Utility Functions and Risk Aversion

Offer someone a coin flip: win $1,000 or lose $1,000, even odds. The expected dollar outcome is exactly $0, yet most people refuse the bet, and some would pay to avoid even being offered it. A model of investing that only tracks expected dollars cannot explain that refusal. Something else is being optimized, and utility functions are the tool for writing it down.

The analogy before any symbols

Think of a hiker on a mountain trail who values altitude but hates cliffs. Every extra hundred feet of climb feels good, but a hundred-foot climb feels less thrilling once she is already near the summit than it did at the trailhead, while a hundred-foot fall near a cliff edge feels catastrophic no matter how high she started. Her "enjoyment of the hike" is not proportional to altitude; it bends. Utility works exactly like that enjoyment curve, except the horizontal axis is wealth instead of altitude, and the bending is what makes a sure $0 sometimes preferable to a risky bet that averages $0.

The mechanics

A utility function $U(W)$ converts wealth $W$ into a single number representing satisfaction. Investors are assumed to maximize expected utility, $E[U(W)]$, not expected wealth. What matters for risk attitude is the function's curvature. If $U$ is a straight line, expected utility equals utility of the expected wealth, and the investor is risk-neutral: only the average matters. If $U$ bends downward as wealth rises, mathematically $U''(W) < 0$, meaning concave, then a certain $0 outcome beats a fair gamble averaging $0, because the pain of the loss branch, evaluated on the curve, outweighs the pleasure of the equal-sized gain branch. That is risk aversion.

A common workhorse is constant relative risk aversion (CRRA):

$U(W) = \frac{W^{1-\gamma}}{1-\gamma}, \qquad \gamma > 0, \; \gamma \neq 1.$

In words: raise wealth to a power controlled by $\gamma$, then rescale. The single parameter $\gamma$ is the coefficient of relative risk aversion: larger $\gamma$ means a more sharply bending, more cautious curve. $\gamma = 1$ is the special case $U(W) = \ln W$, logarithmic utility, which is exactly the criterion behind the Kelly criterion for bet sizing.

Worked example: pricing the coin flip

Take $U(W) = \ln W$ and starting wealth $W_0 = 10{,}000$, i.e. $10,000. For the fair $1,000 coin flip: $E[U] = 0.5\ln(11{,}000) + 0.5\ln(9{,}000) = 0.5(9.3057) + 0.5(9.1050) = 9.2054$. Compare to $\ln(10{,}000) = 9.2103$, the utility of just keeping the $10,000. Expected utility of the bet is lower, so the log-utility investor declines. Converting $9.2054$ back through $e^{9.2054} = 9{,}989$ gives the certainty equivalent: this investor is exactly as happy with a guaranteed $9,989 as with the coin flip, so she would pay about $11 to avoid the gamble entirely. That gap between $10,000 and $9,989 is the risk premium curvature buys.

Worked example: risk aversion sets the risky weight

For CRRA utility and one risky asset with expected excess return $\mu$ and variance $\sigma^2$, the optimal risky weight is approximately $w^\* = \mu / (\gamma\sigma^2)$. With $\mu = 6\%$ and $\sigma = 20\%$ (so $\sigma^2 = 0.04$): an investor with $\gamma = 2$ holds $w^\* = 0.06/(2\times0.04) = 75\%$ in the risky asset. A more cautious investor with $\gamma = 6$ holds $w^\* = 0.06/(6\times0.04) = 25\%$. Same market, same return forecast, wildly different allocation, purely from one parameter's value.

What this means in practice

Mean-variance optimization, the workhorse of two-fund separation, is itself a special case: it is exact expected-utility maximization when returns are normal or utility is quadratic, and only an approximation otherwise. Risk-aversion coefficients are also how "risk tolerance questionnaires" at a robo-advisor get translated into a number that actually drives an allocation, and how a trading desk sizes a single risky bet, rather than a whole portfolio, using the same curvature logic.