Constant relative risk aversion utility ($-c{U}^{\prime \prime }(c)/U^{\prime }(c)=R$)

A constant relative risk aversion utility has $-c{U}^{\prime \prime }(c)/U^{\prime }(c)=R>0$. Up to a positive scale and an additive constant, it is $U(c)=c^{1-R}/(1-R)$ for $R\ne 1$ and $U(c)=\log c$ for $R=1$. Power utility gives multiplicative homogeneity of an investment value; logarithmic utility gives additive scaling.

For a self-financing portfolio earning both running reward $e^{-\rho t}{W}^{\gamma }$ and terminal reward $W_T^{\gamma }$, $0<\gamma <1$, with no consumption withdrawals, Hamilton-Jacobi-Bellman equation gives $J(W,t)=A(t)W^{\gamma }$ and the displayed constant risky fraction. The coefficient satisfies $A^{\prime }+\kappa A+e^{-\rho t}=0$, $A(T)=1$, where $\kappa =\gamma r+\gamma (\mu -r{)}^2/(2(1-\gamma ){\sigma }^2)$. Its positive solution is $e^{\kappa (T-t)}+{\int }_t^T{e}^{\kappa (s-t)-\rho s}ds$. Risk aversion is $1-\gamma$; increasing $\gamma$ raises risky investment and can require borrowing.

In a Brownian complete market with constant coefficients and nonsingular $\sigma$, maximize expected CRRA utility of terminal portfolio wealth divided by a geometric stock index. Writing $p=1/R$ and $q=n^{-1}1$, the pointwise marginal condition gives $w_T^{\ast }={\lambda }^{-p}{\zeta }_T^{-p}{J}_T^{1-p}$. Its state-price budget constraint determines ${\lambda }^{-p}=w_0/\mathbb{E}[({\zeta }_T{J}_T{)}^{1-p}]$. Conditional lognormal moments and replication then give the displayed constant dollar fractions. The first term is ordinary optimal risk exposure; the second hedges the random benchmark. The bank holds the remaining fraction, which can be negative when borrowing is allowed.

For zero cash interest, constant excess drift $\mu$, volatility $\sigma$, and both terminal and consumption utility $2\sqrt{x}$, set $A=\sigma {\sigma }^T$ and assume $\mu$ belongs to the image of a linear map $A$. Let $A^{+}$ be the Moore-Penrose inverse and $q={\mu }^T{A}^{+}\mu$. Then $h^{\prime }+qh+1=0$, $h(T)=1$, so $h(t)=e^{q(T-t)}+(e^{q(T-t)}-1)/q$ for $q>0$, and $h(t)=1+T-t$ for $q=0$. The Hamilton-Jacobi-Bellman equation has the displayed solution. Maximizing dollar investment and consumption are ${\theta }_t=2X_t{A}^{+}\mu$ and $C_t=X_t/h(t)$. The resulting positive portfolio wealth is a geometric Brownian motion with a deterministic time-dependent drift, and finite-horizon moment bounds justify equality in verification.

Logarithmic utility on positive portfolio wealth has derivative $1/x$ and relative risk aversion one. Its utility conjugate is $V(y)=-\log y-1$, since the maximizing wealth is $1/y$. It satisfies the Inada conditions.

Let a portfolio hold $x\in [0,T]$ in a risk-free asset with net return $r$ and $T-x$ in an asset returning $g$ or $b$, with probabilities $p$ and $1-p$. Assume $T>0$, $-1<b<r<g$, $0<p<1$, and $pg+(1-p)b>r$. Its state wealths are $w_g=T(1+g)-(g-r)x$ and $w_b=T(1+b)+(r-b)x$. The expected utility $p\log w_g+(1-p)\log w_b$ has strictly negative second derivative. Setting its first derivative to zero gives the displayed unconstrained root; the derivative at $x=T$ is negative, so only the lower constraint can bind. Thus the displayed formula is the unique optimum. All equity is optimal exactly when $p\ge (r-b)(1+g)/[(g-b)(1+r)]$. The positivity of both state wealths is part of the logarithmic utility domain.

For a stock with observed dynamics $d{S}_t/S_t=\sigma d{\hat{W}}_t+\sigma m_{t}dt$ and constant riskless interest rate $r$, the finite-horizon pricing density for Gaussian drift learning with $k=r/\sigma$ gives ${\zeta }_t=e^{-rt}{D}_t$. The terminal logarithmic utility condition $1/w_T^{\ast }=\lambda {\zeta }_T$ and the state-price budget constraint give $\lambda =1/w_0$. Replication then gives $w_t^{\ast }=w_0/{\zeta }_t$ at every time. The Itô formula for $1/\zeta$ makes the optimal dollar stock fraction $(m_t-r/\sigma )/\sigma$. Its finite expected log return follows from finite posterior second moments on finite horizons.

A fee proportional to terminal portfolio wealth but reduced by accumulated portfolio quadratic variation changes the optimal risk exposure of a manager with constant relative risk aversion utility. Write $A=\sigma {\sigma }^T$, $b=\mu -r1$ and assume $A$ is invertible. The fee process has drift $r+b^T\pi -\epsilon {\pi }^{T}A\pi /2$ and volatility ${\sigma }^T\pi$. Its Hamilton-Jacobi-Bellman equation therefore maximizes $b^T\pi -(R+\epsilon ){\pi }^{T}A\pi /2$, giving ${\pi }^{\ast }=A^{-1}b/(R+\epsilon )$. The fee increases effective relative risk aversion from $R$ to $R+\epsilon$, independently of its positive scale $a$.

There is also a direct global bound. Put $p=1-R$, $v^{\ast }={\sigma }^T{\pi }^{\ast }$, and tilt probability by $\exp (p{v}^{\ast }\cdot W_T-p^2|v^{\ast }{|}^{2}T/2)$. Under this probability let $Z$ be the stochastic exponential of $(1+\epsilon )\int ({\sigma }^T\pi -v^{\ast })\cdot d{W}^{\ast }$. The terminal fee ratio satisfies $(y_T/y_T^{\ast }{)}^p=Z_T^{p/(1+\epsilon )}$. A nonnegative local martingale has expectation at most one. The Jensen inequality, applied to a concave positive power when $p>0$ and a convex negative power when $p<0$, proves that the constant exposure maximizes expected CRRA utility. Effective risk aversion one uses logarithmic utility.

For an interior optimal payoff in a complete two-state market under CRRA utility, $W_u^{-R}/W_d^{-R}=Z_u/Z_d$. If the two positive payoffs differ, taking logarithms gives the displayed coefficient. The multiplier can then be recovered from either state. A positive coefficient requires the price-density and payoff orders to be compatible with decreasing marginal utility.