Expected Utility Theory (EUT) Framework

Foundation of Traditional Finance

Expected Utility Theory (EUT), formalized by John von Neumann and Oskar Morgenstern (1944), provides the mathematical framework for rational decision-making under uncertainty.

Core Principle: Rational individuals choose the option that maximizes their expected utility, not just expected monetary value.

The von Neumann-Morgenstern Axioms

EUT rests on four axioms that define "rational" behavior:

Completeness: For any two alternatives A and B, you can state A > B, B > A, or A ~ B (indifferent)

Transitivity: If A > B and B > C, then A > C

Independence: If A > B, then a lottery mixing A with C is preferred to a lottery mixing B with C

• Critical for expected utility calculation

Continuity: No discontinuous jumps in preferences

The EUT Formula

EU(L) = Σ [p_i × U(x_i)]

Where:

• L = lottery/gamble
• p_i = probability of outcome i
• U(x_i) = utility of outcome i

Decision Rule: Choose the option with highest expected utility.

Example Application

Investment Decision:

Option A: Certain ₹5 lakh return
Option B: 60% chance ₹10 lakh, 40% chance ₹0

Risk-Averse Utility: U(x) = √x

Calculations:

• U(A) = √500,000 = 707
• EU(B) = 0.6×√1,000,000 + 0.4×√0 = 0.6×1,000 + 0 = 600

Choose A (707 > 600) despite B having higher expected monetary value (₹6 lakh vs ₹5 lakh).

This captures risk aversion: certainty premium.

Risk Attitudes Under EUT

Risk Aversion (most common):

• U''(x) < 0 (concave function)
• Prefer certain outcome to equal-expected-value gamble
• Diminishing marginal utility

Risk Neutrality:

• U''(x) = 0 (linear function)
• Indifferent between certain amount and equal-expected-value gamble
• Only expected value matters

Risk Seeking:

• U''(x) > 0 (convex function)
• Prefer gamble to certain amount with same expected value

Certainty Equivalent & Risk Premium

Certainty Equivalent (CE): Amount for certain that gives same utility as gamble

Risk Premium: Expected value of gamble - CE

Example from above:

• Gamble EV = ₹6 lakh
• CE ≈ ₹3.6 lakh (amount where U = 600)
• Risk Premium = ₹6L - ₹3.6L = ₹2.4 lakh

Investor would pay up to ₹2.4 lakh to avoid the risk!

Applications in Finance

Portfolio Theory: Investors maximize expected utility of portfolio returns, balancing risk and return based on their utility function shape.

Insurance Demand: Risk-averse individuals buy insurance even at premium above expected loss (paying for utility of certainty).

Asset Pricing: Risk premium in stock markets compensates for uncertainty—higher risk requires higher expected return to maintain utility.

Limitations of EUT

While elegant, EUT fails to describe actual behavior:

Allais Paradox: People violate independence axiom Ellsberg Paradox: Ambiguity aversion not captured Framing Effects: Identical choices framed differently yield opposite decisions Loss Aversion: Losses hurt more than gains feel good (asymmetric utility)

Prospect Theory addresses these failures by incorporating:

• Reference dependence
• Loss aversion
• Probability weighting

EUT vs Behavioral Models

Key Takeaways

• EUT: Choose option maximizing expected utility EU = Σ [p × U(x)]
• Axioms: Completeness, transitivity, independence, continuity
• Risk aversion: Concave utility function, most investors
• Applications: Portfolio theory, insurance, asset pricing
• Limitations: Violates real behavior; Prospect theory better descriptive model

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