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List of Frequently Used Symbols and Notation a Text Such As Intermediate Financial Theory Is, by Nature, Relatively Nota- Tion I List of Frequently Used Symbols and Notation A text such as Intermediate Financial Theory is, by nature, relatively nota- tion intensive. We have adopted a strategy to minimize the notational burden within each individual chapter at the cost of being, at times, inconsistent in our use of symbols across chapters. We list here a set of symbols regularly used with their speci¯c meaning. At times, however, we have found it more practical to use some of the listed symbols to represent a di®erent concept. In other in- stances, clarity required making the symbolic representation more precise (e.g., by being more speci¯c as to the time dimension of an interest rate). Roman Alphabet a Amount invested in the risky asset; in Chapter 14, fraction of wealth invested in the risky asset or portfolio AT Transpose of the matrix (or vector)A c Consumption; in Chapter 14 only, consumption is represented by C; while c represents ln C k cθ Consumption of agent k in state of nature θ CE Certainty equivalent CA Price of an American call option CE Price of a European call option d Dividend rate or amount ¢ Number of shares in the replicating portfolio (Chapter xx E The expectations operator k eθ Endowment of agent k in state of nature θ f Futures position (Chapter 16); pf Price of a futures contract (Chapter 16) F; G Cumulative distribution functions associated with densities: f; g Probability density functions K The strike or exercise price of an option K(~x) Kurtosis of the random variablex ~ L A lottery L Lagrangian m Pricing kernel M The market portfolio k MUθ Marginal utility of agent k in state θ p Price of an arbitrary asset P Measure of Absolute Prudence q Arrow-Debreu price qb Price of risk-free discount bond, occasionally denoted prf qe Price of equity rf Rate of return on a risk-free asset Rf Gross rate of return on a risk-free asset r~ Rate of return on a risky asset R~ Gross rate of return on a risky asset 1 RA Absolute risk aversion coe±cient RR Relative risk aversion coe±cient s Usually denotes the amount saved S In the context of discussing options, used to denote the price of the underlying stock S(~x) Skewness of the random variablex ~ T Transition matrix U Utility function U von Neuman-Morgenstern utility function V Usually denotes variance-covariance matrix of asset returns; occasionally is used as another utility function symbol; may also signify value as in Vp The value of portfolio P or VF The value of the ¯rm wi Portfolio weight of asset i in a given portfolio Y0 Initial wealth Greek Alphabet ® Intercept coe±cient in the market model (alpha) ¯ The slope coe±cient in the market model (beta) ± Time discount factor ´ Elasticity ¸ Lagrange multiplier ¹ Mean ¼θ State probability of state θ RN ¼θ Risk-neutral probability of state θ ¦ Risk premium ½(~x; y~) Correlation of random variablesx ~ andy ~ ½ Elasticity of intertemporal substitution (Chapter 14) σ Standard deviation σij Covariance between random variables i and j θ Index for state of nature ­ Rate of depreciation of physical capital à Compensating precautionary premium Numerals and Other Terms 1 Vector of ones  Is strictly preferred to  Is preferred to (non strictly, that is allowing for indi®erence) GBM Geometric Brownian Motion stochastic process F SD First-order stochastic dominance SSD Second-order stochastic dominance 2 Preface The market for ¯nancial textbooks is crowded at both the introductory and doctoral levels, but much thinner at the intermediate level. Teaching opportu- nities at this level, however, have greatly increased with the advent of masters of science programs in ¯nance (master in computational ¯nance, in mathematical ¯nance, and the like) and the strengthening demand for higher-level courses in MBA programs. The Master in Banking and Finance Program at the University of Lausanne admitted its ¯rst class in the fall of 1993. One of the ¯rst such programs of its kind in Europe, its objective was to provide advanced training to ¯nance spe- cialists in the context of a one-year theory-based degree program. In designing the curriculum, it was felt that students should be exposed to an integrated course that would introduce the range of topics typically covered in ¯nancial economics courses at the doctoral level. Such exposure could, however, ignore the detailed proofs and arguments and concentrate on the larger set of issues and concepts to which any advanced practitioner should be exposed. Our ambition in this text is, accordingly, ¯rst to review rigorously and con- cisely the main themes of ¯nancial economics (those that students should have encountered in prior courses) and, second, to introduce a number of fron- tier ideas of importance for the evolution of the discipline and of relevance from a practitioner's perspective. We want our readers not only to be at ease with the main concepts of standard ¯nance (MPT, CAPM, etc.) but also to be aware of the principal new ideas that have marked the recent evolution of the discipline. Contrary to introductory texts, we aim at depth and rigor; contrary to higher-level texts, we do not emphasize generality. Whenever an idea can be conveyed through an example, this is the approach we chose. We have, similarly, ignored proofs and detailed technical matters unless a reasonable understanding of the related concept mandated their inclusion. Intermediate Financial Theory is intended primarily for master level stu- dents with a professional orientation, a good quantitative background, and a preliminary education in business and ¯nance. As such, the book is targeted for masters students in ¯nance, but it is also appropriate for an advanced MBA class in ¯nancial economics, one with the objective of introducing students to the precise modeling of many of the concepts discussed in their capital mar- kets and corporate ¯nance classes. In addition, we believe the book will be a useful reference for entering doctoral candidates in ¯nance whose lack of prior background might prevent them from drawing the full bene¯ts of the very ab- stract material typically covered at that level. Finally, it is a useful refresher for well-trained practitioners. As far as prerequisites go, we take the view that our readers will have com- pleted at least one introductory course in Finance (or have read the correspond- ing text) and will not be intimidated by mathematical formalism. Although the mathematical requirements of the book are not large, some con¯dence in the use of calculus as well as matrix algebra is helpful. In preparing the second edition of this text, we have emphasized the overrid- 1 ing concern of modern ¯nance for the valuation of risky cash flows: Intermediate Financial Theory's main focus is thus on asset pricing. (In addition, we exclu- sively consider discrete time methodologies). The new Chapter 2 makes clear this emphasis while simultaneously stressing that asset pricing does not repre- sent the totality of modern ¯nance. This discussion then leads to a new struc- turing of the book into ¯ve parts, and a new ordering of the various chapters. Our goal here is to make a sharper distinction between valuation approaches that rely on equilibrium principles and those based on arbitrage considerations. We have also reorganized the treatment of Arrow-Debreu pricing to make clear how it accommodates both perspectives. Finally, a new chapter entitled \Port- folio Management in the Long Run" is included that covers recent developments that we view as especially relevant for the contemporary portfolio manager. The two appendices providing brief overviews of option pricing and continuous time valuation methods are now assigned to the text website. Over the years, we have bene¯ted from numerous discussions with colleagues over issues related to the material included in this book. We are especially grateful to Paolo Siconol¯, Columbia University, Rajnish Mehra, U. of Cal- ifornia at Santa Barbara, and Erwan Morellec, University of Lausanne, the latter for his contribution to the corporate ¯nance review of Chapter 2. We are also indebted to several generations of teaching assistants { Fran»coisChristen, Philippe Gilliard, Tomas Hricko, Aydin Akgun, Paul Ehling, Oleksandra Hubal and Lukas Schmid { and of MBF students at the University of Lausanne who have participated in the shaping-up of this material. Their questions, correc- tions and comments have lead to a continuous questioning of the approach we have adopted and have dramatically increased the usefulness of this text. Fi- nally we reiterate our thanks to the Fondation du 450`emeof the University of Lausanne for providing \seed ¯nancing" for this project. Jean-Pierre Danthine, Lausanne, Switzerland John B. Donaldson New York City 2 N'estime l'argent ni plus ni moins qu'il ne vaut : c'est un bon serviteur et un mauvais ma^³tre (Value money neither more nor less than it is worth: It is a good servant and a bad master) Alexandre Dumas, ¯ls, La Dame aux Cam¶elias (Pr¶eface) 3 Part I Introduction Chapter 1 : On the Role of Financial Markets and Institutions 1.1 Finance: The Time Dimension Why do we need ¯nancial markets and institutions? We chose to address this question as our introduction to this text on ¯nancial theory. In doing so we touch on some of the most di±cult issues in ¯nance and introduce concepts that will eventually require extensive development. Our purpose here is to phrase this question as an appropriate background for the study of the more technical issues that will occupy us at length. We also want to introduce some important elements of the necessary terminology. We ask the reader's patience as most of the sometimes-di±cult material introduced here will be taken up in more detail in the following chapters.
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