Expected Utility Calculator

What this expected utility calculator helps you decide

Expected value answers a narrow arithmetic question: what payoff do you get on average if the same gamble could be repeated many times? Expected utility answers a more human question: how attractive does that risky choice feel once you acknowledge that an extra 1,000 of upside may matter less than the pain of a downside scenario.

That is why expected utility is useful for savings, investment, insurance, and capital-allocation decisions where the same expected value can hide very different downside profiles. A risky decision can have a positive expected value but still produce a certainty equivalent that is materially lower than the headline weighted average once diminishing marginal utility is applied.

How expected utility, certainty equivalent, and risk premium fit together

The calculator first converts each payoff into utility using the selected utility curve, weights each utility by its probability, and sums the results. It then backs out the certainty equivalent by applying the inverse utility function, which translates that utility score back into the guaranteed payoff that leaves the decision-maker indifferent between certainty and risk.

Risk premium is the gap between expected value and certainty equivalent. For a risk-averse user, the certainty equivalent will usually sit below expected value, and the difference tells you how much expected payoff you would rationally surrender to remove uncertainty from the decision.

This page supports linear utility as a risk-neutral benchmark, log utility for stronger diminishing marginal utility, and square-root utility for a milder risk-averse shape. The result is not telling you which gamble is objectively best for everyone. It is showing how the same distribution changes when the decision rule changes from expected value to expected utility.

Worked example: a four-scenario investment payoff

Suppose you are comparing a risky investment whose payoffs are 12,000 with 45% probability, 7,000 with 35% probability, and 2,500 with 20% probability. The expected value of that distribution is 8,350, which can make the opportunity look attractive if you stop at weighted averages alone.

Under log utility, however, the certainty equivalent is lower than 8,350 because the downside state reduces utility more than the upside state increases it. In other words, the expected utility calculator shows that a guaranteed payoff below expected value may still feel equally acceptable once risk aversion is recognised, and the difference between those two figures is the risk premium.

That interpretation matters in practice. If the guaranteed alternative on the table is above the certainty equivalent, a risk-averse decision-maker may rationally prefer certainty even though the risky option still has the higher expected value. If the guaranteed alternative is below the certainty equivalent, the risky choice may still offer enough compensated upside for that utility profile.

Choosing a utility function and reading the result carefully

Linear utility is best treated as a baseline because it assumes every extra unit of payoff matters equally regardless of the starting level of wealth or consumption. In that case, certainty equivalent and expected value line up, so the calculator becomes an expected value check rather than a risk-aversion screen.

Log utility is commonly used in economics and finance when you want a stronger diminishing-marginal-utility relationship. Square-root utility gives a similar but milder curvature. Neither one is universally correct. They are simple approximations for how the decision-maker trades upside against downside when payoffs are uncertain.

The headline result is most useful when you compare it with an actual sure alternative, a treasury-like fallback, an insured outcome, or a less risky competing project. If the certainty equivalent is materially below the offer's expected value, that is a signal to investigate downside concentration, probability assumptions, and whether taxes, liquidity needs, or sequence risk make the gamble less attractive than the raw average suggests.

Using CRRA sensitivity and scenario quality checks

The CRRA setting adds a constant-relative-risk-aversion curve, which is useful when you want to test how a higher or lower risk-aversion coefficient changes the certainty equivalent. A coefficient near zero behaves closer to a risk-neutral view, while higher values make downside states weigh more heavily in the utility calculation.

Sensitivity matters because expected utility is not only a payoff calculation. It is also an assumption test. If the same scenario set keeps its ranking under linear, square-root, log, and CRRA views, the decision is more robust than a choice whose appeal disappears as soon as the utility curve becomes slightly more conservative.

The payoff standard deviation, best-worst spread, and probability below expected value help you inspect the distribution before trusting the headline answer. A high expected value with a large downside probability can still have a low certainty equivalent, especially when one adverse state would create liquidity pressure or force a decision-maker to abandon the plan.

What this page does not model

This expected utility calculator is a one-period decision screen, not a full portfolio model. It does not estimate diversification benefits, path dependence, drawdown sequencing, rebalancing, leverage constraints, taxes, inflation, or how one gamble interacts with the rest of your balance sheet.

It also does not implement prospect-theory features such as probability weighting, reference dependence, or loss aversion. That means a real person may still reject or prefer a gamble for behavioural reasons that a classical expected utility model cannot capture.

Use the output as a structured comparison tool rather than a stand-alone investment recommendation. If the decision is material, the probability estimates and the utility-function choice both need to be tested against your actual financial circumstances and risk capacity.