing the results for both risk seekers as well as risk averters, and including third-order stochastic dominance. Their results can be used to extend a theorem of Bawa et al. (1985) on comparisons between a convex combinations of several continuous distributions and a single continuous distribution.

Li and Wong (1999) develop some results for the diversification preferences of risk averters and risk seekers. Egozcue and Wong (2010b) incorporate both majorization theory and SD theory to develop a general theory and unifying framework for determining the diversification preferences of risk-averse investors, and conditions under which they would unanimously judge a particular asset to be superior. In particular, they develop a theory for comparing the preferences of different convex combinations of assets that characterize a portfolio to yield higher expected utility by second-order SD.

Egozcue et al. (2011a) analyse the rankings of completely and partially diversified portfolios and also of specialized assets when investors follow so-called Markowitz preferences. Diversification strategies for Markowitz investors are more complex than in the case of risk-averse and risk-inclined investors, whose investment strategies have been investigated extensively in the literature. In particular, they observe that, for Markowitz investors, preferences toward risk vary depending on their sensitivities toward gains and losses.

For example, it can be shown that, unlike the case of risk-averse and risk-inclined investors, Markowitz investors might prefer investing their entire wealth in just one asset. This finding helps us to better understand some financial anomalies and puzzles, such as the well-known diversification puzzle, which notes that investors may concentrate on investing in only a few assets instead of choosing the seemingly more attractive complete diversification.

Lozza et al. (2018) provide a general valuation of the diversification attitude of investors. First, they empirically examine the diversification of mean–variance optimal choices in the US stock market during the 11-year period 2003–2013. Then, they analyze the diversification problem from the perspective of risk-averse investors and risk-seeking investors.

Second, the authors prove that investors' optimal choices will be similar if their utility functions are not too distant, independent of their tolerance (or aversion) to risk. Finally, they discuss investors' attitudes towards diversification when the choices available to investors depend on several parameters.

#### *2.7. Risk Measures*

We have been developing properties for several risk measures to be used in finance, economics, and cognate disciplines, and discuss briefly the properties for some recent risk measures in this section.

#### 2.7.1. VaR and CVaR

Ma and Wong (2010) establish some behavioral foundations for various types of Value-at-Risk (VaR) models, including VaR and conditional-VaR, as measures of downside risk. They establish some logical connections among VaRs, conditional-VaR, SD, and utility maximization. Though supported to some extent by unanimous choices by some specific groups of expected or non-expected-utility investors, VaRs as profiles of risk measures at various levels of risk tolerance are not quantifiable as they can only provide partial and incomplete risk assessments for risky prospects.

They also include in the discussion the relevant VaRs and several alternative risk measures for investors. These alternatives use somewhat weaker assumptions about risk-averse behavior by incorporating a mean-preserving-spread. For this latter group of investors, the authors provide arguments for and against the standard deviation versus VaR and conditional-VaR as objective and quantifiable measures of risk in the context of portfolio choice.

#### 2.7.2. Omega Ratio

Both SD and the Omega ratio can be used to examine whether markets are efficient, whether there is any arbitrage opportunity in the market, and whether there is any anomaly in the market. Guo et al. (2017a) analyse the relationship between SD and the Omega ratio. They find that second-order SD and/or second-order risk-seeking SD (RSD) alone for any two prospects is not sufficient to imply Omega ratio dominance, insofar as the Omega ratio of one asset is always greater than that of the other. They extend the theory of risk measures by proving that the preference of second-order SD implies the preference of the corresponding Omega ratios only when the return threshold is less than the mean of the higher return asset.

On the other hand, the preference of the second-order RSD implies the preference of the corresponding Omega ratios only when the return threshold is larger than the mean of the smaller return asset. Nonetheless, first-order SD does imply Omega ratio dominance. Thereafter, they apply their theory to examine the relationship between property size and property investment in the Hong Kong real estate market, and conclude that the Hong Kong real estate market is not efficient as there are expected arbitrage opportunities and anomalies in the Hong Kong real estate market.

#### 2.7.3. High-Order Risk Measures

Niu et al. (2017) first show the sufficient relationship between the (n + 1)-order SD and the n-order Kappa ratio. They clarify the restrictions for necessarily beating the target for the higher-order SD consistency of the Kappa ratios. Thereafter, the authors show that, in general, a necessary relationship

between SD/RSD and the Kappa ratio cannot be established. They find that when the variables being compared belong to the same location-scale family or the same linear combination of location-scale families, they can obtain the necessary relationship between the (n + 1)-order SD with the n-order Kappa ratio after imposing some conditions on the means.

#### *2.8. Two-Moment Decision Model*

Broll et al. (2006) analyze export production in the presence of exchange rate uncertainty under mean-variance preferences. We present the elasticity of risk aversion, since this elasticity concept permits a distinct investigation of risk and expectation effects on exports. Counterintutitive results are possible, e.g., although the home currency is revaluating (devaluating), exports by the firm increase (decrease). This fact may contribute to the explanation of disturbing empirical results. Broll et al. (2011) use the mean-variance approach to examine a banking firm investing in risky assets and hedging opportunities. They focus on how credit risk affects optimal bank investment in the loan and deposit market when derivatives are available. Furthermore, they explore the relationship among the first- and second-degree stochastic dominance efficient sets and the mean-variance efficient set. Broll et al. (2015) analyze a bank's risk taking in a two-moment decision framework. Their approach offers desirable properties like simplicity, intuitive interpretation, and empirical applicability. The bank's optimal behavior to a change in the standard deviation or the expected value of the risky asset's or portfolio's return can be described in terms of risk aversion elasticities, i.e., the sensitivity of the marginal rate of substitution between risk and return. The bank's investment in a risky asset position goes down when the return risk increases, if and only if the risk aversion elasticity exceeds.

Alghalith et al. (2017a) analyze the impacts of joint energy and output prices uncertainties on input demands in a mean–variance framework. They show that an increase in the expected output price will cause the risk-averse firm to increase input demand, while an increase in expected energy prices will surely cause the risk-averse firm to decrease the demand for energy, but increase the demand for the non-risky inputs.

Furthermore, the authors investigate two cases with only uncertain energy price and only uncertain output price. In the case with only uncertain energy price, they determine that the uncertain energy price has no impact on the demands for the non-risky inputs. They also show that the concepts of elasticity and decreasing absolute risk aversion (DARA) play an important role in the comparative statics analysis.

Alghalith et al. (2017b) analyze the impacts of joint energy and output prices uncertainties on the inputs demands in a mean–variance framework. They find that the concepts of elasticities and variance vulnerability play important roles in the comparative statics analysis. If the firms' preferences exhibit variance vulnerability, increasing the variance of energy price will necessarily cause the risk averse firm to decrease demand for the non-risky inputs.

Furthermore, the authors investigate two special cases with only uncertain energy price and only uncertain output price. In the case with only uncertain energy price, they show that the uncertain energy price has no impact on the demands for the non-risky inputs. If the firms' preferences exhibit variance vulnerability, increasing the variance of energy price will surely cause the risk averse firm to decrease demand for energy.

With multiple additive risks, the mean–variance approach and the expected utility approach of risk preferences are compatible if all attainable distributions belong to the same location–scale family. Under this proviso, Guo et al. (2018) survey existing results on the parallels of the two approaches with respect to risk attitudes, the changes thereof, and comparative statics for simple, linear choice problems under risk.

In the mean–variance approach, all effects can be couched in terms of the marginal rate of substitution between the mean and variance. They apply the theory developed in the paper to examine the behavior of banking firms, and study risk-taking behavior with background risk in the mean–variance model.

#### *2.9. Dynamic Models with Background Risk*

Alghalith et al. (2016) use a general utility function to present two dynamic models of background risk. They present a stochastic factor model with an additive background risk. Thereafter, they present a dynamic model of simultaneous (correlated) multiplicative background risk and additive background risk.
