Expected Utility Calculator

Understanding expected utility theory in financial decision-making

In standard economic analysis and modern portfolio management, evaluating financial choices solely by their expected monetary value often leads to paradoxical conclusions. If people only cared about the mathematical average payout, an individual would treat a guaranteed $50,000 payout identically to a coin flip between $0 and $100,000. In practice, almost everyone prefers the certainty of $50,000 over a high-stakes gamble with the same mathematical mean.

To resolve this dilemma, mathematician John von Neumann and economist Oskar Morgenstern developed Expected Utility Theory (EUT). Rather than assuming people maximize expected dollars, EUT models rational agents as maximizing expected utility: the subjective satisfaction or psychological value derived from different wealth levels. While the expected return calculator helps you find the probability-weighted financial return of an investment portfolio, this Expected Utility Calculator quantifies how different attitudes toward risk transform those raw returns into personal certainty equivalents and risk premiums.

The expected utility formula

Consider an uncertain financial prospect or lottery with $n$ mutually exclusive outcomes. Each outcome yields a final wealth state $W_i$ with an assigned probability $p_i$, where all probabilities sum to 100% ($\sum p_i = 1$).

Under the von Neumann-Morgenstern utility theorem, the expected utility of the risky prospect is the probability-weighted sum of the utility obtained from each possible wealth state:

Here, $U(W)$ represents the individual utility function, which maps dollar wealth to a dimensionless scale of subjective value known as utils.

Utility functions and attitudes toward risk

The mathematical curvature of an individual utility function reveals their underlying attitude toward uncertainty. Economists categorize decision-makers into three distinct risk profiles:

1. Concave Utility Functions (Risk Aversion)

A utility function is concave if its second derivative is negative ($U''(W) < 0$). This reflects the principle of diminishing marginal utility of wealth: each additional dollar gained provides less satisfaction than the previous dollar, while losing a dollar causes more pain than gaining that same dollar brings pleasure.

• Square Root Utility ($U(W) = \sqrt{W}$): A classic utility function modeling moderate risk aversion. Its inverse function is $W = (U)^2$.
• Natural Logarithmic Utility ($U(W) = \ln(W)$): First proposed by Daniel Bernoulli in 1738 to solve the St. Petersburg paradox. Logarithmic utility exhibits Constant Relative Risk Aversion (CRRA) with a coefficient of relative risk aversion equal to 1. Its inverse function is $W = e^U$.

2. Linear Utility Functions (Risk Neutrality)

A linear utility function takes the form $U(W) = aW + b$ (typically simplified to $U(W) = W$), meaning its second derivative is zero ($U''(W) = 0$). A risk-neutral investor evaluates gambles strictly on expected monetary value without demanding any risk discount. Risk neutrality is commonly assumed in pricing corporate options and derivatives via risk-neutral probabilities.

3. Convex Utility Functions (Risk Seeking)

A utility function is convex when its second derivative is positive ($U''(W) > 0$, such as $U(W) = W^2$). A risk-seeking person experiences increasing marginal utility and prefers a risky gamble over receiving its expected value with certainty.

Jensen Inequality: why concavity produces risk aversion

The mathematical foundation of risk aversion is formalized by Jensen Inequality. For any strictly concave function $U(W)$ and non-degenerate random variable $W$:

This equation demonstrates that the expected utility of an uncertain payoff distribution is strictly less than the utility derived from receiving the average payoff with 100% certainty. The gap between $U(E[W])$ and $E[U(W)]$ represents the welfare loss caused by uncertainty.

Certainty equivalent and the risk premium

Two practical metrics allow financial planners and economists to translate abstract utils back into tangible dollars:

Step-by-step worked examples

Example 1: A Standard Fifty-Fifty Gamble

Suppose an individual with zero initial wealth ($W_0 = \$0$) is offered a lottery ticket paying $100 with a 50% chance and $0 with a 50% chance. We assume a square root utility function $U(W) = \sqrt{W}$:

1. Expected Wealth: $E[W] = (0.50 \times \$100) + (0.50 \times \$0) = \$50$.
2. Outcome Utilities: $U(\$100) = \sqrt{100} = 10\text{ utils}$ and $U(\$0) = \sqrt{0} = 0\text{ utils}$.
3. Expected Utility: $E[U(W)] = (0.50 \times 10) + (0.50 \times 0) = 5.0\text{ utils}$.
4. Certainty Equivalent: $\mathrm{CE} = (5.0)^2 = \$25.00$.
5. Risk Premium: $\mathrm{RP} = \$50.00 - \$25.00 = \$25.00$ (50% of expected wealth).

Even though the average payout is $50, this risk-averse person values the lottery at only $25 in guaranteed cash.

Example 2: An Investment Opportunity with Background Wealth

Now consider an investor with existing liquid wealth of $10,000 ($W_0 = \$10,000$). They evaluate a venture opportunity offering a 50% probability of gaining $6,000 (total wealth $16,000) versus a 50% probability of losing $4,000 (total wealth $6,000). The expected gain is $(0.5 \times 6,000) - (0.5 \times 4,000) = +\$1,000$, giving an expected wealth of $11,000.

Comparing square root utility and logarithmic utility for this proposition:

• Under Square Root Utility: $U(\$16,000) = \sqrt{16,000} \approx 126.491$ utils and $U(\$6,000) = \sqrt{6,000} \approx 77.460$ utils. The expected utility is $E[U] = 0.5(126.491) + 0.5(77.460) = 101.975$ utils. The certainty equivalent is $\mathrm{CE} = (101.975)^2 = \$10,398.98$. Because the CE exceeds baseline wealth of $10,000 by $398.98, this investor accepts the opportunity.
• Under Logarithmic Utility: $U(\$16,000) = \ln(16,000) \approx 9.6803$ utils and $U(\$6,000) = \ln(6,000) \approx 8.6995$ utils. The expected utility is $E[U] = 0.5(9.6803) + 0.5(8.6995) = 9.1899$ utils. The certainty equivalent is $\mathrm{CE} = e^{9.1899} = \sqrt{16,000 \times 6,000} \approx \$9,797.96$.

Notice this remarkable result: under logarithmic utility, the certainty equivalent ($9,797.96) is less than the investor starting capital ($10,000). Despite a positive expected monetary gain of $1,000, the more risk-averse logarithmic investor rationally rejects the investment because the pain of losing $4,000 outweighs the joy of making $6,000.

Real-world applications

• Insurance Demand: An individual willingly pays a homeowner premium greater than the actuarial expected claim cost because avoiding catastrophe protects them from the steep negative utility of severe capital destruction.
• Asset Allocation: Long-term compound returns (tracked via the CAGR calculator) matter, but expected utility explains why investors hold bonds and cash even when stocks offer higher historical averages: downside preservation protects utility.
• Executive Compensation: Stock options align incentives by creating convex payoff structures, counteracting the natural risk aversion of corporate executives who might otherwise pass on high-NPV risky ventures.

Frequently asked questions