Constant absolute risk aversion utility ($U(x)=-e^{-\gamma x}$)

Constant absolute risk aversion utility has $-U^{\prime \prime }(x)/U^{\prime }(x)=\gamma >0$. For Gaussian wealth $W$, maximizing $\mathbb{E}[-e^{-\gamma W}]$ is equivalent to maximizing

For unrestricted final portfolio wealth in an $n$-period binomial market, write $R$ for the bank growth factor, $a>0$ for absolute risk aversion, and $Z=dQ/dP$ for the binomial-market probability density. The displayed optimizer has the state-price budget constraint ${\mathbb{E}}_Q{H}^{\ast }=R^n{w}_0$. Its exponential utility derivative satisfies $e^{-a{H}^{\ast }}=\lambda Z$ for a constant $\lambda >0$. The concave supporting-tangent inequality bounds any competing expected utility by this value, because the expected linear error is $\lambda {\mathbb{E}}_Q(H-H^{\ast })=0$. Strict concavity proves uniqueness, and market completeness supplies replication. The formula is altered by a constraint requiring nonnegative final wealth. With physical up probability $p$ and risk-neutral probability $q$, ${\mathbb{E}}_Q\log Z=n[q\log (q/p)+(1-q)\log ((1-q)/(1-p))]$.

For exponential utility $(1-e^{-ax})/a$ on unrestricted terminal portfolio wealth in the Black-Scholes model, the optimal dollar holding in the stock is the displayed deterministic amount. Set $\vartheta =(\mu -\rho )/\sigma$ and let $W^Q=W+\vartheta t$ be the Brownian motion under the risk-neutral measure. The terminal optimizer is $X^{\ast }=w_0{e}^{\rho T}+\vartheta W_T^Q/a$. Its marginal utility is proportional to the state-price density, and the concave supporting-tangent inequality proves global optimality under the state-price budget constraint. The expected utility at the optimum is $(1-\exp (-a{w}_0{e}^{\rho T}-{\vartheta }^{2}T/2))/a$. A nonnegative-wealth constraint changes this optimizer.

With positive welfare weights $a_j$, maximize ${\sum }_j{a}_j{\beta }_j^t[-e^{-{\gamma }_j{c}_j}/{\gamma }_j]$ subject to ${\sum }_j{c}_j=d$. Interior first-order conditions give $a_j{\beta }_j^t{e}^{-{\gamma }_j{c}_j}=\nu$. Put $\Gamma =({\sum }_j{\gamma }_j^{-1}{)}^{-1}$, $\log \bar{\beta }=\Gamma {\sum }_j{\gamma }_j^{-1}\log {\beta }_j$ and $\log A=\Gamma {\sum }_j{\gamma }_j^{-1}\log a_j$. ThenIf consumption must be nonnegative, the interior formula is replaced by $c_j=\max \{0,(\log a_j+t\log {\beta }_j-\log \nu )/{\gamma }_j\}$, with $\nu$ determined by resource clearing. These are Pareto allocations; realizing them as a competitive equilibrium with one productive asset requires feasible financing. For common ${\beta }_j=\beta$ and initial ownership $\Gamma /{\gamma }_j$, constant holdings and $C_t^j=(\Gamma /{\gamma }_j)d_t$ give such an equilibrium whenever fundamental prices and the utility sums are finite.

For Gaussian terminal prices with positive definite covariance $V$, set $M=(\gamma V{)}^{-1}$ and $\eta =(S_0^{T}M\mu -w_0)/(S_0^{T}M{S}_0)$, assuming $S_0\ne 0$. The normal moment-generating function turns expected exponential utility into a strictly concave mean-minus-variance objective. Its free optimum has cash $S_0^{T}M{S}_0(1+r-\eta )$. If this is positive, keep the free optimum; otherwise the cash constraint binds and the multiplier becomes $\eta$. Thus zero optimal cash is equivalent to $\eta \ge 1+r$, including equality.