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In economics and finance, exponential utility refers to 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:
Contents
- Risk aversion characteristic
- Mathematical tractability
- Consumption example
- Multi asset portfolio example
- References
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
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, with coefficient of absolute risk aversion equal to a constant:
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, 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
where E is the expectation operator. With normally distributed noise, i.e.,
E(u(c)) can be calculated easily using the fact that
Thus
Multi-asset portfolio example
Consider the portfolio allocation problem of maximizing expected exponential utility
where the prime sign indicates a vector transpose and where
where
which in turn is equivalent to maximizing
Denoting the covariance matrix of r as V, the variance
This is an easy problem in matrix calculus, and its solution is
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.