Optimal consumption and portfolio strategies when relative risk aversion from consumption differs from relative risk aversion from wealth

Relative risk aversion (RRA) of consumption (RRAC) differs from RRA of wealth (RRAW) is an empirical fact explained in the study of Meyer and Meyer (2005). However dynamic consumption/ investment problems are only solved in the finance literature when both RRA equal (RRAC = RRAW). Following the martingale route, we derive optimal consumption and investment solutions for a (CRRA) investor when both RRA differ. While our framework preserve the usual tractability of the RRAC = RRAW case, it gives the optimal consumption and portfolio problems new and essential features. As far as asset allocation is concerned, the traditional decomposition into a tangent portfolio and two portfolios that replicates the risk of pseudo bonds holds. However, the weights invested in these portfolios are all stochastic in our case but only in so far they depend on a quantity that measures how much importance investors give to satisfaction from future consumptions as opposed to that from terminal wealth. Moreover, RRA, defined in our more general case as the risk aversion from the variation of total utility (the value function), as well as elasticity of optimal wealth to consumption, both known to be constant in the RRAC = RRAW case, are now also random variables only because they depend on this satisfaction measurement quantity. Finally, this satisfaction measurement quantity, which is then the right representative of market conditions, depends also drastically on investors’ initial wealth. The latter, which plays virtually no role in the RRAC = RRAW case, is then a variable of crucial importance. JEL Classification: D81, G11, G12. 1 PRISM, University of Paris 1-Sorbonne, 17, place de la Sorbonne, 75231 Paris Cedex 05, France. E-mail address: [email protected]. Tel: 0140463170. Fax: 0140463177. I am highly indebted to Professor C. Mellios and Professor P. Poncet for discussions on asset allocation and encouraging me writing this article. All remaining errors are mine.