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Multiperiod Portfolio Selection and Capital Asset Pricing
Abstract
The purpose of this paper is to derive a simple capital asset pricing model in a multiperiod setting based on a microeconomic model of multiperiod portfolio selection. Capital asset pricing theory has been developed by Sharpe [1964], Lintner [1965] and Mossin [1966] in a oneperiod context using the microeconomic model of portfolio selection advanced by Markowitz [1959]. The first multiperiod models intended an extension of Markowitz’s results to several periods; the first of these models work with a utility function defined cawealth at the planning horizon (e.g. Smith [1967], Mossin [1968], Chen, Jen and Zionts [1971]). A somewhat newer approach that is more in the tradition of microeconomic theory, uses utility functions defined on the vector of present and future consumption streams (e.g. Hakansson [1969], Samuelson [1969], Fama [197o] and in a continuous time framework Merton [1971]). Usually, these approaches make a profit by the assumption of additive utility or of serial independence of subsequent rates of return on every risky asset (for a more detailed discussion of the literature see Long [1972, pp. 146–15o]).
Praca doktorancka została poświęcona problemowi konstrukcji portfela akcji, który jest aktualnym ważnym problemem ekonomicznym. Problem ten został zaprezentowany jako wielokryterialny problem decyzyjny, a do jego rozwiązania wykorzystano dwuaddytywną miarę rozmytą i opartą na niej całkę Choqueta. Takie podejście pozwoliło dodać do modelu informację o zależnościach między kryteriami oceny. Główny cel pracy to propozycja nowego modelu wielokryterialnego, wspomagającego wybór spółek do portfela inwestycyjnego i jego weryfikacja empiryczna.
Die sog. "Sicherheitsäquivalentmethode" gilt als klassisches Verfahren nutzengestützter Bewertung. Sie bestimmt den Wert eines Stroms unsicherer Zahlungen, indem periodenspezifische Sicherheitsäquivalente gebildet und mit Hilfe des risikofreien Zinssatzes in einer Größe verdichtet werden. Im Anschluss an Kürsten (zfbf, 54. Jg., S. 128–144, 2002) kamen Zweifel an der theoretischen Fundierung dieser Vorgehensweise auf. Es entwickelte sich eine umfangreiche Diskussion, die zu klären versuchte, unter welchen Bedingungen die Sicherheitsäquivalentmethode eingesetzt werden kann und wie alternative nutzengestützte Kalküle aussehen könnten. Vorliegender Beitrag liefert einen systematischen Überblick über die hierbei entstandenen Arbeiten, wobei sowohl Ansätze innerhalb als auch außerhalb des Rahmens der Erwartungsnutzentheorie besprochen werden. Dabei wird die überragende Bedeutung der Annahmen hinsichtlich der Reichhaltigkeit des Kapitalmarkts deutlich. Besondere Beachtung zollt der Beitrag den–in der deutschen Literatur weitgehend außer Acht gelassenen–Verbindungen zum angelsächsischen Schrifttum. Im Vergleich der Literaturstränge zeigen sich Analogien, aber auch gegenseitige Ergänzungen.
This paper provides a categorized bibliography on the application of the techniques of multiple criteria decision making (MCDM) to problems and issues in finance. A total of 265 references have been compiled and classified according to the methodological approaches of goal programming, multiple objective programming, the analytic hierarchy process, etc., and to the application areas of capital budgeting, working capital management, portfolio analysis, etc. The bibliography provides an overview of the literature on “MCDM combined with finance,” shows how contributions to the area have come from all over the world, facilitates access to the entirety of this heretofore fragmented literature, and underscores the often multiple criterion nature of many problems in finance.
Treffen Investoren mit konstanter relativer Risikoaversion auch im Buy-and-Hold-Kontext myopische Portfolioentscheidungen? / Günter Bamberg, Gregor Dorfleitner und Michael Krapp. - In: Kapitalmarkt, Unternehmensfinanzierung und rationale Entscheidungen : Festschrift für Jochen Wilhelm / Wolfgang Kürsten ..., Hrsg. - Berlin [u.a.] : Springer, 2006, - S. 3-14
This article sheds some light on three concepts of risk aversion in a multiattributive decision framework introduced into
the literature by Kihlstrom and Mirman (J Econ Theor 8:361–388, 1974), de Finetti (Giornale degli Economisti e Annali di Economia
11:685–709, 1952), Richard (Manage Sci 22:12–21, 1975), and Meyer (Preferences over time, New York, pp. 473–514, 1976). We
review the three multiattributive risk aversion definitions as well as the notion of partial risk aversion, give a translation
of these concepts into properties of the multiattributive utility function and reveal that Meyer’s, de Finetti/Richard’s concepts
are very closely related. Moreover, it is shown that any additive utility function is risk neutral in the sense of de Finetti/Richard
and Meyer independently of the risk attitude that it expresses in terms of Kihlstrom/Mirman. Additionally we introduce a multiattributive
utility function derived from a one-dimensional function that leads to a coincidence of the differently defined risk attitudes.
An intertemporal model for the capital market is deduced from the portfolio selection behavior by an arbitrary number of investors who act so as to maximize the expected utility of lifetime consumption and who can trade continuously in time. Explicit demand functions for assets are derived, and it is shown that, unlike the one-period model, current demands are affected by the possibility of uncertain changes in future investment opportunities. After aggregating demands and requiring market clearing, the equilibrium relationships among expected returns are derived, and contrary to the classical capital asset pricing model, expected returns on risky assets may differ from the riskless rate even when they have no systematic or market risk.
This paper examines decision problems where the evaluative criteria are interdependent. After a brief study of interaction effects in the statistical analysis of multifactor experiments, a classification of interactions is developed for multiple criteria decision problems. The classification includes artificial, ordinal, configural, and holistic interactions. Some interactions are removable by transforming the response scale or restructuring the factors, while other interactions are not removable and must be considered explicitly. The interaction effects in various nonadditive utility models are discussed, and the method of fractional hypercubes is used to interpret the structural forms of these models.
This paper investigates the properties of a market for risky assets on the basis of a simple model of general equilibrium of exchange, where individual investors seek to maximize preference functions over expected yield and variance of yield on their port- folios. A theory of market risk premiums is outlined, and it is shown that general equilibrium implies the existence of a so-called "market line," relating per dollar expected yield and standard deviation of yield. The concept of price of risk is discussed in terms of the slope of this line.
Testing the two-parameter asset pricing theory is difficult (and currently infeasible). Due to a mathematical equivalence between the individual return/‘beta’ linearity relation and the market portfolio's mean-variance efficiency, any valid test presupposes complete knowledge of the true market portfolio's composition. This implies, inter alia, that every individual asset must be included in a correct test. Errors of inference inducible by incomplete tests are discussed and some ambiguities in published tests are explained.
In this article, the quantitative form of capital market equilibrium is derived for a multi-period economy in which (a) there are many consumption goods whose future prices are uncertain, and (b) the investment opportunities available to consumers include both common stocks and default-free bills of many different maturities. Particular emphasis is placed on consumer reaction to uncertainty about shifts in commodity prices and the term structure of interest rates and on the way one should expect to observe this reaction reflected in portfolio choices and equilibrium stock prices.
This paper is concerned with the valuation of multiperiod cash flows in a world where prices are determined according to the Sharpe-Lintner-Black model of capital market equilibrium. We find that the current market value of any future net cash flow is the current expected value of the flow discounted at risk-adjusted discount rates for each of the periods until the flow is realized. The discount rates are known and non-stochastic, but the rates for the different periods preceding the realization of a cash flow need not to be the same, and the rates relevant for a given period can differ across cash flows. The risk adjustments in the discount rates arise because of uncertainties about reassessments through time of the expected value of a flow and the relationships between these reassessments and the corresponding reassessments of the expected cash flows of all firms.
Thesis (M.S.)--Massachusetts Institute of Technology, Sloan School of Management, 1984. Supervised by Richard S. Ruback. Includes bibliographical references.
This chapter reviews the optimal consumption-investment problem for an investor whose utility for consumption over time is a discounted sum of single-period utilities, with the latter being constant over time and exhibiting constant relative risk aversion (power-law functions or logarithmic functions). It presents a generalization of Phelps' model to include portfolio choice and consumption. The explicit form of the optimal solution is derived for the special case of utility functions having constant relative risk aversion. The optimal portfolio decision is independent of time, wealth, and the consumption decision at each stage. Most analyses of portfolio selection, whether they are of the Markowitz–Tobin mean-variance or of more general type, maximize over one period. The chapter only discusses special and easy cases that suffice to illustrate the general principles involved and presents the lifetime model that reveals that investing for many periods does not itself introduce extra tolerance for riskiness at early or any stages of life.
sumption in period j, such that either the risk aversion index -u"(x)/u'(x), or the risk aversion index -xu"'(x)/u'(x), is a positive constant for all x > 0. In a second paper [6], it was further shown that this model, developed with the individual in mind, also gives rise to an induced theory of the firm under risk for the same class of utility functions. In the foregoing model, it was assumed that the individual's horizon was infinite (or known with certainty). In this paper, we consider the same basic model with three modifications. First, we postulate that the individual's, lifetime is a random variable with a known probability distribution. Second, we introduce a utility function intended to represent the individual's bequest motive. Third, we offer the individual the opportunity to purchase insurance on his life. It is found that when some or all of these modifications are made, all of the more important properties possessed by the optimal consumption and investment strategies under a certain horizon are preserved, albeit only under special conditions. In Section 2, the various components of the decision process are constructed. In the earlier model, the individual's objective was assumed to be the maximization of expected utility from consumption over time. Here, we postulate, more generally, that his objective is to maximize expected utility from consumption as long as he lives and from the bequest left upon his death. As before, the individual's resources are assumed to consist of an initial capital position (which may be negative) and a non-capital income stream. The latter, which may possess any time-shape, is assumed to be known with certainty and to terminate upon his death. In addition to insurance available at a "fair'" rate, the individual faces both financial opportunities (borrowing and lending) and an arbitrary number of productive investment opportunities. The interest rate is presumed to be known but may have any time-shape. The returns from the productive opportunities are assumed to be random variables, whose probability distributions may differ from period to period but are assumed to satisfy the "no-easy-money" condition. While no limit is placed on borrowing, the individual is required to be solvent at the time of his death with
We introduce adaptive learning behavior into a general-equilibrium life-cycle economy with capital accumulation. Agents form forecasts of the rate of return to capital assets using least-squares autoregressions on past data. We show that, in contrast to the perfect-foresight dynamics, the dynamical system under learning possesses equilibria that are characterized by persistent excess volatility in returns to capital. We explore a quantitative case for theselearning equilibria. We use an evolutionary search algorithm to calibrate a version of the system under learning and show that this system can generate data that matches some features of the time-series data for U.S. stock returns and per-capita consumption. We argue that this finding provides support for the hypothesis that the observed excess volatility of asset returns can be explained by changes in investor expectations against a background of relatively small changes in fundamental factors.
The simplest version of the multiperiod consumption-investment problem considers a consumer with wealth w1, defined as the market value of his assets at the beginning of period 1, which must be allocated to consumption c1 and a portfolio investment w1–c1. The portfolio will yield an uncertain wealth level w2 at the beginning of period 2, which must be divided between consumption c2 and investment w2–c2. Consumption-investment decisions must be made at the beginning of each period, until the consumer dies and his wealth is distributed among his heirs. The consumer's objective is to maximize the expected utility of lifetime consumption. This chapter reviews uncertainty models of the multiperiod consumption-investment problem considered by Edmund Phelps, Nils Hakansson, and Jan Mossin and presents a more general multiperiod consumption-investment model but one that nevertheless leads to interesting hypotheses about observable aspects of consumer behavior. The main result is the proposition that if the consumer is risk averse, that is, the utility function for lifetime consumption is strictly concave and markets for consumption goods and portfolio assets are perfect, 3 then the consumer's observable behavior in the market in any period is indistinguishable from that of a risk averse expected utility maximizer who has a one-period horizon.
Capital budgeting of risky projects with ‘imperfect markets’ for physical capital
- M C Bogue
- R Roll
- MC Bogue
Verschuldungs-und Ausschüttungspolitik im Lichte der Portefeuille-Theorie
- G Franke
A transition model for portfolio revision
- R C Smith
- RC Smith
Portfolio analysis, stock valuation and capital budgeting decision rules for risky projects
- R C Stapleton
- RC Stapleton
Market equlibrium in a multiperiod state preference model with logarithmic utility
- A Kraus
- R H Litzenberger
Capital budgeting and the capital asset pricing model: good news and bad news
- S C Myers
- S M Turnbull
The valuation of uncertain income streams and the pricing of options
- M E Rubinstein
- ME Rubinstein
Capital asset prices: a theory of market equilibrium under conditions of risk
- W F Sharpe
- WF Sharpe
- S C Myers
- S M Turnbull
- SC Myers
The Theory of Finance
- E F Fama
- M H Miller
- EF Fama