Dynamic strategies when consumption and wealth risk aversions differ

Abstract

Résumé Cet article analyse l’impact sur l’allocation d’actifs d’un comportement des investisseurs révélé dans une étude empirique récente (Meyer and Meyer, 2005) : les investisseurs ont une aversion pour le risque de la consommation supérieure à celle pour le risque de la richesse. Nous démontrons que ce phénomène s’apprécie simplement à l’aide d’une seule variable financière. Cette variable mesure la part de la richesse que l’investisseur met de côté pour satisfaire sa consommation future par opposition à la richesse dont il souhaite disposer à la fin de l’investissement. Elle ne dépend de la richesse de l’investisseur que lorsque ce phénomène est pris en compte. Pour parvenir à ces résultats, nous nous sommes servis de techniques développées par Karatzas et al . (1987), ainsi que des apports de Wachter (2002) et de Munk et Sørensen (2007) réalisés dans le cadre restreint où l’investisseur a la même aversion pour le risque de la consommation que pour celui de la richesse.

Optimal sharing rule for a household with a portfolio management problem

“…In the classical portfolio management problem, the single agent model is the default representation. It has already been argued [22] that consumption and terminal wealth should be evaluated distinctly, as the nature of the reward is different (empirical works [18] highlight a greater risk aversion toward consumption than toward wealth). In [22], Six makes that distinction, yet for a single agent.…”

“…It has already been argued [22] that consumption and terminal wealth should be evaluated distinctly, as the nature of the reward is different (empirical works [18] highlight a greater risk aversion toward consumption than toward wealth). In [22], Six makes that distinction, yet for a single agent. He shows, as we do, the separability of the problem, and that the allocation of money to consumption (the consumption satisfaction proportion, or CSP) drastically depends-but monotonously-on the initial wealth.…”

“…He shows, as we do, the separability of the problem, and that the allocation of money to consumption (the consumption satisfaction proportion, or CSP) drastically depends-but monotonously-on the initial wealth. By introducing two separate consumption streams, we go further than Six [22], since there is an intricate dependence of each consumption stream with the common resulting wealth. We show that our model, unlike the one of [22], can exhibit a hump shaped consumption satisfaction proportion.…”

“…By introducing two separate consumption streams, we go further than Six [22], since there is an intricate dependence of each consumption stream with the common resulting wealth. We show that our model, unlike the one of [22], can exhibit a hump shaped consumption satisfaction proportion.…”

“…Accordingly, the problem can be separated in three subproblems which are solved explicitely via classical duality techniques, see [13]. We comment also on how [22] extends non trivially from one to two agents, and how to extend the setting to more than two. Section 5 presents the specific case based on power utility functions and a mean reverting price of risk, first introduced in [23].…”