Expected Utility Calculator

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Expected Utility Calculator

Analyze decision-making under uncertainty using expected utility theory and the Constant Relative Risk Aversion (CRRA) fonction.

Last updated: March 2026

Decision Analysis
α = 0: Risk neutral
α = 1: Log utility
α ≥ 1: Risk averse

What is Expected Utility Theory?

Expected Utility Theory is a mathematical framework for making decisions when outcomes are uncertain. Unlike expected value, which simply multiplies outcomes by their probabilities, expected utility accounts for how much satisfaction (utility) you get from each outcome.

The key insight: people don't value money linearly. Gaining 100 dollars provides more satisfaction than losing 50 dollars, even though the expected value difference is -50. This behavior is captured by a utility function, which transforms monetary outcomes into satisfaction levels.

The CRRA (Constant Relative Risk Aversion) utility function is widely used in economics and finance. It models how utility grows with wealth, with a parameter (α) controlling risk preferences. Higher α values mean higher risk aversion—you prefer certainty over gambles with equal expected value.

How to Use Expected Utility Theory

The CRRA Utility Function

The Constant Relative Risk Aversion function:

If α ≠ 1: u(x) = x^(1-α) / (1-α)
If α = 1: u(x) = ln(x)
where x is wealth/outcome, α is risk aversion

Step-by-Step Process

  1. 1. List outcomes: Identify all possible results and their probabilities
  2. 2. Set risk aversion: Choose α (0 = risk neutral, ≥1 = risk averse)
  3. 3. Calculate utilities: Apply CRRA function to each outcome
  4. 4. Expected value: E(U) = Σ(u(x) × probability)
  5. 5. Interpret: Compare certainty equivalent to expected value

Example: 50/50 Gamble

You can either:

  • Option A: Take a gamble: Win $100 with 50% probability, lose $0 with 50% probability
  • Option B: Get a guaranteed $50
Expected Value:
100 × 0.5 + 0 × 0.5 = $50
If α = 2 (risk averse):
u(100) = 100^(1-2) = 100^(-1) = 0.01
u(0) = 0^(-1) = 0
E(U) = 0.01 × 0.5 + 0 × 0.5 = 0.005
Certainty equiv: 0.005 = CE^(-1) → CE = $200^(-1) = $200/200 ≈ lower

Risk-averse person prefers guaranteed $50 over risky $100 expected value

Frequently Asked Questions

What's the difference between expected value and expected utility?

Expected value treats all money equally. Expected utility accounts for diminishing marginal utility—an extra dollar matters less if you're already wealthy. This explains why people buy insurance even with negative expected value.

What does the risk aversion coefficient mean?

Higher α (≥ 1) means more risk aversion. You prefer safer bets. α = 0 means risk neutrality (pure EV maximization). α = 1 means logarithmic utility. In practice, most people have α between 0.5 and 3.

What is the certainty equivalent?

The certainty equivalent is the guaranteed amount you'd accept instead of taking the gamble. If CE ≤ EV, you're risk-averse. If CE = EV, you're risk-neutral. If CE ≥ EV, you're risk-loving (unusual).

What is the risk premium?

The risk premium (EV - CE) is how much you'd pay to avoid risk. For risk-averse people, it's positive: they'll give up expected value for certainty. It measures your willingness to pay for peace of mind.

Can I use this for investment decisions?

Yes! Portfolio optimization uses expected utility to balance risk and return. The risk aversion parameter determines the optimal mix of stocks vs. bonds. Higher α means more bonds, lower α means more stocks.

What if I have negative outcomes?

The CRRA utility function requires positive wealth. If outcomes can include losses, consider alternative utility functions like linear (no risk attitude) or quadratic. Consult a financial advisor for real decisions.

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