What Expected Utility Actually Measures
Expected utility fixes a flaw that dollar averages cannot: the same $50,000 means radically different things depending on who receives it. Daniel Bernoulli made this point in 1738 while analyzing the St. Petersburg paradox, a coin-flip game whose expected dollar payout is infinite yet which nobody would pay more than a few dollars to play. His resolution was to score outcomes in utility, a concave function of wealth, so each additional dollar counts for less than the one before it.
With logarithmic utility and $100,000 of wealth, gaining $50,000 adds about 0.405 utility points while losing $20,000 costs about 0.629 — the loss out-weighs the gain even though the gain is two and a half times larger in dollars. That asymmetry is the entire engine of risk aversion. It is also why two people facing the identical bet can rationally choose differently: the function depends on the wealth base each person starts from.
The same translate-into-personal-value logic runs through other money tools. The willingness-to-pay reasoning behind the consumer surplus calculator likewise prices outcomes by subjective value rather than face amounts. Expected utility pushes that idea one step further by making the value curve explicit and putting probabilities on top of it.
The Formula, Piece by Piece
The core equation is E[U] = p × U(wG) + (1 − p) × U(wB), where wG and wB are your total wealth in the good and bad states. The single most common mistake in hand calculations is applying U to the gain and loss instead of to total wealth — that quietly deletes the wealth base from the math and produces absurd results like a billionaire being equally scared of a $100 loss as a student. Enter the full picture: wealth, then gain, then loss.
The function menu maps to standard textbook choices. Square-root utility is CRRA with gamma = 0.5, a mild aversion level. Logarithmic utility is gamma = 1 and is the classical benchmark dating back to Bernoulli. Power utility lets you set gamma directly — 2 to 4 matches most empirical estimates of household behavior — and linear utility makes U(w) = w, which collapses the whole framework back to expected value for a risk-neutral reference point.
With the defaults ($100,000 wealth, +$50,000 / −$20,000, 40% good chance, log utility), the pieces line up as U($150,000) = 11.9184 and U($80,000) = 11.2898, giving expected utility 0.4 × 11.9184 + 0.6 × 11.2898 = 11.5412. The calculator then inverts the function at that point: exp(11.5412) = $102,870, the certainty equivalent. Each intermediate number is printed in the explanation so you can check the work against your own arithmetic.
Certainty Equivalent and the Decision Rule
The certainty equivalent converts the utility score back into a number you can act on: the guaranteed wealth level exactly as attractive as the gamble. For the default inputs it is $102,870, meaning the risky prospect feels like a sure $102,870 to a log-utility decision maker — even though its dollar average is $108,000. The $5,130 difference is not a rounding artifact; it is the measurable cost of the spread between $150,000 and $80,000.
The decision rule is a straight comparison with current wealth. CE above wealth means accept — the gamble is as good as free money of (CE − wealth). CE below wealth means refuse, and the shortfall tells you how much you would need to be compensated to take the risk on. At a 30% good probability the certainty equivalent is $96,603, below the $100,000 base, so the same bet flips from accept to refuse — the verdict is that sensitive to the probability input.
The break-even logic is worth spelling out. Under risk neutrality you play this gamble whenever the good probability exceeds 28.6%, the point where the dollar average crosses $100,000 — the same threshold reasoning as a break even calculator. Under logarithmic utility the acceptance threshold climbs to 35.5%. That seven-point gap is a clean measure of how much risk aversion tightens your standards once dollars stop being the scorecard.
Risk Premium and What Insurance Is Worth
Risk premium = expected value − certainty equivalent, and it prices the risk itself in dollars. The default gamble carries a $5,130 premium, about 4.7% of its $108,000 expected value. Nothing about the bet changed to create that number — it exists entirely because a concave utility function penalizes the spread between outcomes. Wider spreads and larger stakes relative to wealth push the premium up, which is exactly what real insurance pricing reflects.
Run the insurance case: $100,000 wealth with a 2% chance of a $30,000 loss. The expected loss is $600, but the certainty equivalent of facing that risk is $99,289 against an expected wealth of $99,400 — a $110.81 risk premium. You should therefore rationally pay up to about $710.81 per year for full coverage: $600 of expected loss plus $110.81 for the removal of the spread. Any quoted premium below that number is a buy under log utility; anything above it means you are paying more than the risk personally costs you.
Scale the loss and the premium grows faster than the stakes. At $50,000 wealth with a 10% chance of a $20,000 loss, the expected loss is $2,000 but the risk premium is $490, so full coverage is worth up to $2,490 — nearly 25% on top of the actuarial cost. This is why insurance against catastrophic, wealth-crushing events carries the fattest margins, and why insuring a $300 phone repair rarely passes the same test once you compare premium to your own certainty equivalent.
Expected Utility vs Expected Monetary Value
Expected monetary value is the probability-weighted dollar average — the number an EMV calculator produces for decision trees. Its blind spot is spread: a 50/50 shot at $0 or $100,000 and a guaranteed $50,000 both show $50,000, yet almost nobody treats them as interchangeable. Expected utility layers the concave function on top and makes the distinction, always scoring the risky version lower.
The mathematics behind that penalty is Jensen's inequality: for any concave U, E[U(w)] is at most U(E[w]), so the certainty equivalent never exceeds the expected value and the risk premium never goes negative. When is EMV good enough? Small stakes relative to wealth, and decisions you will repeat many times, where the law of large numbers lets dollar averages dominate — a company screening hundreds of small projects is close to risk neutral and can rationally run on EMV alone.
Investment analysis sits between the two poles. The scenario-weighted averages behind an expected return calculator handle the forecasting side, but how much of your wealth to commit to any single scenario set is a utility question. A useful discipline: compute the expected return first, then run this calculator on the position size to see whether the downside scenario still clears your acceptance threshold.
Choosing a Function and a Gamma
Function choice is really just gamma choice. Square-root is gamma 0.5 — mild aversion suited to small stakes or diversified decision makers. Logarithmic is gamma 1, the benchmark with the cleanest theory behind it. Power utilities from gamma 2 to 5 cover seriously risk-averse profiles; empirical work on household portfolio choices typically estimates gamma between 1 and 4, so values beyond 5 deserve skepticism about whether they describe any real person.
The sweep on the default gamble shows the leverage gamma has. Gamma 1 gives a certainty equivalent of $102,870 with a $5,130 risk premium; gamma 2 gives $98,361 and $9,639; gamma 3 gives $94,691 and $13,309; gamma 5 gives $89,712 and $18,288. Notice that by gamma 3 the certainty equivalent has dropped below the $100,000 wealth base — the identical bet that looked like free money under log utility is now a refusal. The verdict is more sensitive to gamma than most people expect.
Use linear as a sanity anchor: it reproduces the EMV answer exactly, with a risk premium of zero. Markets price risk their own way — through beta and expected returns, as the CAPM calculator approach shows — while a utility function prices risk personally. When the market's implied price of risk and your own diverge sharply, that spread is where insurance and hedging decisions live.
Investing, Kelly Sizing, and Wealth Scaling
Logarithmic utility has a famous investing corollary: maximizing expected log wealth is equivalent to the Kelly criterion, which maximizes long-run compound growth. For an even-money bet you win 60% of the time, the Kelly fraction is f* = 0.60 − 0.40/1 = 20% of wealth per round, compounding at about 2.01% per flip. Bet 10% and growth drops to 1.50%; bet 50% and growth turns negative at −3.40% per flip — overbetting a favorable game can still ruin you.
Risk aversion also shrinks as wealth grows, which these functions capture automatically. The default gamble (+$50,000 / −$20,000 at 40%) costs a $100,000-wealth person a $5,130 risk premium, but the same bet at $1 million of wealth carries a premium of only about $578. The dollar stakes are identical; what changed is their size relative to the base. This decreasing absolute risk aversion is why wealthy households self-insure small risks and buy coverage only for catastrophic ones.
Expected utility slots in around the standard valuation tools rather than replacing them. A single-period profitability check belongs in an ROI calculator, and multi-period cash flow forecasting belongs in a DCF calculator. This tool answers the question those two cannot: given the forecast and the spread, does the bet fit your wealth and your tolerance — or is the certainty equivalent telling you to pass?
Pitfalls That Skew the Answer
The first pitfall is scoring gains instead of wealth. U(+$50,000) is meaningless in this framework — the function needs the terminal wealth level, $150,000, to know where you land. The second is ruin: if the bad outcome leaves wealth at or below zero, log, square-root, and power utility are undefined, and the calculator says so instead of inventing a number. Treat that error message as a result. It means the gamble is unacceptable under any conventional risk-averse function.
Probability calibration matters more than function precision. The break-even analysis above showed the verdict flipping between a 30% and 40% good probability, while moving gamma by a full unit often does less. Anchor your estimate in base rates — historical close rates, failure statistics, actuarial tables — and when the estimate is genuinely uncertain, run the calculator at both ends of the plausible range to see which side of the threshold you are really on.
Know the framework's limits. Economist Matthew Rabin proved that any person with consistently concave utility over wealth would turn down a 50/50 bet to win $200 or lose $100 at every wealth level — yet would then be implied to reject a 50/50 shot at $20 million against losing $1,000. In practice, a $100 coin flip at $100,000 wealth prices a risk premium of only about 5 cents under log utility, which matches intuition: small stakes belong in EMV territory. For pricing systematic risk in options, the risk-neutral valuation inside the Black Scholes option pricing calculator sidesteps personal utility entirely, because replication removes the risk — a reminder that utility functions matter most when a risk genuinely cannot be hedged away.