Decision theory starts with a simple substitution: instead of maximising expected dollars, you maximise expected utility, a subjective value attached to each outcome. This calculator operationalises that idea. Enter the possible results of a bet, a business decision, or a career choice, and set a utility function that describes how you feel about risk. Five standard functions are supported, from logarithmic to exponential CARA. The tool returns the certainty equivalent, the guaranteed amount that would make you equally happy and the risk premium, which quantifies exactly how much you would give up to avoid the uncertainty. It’s used by students learning decision theory, analysts modelling investment preferences, and anyone curious about why insurance exists.
How to Use the Expected Utility Calculator
Choose the currency the outcomes are denominated in.
Pick a utility function: Log, Square Root, Power (CRRA), Exponential (CARA), or Linear.
If you selected Power or Exponential, adjust the risk-aversion parameter in the field that appears.
Enter each outcome’s probability as a percentage and its value.
Confirm that the total probability at the bottom of the outcomes list reads 100%. If it doesn’t, the tool still computes but flags the mismatch.
Optionally enter a guaranteed amount in the Risk-Free Alternative panel to see which option you should prefer.
Load a quick-example scenario, coin flip, lottery ticket, risky investment, insurance, career choice, or certain outcome to see how the output changes.
Read the certainty equivalent at the top, then check the risk premium and risk attitude in the cards below.
How the Expected Utility Calculator Formula Works
The core identity of expected utility theory is a probability-weighted average of utilities.
Formula: Expected Utility = Σ pᵢ × u(xᵢ)
Formula: Certainty Equivalent = u⁻¹(Expected Utility)
Formula: Risk Premium = Expected Value − Certainty Equivalent
Where pᵢ is the probability of outcome i, xᵢ is the monetary value of that outcome, and u(·) is the utility function you selected.
Each utility function interprets the same outcomes differently. Logarithmic (u = ln x) treats the first dollars as far more valuable than the last, which produces strong risk aversion. Square root (u = √x) applies moderate curvature. Power utility (u = x^(1−γ)/(1−γ)) generalises this with a risk-aversion coefficient γ, the higher γ, the more curved the utility function. Exponential utility (u = 1 − e^(−ax)) applies constant absolute risk aversion, which means the dollar amount you’d pay to avoid risk stays the same regardless of wealth. Linear utility (u = x) collapses the whole framework: utility equals dollars, so the certainty equivalent equals the expected value and the risk premium is zero.
The certainty equivalent is the key output. It’s the guaranteed amount that produces the same utility as the gamble, which means it’s the most you would pay to avoid the risk. A $20,000 expected value with a $16,000 certainty equivalent means the decision-maker would accept $16,000 in cash rather than take the uncertain bet, a $4,000 risk premium.
Two practical details matter. First, the tool normalises your probabilities if they don’t sum to 100%, so the math works even with an incomplete input but the warning indicator turns red. Second, logarithmic and square-root utilities require positive values. If any outcome is zero or negative, the utility is undefined and the results become unreliable.
Worked Example
Suppose you’re offered a coin-flip bet: heads pays $25,000, tails pays $5,000. You’re using logarithmic utility, the classic risk-averse assumption.
Step 1 – Expected value:
EV = 0.5 × 25,000 + 0.5 × 5,000 = $15,000
Step 2 – Utility of each outcome:
u(25,000) = ln(25,000) ≈ 10.127
u(5,000) = ln(5,000) ≈ 8.517
Step 3 – Expected utility:
EU = 0.5 × 10.127 + 0.5 × 8.517 = 9.322
Step 4 – Certainty equivalent:
CE = e^9.322 = $11,152
Step 5 – Risk premium:
RP = 15,000 − 11,152 = $3,848
The gamble has an expected value of $15,000, but a log-utility decision-maker treats it as worth only $11,152. That $3,848 gap is the risk premium, the amount they’d be willing to forgo to eliminate the uncertainty. If someone offered them a guaranteed $12,000, they’d take it immediately, because $12,000 in cash produces more utility than the gamble. If the guaranteed amount dropped to $11,000, they’d prefer the gamble.
This is the mechanism behind insurance. Replace the gamble with “your house burns down (1% chance, lose $300,000)” versus “nothing happens (99% chance)”, and the certainty equivalent becomes the amount you’d pay for an insurance premium. The risk premium is the maximum insurance cost you’d accept rather than self-insure.
Frequently Asked Questions
What does expected utility mean in plain language?
Expected utility is the probability-weighted average of how much you value each possible outcome. Unlike expected value, which averages the dollar amounts, expected utility averages a subjective valuation. Risk-averse people value a dollar gained less than a dollar lost, so their expected utility is below the expected dollar value.
What is a certainty equivalent?
The certainty equivalent is the guaranteed amount that would give you the same utility as an uncertain gamble. If a gamble has a certainty equivalent of $8,000, you’d be indifferent between taking the gamble and receiving $8,000 for sure. Anything above $8,000 in guaranteed cash and you’d prefer the cash; anything below and you’d prefer the gamble.
How is the risk premium calculated?
The risk premium is the expected value minus the certainty equivalent. A $15,000 expected value with an $11,000 certainty equivalent produces a $4,000 risk premium. It measures how much you’d give up in expected dollars to avoid the uncertainty.
Can the certainty equivalent be higher than the expected value?
Yes, though it’s unusual. It happens when the utility function is convex, which produces risk-seeking behaviour. For most standard utility functions applied to typical gambles, the certainty equivalent will be below the expected value because risk aversion is the normal assumption.
Which utility function should I use?
For investment decisions, log or power utility with a γ between 1 and 3 are the most common choices. For insurance-style decisions, exponential utility captures how much absolute dollars matter. Square root is a simpler alternative to log. Linear is only appropriate if you genuinely don’t care about risk.
Why do log and square root utilities require positive values?
Because logarithms and square roots are undefined for zero and negative numbers. If your outcomes include a possible loss, you need to shift all values upward by a constant modelling them as “final wealth” rather than “gain or loss.” The calculator treats all values as positive by design.
What is the risk-aversion coefficient in power utility?
It’s the Greek letter γ (gamma), and it controls how curved the utility function is. A γ of 1 produces log utility. Higher values produce stronger risk aversion. Values in the 2–4 range are typical for academic studies, but there’s no single correct number, it’s a preference parameter.
Does the calculator handle probabilities that don’t sum to 100%?
Yes. The tool normalises probabilities so the math still works, and it flags the mismatch with a red indicator. This is useful if you’re entering approximate probabilities or missing an outcome that you’ll add later.
How does the exponential utility function differ from the others?
Exponential utility (u = 1 − e^(−ax)) exhibits constant absolute risk aversion. That means the dollar amount you’d pay to avoid a given risk stays the same whether you’re rich or poor. The other functions exhibit decreasing absolute risk aversion, the more wealth you have, the smaller the dollar amount you’d pay for insurance. Both are legitimate modelling choices depending on the context.
Can I use this calculator to decide whether to take a bet?
Yes. Enter the outcomes and probabilities of the bet, select your utility function, and look at the certainty equivalent. If someone offers you a guaranteed amount above the certainty equivalent, take the sure thing. If they offer less, the gamble has higher utility for you.
What does a negative risk premium mean?
A negative risk premium means the certainty equivalent exceeds the expected value, which signals risk-seeking behaviour. This can happen under certain utility specifications or when the gamble has an appealing upside skew. It’s not the standard case, but it’s mathematically valid.
Why does the risk premium change when I switch utility functions?
Because each utility function applies a different curvature to the same outcomes. Log utility curves more sharply than square root, so it produces a larger risk premium for the same gamble. Power utility with γ = 3 curves more than power utility with γ = 1.5. The utility function is what encodes your risk preference.
