The aim of this article is to analyze the portfolio strategies that are mean-variance efficient when continuous rebalancing is allowed between the current date (0) and the horizon (T). Under very general assumptions, when a zero-coupon bond of maturity T exists, the dynamic efficient frontier is a straight line, the slope of which is explicitly characterized. Every dynamic mean-variance efficient strategy can be viewed as buy and hold combinations of two funds: the zero-coupon bond of maturity T and a continuously rebalanced portfolio. An appropriate dynamic strategy defining the latter is explicitly derived for two particular price processes and comparisons of the Efficient Frontiers (Static versus Dynamic) are provided in these cases.Portfolio Selection Model, Mean-Variance Analysis, Dynamic Strategies

The numeraire portfolio, also called the optimal growth portfolio, allows simple derivations of the main results of financial theory. The prices of self financing portfolios, when the optimal growth portfolio is the numeraire, are martingales in the 'true' (historical) probability. Given the dynamics of the traded securities, the composition of the numeraire portfolio as well as its value are easily computable. Among its numerous properties, the numeraire portfolio is instantaneously mean variance efficient. This key feature allows a simple derivation of standard continuous time CAPM, CCAPM, APT and contingent claim pricing. Moreover, since the Radon-Nikodym derivatives of the usual martingale measures are very simple functions of the numeraire portfolio, the latter provides a convenient link between the standard Capital Market Theory a la Merton and the probabilistic approach a la Harrison-Kreps-Pliska.Martingale Pricing Equilibrium Pricing Numeraire Portfolio Theory,

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Closed-form solutions for HARA optimal portfolios are obtained in a dynamic portfolio optimization model in three assets (stocks, bonds and cash) with stochastic interest rates. A Vasicek-type model of stochastic interest rates with a correlated stock price is assumed. The HARA solution can be expressed as a buy and hold combination of a zero-coupon bond with maturity matching the investor's horizon and a "CRRA mutual fund," which is the optimal portfolio for a CRRA investor expressed in terms of the weights on cash, stock, a constantduration bond fund and the (redundant) bond with maturity matching the investor's horizon (a generalization of the Merton's (1971) result of constant weight in stock for a CRRA investor, derived for two assets with constant interest rates). This simple characterization facilitates insights about investor behavior over time and under different economic scenarios and allows fast computation time (without simulation or other numerical methods). We use the model to provide explanations of the Canner-Mankiw-Weil (1997) asset allocation puzzle, the use of "convex" (momentum) and "concave" (contrarian) investment strategies and other features of popular investment advice. The model illuminates clearly the role of the different market parameters, the composition of initial investor wealth, and relative risk aversion in portfolio strategies.

Ross (1976) has shown, in a static framework, how options can complete financial markets. This paper examines the possible extensions of Ross's idea in a dynamic setup. Surprisingly enough, we find that the answer is very sensitive to the choice of the stochastic model for the underlying security returns. More specifically we obtain the following results: In a discrete-time model, classical European options typically become redundant with some probability (Proposition 2.1). Obnly path dependent ("exotic") options may generate dynamic spanning (Proposition 4.1). In a continuous-time model with stochastic volatility of the underlying security, and under reasonable assumptions, a European option is "always" a good instrument for completing markets (Proposition 5.2). Copyright 1996 Blackwell Publishers.