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Constant relative risk aversion coefficient means that $\alpha =-W{U}^{\prime \prime }(W)/U^{\prime }(W)$ does not depend on wealth, for $W>0$ and $U^{\prime }(W)>0$. The defining differential equation integrates asA second integration therefore givesThe additive constant and positive scale do not affect expected utility preferences. Since $U^{\prime \prime }(W)=-\alpha c{W}^{-\alpha -1}$, positive $\alpha$ gives risk aversion, zero gives risk neutrality, and negative $\alpha$ gives risk seeking. These are respectively concave, affine and convex utility functions. The usual CRRA utility class is the positive-$\alpha$ case.

The power-wealth investment with running utility objective is homogeneous of degree $\gamma$ in wealth: scaling both wealth and positions by $a>0$ scales the reward by $a^{\gamma }$. Accordingly set $J(w,t)=A(t)w^{\gamma }$. ThenFor completeness the remaining equation is $A^{\prime }+\kappa A+e^{-\rho t}=0$, $A(T)=1$, whereThis confirms $J_{ww}<0$. The constant optimal risky fraction also makes wealth a positive geometric Brownian motion when started positive.

Financially the relative risk aversion coefficient is $1-\gamma$. Increasing $\gamma$ reduces risk aversion, raises the risky fraction, and decreases the bond fraction: its derivative is $-(\mu -r)/[{\sigma }^2(1-\gamma {)}^2]<0$. If the risky fraction is below one, some wealth is lent; at one, none is invested in bonds; above one, $B^{\ast }<0$ represents borrowing to leverage the risky position. As $\gamma \uparrow 1$ the risky fraction becomes unbounded, consistent with the absence of a finite unconstrained optimum for linear utility and positive excess drift. As $\gamma \downarrow 0$ the limiting fraction is $(\mu -r)/{\sigma }^2$, the logarithmic-utility allocation obtained by the appropriately rescaled power-utility limit; the literal objective at $\gamma =0$ would be constant and would not select a unique strategy.

An increasing convex utility function prefers a nondegenerate lottery to its sure mean by the Jensen inequality. Its relative risk aversion coefficient is negative when its second derivative is strictly positive. This differs from risk neutrality, which corresponds to affine utility.