The expected utility calculator above takes a risky prospect — a list of payoffs with their probabilities — passes each payoff through a utility function that you choose, and returns the probability-weighted average of those utilities. It then inverts the function to give the certainty equivalent: the sure amount that would leave you exactly as well off as the gamble, according to the model. The gap between the expected value and the certainty equivalent is the risk premium.
Expected utility is a normative model of choice under risk, not a measurement of anything. Arb Digital publishes it with every parameter as an input, including the utility function and its risk-aversion coefficient, because none of those are observable facts. Our expected value calculator handles the different job of computing the mean, variance and standard deviation of a draw in money units; this page transforms the payoffs first and reports what that transformation implies.
What This Expected Utility Calculator Does
It computes four things. Expected value is the plain probability-weighted average of the payoffs, with no utility function involved. Expected utility is the probability-weighted average of the transformed payoffs. The certainty equivalent is the payoff whose utility equals that expected utility, found by inverting the function in closed form. The risk premium is the expected value minus the certainty equivalent: what a decision maker with these preferences would pay to avoid the risk.
Five utility functions are offered, spanning the standard families. The risk-neutral case is the identity, so the certainty equivalent equals the expected value and the premium is zero — a useful control. Logarithmic and square-root utility are the classical concave choices. The constant relative risk aversion family covers both as special cases and lets you vary the coefficient continuously. Constant absolute risk aversion is the exponential form, whose defining property is that the risk premium does not change when you add a constant to every payoff.
Probabilities are normalised if they do not sum to one, and the tool says so rather than silently rescaling. Payoffs that fall outside a function's domain — a zero or negative amount under logarithmic or square-root utility — produce a written explanation rather than a NaN, because the failure is a real statement about the model, not an arithmetic accident.
How to Use It
- List the payoffs as final wealth or consumption levels, one per outcome. Most utility functions are defined over levels, not over gains and losses from a reference point.
- List the probabilities in the same order, one per payoff.
- Choose a utility function and, for the two parametric families, a risk-aversion coefficient.
- Read the certainty equivalent, not the raw utility number. That is the output with a meaning.
- Vary the coefficient and watch the risk premium move. The sensitivity is usually the interesting finding.
The Formula and How It Is Calculated
For a prospect with payoffs x₁ … xₙ occurring with probabilities p₁ … pₙ, the expected utility is
EU = ∑i pi · u(xi)
and the certainty equivalent CE solves u(CE) = EU, so CE = u⁻¹(EU). The risk premium is E[x] − CE. Because the utility functions here are all strictly increasing, the inverse exists and is computed in closed form rather than searched for numerically.
Work the default values through by hand. The prospect is an even chance of 100 or 400, and the function is logarithmic. Expected value is 0.5(100) + 0.5(400) = 250. Expected utility is 0.5 ln 100 + 0.5 ln 400 = 0.5(4.60517) + 0.5(5.99146) = 5.29832. Inverting, CE = e5.29832 = 200 exactly, which is the geometric mean √(100 × 400). The risk premium is 250 − 200 = 50.
That exact result is a property of logarithmic utility: the certainty equivalent is always the geometric mean of the payoffs weighted by their probabilities. Switch to square-root utility and the same prospect gives EU = 0.5(10) + 0.5(20) = 15, so CE = 15² = 225 and the premium falls to 25 — square-root utility is less risk averse than logarithmic. Switch to relative risk aversion with γ = 2 and the certainty equivalent becomes the harmonic mean, 2 ÷ (1/100 + 1/400) = 160, with a premium of 90.
Why the Utility Number Itself Means Nothing
This is the most common misunderstanding, and it is worth being blunt about. A von Neumann–Morgenstern utility function is unique only up to a positive affine transformation: if u represents your preferences then so does 3u + 17, and every choice implied by one is implied by the other. The number 5.29832 above would become 32.895 under that transformation without a single decision changing.
Two consequences follow. First, an expected utility figure cannot be compared across different utility functions, so the hero number above is only meaningful against another run using the identical function. Second, utilities cannot be compared or added across people. “This policy produces 400 utils” is not a statement with content, and any welfare argument built that way needs an additional assumption it usually does not admit to. The Stanford Encyclopedia of Philosophy's entry on normative theories of rational choice: expected utility sets out exactly what the representation does and does not license.
The certainty equivalent has none of these problems. It is measured in money, it is invariant under affine transformations of the utility function, and it can be compared directly against the expected value and against other prospects. That is why it is in the grid and why the tip panel points at it.
The Assumptions Behind the Model
Expected utility is a representation theorem, not an empirical law. It says that if a decision maker's preferences over lotteries satisfy four axioms — completeness, transitivity, continuity and independence — then those preferences can be represented as maximising the expected value of some utility function. The Stanford Encyclopedia's entry on decision theory gives the axioms and the theorem in full.
The independence axiom is the one that fails experimentally. The Allais paradox constructs two pairs of lotteries that differ by a common consequence; most people reverse their preference between the pairs, which independence forbids. The Ellsberg paradox shows people distinguishing known probabilities from unknown ones, which the model treats as identical. Prospect theory was built to accommodate these, using a reference point, loss aversion and non-linear probability weighting — none of which this page implements.
The model also assumes the probabilities are known. That is risk, in the technical sense. Where they are not known, the situation is uncertainty, and applying this arithmetic to invented probabilities produces a precise number resting on nothing. Our probability calculator is for the cases where the probabilities genuinely are calculable, and our risk-reward ratio calculator gives a simpler measure that makes fewer claims.
Why Concavity Is the Whole Story
Risk aversion in this framework is exactly concavity of the utility function, and nothing else. A concave function's chord lies below the curve, so by Jensen's inequality the expected utility of a gamble is below the utility of its expected value, which means the certainty equivalent is below the expected value and the risk premium is positive. Change the curvature and you change the premium; that is the entire mechanism.
The two parametric families differ in how the premium scales. Under constant relative risk aversion the premium is proportional to wealth, so a person facing a gamble over 1% of their wealth behaves the same whether they are rich or poor. Under constant absolute risk aversion the premium is fixed in money terms regardless of wealth, which is analytically convenient and empirically implausible.
The St Petersburg prospect is the historical reason any of this exists. A gamble paying two to the power n with probability one over two to the power n has an infinite expected value, and yet nobody will pay much for it. Daniel Bernoulli's resolution in 1738 was to propose that people maximise the expectation of the logarithm of wealth rather than of wealth itself, which makes the value finite. The Stanford Encyclopedia's entry on the St Petersburg paradox covers the argument and the objections to it, including the fact that a modified version defeats logarithmic utility too. The preset button loads a truncated version so you can see the expected value and the certainty equivalent diverge; our logarithm calculator handles the underlying transformation directly.
Arb Digital builds free tools that put the interpretable output in front, not the intermediate one.
Browse All Free Tools Talk To Our TeamCommon Mistakes to Avoid
- Quoting the expected utility number — it is meaningful only up to a positive affine transformation, so it says nothing on its own. Quote the certainty equivalent.
- Adding utilities across people — the representation gives no basis for interpersonal comparison, and welfare arguments built that way smuggle in an extra assumption.
- Entering gains rather than final wealth — the standard functions are defined over levels, and using differences changes the curvature the model sees.
- Inventing probabilities — the model requires known probabilities, and feeding it guesses produces a precise answer resting on nothing.
- Treating a fitted risk-aversion coefficient as a personality trait — measured values vary enormously with the size and framing of the gamble.
Related Free Tools From Arb Digital
Get the mean, variance and spread of a draw with the expected value calculator, work out the probabilities themselves with the probability calculator, compare upside against downside with the risk-reward ratio calculator, weight a set of values with the weighted average calculator, measure dispersion with the variance calculator, or transform values directly with the logarithm calculator. The full free online tools hub lists every mathematics tool we publish.
Frequently Asked Questions
It is the probability-weighted average of a utility function applied to each possible payoff. It is a normative model of choice under risk, not a measurement of anything observable.
The sure amount whose utility equals the expected utility of the gamble. Unlike the utility number itself it is measured in money and can be compared directly against the expected value.
Because a utility function is unique only up to a positive affine transformation. Multiplying it by three and adding seventeen represents identical preferences and changes no decision at all.
The expected value minus the certainty equivalent, which is what a decision maker with these preferences would give up to replace the gamble with a sure amount.
That is a modelling choice, not a fact. Logarithmic and constant relative risk aversion are the common defaults in finance, but the answer depends on the function, so state which one you used.
Completeness, transitivity, continuity and independence. If preferences over lotteries satisfy all four they can be represented as maximising expected utility, which is a theorem rather than an empirical claim.
The Allais paradox violates independence and the Ellsberg paradox shows people treating known and unknown probabilities differently. Prospect theory was built to accommodate both.
Logarithmic and square-root utility are undefined at or below zero, and the relative risk aversion family is too for most coefficients. The tool says so rather than returning a meaningless value.
This page explains a standard model in decision theory for educational purposes and is not financial or investment advice. Every parameter, including the utility function and its coefficient, is a modelling assumption you supply rather than an observed fact.
