Utility Theory and Risk Preferences

Imagine a coin toss: heads you win 90? Your answer shows how you feel about risk. This chapter gives you a clear way to talk about those feelings — a language that is the foundation of all modern asset pricing.

The Big Picture#

Every financial decision — buying a stock, saving for retirement — is a choice under uncertainty. To understand market prices and investor behavior, we first need a clear way to describe what people want and how they trade off risk and reward. Expected utility theory gives us that: a solid but easy-to-understand framework for preferences over uncertain outcomes. In this chapter, we’ll build that framework from its basic rules, learn how to measure risk aversion, look at the most important utility functions, and introduce prudence and impatience. By the end, you’ll have the lens we’ll use to look at every pricing model later.

Decision-Making Under Uncertainty: The Expected Utility Paradigm#

When outcomes are uncertain, a sensible person can’t just compare dollar amounts. Instead, she must weigh the satisfaction each possible outcome would bring, and average those satisfactions across all possible scenarios. The standard way to do this is called expected utility.

The idea is to give a utility number to every possible wealth level . Then we judge a risky prospect (a lottery) by the probability-weighted average of those utility numbers. If a lottery pays with probability , with , and so on, its expected utility is

If outcomes can be any number (continuous), we use an integral: .

Why is this a sensible way to rank gambles? In the 1940s, von Neumann and Morgenstern showed that if a person’s preferences over lotteries obey a few simple consistency rules (called axioms), then there must exist a utility function such that the person acts as if she maximizes expected utility. The axioms are:

1. Completeness: For any two lotteries, you can say which you prefer, or that you are indifferent.
2. Transitivity: If you prefer A over B and B over C, you must prefer A over C.
3. Continuity: Small changes in probabilities shouldn’t flip your preferences wildly.
4. Independence (Substitution): If you like A more than B, then mixing both with the same third lottery shouldn’t change your ranking.

The independence axiom is the key one. It says that if you mix two choices with the same third option, your preference between the original two shouldn’t change. This property forces the utility function to appear linearly in the expected-value formula.

Expected Utility Representation: When preferences satisfy the von Neumann–Morgenstern axioms, there exists a utility function , unique up to positive linear transformations ( with ), such that the person ranks lotteries by expected utility .

In this framework, the shape of encodes everything about attitudes toward risk. A linear means the person cares only about the expected payoff — she is risk-neutral. If is curved, the person is risk-averse (concave) or risk-seeking (convex). We therefore turn next to measuring that curvature.

📝 Section Recap: A few simple rules lead to expected utility: we judge a gamble by the average utility of its outcomes, weighted by their probabilities.

Measuring Risk Aversion: Certainty Equivalents and Arrow-Pratt#

A risk-averse investor has a concave utility function: . The concavity captures the idea of diminishing marginal utility — an extra dollar when you are poor adds more happiness than an extra dollar when you are rich. This makes a sure outcome more attractive than a risky one with the same expected value.

We can quantify this aversion with two simple concepts. Suppose you face a gamble that will leave you with random wealth . The certainty equivalent (CE) is the guaranteed amount of wealth that you would accept instead of the gamble:

The risk premium (π) is the difference between the expected value of the gamble and the certainty equivalent: . It is the dollar amount you are willing to “pay” to swap the risky prospect for a sure thing.

For a very small gamble — say a tiny zero-mean risk added to initial wealth — the local curvature of determines how much compensation you demand. The Arrow-Pratt measure of absolute risk aversion is

Intuitively, tells us how many extra dollars you need to be paid to accept a small fair bet when your wealth is . Higher means more risk-averse.

A related measure, relative risk aversion, scales by wealth:

It captures your dislike for a gamble that is proportional to wealth, such as a bet that pays of your holdings. A common pattern observed in the real world is decreasing absolute risk aversion (DARA): as you get richer, you become less nervous about a given dollar-sized bet, so falls. Many utility functions also exhibit constant or varying relative risk aversion.

Example: For we have , , so (DARA) and (constant relative risk aversion — CRRA).

These numbers let us compare risk attitudes across people and wealth levels.

📝 Section Recap: Risk aversion comes from a curved utility function. The Arrow-Pratt numbers turn that curvature into a risk premium for small bets. The certainty equivalent tells you the sure amount you’d accept instead of a gamble.

Common Utility Functions: The HARA Class and CRRA#

In finance models, we need concrete formulas for that are easy to work with and capture key behaviors. A very flexible family is the hyperbolic absolute risk aversion (HARA) class. It’s defined by making risk tolerance (the inverse of absolute risk aversion) a straight line in wealth:

with constants and such that .

Integrating gives, up to an affine transformation,

This family includes all the workhorse utility functions found in asset pricing. Let’s look at the most important special cases.

Constant Relative Risk Aversion (CRRA)
When and with , we get

with the limiting case when . Here relative risk aversion is constant; absolute risk aversion decreases with wealth. CRRA is the standard benchmark in macro-finance because it implies that individuals spend a constant fraction of their wealth over time and their risk attitudes scale naturally with wealth. The parameter is often called the coefficient of relative risk aversion. Higher means more cautious behaviour.

Constant Absolute Risk Aversion (CARA)
Setting and yields

with absolute risk aversion constant. CARA implies that the dollar risk premium for a given bet is independent of wealth — a property that is convenient for models but unrealistic at very high or low wealth. It also has the feature that relative risk aversion increases with wealth.

Quadratic Utility
Taking and (with ) gives

Quadratic utility is famous because, under normally distributed returns, expected utility depends only on the mean and variance of wealth. This leads directly to the classic mean-variance portfolio analysis. However, it has two important drawbacks:

• Satiation: Utility reaches a maximum at ; beyond that, more wealth actually reduces utility.
• Increasing absolute risk aversion (IARA): rises with , implying richer people become more scared of a given dollar gamble — the opposite of what we observe.

Because of these quirks, quadratic utility is rarely used today outside its historical connection to mean-variance optimization. Most modern models rely on CRRA or broader HARA specifications.

HARA Class: A family of utility functions for which risk tolerance is linear in wealth, , encompassing CRRA, CARA, and quadratic utility as special cases.

📝 Section Recap: The HARA class brings together most utility functions in finance. CRRA is the most popular because risk attitudes scale with wealth, making models easy to solve.

Higher-Order Risk Preferences: Prudence and Precautionary Saving#

Risk aversion is about how you react to a gamble right now. But what if the risk is in your future income, not your current wealth? That leads to the idea of prudence.

Suppose you do not know exactly how much you will earn next year. To guard against a bad outcome, you might save a little more today — this extra saving is called precautionary saving. Prudence is the preference that drives this behaviour. It is linked to the third derivative of the utility function.

A risk-averse person has (diminishing marginal utility). If, on top of that, marginal utility is convex — — then uncertainty about future consumption makes the expected marginal utility of future consumption higher. Why? Because in bad states (low consumption), marginal utility is very high, and convexity gives those states extra weight. Higher expected future marginal utility means you value future consumption more, so you save more today.

We measure prudence with the absolute prudence coefficient:

A positive (which reflects ) implies precautionary saving. For the ubiquitous CRRA utility with , we have and . So CRRA investors are both risk-averse and prudent. In contrast, quadratic utility has , so there is no precautionary saving motive — an important limitation.

Prudence: A preference for shifting resources toward the future in the face of income uncertainty, driven by a convex marginal utility ().

📝 Section Recap: Prudence is the reason we save extra when future income is uncertain. It comes from the third derivative of utility and is positive for standard functions like CRRA.

Time Impatience and Inada Conditions#

So far we’ve looked at one-time gambles. But finance often deals with choices over time: how much to spend today vs. invest for tomorrow. To model these decisions, we need a utility function over a stream of consumption.

The standard approach is to use time-separable expected utility with a constant discount rate. If an investor lives from to , her lifetime utility is

where is consumption at time and is the pure rate of time preference, or subjective discount rate. The exponential factor means that utility from consumption in the distant future is worth less today than the same utility from current consumption. A larger means greater impatience — the investor discounts the future more heavily. Exponential discounting has a nice property: time consistency — a plan that looks best today will still look best tomorrow.

To make sure our optimization problems have well-behaved interior solutions, we usually impose two Inada conditions on the instantaneous utility function :

The first condition says that when consumption approaches zero, marginal utility becomes infinite — you would do almost anything to avoid starvation. This guarantees that optimal consumption is always strictly positive. The second says that when you have more than enough, an extra unit adds almost nothing, so consumption is finite. CRRA utility with automatically satisfies both Inada conditions. Exponential (CARA) utility, by contrast, does not satisfy the first condition because is finite; this can lead to corner solutions where optimal consumption hits zero in some states.

Time Preference Rate (): The rate at which a person discounts future utility; a higher means stronger preference for present consumption.

Inada Conditions: Restrictions on that ensure optimal consumption stays away from boundaries: and .

📝 Section Recap: Time preferences are captured by exponential discounting at rate . Inada conditions keep consumption away from zero and infinity, giving well-behaved solutions.

Summary#

We’ve built the tools to talk clearly about investor preferences under uncertainty. Starting from a few simple rules, we got to expected utility. We learned to measure risk aversion with certainty equivalents and Arrow-Pratt numbers, looked at the most important utility functions (especially CRRA), and introduced prudence to explain precautionary saving. Finally, we added time with discounting and Inada conditions. These ideas are the language of every portfolio choice and asset pricing model.