Chapter 2: Measuring Risk Aversion and Utility Representations#

We all know that most people dislike risk. But how much? In this chapter, we build a toolkit to measure risk aversion—turning a fuzzy feeling into concrete numbers. You’ll learn to read utility functions like a doctor reads a heartbeat. You’ll see whether someone gets more or less cautious as they grow wealthier, and why that matters for everything from investing to insurance.

The Big Picture#

Last chapter introduced expected utility as a way to make choices under uncertainty. But simply knowing that “people are risk averse” isn’t enough. We need to measure how risk averse someone is—how steeply their happiness curve bends as wealth changes. This chapter builds local curvature measures (absolute and relative risk aversion), shows how they predict the price someone would pay to avoid a small gamble, and then tours the most important families of utility functions economists actually use. By the end, you’ll be able to describe someone’s risk feelings with just one or two numbers, and understand why a billionaire and a student might hold totally different portfolios.

The Arrow-Pratt Measures: Absolute and Relative Risk Aversion#

Imagine you have a utility function that tells you how much satisfaction (utility) you get from wealth . If you’re risk averse, is concave—each extra dollar adds less happiness than the last one. But concavity alone doesn’t tell you how much you’d pay to avoid a given risk. For that we need a number.

Local curvature scaled by marginal utility#

Think about driving along a winding road. How hard you steer depends on two things: how sharp the bend is (curvature) and how fast you’re going (slope). In utility language, speed is your marginal utility —how much an extra dollar matters right now. So a useful measure of risk attitude at wealth is the curvature divided by the slope:

Coefficient of absolute risk aversion (ARA):

We put a minus sign because is negative for a risk‑averse (concave) utility, making positive. A larger means you’re more frightened of a given dollar risk at that wealth level.

Why divide by marginal utility? If you’re very poor, is huge—losing a dollar stings a lot. captures this “dollar‑risk attitude” cleanly, no matter the units we use to measure happiness.

Absolute refers to a fixed‑dollar change, say losing 100$100$200$. So we also define:

Coefficient of relative risk aversion (RRA):

measures aversion to a proportional gamble—like a risk that changes your wealth by 1%. If stays constant as wealth grows, your dislike for a given percentage bet stays the same.

Example: Take log utility . Then , , so , and . So log utility has constant relative risk aversion equal to 1: no matter how rich you become, you dislike a proportional gamble just as much.

📝 Section Recap: Absolute risk aversion measures how much you hate a fixed‑dollar gamble at a given wealth. Relative risk aversion measures how much you hate a percentage gamble. Both are local curvature divided by marginal utility.

Risk Tolerance and Small Gambles#

A very handy number is risk tolerance, which is simply the inverse of absolute risk aversion:

Risk tolerance:

It answers: “At my current wealth, how large a dollar risk would I be comfortable with before I start feeling uneasy?” The units are dollars, making it easy to grasp. High risk tolerance means you’re less risk averse.

How much for a tiny gamble? A variance‑based premium#

Imagine someone with wealth faces a small, fair gamble with mean zero and variance . How much would she pay (the risk premium ) to get rid of that risk? Arrow and Pratt found a neat approximation:

The premium is proportional to the variance and to the local absolute risk aversion. That’s why we call it second‑order risk aversion: for a tiny gamble, the amount you dislike it scales with the square of the risk’s size (the variance). If you halve the standard deviation, the premium drops to one‑quarter, not one‑half. So people are almost risk‑neutral toward very tiny risks, but risk aversion kicks in more noticeably as the gamble grows.

Example: Suppose you have log utility, so , and wealth 100{,}000A=0.00001$5{,}000(5000)^2 = 25{,}000{,}000$. The approximate premium is

So you’d pay about 125$5{,}000$125.3$, confirming the approximation works beautifully for small risks.

📝 Section Recap: Risk tolerance is the inverse of ARA and tells you how many dollars of risk you can handle comfortably. For any small fair gamble, the risk premium you’d pay is about times the gamble’s variance—a cornerstone of risk analysis.

The HARA Family: A Spectrum of Risk Preferences#

We want utility functions that are easy to work with and can be tuned to fit real people. The most flexible and widely used family is the hyperbolic absolute risk aversion (HARA) family, also called linear risk tolerance (LRT) utilities. Its defining property: risk tolerance is a straight‑line function of wealth.

where and are constants. Equivalently, . This simple form contains almost every famous utility function.

Special case 1: CARA — constant absolute risk aversion#

Set . Then is constant, so is constant. The utility function that gives this is the negative exponential:

Here . This is called CARA (constant absolute risk aversion). A CARA person pays the same dollar amount to avoid a given absolute risk whether she’s rich or poor—there is no wealth effect. If you’d pay 50$100$1{,}000$50$1{,}000{,}000$. That’s rarely realistic for everyday decisions, but it makes many theoretical models very tidy.

Special case 2: CRRA — constant relative risk aversion#

Set . Then , risk tolerance grows with wealth, and ARA falls as you get richer: . Relative risk aversion is , a constant. The utility is

(or when ). Here is exactly the constant RRA. This is the CRRA (constant relative risk aversion) family—also called power utility. A CRRA investor keeps the same fraction of wealth in risky assets no matter how rich she is. For example, if , she might always put 40% of her money in stocks and 60% in bonds, whether she has 10{,}000$10$ million. This matches the rough real‑world fact that portfolio shares, not dollar amounts, seem to drive asset allocation.

Shifted CRRA: a subsistence level#

A small tweak: let risk tolerance be , so . The utility becomes the shifted power utility:

and shifted log for . Here acts as a subsistence or minimum required wealth. As your wealth approaches from above, ARA blows up—you become extremely risk averse. This captures the idea that you need a certain living standard just to survive.

Quadratic utility: the mean–variance workhorse#

Set . Then , so . The utility that fits this is the quadratic:

where . A problem: after wealth goes past , marginal utility turns negative—more money actually makes you unhappier. So quadratic utility only makes sense for wealth below a certain “bliss point.” Its big appeal is that expected utility depends only on the mean and variance of returns (since ). This connection makes quadratic utility the foundation of classic mean–variance portfolio theory. However, its increasing absolute risk aversion ( goes up with wealth) is strongly counterfactual—in reality, richer people typically take more absolute risk, not less.

📝 Section Recap: The HARA class, with linear risk tolerance , generates all the classic utility families: CARA (, constant dollar‑risk aversion), CRRA (, constant percentage‑risk aversion), shifted CRRA (with a subsistence point), and quadratic (, mean–variance preferences). Just two numbers— and —describe a rich range of risk behaviors.

Signs and Insights from Higher Derivatives#

So far we only looked at the first two derivatives. But real risk attitudes often need derivative number three.

Prudence and decreasing absolute risk aversion#

If , you show decreasing absolute risk aversion (DARA)—as you grow wealthier, you become willing to take larger absolute gambles. This is widely regarded as normal. DARA is linked to a positive third derivative, . A positive means your marginal utility curve is convex: a given loss reduces your marginal utility more than an equal gain increases it. This trait is called prudence:

Absolute prudence:

If , then . A prudent person, facing uncertain future income, will save extra today as a safety buffer—the phenomenon of precautionary saving. Many people also show decreasing absolute prudence (DAP), meaning , so the extra saving motive gets weaker at higher wealth.

DARA, positive prudence, and decreasing prudence form a standard package of plausible assumptions: you become less risk averse as you get richer, but also less worried about future uncertainties.

Expected utility as a moment series#

We can also express expected utility in terms of the central moments (variance, skewness, etc.) of the wealth distribution. Start with a Taylor expansion of around expected wealth :

Take expectations. The first‑order term disappears. So

For a risk‑averse person (), higher variance hurts. But if the distribution has positive skewness (like a lottery ticket) and your third derivative is positive (), you actually like that skewness. More generally, this expansion shows that mean–variance is only an exact description of preferences if utility is quadratic (so all higher derivatives vanish). For other utilities, skewness and even higher moments matter.

📝 Section Recap: Higher derivatives capture richer attitudes: (positive prudence) explains precautionary saving and goes hand‑in‑hand with DARA. Expanding expected utility in moments reveals that investors care about variance, skewness, and beyond—a reminder that mean–variance analysis is an approximation, exact only for quadratic utility.

Mean Independence and the Anatomy of Risk Aversion#

Sometimes a gamble has nothing to do with your other wealth. A coin flip’s outcome doesn’t depend on your salary, for example. This is mean independence: the expected value of the gamble doesn’t change when you know your background wealth. Under that condition, concavity alone is enough to guarantee you dislike a fair gamble.

In symbols, suppose your total wealth is , where is a fair gamble (zero mean) that is mean‑independent of . Then

if is concave. The inequality holds because, conditional on , you face a fair gamble and would reject it. Averaging over all possible values of , your expected utility falls.

This result is comforting: under mild independence conditions, the local curvature measures and cleanly capture the risk premium—no extra complications from background risk.

📝 Section Recap: Mean independence of a gamble from background wealth guarantees that a concave utility function always lowers welfare compared to the expected wealth. The risk premium then depends only on the local curvature measure—no extra complexities.

Summary#

In this chapter, we built a language to talk about risk aversion in numbers. Absolute risk aversion tells you how much dollar risk someone hates; relative risk aversion measures fear of percentage gambles. The HARA family gives us useful shapes like CARA and CRRA, each described by just two parameters. Prudence explains why people save extra for a rainy day. And the moment expansion reminds us that mean and variance aren’t the whole story. These tools are the foundation for making smart portfolio choices and understanding asset prices.