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... Even obtaining accurate estimates of them is very complicated. Indeed, many possible sources of errors (e.g., impossibility to obtain a sufficient number of data samples, instability of data, differing personal views of decision makers on the future returns [24], etc.) affect their estimation, and lead to what Bawa et al. [3] call estimation risk in portfolio selection. This estimation risk has been shown to be the source of very erroneous decisions, for, as pointed in [7,10], the composition of the optimal portfolio is very sensitive to the mean and the covariance matrix of the asset returns, and minor perturbations in the moments of the random returns can result in the construction of very different portfolios. ...

... of problem (24). The relaxation is obtained by transforming the portfolio return constraint into an equality constraint from which the non-linear component is dropped, and by removing the non-negativity constraints. ...

... The optimal portfolio with buy-in constraints has positions in 16, 24 and 10 assets for 50-, 100-100-, and 200-stock instances, respectively. These number must be contrasted to those of the optimal portfolios without any integer constraints (24,30,34), with diversification constraints (26,37,41), and with round lot constraints (24,28,30). ...

In this paper, we study extensions of the classical Markowitz’ mean-variance portfolio optimization model. First, we consider that the expected asset returns are stochastic by introducing aprobabilistic constraint imposing that the expected return of the constructed portfolio must exceeda prescribed return level with a high conﬁdence level. We study the deterministic equivalents ofthese models. In particular, we deﬁne under which types of probability distributions the deterministic equivalents are second-order cone programs, and give exact or approximate closed-form formulations. Second, we account for real-world trading constraints, such as the need to diversify theinvestments in a number of industrial sectors, the non-proﬁtability of holding small positions and theconstraint of buying stocks by lots, modeled with integer variables. To solve the resulting problems,we propose an exact solution approach in which the uncertainty in the estimate of the expected returns and the integer trading restrictions are simultaneously considered. The proposed algorithmicapproach rests on a non-linear branch-and-bound algorithm which features two new branching rules.The ﬁrst one is a static rule, called idiosyncratic risk branching, while the second one is dynamic andcalled portfolio risk branching. The proposed branching rules are implemented and tested using theopen-source framework of the solver Bonmin. The comparison of the computational results obtainedwith standard MINLP solvers and with the proposed approach shows the effectiveness of this latterwhich permits to solve to optimality problems with up to 200 assets in a reasonable amount of time.

... A trade-off between safe liquidity conditions, key to shareholders' short-mediumterm returns, and long-term business sustainability has emerged and lead to strategies based on long-term horizons. This motivates the adoption of a multistage stochastic programming (MSP) problem formulation (Birge and Louveaux, 1997;Cariño et al., 1994;Consigli and Dempster, 1998;Mulvey and Erkan, 2005;Zenios and Ziemba, 2007a) able to capture both short-and long-term goals. Contrary to current standards in insurance-based investment divisions which largely rely on one-period static approaches (de Lange et al., 2003;Mulvey and Erkan, 2003;Zenios and Ziemba, 2007b), the adoption of dynamic approaches allows both the extension of the decision horizon and a more accurate short-term modelling of P&C variables. ...

... In this chapter we present an asset-liability management (ALM) problem integrating the definition of an optimal asset allocation policy over a 10-year planning horizon with the inclusion of liability constraints generated by an ongoing P&C business (de Lange et al., 2003;Dempster et al., 2003;Mulvey and Erkan, 2005;Mulvey et al., 2007). Relying on a simplified P&C income statement we clarify the interaction between the investment and the classical insurance business and introduce an objective function capturing short-, medium-, and long-term goals within a multistage model. ...

Recent trends in the insurance sector have highlighted the expansion of large insurance firms into asset management. In addition
to their historical liability risk exposure associated with statutory activity, the growth of investment management divisions
has caused increasing exposure to financial market fluctuations. This has led to stricter risk management requirements as
reported in the Solvency II 2010 impact studies by the European Commission. The phenomenon has far-reaching implications for
the definition of optimal asset–liability management (ALM) strategies at the aggregate level and for capital required by insurance
companies. In this chapter we present an ALM model which combines in a dynamic framework an optimal strategic asset allocation
problem for a large insurer and property and casualty (P&C) business constraints and tests it in a real-world case study.
The problem is formulated as a multistage stochastic program (MSP) and the definition of the underlying uncertainty model,
including financial as well as insurance risk factors, anticipates the model’s application under stressed liability scenarios.
The benefits of a dynamic formulation and the opportunities arising from an integrated approach to investment and P&C insurance
management are highlighted in this chapter.
KeywordsProperty and casualty insurance-Asset–liability management-Multistage stochastic programming-Insurance liabilities

... Managing a global insurance company via a centralized DFA system requires, however, the close coordination between the divisions (groups) and the company's headquarters. Mulvey et al. (2005) describes in detail a DFA system for the Towers Perrin Company and its Tillinghast business. This system has been implemented by global insurance companies, including AXA. ...

... Hence, for simplicity, the DFA model consists of a single time-period in this paper. Mulvey et al. (2005) provides a comprehensive description of a full enterprise DFA for an insurance company. ...

Over the past decade, financial companies have merged diverse areas including investment banking, insurance, retail banking, and trading operations. Despite this diversity, many global financial firms suffered severe losses during the recent recession. To reduce enterprise risks and increase profits, we apply a decentralized risk management strategy based on a stochastic optimization model. We extend the decentralized approach with the CVaR risk-metric, showing the advantages of CVaR over traditional risk measures such as value-at-risk. An example taken from the earthquake insurance area illustrates the concepts.

... However, these returns are unobservable and unknown. The impossibility to obtain a sufficient number of data samples, instability of data, and differing personal views of decision makers on the A. I. Logubayom, T. A. Victor Journal of Financial Risk Management future returns affect their estimation and have led to what call estimation risk in portfolio selection (Mulvey & Erkan, 2003;Bawa, Brown, & Klein, 1979). This estimation risk has shown to be the source of very erroneous decisions; as pointed in (Ceria & Stubbs, 2006;Cornuejols & Tutuncu, 2007), the composition of the optimal portfolio is very sensitive to the mean and the covariance matrix of the asset returns and agitation in the moments of the random returns can result in the difficulties in constructing different optimization. ...

... After the seminal paper by Cariño et al. (1994) focusing, as here and in Consigli et al. (2011), on a real-world application of SP techniques, a rich set of contributions originated from the extended cooperation with the insurance sector by Mulvey, who concentrates in Mulvey and Erkan (2005) and Mulvey et al. (2007) on the general structure of the decision process for multinational insurers operating in global markets and the implications on the optimal capital allocation decisions. The general relationship between risk measures and capital allocation was considered in Dhaene et al. (2003), while in de Lange et al. (2004) a stochastic programming approach was adopted to model a PC reinsurance problem. ...

The practical adoption of the Solvency II regulatory framework in 2016, together with increasing property and casualty (PC) claims in recent years and an overall reduction of treasury yields across more developed financial markets have profoundly affected traditional risk management approaches by insurance institutions. The adoption of firm-wide risk capital methodologies to monitor the companies’ overall risk exposure has further consolidated the introduction of risk-adjusted performance measures to guide the management medium and long-term strategies. Relying on a dynamic stochastic programming formulation of a 10 year asset-liability management (ALM) problem of a PC company, we analyse in this article the implications on capital allocation and risk-return trade-offs of an optimization problem developed for a global insurance company based on a pair of risk-adjusted return functions. The analysis is relevant for any institutional investor seeking a high risk-adjusted performance as for regulators in their structuring of stress-tests and effective regulatory frameworks. The introduction of the concept of risk capital, or economic capital, in the definition of medium and long term insurance strategies poses a set of modeling and methodological issues tackled in this article. Of particular interest is the study of optimal ALM policies under different assets’ correlation assumptions. From a computational viewpoint it turns out that, depending on the assumed correlation matrix, the stochastic program is linear or of second order conic type. A case study from a real-world company development is presented to highlight the effectiveness of applied stochastic programming in capturing complex risk and return dynamics arising in modern corporate finance and lead to an efficient long-term financial allocation process.

... However, these returns are unobservable and unknown. The impossibility to obtain a sufficient number of data samples, instability of data, differing personal views of decision makers on the future returns [13] affect their estimation and has led to what [1] call estimation risk in portfolio selection. This estimation risk has shown to be the source of very erroneous decisions, for, as pointed in ( [2], [6]), the composition of the optimal portfolio is very sensitive to the mean and the covariance matrix of the asset returns and agitation in the moments of the random returns can result in the difficulties in constructing different optimization. ...

Analyzing risk has been a principal concern of actuarial and insurance professionals which plays a fundamental role in the theory of portfolio selection where the prime objective is to find a portfolio that maximizes expected return while reducing risk. Portfolio optimization has been applied to asset management and in building strategic asset allocation. The purpose of this paper is to construct optimal and efficient portfolios using the matrix approach. This paper used secondary data on 13 stocks (ETI, GCB, GOIL, TOTAL, FML, GGBL, CLYD, EGL, PZC, UNIL, TLW, AGA and BOPP) from the Ghana Stock Exchange (GSE) database comprising the monthly closing prices from the period 02/01/2004 to 16/01/2015. The results revealed that, all the portfolios were optimal and that portfolios 1, 2, 4, 5, 6, 9, 10, 11 and 12 with expected return 2.523, 2.593, 2.827, 3.642, 2.405, 2.812, 5.229, 3.559 and 5.928 respectively were efficient portfolios whereas portfolios 3, 7 and 8 with expected return 0.377, 0.699 and 0.152 respectively were inefficient portfolios with reference to the expected return of the global minimum variance portfolio (2.360). GGBL was seen as the stock with the highest allocation of wealth in most of the portfolios. Six out of the 12 portfolios had CLYD exhibiting the least asset allocation.

... Even obtaining accurate estimates of them is very complicated. Indeed, many possible sources of errors (e.g., impossibility to obtain a sufficient number of data samples, instability of data, differing personal views of decision makers on the future returns; e.g., Mulvey and Erkan 2003) affect their estimation leading to the so-called estimation risk (Bawa et al. 1979) in portfolio selection. The estimation risk has been shown to be the source of very erroneous decisions for, as pointed out in Ceria and Stubbs (2006) and Cornuejóls and Tütüncü (2007), the composition of the optimal portfolio is very sensitive to the mean and the covariance matrix of the asset returns, and minor perturbations in the moments of the random returns can result in the construction of very different portfolios. ...

In this paper, we study extensions of the classical Markowitz’ mean-variance portfolio optimization model. First, we consider that the expected asset returns are stochastic by introducing a probabilistic constraint imposing that the expected return of the constructed portfolio must exceed a prescribed return level with a high conﬁdence level. We study the deterministic equivalents of these models. In particular, we deﬁne under which types of probability distributions the deterministic equivalents are second-order cone programs, and give exact or approximate closed-form formulations. Second, we account for real-world trading constraints, such as the need to diversify the investments in a number of industrial sectors, the non-proﬁtability of holding small positions and the constraint of buying stocks by lots, modeled with integer variables. To solve the resulting problems, we propose an exact solution approach in which the uncertainty in the estimate of the expected returns and the integer trading restrictions are simultaneously considered. The proposed algorithmic approach rests on a non-linear branch-and-bound algorithm which features two new branching rules. The ﬁrst one is a static rule, called idiosyncratic risk branching, while the second one is dynamic and called portfolio risk branching. The proposed branching rules are implemented and tested using the open-source framework of the solver Bonmin. The comparison of the computational results obtained with standard MINLP solvers and with the proposed approach shows the effectiveness of this latter which permits to solve to optimality problems with up to 200 assets in a reasonable amount of time.

... After the seminal paper by Cariño et al. (1994) focusing, as here and in Consigli et al. (2011), on a real-world application of DSP techniques, a discussion on the role of reinsurance decisions by P/C managers can be found in de Lange et al. (2003). A rich set of contributions originates from the extended cooperation with the insurance sector by Mulvey, who concentrates in Mulvey and Erkan (2005), Mulvey et al. (2007) on the general structure of the decision process for multinational insurers operating in global markets and the implications on the optimal capital allocation decisions. The general relationship between risk measures and capital allocation is considered in Dhaene et al. (2003). ...

The introduction of the Solvency II regulatory framework in 2011 and unprecendented property and casualty (P/C) claims experienced in recent years by large insurance firms have motivated the adoption of risk-based capital allocation policies in the insurance sector. In this article, we present the key features of a dynamic stochastic program leading to an optimal asset-liability management and capital allocation strategy by a large P/C insurance company and describe how from such formulation a specific, industry-relevant, stress-testing analysis can be derived. Throughout the article the investment manager of the insurance portfolio is regarded as the relevant decision-maker: he faces exogenous constraints determined by the core insurance division and is subject to the capital allocation policy decided by the management, consistently with the company's risk exposure. A novel approach to stress-testing analysis by the insurance management, based on a recursive solution of a large-scale dynamic stochastic program, is presented.

... Decentralized decision making has also been long recognized as an important decision-making problem. Decentralization among the divisions of a global firm requires a decomposition pattern that would also reduce the number of the steps required for the convergence to the centralized one (see [8], [9] for decentralized risk management theory and application in financial services industry). The comprehensive reviews published so far suggest that the large-scale models have proven to be extremely difficult to solve to optimality without the application of Benders' decomposition or factorization methods. ...

The design of an optimal supply chain rarely considers uncertainty within the modeling framework. This omission is due to
several factors, including tradition, model size, and the difficulty in measuring the stochastic parameters. We show that
a stochastic program provides an ideal framework for optimizing a large supply chain in the face of an uncertain future. The
goal is to reduce disruptions and to minimize expected costs under a set of plausible scenarios. We illustrate the methodology
with a global production problem possessing currency movements.

... Even obtaining accurate estimates of them is very complicated. Indeed, many possible sources of errors (e.g., impossibility to obtain a sufficient number of data samples, instability of data, differing personal views of decision makers on the future returns[24], etc.) affect their estimation, and lead to what Bawa et al.[3]call estimation risk in portfolio selection. This estimation risk has been shown to be the source of very erroneous decisions, for, as pointed in[7,10], the composition of the optimal portfolio is very sensitive to the mean and the covariance matrix of the asset returns, and minor perturbations in the moments of the random returns can result in the construction of very different portfolios. ...

In this paper, we study extensions of the classical Markowitz mean-variance portfolio optimization model. First, we consider that the expected asset returns are stochastic by introducing a probabilistic constraint, which imposes that the expected return of the constructed portfolio must exceed a prescribed return threshold with a high confidence level. We study the deterministic equivalents of these models. In particular, we define under which types of probability distributions the deterministic equivalents are second-order cone programs and give closed-form formulations. Second, we account for real-world trading constraints (such as the need to diversify the investments in a number of industrial sectors, the nonprofitability of holding small positions, and the constraint of buying stocks by lots) modeled with integer variables. To solve the resulting problems, we propose an exact solution approach in which the uncertainty in the estimate of the expected returns and the integer trading restrictions are simultaneously considered. The proposed algorithmic approach rests on a nonlinear branch-and-bound algorithm that features two new branching rules. The first one is a static rule, called idiosyncratic risk branching, while the second one is dynamic and is called portfolio risk branching. The two branching rules are implemented and tested using the open-source Bonmin framework. The comparison of the computational results obtained with state-of-the-art MINLP solvers ( MINLP_BB and CPLEX ) and with our approach shows the effectiveness of the latter, which permits to solve to optimality problems with up to 200 assets in a reasonable amount of time. The practicality of the approach is illustrated through its use for the construction of four fund-of-funds now available on the major trading markets.

... SP models can be used for capacity expansion of power systems to model the unexpected growth in demand, fuel prices and other financial constraints [97]- [99]. Several SP models are developed for trading electricity in the deregulated power market [100]- [103]. ...

The primary objective of this dissertation is to develop a black box optimization
tool. The algorithm should be able to solve complex nonlinear, multimodal, discontinuous
and mixed-integer power system optimization problems without any
model reduction. Although there are many computational intelligence (CI) based
algorithms which can handle these problems, they require intense human intervention
in the form of parameter tuning, selection of a suitable algorithm for a given
problem etc. The idea here is to develop an algorithm that works relatively well on
a variety of problems with minimum human effort. An adaptive particle swarm
optimization algorithm (PSO) is presented in this thesis. The algorithm has special
features like adaptive swarm size, parameter free update strategies, progressive
neighbourhood topologies, self learning parameter free penalty approach etc.
The most significant optimization task in the power system operation is the
scheduling of various generation resources (Unit Commitment, UC). The current
practice used in UC modelling is the binary approach. This modelling results in a
high dimension problem. This in turn leads to increased computational effort and
decreased efficiency of the algorithm. A duty cycle based modelling proposed in
this thesis results in 80 percent reduction in the problem dimension. The stern uptime
and downtime requirements are also included in the modelling. Therefore,
the search process mostly starts in a feasible solution space. From the investigations
on a benchmark problem, it was found that the new modelling results in high
quality solutions along with improved convergence.
The final focus of this thesis is to investigate the impact of unpredictable nature
of demand and renewable generation on the power system operation. These quantities
should be treated as a stochastic processes evolving over time. A new PSO
based uncertainty modelling technique is used to abolish the restrictions imposed
by the conventional modelling algorithms. The stochastic models are able to incorporate
the information regarding the uncertainties and generate day ahead UC
schedule that are optimal to not just the forecasted scenario for the demand and
renewable generation in feed but also to all possible set of scenarios. These models
will assist the operator to plan the operation of the power system considering
the stochastic nature of the uncertainties. The power system can therefore optimally
handle huge penetration of renewable generation to provide economic operation
maintaining the same reliability as it was before the introduction of uncertainty.

This paper considers the worst-case CVaR in the case where only partial information on the underlying probability distribution is given. The minimization of worst-case CVaR under the mixture distribution uncertainty, componentwise bounded uncertainty and ellipsoidal uncertainty are investigated. The application of worst-case CVaR to robust portfolio op- timization is proposed, and the corresponding problems are cast as linear programs and second-order cone programs which can be e-ciently solved. Market data simulation and Monte Carlo simulation examples are presented to illustrate the methods. Our approaches can be applied in many situations, including those outside of flnancial risk management. Subject Classiflcations: Finance, portfolio: conditional VaR, portfolio optimization; Pro- gramming, linear, nonlinear: robust optimization, second-order cone programming.

A large conglomerate such as a property/casualty insurance firm in this case, can be divided along business boundaries. This division might be along commercial lines, homeowner lines and perhaps across countries. An insurance firm's capital can be interpreted as a buffer that protects the company from insolvency and its inability to pay policyholder losses. Rare events have been simulated over the two divisions of an insurance firm. Different risk measures like conditional value at risk (CVaR) have been implemented into the optimization model. Decomposition methods will be applied in the context of decentralized decision making of a multi-divisional firm.

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