The fallacy of large numbers revisited: The construction of a utility function that leads to the acceptance of two games
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Philippe De Brouwer* holds degrees in Physics and Economics, and is currently Director at Fortis Investment Management. Philippe De Brouwer graduated in theoretical physics (University of Brussels) and published some work in the field of non-linear dynamics and chaos theory. Afterwards, he oriented his career towards financial economics and specialised in commercial engineering (University of Brussels). Now after the recent merger with Fimagen, and the creation of FIM (Fortis Investment Management), he is Director and is responsible for strategy and co-ordination.
Freddy Van den Spiegel holds a PhD in Economics, and is currently Chief Economist of Fortis Bank, Professor at the Free University of Brussels, and Member of the Executive Committee of Association Belge des OPC. He has an impressive track record in asset management and was the founder of Fortis Investments Belgium. Professor Van den Spiegel is an active academician and a speaker at many conferences. *Fortis Investment Management, Boulevard Roi Albert II, 1, 1210 Brussels, Belgium. Tel: ⫹32 2274 8487; Fax: ⫹32 2274 8206; e-mail: [email protected]
Abstract Should the composition of an investment portfolio differ depending on whether the investment horizon is longer or shorter? Practitioners have always allocated more risky investments to portfolios with longer investment horizons. But in 1963, P. A. Samuelson put forward strong arguments that this is an erroneous approach. Since then, the financial community has lived with this conflict and searched for plausible explanations. In this paper, however, we construct a counterexample to the thesis of 1963. Keywords: law of large numbers; investment horizon; portfolio construction; decisions under uncertainty; asset allocation
Introduction One of the most intriguing debates of recent years is without any doubt the question of whether the composition of an investment portfolio should vary with the investment horizon. According to the theory of Markowitz (1952), the answer seemed to be ‘yes’; however, this was
䉷 Henry Stewart Publications 1470-8272 (2001)
only until Samuelson added two other important papers to his already impressive list of publications: one tackling the generally accepted ‘law of large numbers’ (1963), and a second (1969) showing that the composition of an optimal portfolio under certain conditions is constant during the lifetime. The conditions
Vol. 1, 3, 257-266
Journal of Asset Management
257
De Brouwer and Van den Spiegel
necessary for this result are thoroughly studied, for example by Samuelson himself (1994), or Fisher and Statman (1999). In our opinion, these conditions do not seem to be the generic case, and therefore their importance is rather academic. The thesis of the first article (Samuelson, 1963), however, remained unchanged: ‘a person whose utility schedule prevents him from ever taking a specific favourable bet when offered only once can never rationally take a large sequence of such fair bets, if expected utility is maximised’. This is of utmost importa
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