# Expected Utility Calculator

URL: https://speedycalculator.com/calculators/finance/expected-utility/
Last updated: 2026-10-25

> The expected utility calculator compares a risky gamble with a certain amount using a utility function – linear (risk-neutral), square-root or logarithmic (risk-averse) – and gives the expected value, expected utility and certainty equivalent.

## Inputs

- Outcome 1 – probability (%)
- Outcome 1 – amount
- Outcome 2 – probability (%)
- Outcome 2 – amount
- Outcome 3 – probability (%)
- Outcome 3 – amount
- Utility function

## Results

- Expected value
- Expected utility
- Certainty equivalent
- Risk premium

## Formula

```
Expected value = Σ p × x
Expected utility = Σ p × u(x); u(x) = x, √x or ln(x)
Certainty equivalent = u⁻¹(expected utility); risk premium = EV − CE
```

## Explanation

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

FieldWhat to enterOutcome 1 – probabilityIn %Outcome 1 – amountAmount in your currencyOutcome 2 – probabilityIn %Outcome 2 – amountAmount in your currencyOutcome 3 – probabilityIn %Outcome 3 – amountAmount in your currencyUtility functionChoose 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 equivalent252,756.2537.52,889.06503,02562.53,164.06753,306.251003,600

## Worked 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.

## FAQ

**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.