Expected utility theory explains why people buy insurance and turn down fair bets: with diminishing marginal utility, a certain amount is worth more than a gamble with the same expected value. The risk premium measures how much value the risk costs.
How to use the Expected Utility Calculator
- Outcome 1 – probability
- Outcome 1 – amount
- Outcome 2 – probability
- Outcome 2 – amount
- Outcome 3 – probability
- Outcome 3 – amount
- Utility function
Results update as you type, and every field accepts the units shown in its menu.
How it works
Enter the probability and payoff of up to three outcomes and choose a utility function.
Tips
- Use the nominal annual rate and pick the compounding frequency your bank or broker states.
- Small differences in return or fees compound into large differences over decades.
- Run the calculation with a cautious, a central and an optimistic return to see the range of outcomes.
- Past returns do not guarantee future returns – nothing here is investment advice.
Common mistakes to avoid
- Comparing an APR with an APY or effective rate.
- Ignoring inflation, taxes and fees when judging a real return.
- Entering a percentage as a decimal (5 instead of 0.05 is correct here).
Inputs explained
| Field | What to enter |
|---|---|
| Outcome 1 – probability | In % |
| Outcome 1 – amount | Amount in your currency |
| Outcome 2 – probability | In % |
| Outcome 2 – amount | Amount in your currency |
| Outcome 3 – probability | In % |
| Outcome 3 – amount | Amount in your currency |
| Utility function | Choose one: Linear u = x (risk-neutral), Square root u = √x (risk-averse), Logarithmic u = ln x (more risk-averse) |
What each result means
- Expected value = ([Outcome 1 – probability]×[Outcome 1 – amount] + [Outcome 2 – probability]×[Outcome 2 – amount] + [Outcome 3 – probability]×[Outcome 3 – amount])÷100
- Risk premium = [Expected value] – [Certainty equivalent]
Example values
How the main result changes when Outcome 1 – probability changes and everything else stays at the example values above:
| Outcome 1 – probability (%) | Certainty equivalent |
|---|---|
| 25 | 2,756.25 |
| 37.5 | 2,889.06 |
| 50 | 3,025 |
| 62.5 | 3,164.06 |
| 75 | 3,306.25 |
| 100 | 3,600 |
Formula
Expected value = Σ p × xExpected utility = Σ p × u(x); u(x) = x, √x or ln(x)Certainty equivalent = u⁻¹(expected utility); risk premium = EV − CEWorked example
Inputs — Outcome 1 – probability: 50 %; Outcome 1 – amount: $100; Outcome 2 – probability: 50 %; Outcome 2 – amount: $10,000; Outcome 3 – probability: 0 %; Outcome 3 – amount: $0; Utility function: Square root u = √x (risk-averse).
Results — Expected value = $5,050.00; Expected utility = 55; Certainty equivalent = $3,025.00; Risk premium = $2,025.00.
Frequently asked questions
What is expected utility?
The probability-weighted average of the utility (satisfaction) of each outcome.
What is a certainty equivalent?
The sure amount you'd accept instead of the gamble.
Why do people buy insurance?
Risk-averse utility makes a certain small loss better than a small chance of a large one.
What formula does the Expected Utility Calculator use?
Expected value = Σ p × x.