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This chapter presents selected multiobjective methods for multiperiod portfolio optimization problem. Portfolio models are formulated as multicriteria mixed integer programs. Reference point method together with weighting approach is proposed. The portfolio selection problem considered is based on a multiperiod model of investment, in which the investor buys and sells securities in successive investment periods. The problem objective is to allocate the wealth on different securities to optimize the portfolio expected return, the probability that the return is not less than a required level. Multiobjective methods were used to find tradeoffs between risk, return, and the number of securities in the portfolio. In computational experiments the data set of daily quotations from the Warsaw Stock Exchange were used.

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 determination of the capital requirements represents the first Pillar of Solvency II. The main purpose of the new solvency regulation is to obtain more realistic modelling and assessment of the different risks insurance companies are exposed to in a balance‐sheet perspective. In this context, the Solvency Capital Requirement (SCR) standard calculation is based on a modular approach, where the overall risk is split into several modules and submodules. In Solvency II, standard formula longevity risk is explicitly considered. The purpose of this paper is to look at the backtesting approach for measuring the consistency of SCR calculations for life insurance policies.

A multiperiod approach is suggested for correctly calculating the SCR in a risk management perspective, in the sense that the amount of capital necessary to meet company future obligations year by year until the contract will be in force has to be assessed. The backtesting approach for measuring the consistency of SCR calculations for life insurance policies represents the main contribution of the research. In fact this kind of model performance is generally specified in the VaR validation analysis. In this paper, this approach is considered for testing the ex post performance of SCR calculation methodology.

The backtesting framework is able to measure, from time to time, if the insurer has allocated more or less capital to support his in‐force business, with adverse effects on free reserves and profitability or solvency.

The paper shows that the forecasting performance is an important aspect to assess the effectiveness of the model, a poor performance corresponding to a biased allocation of capital.

The backtesting approach for measuring the consistency of SCR calculations for life insurance policies represents the main contribution of the research. In fact this kind of model performance is generally specified in the VaR validation analysis. Recently, Dowd et al. have proposed it for verifying the goodness of mortality models and now, in this paper, this approach is considered for testing the ex post performance of SCR calculation methodology.

This paper aims to examine from commodity portfolio managers’ perspective the performance of liquidity adjusted risk modeling in assessing the market risk parameters of a large commodity portfolio and in obtaining efficient and coherent portfolios under different market circumstances.

The implemented market risk modeling algorithm and investment portfolio analytics using reinforcement machine learning techniques can simultaneously handle risk-return characteristics of commodity investments under regular and crisis market settings besides considering the particular effects of the time-varying liquidity constraints of the multiple-asset commodity portfolios.

In particular, the paper implements a robust machine learning method to commodity optimal portfolio selection and within a liquidity-adjusted value-at-risk (LVaR) framework. In addition, the paper explains how the adapted LVaR modeling algorithms can be used by a commodity trading unit in a dynamic asset allocation framework for estimating risk exposure, assessing risk reduction alternates and creating efficient and coherent market portfolios.

The optimization parameters subject to meaningful operational and financial constraints, investment portfolio analytics and empirical results can have important practical uses and applications for commodity portfolio managers particularly in the wake of the 2007–2009 global financial crisis. In addition, the recommended reinforcement machine learning optimization algorithms can aid in solving some real-world dilemmas under stressed and adverse market conditions (e.g. illiquidity, switching in correlations factors signs, nonlinear and non-normal distribution of assets’ returns) and can have key applications in machine learning, expert systems, smart financial functions, internet of things (IoT) and financial technology (FinTech) in big data ecosystems.

This article is based on responses to the 1982 general invitation from the Review Board for Government Contracts. The authors' main contention is that the pricing of government contracts has hitherto been deficient in at least two fundamental respects — the use of the historic cost accounting model as a computational framework for the costing and pricing of non‐competitive Government contracts and the use of ex post accounting rates of return for estimating target rates of return.

Though of fairly recent origin, the capital‐asset pricing model (CAPM) is becoming a dominant influence in the analysis of financial and investment decisions. While continuing to undergo stringent theoretical and empirical examination, the demonstrable explanatory and predictive ability of the CAPM have led to its widespread recognition as the foundation of modern financial management. Though usually attributed to Sharpe, Lintner and Mossin, the origins of the CAPM can be traced back to the celebrated work of Harry Markowitz on portfolio selection.

The main purpose of this research is to find an optimal allocation of marketing budgets which maximizes customer equity in an uncertain environment. Since markets are naturally uncertain environments, the aim is to incorporate uncertainty into the model.

Researchers have developed a mathematical programming model which employs customer equity as an objective function in order to allocate marketing budgets. The robust optimization approach is employed to tackle the proposed model, which deals with uncertainty.

The solution of the robust model is shown to be feasible and satisfactory in all uncertain situations. The robust solutions (of the presented model) are stable in volatile situations; while if the solution of deterministic models is used, it may be suboptimal or even infeasible. Sensitivity analysis of the deterministic solution only describes how stable is the suggested solution, but a robust optimization approach always provides a stable solution.

The presented model will be most effective where uncertainty is high; if uncertainty is not a matter of concern or estimates are reliable, deterministic models are also effective.

Companies periodically decide on marketing budgets in order to achieve predefined marketing targets in future periods. The results of this research may be useful and applicable in marketing departments for allocating marketing budgets, especially in uncertain environments.

The main contribution of this research lies in providing an approach to allocate marketing budgets in uncertain environments. Unlike previous studies, the presented method takes into account the uncertainty of parameters in a systematic way. Hence, in case of high degrees of uncertainty, the use of robust optimization is strictly recommended.

This paper attempts to review briefly the current position of capital asset pricing theory and testing, and to consider the implications for portfolio managers. Inevitably in a paper of this length, it is not possible to include all the material that is relevant in such a large and important area, but most of the important matters have hopefully been covered.

The new Basel Accord (known as Basel II) attempts to introduce more risk-sensitive capital requirements. We propose a multiperiod deposit insurance pricing model that incorporates specific regulatory capital requirements and the possibility of capital forbearance and moral hazard. We estimate the cost of deposit insurance under alternative regulation regimes based on the building block approach of the 1988 Basel Accord (known as Basel I) and internal model-based (IMB) capital regulation. In contrast to the building block of Basel I, Basel II's IMB capital regulation links more closely the capital requirement to a bank's actual risk. We develop a multiperiod pricing model while incorporating the effects of capital forbearance and moral hazard. The fairly-priced premium rates are computed by assuming that a bank's asset value follows a GARCH process. In contrast to previous studies based on the building block capital standard, we find that forbearance and the potential moral hazard behavior will not increase the cost of deposit insurance in the scheme of Basel II's IMB capital regulation.