Exponential-utility risk measures

entropic_risk() is (1 / theta) log E[exp(theta X)], the certainty equivalent of a loss under exponential utility: it rises from the mean (theta -> 0) to the largest value (theta -> Inf), and is mu + theta sigma^2 / 2 for a normal loss. esscher_premium() is E[X exp(h X)] / E[exp(h X)], the mean after tilting probability towards large losses: the mean at h = 0, mu + h sigma^2 for a normal loss. Both treat the draws as equally likely.

Usage

entropic_risk(x, theta)

esscher_premium(x, h)

Arguments

x

A numeric vector of draws, a sampled or a predictive_distribution (measured on its total).

theta

Risk aversion, positive.

h

Esscher parameter.

Value

A single number.

Examples

entropic_risk(c(0, 1), log(2))
#> [1] 0.5849625
esscher_premium(c(0, 1), log(3))
#> [1] 0.75