In economics and finance, exponential utility is a specific form of the utility function, used in some contexts because of its convenience when risk (sometimes referred to as uncertainty) is present, in which case expected utility is maximized. Formally, exponential utility is given by:
formula_1
formula_2 is a variable that the economic decision-maker prefers more of, such as consumption, and formula_3 is a constant that represents the degree of risk preference (formula_4 for risk aversion, formula_5 for risk-neutrality, or formula_6 for risk-seeking). In situations where only risk aversion is allowed, the formula is often simplified to formula_7.
Note that the additive term 1 in the above function is mathematically irrelevant and is (sometimes) included only for the aesthetic feature that it keeps the range of the function between zero and one over the domain of non-negative values for "c". The reason for its irrelevance is that maximizing the expected value of utility formula_8 gives the same result for the choice variable as does maximizing the expected value of formula_9; since expected values of utility (as opposed to the utility function itself) are interpreted ordinally instead of cardinally, the range and sign of the expected utility values are of no significance.
The exponential utility function is a special case of the hyperbolic absolute risk aversion utility functions.
Risk aversion characteristic.
Exponential utility implies constant absolute risk aversion (CARA), with coefficient of absolute risk aversion equal to a constant:
formula_10
In the standard model of one risky asset and one risk-free asset, for example, this feature implies that the optimal holding of the risky asset is independent of the level of initial wealth; thus on the margin any additional wealth would be allocated totally to additional holdings of the risk-free asset. This feature explains why the exponential utility function is considered unrealistic.
Mathematical tractability.
Though isoelastic utility, exhibiting constant "relative" risk aversion (CRRA), is considered more plausible (as are other utility functions exhibiting decreasing absolute risk aversion), exponential utility is particularly convenient for many calculations.
Consumption example.
For example, suppose that consumption "c" is a function of labor supply "x" and a random term formula_11: "c" = "c"("x") + formula_11. Then under exponential utility, expected utility is given by:
formula_13
where E is the expectation operator. With normally distributed noise, i.e.,
formula_14
E("u"("c")) can be calculated easily using the fact that
formula_15
Thus
formula_16
Multi-asset portfolio example.
Consider the portfolio allocation problem of maximizing expected exponential utility formula_17 of final wealth "W" subject to
formula_18
where the prime sign indicates a vector transpose and where formula_19 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 formula_20 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
formula_21
where formula_22 is the mean vector of the vector "r" and formula_23 is the variance of final wealth. Maximizing this is equivalent to minimizing
formula_24
which in turn is equivalent to maximizing
formula_25
Denoting the covariance matrix of "r" as "V", the variance formula_26 of final wealth can be written as formula_27. Thus we wish to maximize the following with respect to the choice vector "x" of quantities to be placed in the risky assets:
formula_28
This is an easy problem in matrix calculus, and its solution is
formula_29
From this it can be seen that (1) the holdings "x"* of the risky assets are unaffected by initial wealth "W"0, 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.
References.
 ;templatestyles src="Reflist/styles.css" / ;