Expected Utility Calculator

Understanding Expected Utility Theory

Expected Utility Theory (EUT), established by John von Neumann and Oskar Morgenstern, models decision-making under uncertainty. Unlike expected financial value which measures linear dollar outcomes, expected utility accounts for an individual's psychological risk preferences using concave, linear, or convex utility functions.

Expected Utility Formula

For a gamble or decision with potential wealth outcomes $w_i$ occurring with probabilities $p_i$, expected utility is defined as:

$$E[U(w)] = \sum_{i=1}^{n} p_i \cdot U(w_i)$$

Common Utility Functions & Risk Aversion

Different utility functional forms capture distinct attitudes toward financial risk:

• Square Root Utility ($U(w) = \sqrt{w}$): Represents risk-averse investors who experience diminishing marginal utility of wealth.
• Logarithmic Utility ($U(w) = \ln(w)$): Represents risk-averse behavior with constant relative risk aversion (CRRA).
• Linear Utility ($U(w) = w$): Represents risk-neutral decision makers who evaluate gambles solely on expected monetary return.

Certainty Equivalent & Risk Premium

Two central concepts derived from expected utility:

• Certainty Equivalent (CE): The guaranteed dollar amount that yields the exact same utility as the risky outcome: $$CE = U^{-1}(E[U(w)])$$
• Risk Premium (RP): The amount of expected wealth an investor willingly surrenders to eliminate uncertainty: $$RP = E[w] - CE$$

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