Exponential utility

For example, suppose that consumption c is a function of labor supply x and a random term ${\displaystyle \epsilon }$: c = c(x) + ${\displaystyle \epsilon }$. Then under exponential utility, expected utility is given by:

${\displaystyle {\text{E}}(u(c))={\text{E}}[1-e^{-a(c(x)+\epsilon )}],}$

where E is the expectation operator. With normally distributed noise, i.e.,

${\displaystyle \varepsilon \sim N(\mu ,\sigma ^{2}),\!}$

E(u(c)) can be calculated easily using the fact that

${\displaystyle {\text{E}}[e^{-a\varepsilon }]=e^{-a\mu +{\frac {a^{2}}{2}}\sigma ^{2}}.}$

Thus

${\displaystyle {\text{E}}(u(c))={\text{E}}[1-e^{-a(c(x)+\epsilon )}]={\text{E}}[1-e^{-ac(x)}e^{-a\varepsilon }]=1-e^{-ac(x)}{\text{E}}[e^{-a\epsilon }]=1-e^{-ac(x)}e^{-a\mu +{\frac {a^{2}}{2}}\sigma ^{2}}.}$

Consider the portfolio allocation problem of maximizing expected exponential utility ${\displaystyle {\text{E}}[-e^{-aW}]}$ of final wealth W subject to

${\displaystyle W=x'r+(W_{0}-x'k)\cdot r_{f}}$

where the prime sign indicates a vector transpose and where ${\displaystyle W_{0}}$ is initial wealth, x is a column vector of quantities placed in the n risky assets, r is a random vector of stochastic returns on the n assets, k is a vector of ones (so ${\displaystyle W_{0}-x'k}$ is the quantity placed in the risk-free asset), and r_f is the known scalar return on the risk-free asset. Suppose further that the stochastic vector r is jointly normally distributed. Then expected utility can be written as

${\displaystyle {\text{E}}[-e^{-aW}]=-{\text{E}}[e^{-a[x'r+(W_{0}-x'k)\cdot r_{f}]}]=-e^{-a[(W_{0}-x'k)r_{f}]}{\text{E}}[e^{-a\cdot x'r}]=-e^{-a[(W_{0}-x'k)r_{f}]}e^{-a\cdot x'\mu +{\frac {a^{2}}{2}}\sigma ^{2}}}$

where ${\displaystyle \mu }$ is the mean vector of the vector r and ${\displaystyle \sigma ^{2}}$ is the variance of final wealth. Maximizing this is equivalent to minimizing

${\displaystyle e^{ar_{f}(x'k)-a\cdot x'\mu +{\frac {a^{2}}{2}}\sigma ^{2}},}$

which in turn is equivalent to maximizing

${\displaystyle x'(\mu -r_{f}\cdot k)-{\frac {a}{2}}\sigma ^{2}.}$

Denoting the covariance matrix of r as V, the variance ${\displaystyle \sigma ^{2}}$ of final wealth can be written as ${\displaystyle x'Vx}$. Thus we wish to maximize the following with respect to the choice vector x of quantities to be placed in the risky assets:

${\displaystyle x'(\mu -r_{f}\cdot k)-{\frac {a}{2}}\cdot x'Vx.}$

This is an easy problem in matrix calculus, and its solution is

${\displaystyle x^{*}={\frac {1}{a}}V^{-1}(\mu -r_{f}\cdot k).}$

From this it can be seen that (1) the holdings x* of the risky assets are unaffected by initial wealth W₀, an unrealistic property, and (2) the holding of each risky asset is smaller the larger is the risk aversion parameter a (as would be intuitively expected). This portfolio example shows the two key features of exponential utility: tractability under joint normality, and lack of realism due to its feature of constant absolute risk aversion.