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This chapter presents the survey of selected linear and mixed integer programming multi-objective portfolio optimization. The definitions of selected percentile risk measures are presented. Some contrasts and similarities of the different types of portfolio formulations are drawn out. The survey of multi-criteria methods devoted to portfolio optimization such as weighting approach, lexicographic approach, and reference point method is also presented. This survey presents the nature of the multi-objective portfolio problems focuses on a compromise between the construction of objectives, constraints, and decision variables in a portfolio and the problem complexity of the implemented mathematical models. There is always a trade-off between computational time and the size of an input data, as well as the type of mathematical programming formulation with linear and/or mixed integer variables.

The authors discuss the econometric underpinnings of Barro (2006)'s defense of the rare disaster model as a way to bring back an asset pricing model “into the right ballpark for explaining the equity-premium and related asset-market puzzles.” Arbitrarily low-probability economic disasters can restore the validity of model-implied moment conditions only if the amplitude of disasters may be arbitrary large in due proportion. The authors prove an impossibility theorem that in case of potentially unbounded disasters, there is no such thing as a population empirical likelihood (EL)-based model-implied probability distribution. That is, one cannot identify some belief distortions for which the EL-based implied probabilities in sample, as computed by Julliard and Ghosh (2012), could be a consistent estimator. This may lead to consider alternative statistical discrepancy measures to avoid the problem with EL. Indeed, the authors prove that, under sufficient integrability conditions, power divergence Cressie-Read measures with positive power coefficients properly define a unique population model-implied probability measure. However, when this computation is useful because the reference asset pricing model is misspecified, each power divergence will deliver different model-implied beliefs distortion. One way to provide economic underpinnings to the choice of a particular belief distortion is to see it as the endogenous result of investor's choice when optimizing a recursive multiple-priors utility a la Chen and Epstein (2002). Jeong et al. (2015)'s econometric study confirms that this way of accommodating ambiguity aversion may help to address the Equity Premium puzzle.

This chapter presents a multi-criteria portfolio model with the expected return as a performance measure and the expected worst-case return as a risk measure. The problems are formulated as a single-objective linear program, as a bi-objective linear program, and as a triple-objective mixed integer program. The problem objective is to allocate the wealth on different securities to optimize the portfolio return. The portfolio approach has allowed the two popular financial engineering percentile measures of risk, value-at-risk (VaR) and conditional value-at-risk (CVaR) to be applied. The decision-maker can assess the value of portfolio return, the risk level, and the number of assets, and can decide how to invest in a real-life situation comparing with ideal (optimal) portfolio solutions. The concave efficient frontiers illustrate the trade-off between the conditional value-at-risk and the expected return of the portfolio. Numerical examples based on historical daily input data from the Warsaw Stock Exchange are presented and selected computational results are provided. The computational experiments prove that both proposed linear and mixed integer programming approaches provide the decision-maker with a simple tool for evaluating the relationship between the expected and the worst-case portfolio return.

Innovation is the key to competitive advantage, and attaining innovation often requires taking on higher-than-usual levels of risk. Yet, while managers commonly profess support for efforts in innovation, they often emphasize safe, short-term results over more risky, long-term outcomes. As a result, a major challenge to firms is increasing employees’ willingness to adopt risky yet more profitable alternatives.

This study uses an experiment to test how the level of performance standard, per se, affect employees’ propensity to take on (more) risky projects. Using participants from the U.S. and Taiwan to represent higher versus lower individualism national cultures, it also examines the effects of national culture on employee actions. The findings are consistent with expectations from combining goal and prospect theories that a specific high standard motivates greater risk taking than a low standard. We find only limited difference between the U.S. and Taiwanese samples’ individualism/collectivism scores, which may help to explain the lack of significant differences between their reactions to the performance standard treatment.