Expected utility hypothesis

The expected utility hypothesis ranks uncertain outcomes by the expected value of a utility function. When these preference assumptions hold, expected utility maximization selects an admissible outcome with greatest expected utility.

Expected utility maximization chooses an admissible random payoff $X$ to maximize $\mathbb{E}[U(X)]$.

If a positive terminal state-price density has essential infimum zero, let $A_n=\{{\zeta }_T<1/n\}$ and $X_n=w_0{1}_{A_n}/\mathbb{E}[{\zeta }_T{1}_{A_n}]$. Whenever $\mathbb{P}(A_n)>0$, these claims cost $w_0$ and satisfy $\mathbb{E}{X}_n>n{w}_0$. In a Brownian complete market with constant nonzero market price of risk, ${\zeta }_T$ has a lognormal distribution and all these events have positive probability. Thus bounded market coefficients and an increasing concave utility do not alone ensure a finite attained optimum: the linear utility $U(x)=x$ already fails. Usual inverse-marginal-utility formulas require further utility and integrability hypotheses.

For an increasing strictly concave utility function satisfying the Inada conditions, put $I=(U^{\prime }{)}^{-1}$ and let $\zeta$ be the positive terminal state-price density. In a finite complete market, select $y>0$ so that $\mathbb{E}[\zeta I(y\zeta )]$ equals initial wealth. The displayed payoff is optimal under that budget: the supporting-line inequality for utility has marginal slope $y\zeta$, and its expected linear error vanishes on the budget constraint. Market completeness supplies its replicating strategy.

In a complete $n$-period binomial market with riskless gross return $R$, pricing density $Z$ and utility function $U(x)=\gamma x^{1/\gamma }$, put $\eta =\gamma /(\gamma -1)$. The optimal terminal payoff is $X^{\ast }=R^n{w}_0{Z}^{-\eta }/\mathbb{E}[Z^{1-\eta }]$. Its marginal utility is proportional to $Z$; the supporting-line inequality for the strictly concave utility function proves optimality using the pricing budget constraint.

A holding-dependent cost proportional to the absolute position size. In a one-period gain objective it changes the sample payoff to $\theta X-\epsilon |\theta |$, giving gain parameter $X-\epsilon$ for positive holdings and $X+\epsilon$ for negative holdings. The objective is concave but can have a kink at zero.

If a negative differentiable concave utility function satisfies $\mathbb{E}U(\theta X)>-\infty$ on an open interval, outer secants at four points bound the sample derivative on a smaller compact interval. Negativity makes the endpoint values absolutely integrable. The dominated convergence theorem then gives $d\mathbb{E}U(\theta X)/d\theta =\mathbb{E}[X{U}^{\prime }(\theta X)]$ and continuous derivative when $U^{\prime }$ is continuous. This is a justification of differentiation, rather than an assumption of integrability of marginal utility.

A terminal portfolio wealth floor is a prescribed lower bound, possibly a random variable, on terminal portfolio portfolio wealth. In a complete market, its state-price density cost cannot exceed initial portfolio wealth. Nonnegative admissibility replaces a possibly negative floor by its positive part.

For a nonnegative terminal wealth floor whose state-price density cost is strictly below the available budget, pointwise maximization of $U(y)-\lambda {\zeta }_{T}y$ over $y\ge \xi$ gives the displayed payoff. The positive multiplier exhausts the state-price budget constraint. This requires the usual utility and integrability hypotheses ensuring that the multiplier exists and the payoff is replicable.

Finite lower and upper utility bounds together with a positive discount rate bound the absolute time integral by $\max (|L|,|R|)/b$. This justifies expectation and integration in marginal-utility verification. Bounded utility alone on an undiscounted infinite horizon does not prevent both positive and negative parts from having infinite integral.

An incentive fee above a hurdle changes a manager’s utility function into $F(y)=U(\epsilon y+\alpha (y-w_0{)}_{+})$. For constant absolute risk aversion utility, each branch is strictly concave, but the marginal reward jumps upward at the hurdle. Thus the effective terminal-wealth utility is not concave, and concavification of incentive utility can reveal optimal risk-taking lotteries.

Concavification replaces a nonconcave incentive utility function by its least concave majorant. In a complete market with an atomless state-price density, the optimizing payoff may avoid all intervals where the majorant strictly exceeds the original utility, making the relaxed optimum exactly attainable. At a pricing atom corresponding to a linear segment, a lottery between contact points can attain the same value at unchanged cost.

With zero interest and zero risk premium, a manager whose current wealth lies between common-tangent contacts $\ell ,h$ can improve expected utility maximization through a fair lottery paying those two values. The probability of $h$ is $(w_0-\ell )/(h-\ell )$. In a Brownian filtration, replicating a bounded terminal lottery gives a nonnegative wealth martingale throughout.

For $F(y)=-e^{-ay}$ below hurdle $w_0$ and $F(y)=-e^{(b-a)w_0-by}$ above it, with $b>a>0$, an interior common tangent touches at $\ell =w_0-[b/a-1-\log (b/a)]/(b-a)$ and $h=w_0+[\log (b/a)-1+a/b]/(b-a)$. The slope is $a{e}^{-a\ell }$ and $h-\ell =1/a-1/b$. The formula requires $\ell \ge 0$; a smaller hurdle gives a different endpoint-at-zero concavification.

An investment-consumption problem chooses portfolio holdings and consumption to maximize discounted expected utility maximization subject to a self-financing portfolio wealth equation and admissibility constraints. A Hamilton-Jacobi-Bellman equation or utility duality with martingale deflators characterizes the optimum when the value is finite.

For constant relative risk aversion utility with parameter $\gamma >0$, terminal weight $\eta >0$ and utility discount rate $\delta$, in a Black-Scholes model put $\lambda =(\mu -r)/\sigma$, $k=[\delta +(\gamma -1)(r+{\lambda }^2/(2\gamma ))]/\gamma$ and $b(\tau )={\int }_0^{\tau }{e}^{-ks}ds+{\eta }^{1/\gamma }{e}^{-k\tau }$. The displayed optimal consumption and dollar stock position follow by inverse marginal utility and the state-price budget constraint. For $\gamma \ne 1$ the value is $b(T-t{)}^{\gamma }{W}_t^{1-\gamma }/(1-\gamma )$ with discount measured from $t$. Finite horizons do not require $k>0$. The concave supporting-tangent inequality and replication of the optimal budget prove global optimality, rather than only a formal differential-equation solution.

For terminal logarithmic utility $a\log V_T$ and running logarithmic utility $b\log c_t$, with $a,b,w_0>0$, constant Black-Scholes model coefficients give $\lambda =(a+bT)/w_0$ and $V_t=[a+b(T-t)]/(\lambda {\xi }_t)$, where $\xi$ is the state-price density. The optimal terminal wealth and consumption are $a/(\lambda {\xi }_T)$ and $b/(\lambda {\xi }_t)$. Their discounted budget adds to $(a+bT)/\lambda =w_0$. The concave supporting-tangent inequality, integrated over time, proves optimality; the Itô formula gives the displayed dollar stock holding. The time-dependent consumption-to-wealth ratio reflects both remaining consumption opportunities and the terminal utility weight.

A fixed loan principal contributes a constant interest outflow $h$. If the objective stops when available portfolio wealth reaches zero, the value function has an absorbing boundary $V(0)=0$. This differs from requiring the portfolio to finance the debt service forever.

For the wealth-variable Legendre dual $J(z)={\sup }_{w\ge 0}[V(w)-zw]$, a finite slope $z_{\ast }=V^{\prime }(0+)$ at an absorbing zero-wealth boundary makes $J$ identically zero for $z\ge z_{\ast }$. Matching gives $J(z_{\ast })=J^{\prime }(z_{\ast })=0$. It does not generally give $J^{\prime \prime }(z_{\ast }^{-})=0$: killing at ruin permits the dual curvature to jump.

When the market price of risk is constant and nonzero spot volatility gives access to the driving Brownian motion, changes in volatility only rescale the dollar holding needed for fixed Brownian portfolio exposures. With unrestricted holdings and constant interest and discount rates, the CRRA utility value is consequently independent of the factor state. Zero volatility requires separate treatment.

A consumption satisfaction stock smooths past consumption by ${\xi }_t=e^{-\lambda t}{\xi }_0+{\int }_0^t{e}^{-\lambda (t-s)}{c}_{s}ds$. Unlike multiplicative habit utility, the running utility function may depend on this stock alone. A unit of present consumption raises future consumption satisfaction while costing one unit of financial portfolio wealth.

In singular stochastic control, an unrestricted consumption rate can approximate instantaneous transfers from portfolio wealth into a consumption satisfaction stock. In the relaxed problem a nondecreasing control $C$ records these transfers. A transfer $\Delta C$ decreases portfolio wealth and increases consumption satisfaction by the same amount. Rate controls need not attain the supremum when the relaxed optimum has a jump.

If consumption contributes $c(V_{\xi }-V_w)$ to the Hamilton-Jacobi-Bellman equation and has no finite upper bound, finite value requires $V_{\xi }\le V_w$. Strict inequality makes zero consumption optimal locally. Equality describes an active transfer boundary, often reached through singular consumption control.

For CRRA utility of a consumption satisfaction stock, jointly scaling initial portfolio wealth, consumption satisfaction and controls gives homogeneity of degree $1-R$. The value function therefore depends on one dimensionless ratio after factoring out ${\xi }^{1-R}$. The effective consumption coefficient becomes ${\xi }^{-R}[(1-R)v-(x+1)v^{\prime }]$.

A discounted-value transversality condition rules out residual value at infinity in an infinite-horizon Hamilton-Jacobi-Bellman equation verification. Under appropriate integrability, admissibility, and localization assumptions, $\mathbb{E}[e^{-\rho T}V(w_T,s_T)]\to 0$ identifies the economic value among formal differential-equation solutions.

When asset correlation varies with a traded market index, the index is both an asset and a state variable. Writing $m=\log M$ and optimizing Brownian portfolio exposures accounts for the cross derivative between wealth and $m$. Power homogeneity reduces the Hamilton-Jacobi-Bellman equation to an ordinary differential equation in $m$.

For a complete-market investment-consumption problem with constant relative risk aversion utility, the nonlinear wealth-homogeneity coefficient equation may contain $f^{\prime 2}/f$. Writing $f=g^R$ cancels this gradient square against the one from $f^{\prime \prime }$. In the index-driven correlation model the result is ${\sigma }_0^2{g}^{\prime \prime }/2+B{g}^{\prime }-\delta (m)g+1=0$, a linear differential equation; the positive economic solution gives consumption $w/g$.

Intertemporal hedging demand adjusts the myopic risky position when investment opportunities depend on a stochastic state. Shared noise between wealth and that state contributes a cross derivative to the Hamilton-Jacobi-Bellman equation. In a complete diffusion market the optimal exposures combine the myopic market price of risk term and the value-gradient hedge term.

With the binary Brownian drift filter, a stock of observed excess drift $b(x)=\mu -r+\sigma a\tanh (ax)$ is driven by the same Brownian motion as its information state $X$. The Hamilton-Jacobi-Bellman equation for portfolio wealth $w$, dollar stock holding $\theta$ and value $J(w,x)$ includes $\sigma \theta J_{wx}$. Consequently ${\theta }^{\ast }=-[b(x)J_w+\sigma J_{wx}]/({\sigma }^2{J}_{ww})$. For CRRA utility, homogeneity gives $J=K(x)w^{1-R}/(1-R)$ and hence the displayed stock fraction. The second term is intertemporal hedging demand; replacing the unknown drift by its posterior mean while omitting this term does not generally give the optimal policy.

A high-water mark investment tax charges increments of historical maximum wealth ${\bar{w}}_t=\max ({\bar{w}}_0,{\sup }_{s\le t}{w}_s)$ rather than every positive instantaneous return. For the convention charging $\tau d\bar{w}$, the high-water mark tax boundary condition is $V_{\bar{w}}=\tau V_w$ when the optimal policy raises the maximum. At sufficiently high tax, a wealth-cap investment boundary can make further maximum increases suboptimal.

A wealth-cap investment boundary prevents wealth from exceeding a fixed historical maximum. In the wealth-variable Legendre dual for wealth normalized by that maximum, $J^{\prime }(z_{\ast })=-1$ and vanishing optimal portfolio volatility gives $J^{\prime \prime }(z_{\ast })=0$. For power utility and a high-water tax, the smooth tax-paying solution has boundary dual curvature $(1-\alpha \tau )/(R{z}_{\ast })$, so it cannot be admissible when $\alpha \tau >1$.

At an active maximum-raising boundary in high-water mark investment taxation, the finite-variation part of the Itô formula is $(V_{\bar{w}}-\tau V_w)d\bar{w}$, so smooth optimality sets this coefficient to zero. If the maximizing policy avoids increasing the maximum, the corresponding inequality $V_{\bar{w}}-\tau V_w\le 0$ is compatible with a wealth-cap investment boundary instead.

For a constant-coefficient single-asset investment-consumption problem with constant relative risk aversion utility, put $\kappa =(\mu -r)/\sigma$ and $\delta =[\rho -(1-R)(r+{\kappa }^2/(2R))]/R$. When $\delta >0$, the infinite-horizon value is ${\delta }^{-R}{w}^{1-R}/(1-R)$ and the optimal controls are $c^{\ast }=\delta w$ and ${\theta }^{\ast }=(\mu -r)w/(R{\sigma }^2)$. The case $R=1$ uses logarithmic utility.

For an observed finite-state continuous-time Markov chain with rate matrix $Q$, constant bank rate $r$, and state-dependent stock drifts ${\mu }_i$ and positive volatilities ${\sigma }_i$, constant relative risk aversion utility gives $V_i(w)=A_i{w}^{1-R}/(1-R)$. Set ${\kappa }_i=({\mu }_i-r)/{\sigma }_i$ and $d_i=\rho -(1-R)(r+{\kappa }_i^2/(2R))$. The Hamilton-Jacobi-Bellman equation reduces to the displayed nonlinear algebraic system with $A_i>0$. Its feedback controls are $c_i=A_i^{-1/R}w$ and ${\theta }_i=({\mu }_i-r)w/(R{\sigma }_i^2)$.

A damped Newton method in the logarithmic variables $A_i=e^{x_i}$ preserves positivity. Solving these equations produces a candidate value; admissibility, stochastic-integral bounds and the investment value transversality condition still need verification.

Consider irreversible optimal stopping of employment paying income $\epsilon >0$ and imposing disutility $\lambda >0$. Assume $r,\rho >0$, nonzero market price of risk $\kappa$, a positive Merton consumption constant, and solvency allowing borrowing against perpetual income up to $\epsilon /r$ before retirement. Let $m<0$ solve ${\kappa }^{2}m(m-1)/2+(\rho -r)m-\rho =0$. The retirement threshold is the displayed wealth level.

To derive it, apply the wealth-variable Legendre dual. The working dual equals the retired dual plus $h(z)=\epsilon z/r-\lambda /\rho +D{z}^m$. Excluding the growing homogeneous solution enforces the solvency asymptotic. Value matching and smooth fit at the retirement marginal value $z_{\ast }$ give $h(z_{\ast })=h^{\prime }(z_{\ast })=0$, hence $z_{\ast }=(\lambda r/(\epsilon \rho ))m/(m-1)$ and $D=-\epsilon z_{\ast }^{1-m}/(rm)>0$. The retired inverse marginal value gives $\overline{w}={\gamma }_M^{-1}{z}_{\ast }^{-1/R}$.

The root equation implies $z_{\ast }/(\lambda /\epsilon )=1+{\kappa }^{2}m/(2\rho )<1$. Thus the boundary exceeds the instantaneous gain-zero level ${\gamma }_M^{-1}(\epsilon /\lambda {)}^{1/R}$. A tighter borrowing constraint changes this free-boundary problem. Smooth fit matches the first derivative; matching the second derivative would incorrectly remove the income option.

In the infinite-horizon Merton consumption-investment problem with constant relative risk aversion utility, this is the optimal ratio of consumption to wealth. A finite value requires ${\gamma }_M>0$. The state-price budget constraint then fixes consumption's multiplier because $\mathbb{E}[{\xi }_t(y{e}^{\rho t}{\xi }_t{)}^{-1/R}]=y^{-1/R}{e}^{-{\gamma }_{M}t}$, whose time integral is $y^{-1/R}/{\gamma }_M$. When this integral diverges, the value is $+\infty$ for $0<R<1$ and $-\infty$ for $R>1$.

For $0<R<1$, common constant interest and volatility, and observed excess drift bounded in absolute value by $b_{+}>0$, let $V_{+}$ be the finite Merton consumption-investment problem value for excess drift $b_{+}$. In its Hamilton-Jacobi-Bellman equation, optimizing the stock holding contributes $-b^2(V_{+}^{\prime }{)}^2/(2{\sigma }^2{V}_{+}^{\prime \prime })$. Since $V_{+}^{\prime \prime }<0$, this contribution increases with $b^2$. Thus every admissible policy makes accumulated discounted CRRA utility plus discounted $V_{+}$ a nonnegative local supermartingale, hence a supermartingale. Dropping its nonnegative terminal term and using the monotone convergence theorem proves the value bound. The condition $[\rho -(1-R)(r+b_{+}^2/(2R{\sigma }^2))]/R>0$ is essential for the finite supermartingale construction.

Consider an investment-consumption problem with an independent one-way rate change of intensity $\lambda$ from $r_1$ to $r_0$. For a stock of constant drift $\mu$ and positive volatility $\sigma$, set $B_j=r_j+(\mu -r_j{)}^2/(2R{\sigma }^2)$, ${\gamma }_0=[\rho -(1-R)B_0]/R$, and $D_1=\rho +\lambda -(1-R)B_1$. In the finite-value regime ${\gamma }_0,D_1>0$, the prechange consumption coefficient is the unique positive root displayed above. The left side increases continuously from zero to infinity. Before the change, the value is ${\gamma }_1^{-R}{w}^{1-R}/(1-R)$, the stock fraction is $(\mu -r_1)/(R{\sigma }^2)$, and consumption is ${\gamma }_{1}w$. At the change these become the ordinary Merton consumption-investment problem controls for $r_0$. The jump term in the Hamilton-Jacobi-Bellman equation is $\lambda (V_0-V_1)$; knowing only the marginal law of the change time does not justify that term without a conditional intensity assumption.

A deterministic wealth level $c$ satisfying $U(c)=\mathbb{E}U(W)$. For normal wealth with exponential utility $U(x)=-e^{-\gamma x}$ it is $c=\mathbb{E}W-\gamma \mathrm{Var}(W)/2$.

For $X=\mu +\sigma Z$ with $Z$ having the standard normal distribution and an increasing concave utility function $U$, define $f(\mu ,\sigma )=\mathbb{E}[U(X)]$. Then $f_{\mu }=\mathbb{E}[U^{\prime }(X)]>0$ and Gaussian integration by parts gives $f_{\sigma }=\sigma \mathbb{E}[U^{\prime \prime }(X)]\le 0$. Moreoveris negative semidefinite, so $f$ is jointly concave in $\mu$ and $\sigma$.

A utility indifference price for a claim is the cash amount at which optimally buying the claim leaves the investor with the same maximal expected utility as trading without it.

For Gaussian income of mean $a$, standard deviation $b$ and tradable correlation $\rho$, exponential utility makes the maximum acceptable per-period fee equal to the mean minus the correlated-income hedge cost and the residual variance penalty. Compare optimized utilities with and without the contract; a fee above this level makes the contract strictly inferior. At perfect correlation the residual penalty vanishes, leaving only the mean and hedge-cost terms.

Let $\mathcal{X}$ be a vector space of random variables and supposehas an optimizer for every $y$. If $U$ is increasing and concave, then $F$ is increasing and concave. For concavity, combine optimizers at two values using the same convex coefficient as the values themselves.

If $U$ is concave and $\mathbb{E}Z=0$, thenis concave in $s$. By Jensen inequality, $G(s)\le G(0)$; concavity then makes $G$ nonincreasing on $[0,\infty )$.

The concave conjugate $\hat{u}(y)={\sup }_{x>0}\{u(x)-xy\}$ gives the weak-duality bound $\mathbb{E}u(X)\le \mathbb{E}\hat{u}(Y)+X_0{Y}_0$. Equality holds when $u^{\prime }(X)=Y$ and the payoff exhausts the deflated budget.

In a complete market with a unit cash asset and normalized strictly positive state-price density $Z$, the optimal positive terminal portfolio wealth is $x/Z$ whenever it is replicable in the admissible class. The state-price budget constraint is an equality: $\mathbb{E}[Z(x/Z)]=x$. The utility conjugate inequality gives $\mathbb{E}\log H\le \log x-\mathbb{E}\log Z$ for every affordable positive claim. This value can be infinite, because $\mathbb{E}(\log Z{)}^{+}<\infty$ while $\mathbb{E}(\log Z{)}^{-}$ need not be finite. Even completeness for bounded claims suffices to approximate the value: set $b_n=\mathbb{E}[Z/(Z\vee n^{-1})]$ and $H_n=x/[b_n(Z\vee n^{-1})]$. Each claim is bounded and costs $x$, while its expected logarithmic utility tends to the displayed value by monotone convergence after an integrable lower bound is subtracted.

For a negative, strictly increasing, concave differentiable utility with finite expected utility at every holding, a gain having both signs makes the objective coercive at both ends. At a finite optimum its derivative is zero. Secant domination for expected utility derivatives gives integrability of $|X|U^{\prime }({\theta }_{\ast }X)$; boundedness of $U^{\prime }$ on the remaining compact set gives integrability of the positive marginal utility itself. Its normalization has expectation one and prices the gain at zero.

In a one-period utility problem with gross riskless return $R$, the marginal utility price is the infinitesimal claim price leaving optimized expected utility unchanged. If optimal terminal wealth is $W^{\ast }$, the pricing density is $Z=U^{\prime }(W^{\ast })/(R\mathbb{E}{U}^{\prime }(W^{\ast }))$, and $p_H=\mathbb{E}[ZH]$. The denominator is the marginal utility of initial wealth. Under the standard first-order optimality conditions this density prices the traded assets and gives the state-price density associated with the investor.

At a nonzero optimum, the derivative of expected utility of a proportional transaction cost payoff is zero. Normalized marginal utility therefore prices $X$ at $\epsilon$ for a positive optimum or $-\epsilon$ for a negative optimum. At a zero optimum the one-sided derivative inequalities put $\mathbb{E}X$ inside that interval, so density one works. The competing one-sided-payoff alternative prevents the coercive-maximizer argument from being assumed when a net gain has only one sign.

For a positive state-price density $Z$ pricing every attainable payoff, a feasible positive terminal payoff $W^{\ast }$ is optimal at its budget if its marginal utility is $yZ$ for some scalar $y>0$. The pointwise conjugate inequality gives $\mathbb{E}U(W)\le \mathbb{E}\hat{U}(yZ)+yx$ for every competitor; the marginal-utility identity makes equality hold for $W^{\ast }$. Finite states make integrability straightforward. In a complete market an interior optimum also necessarily has this proportionality, by variations preserving the state-price budget.

For concave increasing utility, the concave supporting-tangent inequality gives $e^{-bt}[U({\hat{c}}_t)-U(c_t)]\le Y_t({\hat{c}}_t-c_t)$. If the proposed consumption exhausts the state-price budget and every competitor has no larger budget, integration proves optimality. The objective must be well defined; a positive discount rate and finite utility bounds are a convenient sufficient condition.

For increasing strictly concave wealth value $v$, the dual $J(z)={\sup }_x[v(x)-xz]$ is convex. At an interior optimum $z=v^{\prime }(x)$, $J^{\prime }=-x$, $J^{\prime \prime }=-1/v^{\prime \prime }>0$, and $v=J-z{J}^{\prime }$. This sign convention is the negative of the concave Legendre dual. It can turn the optimized portfolio term of a Hamilton-Jacobi-Bellman equation into a linear second derivative.