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 Dec13 awarded Popular Question Sep24 awarded Autobiographer May28 awarded Supporter Feb4 comment Maximization of CARA utility function: unique solution with an unbounded parameter? Sure, it tends asymptotically to zero but in fact we have no maxima. $a^*$ strongly dependes from your pdf assumption, in case of a normal you have that $a^*$ is approximatively the Sharpe ratio. Feb4 awarded Student Feb3 comment Maximization of CARA utility function: unique solution with an unbounded parameter? Probably I looked at it in a wrong perspective. $E(x)$ is a the expected value of the pdf returns. So We have to evaluate this: $E[U(w)] = E[-e^{-\lambda a(x-r_f)}]=\max_a\int_\infty^\infty -e^{-\lambda a(x-rf)}f(x)$ where $f(x)$ is the pdf of the returns. In this case we should have an unique optimal solution for $a$. Feb3 comment Maximization of CARA utility function: unique solution with an unbounded parameter? They assume that the investor has a wealth of $w_0$ and invests $a$ in the risky asset and, therefore, $w-a$ in the risk-free asset. As shown above the maximization of the expected utility depends on $a$ and not on $w_0$. So, many authors set $w_0$ = 0 for convenience, so what is the implications for $a$? I don't get this point, it seems as $a$ exists and is unique without any assumption. If we bounded $a$ the maxima of the expected utility will be on the bound of $a$. Feb3 awarded Editor Feb3 revised Maximization of CARA utility function: unique solution with an unbounded parameter? added 5 characters in body Feb3 asked Maximization of CARA utility function: unique solution with an unbounded parameter?