This paper is concerned with the problem of determining the optimal portfolio choice for an investor with constant risk aversion, that is, for an investor with an utility function of the form u(x)=1-exp(-¿x), ¿>0. It is assumed he is required to make his choice from amongst n different assets whose returns per period may be described by n independent random variables. The optimal choice depends on the distribution of the random variables and in this paper the optimal choice is determined for certain familiar distributions, including the Gamma and the certain positive stable ones. For other distributions, including the log normal, only the asymptotic (for large initial capital) choice is obtainable. Finally, the effect of including money as a possible asset is studied. It is shown that for a wide class of utility functions (including the one mentioned above) one haslimY ® ¥ yo* /Y = 1limYyoY=1 where y*0 (0=y*0=Y) is the amount held in cash and Y is the initial capital.