[pt] A alocação de ativos é uma das mais importantes decisões financeiras para investidores. No entanto, as decisões humanas não são totalmente racionais. Sabemos que as pessoas cometem muitos erros sistemáticos como, excesso de confiança, aversão à perda irracional e mau uso da informação entre outros. Nesta tese desenvolvemos duas metodologias distintas para enfrentar esse problema. A primeira abordagem é qualitativa, utiliza o modelo de Black-Litterman e tenta mapear a visão que o investidor tem do mercado. Esse método tenta mitigar a irracionalidade na tomada de decisão tornando mais fácil para um investidor demonstrar suas preferências em relação aos ativos. Black e Litterman desenvolveram um método para otimização de carteiras com a proposta de melhorar o modelo Markowitz, utilizando a construção de visões para representar a opinião do investidor sobre o futuro. No entanto, a forma de construir essas visões é bastante confusa e exige que o investidor estime vários parâmetros que são subjetivos. Assim, propomos uma nova forma de criar essas visões, utilizando Análise Verbal de Decisão. A segunda pesquisa envolve métodos quantitativos para resolver o problema de alocação de ativos com múltiplos estágios com premissas mais realistas. Embora a Programação Dinâmica Dual Estocástica (PDDE) seja uma técnica promissora para a solução de problemas de grande porte, não é adequada para o problema de alocação de ativos devido à dependência temporal associada aos retornos dos ativos. PDDE assume que o processo estocástico tem independência por estágio assegurando uma função única de custo futuro para cada estágio. No problema de alocação de ativos, a dependência do tempo é tipicamente não-linear e no lado esquerdo, o que torna PDDE tradicional não aplicável. Propomos uma variação do PDDE usando modelo oculto de Markov com estados discretos para resolver problemas reais de alocação de ativos com múltiplos períodos e dependência no tempo. Ambas as abordagens foram testadas em dados reais e empiricamente analisadas. As principais contribuições são as metodologia desenvolvidas para simplificar a construção de portfólios e para resolver o problema de alocação de ativos com múltiplos estágios. / [en] Asset allocation is one of the most important financial decisions made by investors. However, human decisions are not fully rational, and people make several systematic mistakes due to overconfidence, irrational loss aversion and misuse of information, among others. In this thesis, we developed two distinct methodologies to tackle this problem. The first approach has a more qualitative view, trying to map the investor s vision of the market. It tries to mitigate irrationality in decision-making by making it easier for an investor to demonstrate his/her preferences for specirfic assets. This first research uses the Black-Litterman model to construct portfolios. Black and Litterman developed a method for portfolio optimization as an improvement over the Markowitz model. They suggested the construction of views to represent an investor s opinion about future stocks returns. However, constructing these views has proven difficult, as it requires the investor to quantify several subjective parameters. This work investigates a new way of creating these views by using Verbal Decision Analysis. The second research focuses on quantitative methods to solve the multistage asset allocation problem. More specifically, it modifies the Stochastic Dynamic Dual Programming (SDDP) method to consider real asset allocation models. Although SDDP is a consolidated solution technique for large-scale problems, it is not suitable for asset allocation problems due to the temporal dependence of returns. Indeed, SDDP assumes a stagewise independence of the random process assuring a unique cost-to-go function for each time stage. For the asset allocation problem, time dependency is typically nonlinear and on the left-hand side, which makes traditional SDDP inapplicable. This thesis proposes an SDDP variation to solve real asset allocation problems for multiple periods, by modeling time dependence as a Hidden Markov Model with concealed discrete states. Both approaches were tested in real data and empirically analyzed. The contributions of this thesis are the methodology to simplify portfolio construction and the methods to solve real multistage stochastic asset allocation problems.

Le souhait d’augmenter la part des énergies renouvelables dans le mix énergétique entraine une augmentation des parts des énergies volatiles et non pilotables, et rend donc l’équilibre offre-demande difficile à satisfaire. Une façon d’intégrer ces énergies dans le réseau électrique actuel est d’utiliser de petits moyens de production, de consommation et de stockage répartis sur tout le territoire pour compenser les sous ou sur productions. Afin que ces procédés puissent être intégrés dans le processus d’équilibre offre-demande, ils sont regroupés au sein d’une centrale virtuelle d’agrégation de flexibilité, qui est vue alors comme une centrale virtuelle. Comme pour tout autre moyen de production du réseau, il est nécessaire de déterminer son plan de production. Nous proposons dans un premier temps dans cette thèse une architecture et un mode de gestion pour une centrale d’agrégation composée de n’importe quel type de procédés. Dans un second temps, nous présentons des algorithmes permettant de calculer le plan de production des différents types de procédés respectant toutes leurs contraintes de fonctionnement. Et enfin, nous proposons des approches pour calculer le plan de production de la centrale d’agrégation dans le but de maximiser son gain financier en respectant les contraintes réseau. / The desire to increase the share of renewable energies in the energy mix leads to an increase inshare of volatile and non-controllable energy and makes it difficult to meet the supply-demand balance. A solution to manage anyway theses energies in the current electrical grid is to deploy new energy storage and demand response systems across the country to counter balance under or over production. In order to integrate all these energies systems to the supply and demand balance process, there are gathered together within a virtual flexibility aggregation power plant which is then seen as a virtual power plant. As for any other power plant, it is necessary to compute its production plan. Firstly, we propose in this PhD thesis an architecture and management method for an aggregation power plant composed of any type of energies systems. Then, we propose algorithms to compute the production plan of any types of energy systems satisfying all theirs constraints. Finally, we propose an approach to compute the production plan of the aggregation power plant in order to maximize its financial profit while complying with all the constraints of the grid.

L’hybridation électrique de la chaîne de traction automobile est l’une des solutions adoptées pour respecter les règlementations futures sur ses émissions. La stratégie de supervision de la chaîne de traction hybride répartit la puissance produite par le moteur à combustion interne et la machine électrique. Elle répond habituellement à un problème d’optimisation où l’objectif est de réduire la consommation de carburant mais nécessite à présent d’y ajouter les émissions polluantes. La chaîne de dépollution, placée à l’échappement du moteur, permet de diminuer la quantité de polluants émise dans l’atmosphère. Cependant, elle n’est efficace qu’à partir d’un seuil de température, et dépend de la chaleur apportée par les gaz d’échappement du moteur thermique. La première partie de ce travail est donc consacrée à la modélisation de la consommation énergétique et des émissions polluantes de la chaine de traction hybride. La modélisation de l’efficacité de la chaîne de dépollution est réalisée selon deux contextes. Le modèle zéro-dimensionnel est adapté aux contraintes de calcul de la commande optimale. Le modèle unidimensionnel associé à un estimateur d’état permet d’être embarqué et calculé en temps réel. À partir de ces travaux, la seconde partie de cette thèse déduit des stratégies de supervision à l’aide de la théorie de la commande optimale. Dans un premier cas, le principe de Bellman permet de calculer la commande optimale d’un véhicule hybride Diesel selon des critères de supervision ayant plus ou moins connaissance de l’efficacité de la chaîne de dépollution des émissions de NOX. Dans un second cas, une stratégie issue du Principe du Minimum de Pontryagin, embarquée sur un véhicule hybride essence, fonctionnant en temps réel et calibrée selon deux paramètres est proposée. L’ensemble de ces travaux est validé expérimentalement au banc moteur et montre une réduction significative des émissions polluantes pour une faible pénalité de carburant. / Powertrain hybridization is a solution that has been adopted in order to conform to future standards for emissions regulations. The supervisory strategy of the hybrid powertrain divides the power emitted between the internal combustion engine and the electric machine. In past studies, this strategy has typically responded to an optimization problem with the objective of reducing consumption. However, in addition to this, it is now necessary to take pollutant emissions into account as well. The after-treatment system, placed in the exhaust of the engine, is able to reduce pollutants emitted into the atmosphere. It is efficient from a certain temperature threshold, and the temperature of the system is dependent on the heat brought by the exhaust gas of the engine. The first part of this dissertation is aimed at modelling the energy consumption and pollutant emissions of the hybrid powertrain. The efficiency model of the after-treatment system is adapted for use in two different contexts. The zero-dimensional model conforms to the constraints of the optimal control calculation. The one-dimensional model associated with a state estimator can be embedded in a vehicle and calculated in real time. From this work, the second part of this dissertation deduces supervisory strategies from the optimal control theory. On the one hand, Bellman’s principle is used to calculate the optimal control of a Diesel hybrid vehicle using different supervisory criteria, each having more or less information about the after-treatment system efficiency over NOX emissions. On the other hand, a strategy from Pontryagin’s minimum principle, embedded in a gasoline hybrid vehicle, running in real time and calibrated with two parameters, is proposed. The whole of this work is validated experimentally on an engine test bed and shows a significant reduction in pollutant emissions for a slight fuel consumption penalty.

Dans cette thèse, nous nous intéressons à la résolution exacte de problèmes NP-difficiles sur les graphes et les hypergraphes. Les problèmes que nous étudions regroupent dans un premier temps des variantes du problème classique du nombre chromatique. Les variantes de ce problème se distinguent par la difficulté introduite par les relations entre les classes de couleurs, ou la difficulté de reconnaissance des classes de couleurs elles-mêmes. Puis nous ferons le lien avec les problèmes de transversaux sur les hypergraphes. Plus particulièrement, il s’agira de s’intéresser à l’énumération de transversaux minimaux dans un hypergraphe de rang borné. Outre la résolution exacte, nous nous intéressons à la résolution à paramètre fixe. Le problème de racine carrée de graphe est un problème important en théorie des graphes. Nous proposons et montrons la solubilité à paramètre fixe de deux problèmes d’optimisation reliés. Finalement, nous nous intéresserons à la résolution de problèmes de graphe, soit en lien avec les problèmes de colorations, soit pour montrer les performances possibles de différents algorithmes en fonction de l’espace mémoire disponible. Dans cette thèse, nous aurons à cœur d’appliquer judicieusement la grande majorité des techniques essentielles en algorithmique exacte exponentielle. Principalement, nous appliquerons la programmation dynamique ou le principe d’inclusion-exclusion pour les problèmes de coloration. La technique de programmation dynamique se retrouvera pour d’autres problèmes de cette thèse, aux côtés d’autres méthodes comme la technique de branchement ou de mesurer et conquérir / In this thesis, we are interested in the exact computation of np-hard problems on graphs and hypergraphs. Firstly, we study several variants of colorings. Those variants appear harder than the famous chromatic number problem, by adding difficulty in recognizing the color classes, or more often by introducing various relationships between them. Then we link to problems of transversals in hypergraphs. More precisely, we are interested in enumerating minimal transversals in bounded ranked hypergraphs. Besides the exact computation, we are also interested in fixed parameter tractability. For this area, we study two optimization versions of the famous square root of graphs problem. Finally, we will be interested in solving other problems of graphs related to colorings, or in order to compare efficiencies of algorithms depending on the memory space available. In this thesis, we will apply most of major techniques in designing exact exponential algorithms. The main techniques we use are dynamic programming, inclusion-exclusion, branching, or measure and conquer

Este trabalho analisa três estratégias de hedge de opção, buscando identificar a importância da escolha da estratégia para a obtenção de um bom desempenho do hedge. O conceito de hedge é analisado de forma retrospectiva e uma teoria geral de hedge é apresentada. Em seguida são descritos alguns estudos comparativos de desempenho de estratégias de hedge de opção e suas metodologias de implementação. Para esta análise comparativa são selecionadas três estratégias de hedge de opção de compra do tipo européia: a primeira utiliza o modelo Black-Scholes-Merton de precificação de opções, a segunda utiliza uma solução de programação dinâmica para hedge dinâmico multiperiódico e a terceira utiliza um modelo GARCH para precificação de opções. As estratégias são comentadas e comparadas do ponto de vista de suas premissas teóricas e por meio de testes comparativos de desempenho. O desempenho das estratégias é comparado sob uma perspectiva dinâmicamente ajustada, multiperiódica e autofinanciável. Os dados para comparação de desempenho são gerados por simulação e o desempenho é avaliado pelos erros absolutos médios e erros quadráticos médios, resultantes na carteira de hedge. São feitas ainda considerações a respeito de alternativas de estimação e suas implicações no desempenho das estratégias. / This work analyzes three option hedging strategies, to identify the importance of choosing a strategy in order to achieve a good hedging performance. A retrospective analysis of the concept of hedging is conducted and a general hedging theory is presented. Following, some comparative papers of hedging performance and their implementation methodologies are described. For the present comparative analysis, three hedging strategies for European options have been selected: the first one based on the Black-Scholes-Merton model for option pricing, the second one based on a dynamic programming solution for dynamic multiperiod hedging and the third one based on a GARCH model for option pricing. The strategies are compared under their theoric premisses and through comparative performance testes. The performances of the strategies are compared under a dynamically adjusted multiperiodic and self-financing perspective. Data for performance comparison are generated by simulation and performance is evaluated by mean absolute errors and mean squared errors resulting on the hedging portfolio. An analysis is also done regarding estimation approaches and their implications over the performance of the strategies.

Pour résoudre les problèmes environnementaux et énergétiques liés au nombre croissant de véhicules en circulation, deux approches sont envisageables : l'une est technologique et vise à améliorer les composants du véhicule ou son architecture, l'autre est comportementale et cherche à changer la manière d'utiliser les véhicules. Dans ce contexte, l'éco-conduite représente une méthode, applicable immédiatement, permettant à chaque conducteur de réduire sa consommation. L'objectif de cette thèse est donc l'analyse des gains potentiels de l'éco-conduite pour les différents types de véhicules existant : thermique, électrique et hybride. Ainsi, la première partie de ce travail se focalise sur une étude théorique visant à calculer les gains potentiels et à déterminer les règles d'éco-conduite, avant d'aborder dans un second temps une mise en situation plus réaliste et une intégration des algorithmes dans un système d'assistance pour le conducteur. En s'appuyant sur une modélisation énergétique des différents types de véhicules, la détermination et la comparaison du fonctionnement optimal se base sur l'optimisation du profil de vitesse pour des trajets connus. La programmation dynamique a été mise enoeuvre pour calculer la trajectoire optimale énergétique en tenant compte de la contrainte temporelle afin de ne pas pénaliser l'intérêt d'une conduite économe. Evidemment, l'intégration de l'éco-conduite doit, d'une part, tenir compte du trafic à proximité du véhicule et d'autre part, ne pas aboutir à une augmentation des émissions de polluants. Ainsi, en nous appuyant sur des modèles de suivi de véhicules (trafic), nous avons montré que les principes d'éco-conduite restent valables et conduisent de toute façon à des gains énergétiques. Concernant les contraintes d'émissions, des résultats expérimentaux nous ont conduit à adapter nos algorithmes pour répondre simultanément aux aspects écologiques et économiques. Enfin, les connaissances acquises ont été appliquées à la conception d'un système d'assistance testé sur un simulateur de conduite. / The transportation sector has been identified as one of many sources of today's energetic and environmental problems. With constantly increasing numbers of vehicles on the road, non-renewable fossil fuels are becoming scarce and expensive. In addition, due to the pollutant emissions of internal combustion engines, the transportation sector is a major producer of greenhouse gas emissions. To resolve these problems researcher are looking for technological solutions, such as more efficient components and alternative drive train technologies, on one hand. On the other hand, work is being done to ensure the most efficient utilization of available technological resources. Eco driving is one way to immediately reduce a driver's energy consumption. In this thesis the potential gains of eco driving for passenger vehicles will be discussed. The main objective of this work is to, first, identify and compare drive train specific, optimal vehicle operation. Secondly, the effect of real-life constraints on potential gains of eco driving is evaluated. In addition, an approach to integrate mathematical optimization algorithms in an advanced driver assist system for eco driving is proposed. Physical vehicle models are developed for three representative vehicles: the conventional, electric and power-split hybrid vehicle. Using real-life and standard drive cycles a baseline mission is defined by specifying trip and road constraint. Applying the dynamic programming algorithms the trajectory optimization problem is solved, minimizing energy consumption for the trip. The effect of traffic on potential gains of eco driving is discussed, considering a vehicle following situation. Integrating emission constraints in the optimization algorithm the environmental advantages of eco driving are discussed. Finally, the developed algorithms were integrated in a driver assist system. Experimental tests on a driving simulator were used to verify the effectiveness of the system, as well as driver acceptance.

Au moins 30% de l'énergie produite par les moteurs à combustion interne est dissipée sous forme de chaleur dans les gaz d'échappement. L'intérêt des constructeurs pour les systèmes de récupération de chaleur bases sur le cycle thermodynamique de Rankine est justifié par des réductions de consommation espérées entre 5 et 10%. L'ambition de cette thèse est de contribuer à lever les principaux verrous liés à la gestion des procédés Rankine pour des applications ≪ mobiles ≫. Ce manuscrit s'appuie sur trois cas d'étude avec, pour chacun, un procédé pilote destiné à être intégré respectivement sur des véhicules légers à moteur essence, sur des camions poids-lourds et sur des trains à propulsion hybride Diesel électrique. Pour cela, des approches de l'automatique à base de modèle ont été développées. Une nouvelle loi de commande non-linéaire, permettant l'asservissement de la température et de la pression en sortie d'évaporateur, est proposée. Il est montré expérimentalement que le système peut être maintenu dans des conditions permettant la récupération d'énergie sans discontinuer, même sur des cycles routiers très dynamiques. La supervision énergétique du cycle de Rankine à bord d'un véhicule est ensuite abordée. Il s'agit de trouver les consignes pour la commande rapprochée qui permettent de maximiser l'efficacité énergétique d'un véhicule équipé d'un système de récupération d'énergie par cycle de Rankine. Il est montré que le gain énergétique apporté par l'optimisation dynamique temps réel proposée est important, comparé à une stratégie basée sur l'optimisation statique du système habituellement employée dans la littérature / More than 30% of the energy produced by internal combustion engines (ICE) is dissipated as heat through the exhaust gases. The interest of manufacturers in heat recovery systems based on the thermodynamic Rankine cycle is justified by announced reductions in fuel consumption ranging from 5 and 10% depending on the system and the driving cycle. The aim of this thesis is to help remove the main barriers associated with supervising and controlling Rankine processes for ≪ mobile ≫ applications. This dissertation is based on three study cases, each corresponding to a pilot process installed in engine test benches at IFP Energies nouvelles (IFPEN). These are applications to be integrated respectively on board light-duty vehicles with spark-ignition engine, heavy-duty trucks and trains with Diesel-electric propulsion. An original nonlinear (model-based) control law for the temperature and the pressure tracking at the evaporator outlet is proposed. It is shown experimentally that the system can be maintained under conditions allowing continuous energy recovery, even during highly transient road cycles. Then the supervision of Rankine systems is addressed, resulting in the choice of optimal set-points (in term of energy management) for the low-level controller. An optimal control problem is formulated, allowing online implementation via dynamic real-time optimization.The proposed approach is validated on a realistic simulator, showing significant benefits in the amount of energy recovered when compared with the classical (static) approach found in Rankine cycle literature

Cette thèse est composée de deux parties, chacune portant sur un sous-domaine de l'optimisation combinatoire a priori distant de l'autre. Le premier thème de recherche abordé est la programmation biniveau stochastique. Se cachent derrière ce terme deux sujets de recherche relativement peu étudiés conjointement, à savoir d'un côté la programmation stochastique, et de l'autre la programmation biniveau. La programmation mathématique (PM) regroupe un ensemble de méthodes de modélisation et de résolution, pouvant être utilisées pour traiter des problèmes pratiques que se posent des décideurs. La programmation stochastique et la programmation biniveau sont deux sous-domaines de la PM, permettant chacun de modéliser un aspect particulier de ces problèmes pratiques. Nous élaborons un modèle mathématique issu d'un problème appliqué, où les aspects biniveau et stochastique sont tous deux sollicités, puis procédons à une série de transformations du modèle. Une méthode de résolution est proposée pour le PM résultant. Nous démontrons alors théoriquement et vérifions expérimentalement la convergence de cette méthode. Cet algorithme peut être utilisé pour résoudre d'autres programmes biniveaux que celui qui est proposé.Le second thème de recherche de cette thèse s'intitule "problèmes de coupe et de couverture partielles dans les graphes". Les problèmes de coupe et de couverture sont parmi les problèmes de graphe les plus étudiés du point de vue complexité et algorithmique. Nous considérons certains de ces problèmes dans une variante partielle, c'est-à-dire que la propriété de coupe ou de couverture dont il est question doit être vérifiée partiellement, selon un paramètre donné, et non plus complètement comme c'est le cas pour les problèmes originels. Précisément, les problèmes étudiés sont le problème de multicoupe partielle, de coupe multiterminale partielle, et de l'ensemble dominant partiel. Les versions sommets des ces problèmes sont également considérés. Notons que les problèmes en variante partielle généralisent les problèmes non partiels. Nous donnons des algorithmes exacts lorsque cela est possible, prouvons la NP-difficulté de certaines variantes, et fournissons des algorithmes approchés dans des cas assez généraux. / This thesis is composed of two parts, each part belonging to a sub-domain of combinatorial optimization a priori distant from the other. The first research subject is stochastic bilevel programming. This term regroups two research subject rarely studied together, namely stochastic programming on the one hand, and bilevel programming on the other hand. Mathematical Programming (MP) is a set of modelisation and resolution methods, that can be used to tackle practical problems and help take decisions. Stochastic programming and bilevel programming are two sub-domains of MP, each one of them being able to model a specific aspect of these practical problems. Starting from a practical problem, we design a mathematical model where the bilevel and stochastic aspects are used together, then apply a series of transformations to this model. A resolution method is proposed for the resulting MP. We then theoretically prove and numerically verify that this method converges. This algorithm can be used to solve other bilevel programs than the ones we study.The second research subject in this thesis is called "partial cut and cover problems in graphs". Cut and cover problems are among the most studied from the complexity and algorithmical point of view. We consider some of these problems in a partial variant, which means that the cut or cover property that is looked into must be verified partially, according to a given parameter, and not completely, as it was the case with the original problems. More precisely, the problems that we study are the partial multicut, the partial multiterminal cut, and the partial dominating set. Versions of these problems were vertices are

Preferences are very important in research fields such as decision making, recommendersystemsandmarketing. The focus of this thesis is on preferences over combinatorial domains, which are domains of objects configured with categorical attributes. For example, the domain of cars includes car objects that are constructed withvaluesforattributes, such as ‘make’, ‘year’, ‘model’, ‘color’, ‘body type’ and ‘transmission’.Different values can instantiate an attribute. For instance, values for attribute ‘make’canbeHonda, Toyota, Tesla or BMW, and attribute ‘transmission’ can haveautomaticormanual. To this end,thisthesis studiesproblemsonpreference visualization and learning for lexicographic preference trees, graphical preference models that often are compact over complex domains of objects built of categorical attributes. Visualizing preferences is essential to provide users with insights into the process of decision making, while learning preferences from data is practically important, as it is ineffective to elicit preference models directly from users. The results obtained from this thesis are two parts: 1) for preference visualization, aweb- basedsystem is created that visualizes various types of lexicographic preference tree models learned by a greedy learning algorithm; 2) for preference learning, a genetic algorithm is designed and implemented, called GA, that learns a restricted type of lexicographic preference tree, called unconditional importance and unconditional preference tree, or UIUP trees for short. Experiments show that GA achieves higher accuracy compared to the greedy algorithm at the cost of more computational time. Moreover, a Dynamic Programming Algorithm (DPA) was devised and implemented that computes an optimal UIUP tree model in the sense that it satisfies as many examples as possible in the dataset. This novel exact algorithm (DPA), was used to evaluate the quality of models computed by GA, and it was found to reduce the factorial time complexity of the brute force algorithm to exponential. The major contribution to the field of machine learning and data mining in this thesis would be the novel learning algorithm (DPA) which is an exact algorithm. DPA learns and finds the best UIUP tree model in the huge search space which classifies accurately the most number of examples in the training dataset; such model is referred to as the optimal model in this thesis. Finally, using datasets produced from randomly generated UIUP trees, this thesis presents experimental results on the performances (e.g., accuracy and computational time) of GA compared to the existent greedy algorithm and DPA.

附最低保證投資型保險商品的特色在於無論投資者的投資績效好壞，保險金額皆享有一最低投資保證，過去關於此類商品的研究皆假設標的資產為單一資產，或依固定比例之投資組合，並沒有考慮到投資人自行配置投資組合的效果，但大部分市售商品中，投資人可以自行配置投資標，此情況之下，保險公司如何衡量適當的保證成本即為一相當重要之課題。 本研究假設投資人風險偏好服從冪次效用函數，並假設與保單所連結之投資標的有兩種資產，一為具有高風險高報酬的資產，另一為具有低風險低報酬之資產，在每個保單年度之初，投資人可以選擇配置在兩種資產之比例，我們運用黃迪揚(2009)所提出的動態規劃數值解之方法，計算出在考慮投資人自行配置資產之下，保證成本將會比固定比例之投資高出12個百分點。 此外，為了瞭解在不同資產報酬率的模型之下，保證成本是否會有不一樣的結論，除了對數常態模型之外，我們假設高風險資產與低風險資產服從ARIMA-GARCH(Autoregressive Integrated Moving Average-Generalized Autoregressive Conditional Heteroscedastic )模型，並得到較高的保證成本。 / The main characteristic of variable annuities (VA) with minimum benefits is that the benefit will be guaranteed. Previous literatures assume a specific underling asset return process when considering the guaranteed cost of VA; but they do not consider the portfolio choice opportunity of the policyholders. However, it is common for policyholders to rebalance his portfolio in many types of VA products. Therefore it’s important for insurance companies to apply an approximate method to measure the guaranteed cost. In this research, we assume that there are two potential assets in policyholders’ portfolio; one with high risk and high return and the other one with low risk and low return. The utility function of the policyholder is assumed to follow a power utility. We consider the asset allocation effect on the guaranteed cost for a VA with guaranteed minimum withdrawal benefits, finding that the guaranteed cost will increase 12% compared with a specific underling asset. The model effect of the asset return process is also examined by considering two different asset processes, the lognormal model and ARIMA-GARCH model. The solution of dynamic programming problem is solved by the numerical approach proposed by Huang (2009). Finally we get the conclusion which the guaranteed cost given by the ARIMA-GARCH model is greater than the lognormal model.