How the Expected Utility Calculator works
Expected utility theory (the von Neumann-Morgenstern model) says a rational decision-maker facing a risky choice should not just look at the average dollar payoff — they should weigh each outcome by how much satisfaction, or utility, it provides, then average those utilities by probability. This calculator applies that theory to a simple two-outcome gamble: outcome 1 happens with probability p and pays x1; outcome 2 happens with probability (1 − p) and pays x2.
The formula
Expected utility is the probability-weighted average of the utility of each outcome:
EU = p × U(x1) + (1 − p) × U(x2)
The calculator uses the CRRA (constant relative risk aversion) utility function, the standard textbook choice for this kind of problem:
U(x) = x^(1−γ) / (1−γ) for γ ≠ 1, or U(x) = ln(x) when γ = 1
where γ (gamma) is the risk aversion coefficient. Larger γ means the utility function is more strongly curved (more concave), which represents a decision-maker who dislikes uncertainty more.
Certainty equivalent and risk premium
The expected utility number itself is not directly meaningful in dollars — utility is only defined up to a positive rescaling. What is meaningful is the certainty equivalent (CE): the guaranteed, risk-free amount that would give the decision-maker exactly the same utility as the gamble. It is found by inverting the utility function:
CE = U⁻¹(EU) = [EU × (1−γ)]^(1 / (1−γ)) for γ ≠ 1, or CE = e^EU when γ = 1
Because a concave utility function (γ > 0) always sits below the straight line connecting two points on it (Jensen's inequality), the certainty equivalent is always less than or equal to the expected monetary value (EMV) when γ > 0. The difference, EMV − CE, is called the risk premium: it is the amount of expected value the decision-maker would willingly give up to trade the gamble for a guaranteed payment.
Worked example
With the default inputs — a 50% chance of $10,000 and a 50% chance of $20,000, at γ = 2 — the expected value is EMV = 0.5 × 10,000 + 0.5 × 20,000 = $15,000. Using U(x) = −1/x for γ = 2, the expected utility works out to about −0.000075, which inverts to a certainty equivalent of roughly $13,333. The risk premium is about $1,667: this risk-averse decision-maker would rather have a guaranteed $13,333 than a 50/50 shot at $10,000 or $20,000.
Reading the risk aversion coefficient
- γ = 0: risk-neutral. Utility equals the dollar amount, so CE = EMV and the risk premium is zero.
- 0 < γ < 1: mildly risk-averse.
- γ = 1: logarithmic utility, a common default in economics for moderate risk aversion.
- γ between roughly 1 and 4: the range most often cited in economics research as plausible estimates of investor risk aversion.
- γ > 4: strongly risk-averse — a large risk premium relative to the EMV.
Assumptions and limits
This is a two-outcome, single-decision model: it assumes the decision-maker's utility depends only on the dollar amount received (not on existing wealth, other assets, or how outcomes correlate with the rest of a portfolio), and that both payoffs are positive amounts (CRRA utility is undefined for zero or negative wealth). Real decisions often involve more than two outcomes, non-CRRA preferences, or path-dependent effects that this simplified model does not capture — treat the output as a teaching and estimation tool, not personalized financial advice.