OPTIMAL MULTIPERIOD PORTFOLIO POLICIES+ JAN MOSSINT mulation of the decision problem(even in the A. BACKGROUND e)in terms of por Most of the work in portfolio theory to folio rate of return tends to obscure an date! has taken what may be called a important aspect of the problem, namely, the role of the absolute size of the port- mean variability approach-that is, the folio. In a multiperiod theory the devel investor is thought of as choosing among opment through time of total wealth be- alternative portfolios on the basis of the mean and variance of the portfolios'rate comes crucial and must be taken into nt. a formula lecting this of return. A recent contribution by Ar- can easily become misleading row prepares the ground for a consider- In order to bring out and resolve the ably more general approach. 2 problems connected with a rate-of-return Although there would seem to be an formulation, it is therefore necessary to obvious need for extending the one-peri- start with an analysis of the one-period od analysis to problems of portfolio man agement over several periods, Tobin problem. Thus prepared, the extension to appears to be one of the first to make an plished, essentially by means of a dy will be demonstrated in this article, the validity of portions of this analysis ap- B. RISK-AVERSION FUNCTIONS pears to be doubtful. The explanation is The Pratt-Arrow measures An earlier as CORE Discus. in the analysis. They are abse the au- aversion Research and Econometrics, Unive Louvain, Belgium. Ra(r)=-v(yy, nomics and Business Administration, Bergen, Nor- relative risk aversion, way uidity Preference as behavior R+(Y)= U(Y) New York: Wiley, 1959);J. Tobin, "The Theory of where U is a utility function representing Portfolio selection in F H. Hahn and e. P. r. preferences over probability distribu don: Macmillan,1965),. Mossin,"Equilibrium in tions for wealth Y. Discussions of the a Capital Asset Market, "Econometrica(1966), pp. significance of these functions are found 768-83. Arrow and pratt. 4 a K. J. Arrow, Aspects of the Theory of risk Bearing (Yrjo Jahnsson Lectures [Helsinki: The rjo Jansson Foundation, 1965) 4 Arrow, op. cil. i J. Pratt, "Risk Aversion in the Small and in the Large, " Econometrica(1964), pp 3 Tobin, " Theory of Portfolio Selection. " 122-36. his content downloaded from 202.. 18.13 on Wed, 1 1 Sep 2013 02: 33: 00 AM All use subject to JSTOR Terms and ConditionsOPTIMAL MULTIPERIOD PORTFOLIO POLICIES* JAN MOSSINt I. INTRODUCTION A. BACKGROUND Most of the work in portfolio theory to date' has taken what may be called a mean variability approach-that is, the investor is thought of as choosing among alternative portfolios on the basis of the mean and variance of the portfolios' rate of return. A recent contribution by Ar￾row prepares the ground for a consider￾ably more general approach.2 Although there would seem to be an obvious need for extending the one-peri￾od analysis to problems of portfolio man￾agement over several periods, Tobin appears to be one of the first to make an attempt in this direction.3 However, as will be demonstrated in this article, the validity of portions of this analysis ap￾pears to be doubtful. The explanation is partly to be found in the fact that a for￾mutation of the decision problem (even in the one-period case) in terms of port￾folio rate of return tends to obscure an important aspect of the problem, namely, the role of the absolute size of the port￾folio. In a multiperiod theory the devel￾opment through time of total wealth be￾comes crucial and must be taken into account. A formulation neglecting this can easily become misleading. In order to bring out and resolve the problems connected with a rate-of-return formulation, it is therefore necessary to start with an analysis of the one-period problem. Thus prepared, the extension to multiperiod problems can be accom￾plished, essentially by means of a dy￾namic programing approach. B. RISK-AVERSION FUNCTIONS The Pratt-Arrow measures of risk aversion are employed at various points in the analysis. They are absolute risk aversion, Ra( Y) U" ( Y) relative risk aversion, Rr( Y) = U( Y) Y U'(Y)I where U is a utility function representing preferences over probability distribu￾tions for wealth Y. Discussions of the significance of these functions are found in Arrow and Pratt.4 * An earlier version appeared as CORE Discus￾sion Paper No. 6702. It was written during the au￾thor's stay as visitor to the Center for Operations Research and Econometrics, University of Louvain, Louvain, Belgium. t Assistant professor, Norwegian School of Eco￾nomics and Business Administration, Bergen, Nor￾way. IJ. Tobin, "Liquidity Preference as Behavior towards Risk," Review of Economic Studies (1957- 58), pp. 65-86; H. Markowitz, Portfolio Selection (New York: Wiley, 1959); J. Tobin, "The Theory of Portfolio Selection," in F. H. Hahn and F. P. R. Brechling (eds.), The Theory of Interest Rates (Lon￾don: Macmillan, 1965), J. Mossin, "Equilibrium in a Capital Asset Market," Econometrica (1966), pp. 768-83. 2K. J. Arrow, Aspects of the Theory of Risk￾Bearing (Yrj6 Jahnsson Lectures [Helsinki: The Yrj6 Jahnsson Foundation, 1965]). 3 Tobin, "Theory of Portfolio Selection." 4 Arrow, op. cit.; J. Pratt, "Risk Aversion in the Small and in the Large," Econometrica (1964), pp. 122-36. 215 This content downloaded from 202.115.118.13 on Wed, 11 Sep 2013 02:33:00 AM All use subject to JSTOR Terms and Conditions