Calculate the expected utility of a risky decision from outcomes, probabilities, and a chosen utility function. Use it to test different inputs quickly, compare outcomes, and understand the main factors behind the result before moving on to related tools or deeper guidance.
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Decision Theory
Expected utility theory extends expected value by accounting for risk preferences. This calculator computes expected utility, certainty equivalent, and risk premium for a set of probabilistic outcomes under linear, logarithmic, or square root utility functions.
Expected value treats every dollar equally regardless of wealth level. Expected utility applies a concave function (for risk-averse agents) that gives diminishing marginal value to each additional dollar.
The certainty equivalent is the guaranteed amount a decision-maker would accept instead of the risky gamble. The risk premium is how much expected value they would sacrifice to eliminate the uncertainty.
Expected utility and certainty equivalent.
EU = Σ pᵢ × U(xᵢ)
pᵢ = probability of outcome i, U(xᵢ) = utility of outcome i.
CE = U⁻¹(EU)
Certainty equivalent = inverse utility of expected utility.
Risk Premium = EV − CE
Positive risk premium indicates risk aversion.
Requires choosing a specific utility function, which is subjective. Real preferences may not fit standard functional forms. Does not model probability weighting (prospect theory) or loss aversion.
Frequently asked questions
Logarithmic utility (Bernoulli) is the most common for modelling risk aversion. Square root is a milder form. Linear means risk-neutral (expected utility = expected value). The choice depends on the decision-maker’s actual risk preferences.
The risk premium is the maximum amount you would pay to eliminate uncertainty. A risk premium of 1,200 means you would accept a guaranteed 5,600 instead of a gamble with an expected value of 6,800.
Probabilities represent the complete set of possible outcomes. If they don’t sum to 1, the model is incomplete — either an outcome is missing or the probabilities are incorrectly estimated.
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