Expected Utility Calculator — Certainty Equivalent & Risk Premium
Quantify how much a risky lottery is worth to you by computing expected utility E[U] and the certainty equivalent CE — the guaranteed wealth you would accept instead of the gamble. Choose logarithmic, square-root, linear, or CRRA power utility to model different attitudes to risk.
Utility function
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Guaranteed wealth you would accept instead of taking the risky lottery
- 1
Expected wealth E[W]
25% × 140,000 + 50% × 108,000 + 25% × 70,000 = 106,500 - 2
Expected utility E[U]
25% × 11.8494 + 50% × 11.5899 + 25% × 11.1563 = 11.5464Each scenario utility is weighted by its probability. - 3
Certainty Equivalent CE
exp(11.5464) = 103,399.49The sure wealth that yields the same utility as the lottery — below E[W] for risk-averse utility.
How does this calculator work?
E[U] = Σ pᵢ U(W₀ + gᵢ) where gains/losses in each scenario add to initial wealth W₀. The certainty equivalent CE = U⁻¹(E[U]) is the guaranteed wealth equally preferred to the lottery. Risk premium E[W] − CE shows how much the decision-maker is willing to pay to eliminate uncertainty.
Formula
How this is calculated
Expected utility theory (von Neumann & Morgenstern, 1944) provides a mathematically consistent framework for ranking risky choices. Instead of maximising expected wealth E[W] = Σ pᵢ Wᵢ, a rational decision-maker with a concave utility function U maximises expected utility E[U] = Σ pᵢ U(Wᵢ), where Wᵢ = initial wealth + gain/loss in scenario i. Because a concave U has diminishing marginal utility, the same £ gained matters less than £ lost — capturing risk aversion.
The certainty equivalent CE is the guaranteed wealth that yields the same utility as the lottery: U(CE) = E[U], so CE = U⁻¹(E[U]). For logarithmic utility U(W) = ln(W), CE = exp(E[U]); for square-root utility CE = E[U]². The risk premium E[W] − CE is how much the decision-maker would pay to avoid uncertainty. A more concave U (higher γ in CRRA) produces a larger risk premium and a CE further below E[W]. If CE = E[W], utility is linear (risk-neutral).
This model assumes preferences are stable and well-defined across the three scenarios, which must be mutually exclusive and cover all outcomes (probabilities sum to 100%). It does not capture framing effects, loss aversion (Kahneman–Tversky), or dynamic inconsistency. Utility values are only ordinal — the numbers matter only in their relative ordering, not their scale.
Frequently asked questions
The certainty equivalent (CE) is the sure amount of money that you value exactly as much as the risky lottery. If CE < E[W], you are risk-averse and would pay a premium to avoid the gamble; if CE = E[W], risk-neutral; if CE > E[W], risk-seeking.
Logarithmic utility (γ = 1 CRRA) is the classic starting point and underlies the Kelly criterion. Power CRRA with γ ≈ 2–4 fits observed household risk behaviour. Linear utility represents a risk-neutral agent who only cares about E[W]. Square-root utility is a mild risk-averse alternative to log.
Both U(W) = ln(W) and U(W) = √W are undefined for W ≤ 0. If a scenario would reduce wealth below zero (e.g. a loss larger than initial wealth), these utility functions cannot be applied — either use linear utility or choose scenarios where final wealth stays positive.
TG we-Calculate Editorial Team. (2026). Expected Utility Calculator — Certainty Equivalent & Risk Premium [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/expected-utility-calculator
TG we-Calculate Editorial Team. "Expected Utility Calculator — Certainty Equivalent & Risk Premium." TG we-Calculate. 2026. https://we-calculate.com/calculator/expected-utility-calculator.
TG we-Calculate Editorial Team, "Expected Utility Calculator — Certainty Equivalent & Risk Premium," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/expected-utility-calculator
@misc{wecalculate_expected_utility_calculator, title = {Expected Utility Calculator — Certainty Equivalent & Risk Premium}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/expected-utility-calculator}}, year = {2026}, note = {TG we-Calculate} }
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