Uncertainty, Risk, and Expected Utility Theory in Finance

Chapter 5 of Hilpisch’s book is called “Normative Finance.” And it opens with a quote from Fama and French admitting that the CAPM is built on “many unrealistic assumptions.” That’s a bold way to kick things off. Basically saying: here are the theories that shaped modern finance, and by the way, they don’t quite match reality.

But that’s the whole point. Normative theories aren’t about describing what is. They describe what should be. What rational people should do if they follow certain logical rules. Whether actual humans follow those rules is a completely different question.

This distinction matters a lot. And it’s one of those things finance textbooks sometimes gloss over.

Uncertainty and Risk: Setting the Stage

Hilpisch starts with the absolute basics of financial modeling. Picture a super simple economy. There’s today, and there’s tomorrow. Today, everything is known. Tomorrow, the world can land in one of several possible states.

That’s it. That’s the starting model.

Each possible future state gets a probability. The set of all states, plus the events we can observe, plus the probabilities assigned to them, form what’s called a probability space. This is the math foundation for everything that comes next.

Here’s a key distinction the book makes. If the probabilities are fixed and known, you’re dealing with risk. If they aren’t known, that’s uncertainty. Finance theory mostly works with risk. It assumes you know the odds. Real life, of course, is more about uncertainty. You don’t know the odds. You’re guessing.

This gap between risk and uncertainty is honestly one of the biggest blind spots in traditional finance. The models assume you can assign a number to every possible outcome. But anyone who’s lived through a black swan event knows that’s a stretch.

The Simple Two-State Economy

To make things concrete, Hilpisch uses a toy economy with just two possible future states: up and down. Each happens with 50/50 probability.

Two assets trade in this economy. A stock costs 10 today and either doubles to 20 (up state) or drops to 5 (down state). A bond also costs 10 today but pays 11 no matter what. The bond is risk-free. The stock is not.

Simple? Yes. But you can already do interesting things with this setup.

The book shows how to price a European call option using a replication argument. You find a portfolio of the stock and bond that exactly copies the option’s payoff in every state. The cost of that portfolio must be the fair price of the option. Otherwise someone could make free money, and the model assumes that can’t happen.

What’s cool about this approach is that you don’t even need the probabilities to price the option. The no-arbitrage argument works on its own. Hilpisch calls arbitrage pricing “one of the strongest financial theories” for exactly this reason. It needs very few assumptions. Just the idea that free money shouldn’t exist.

Expected Utility Theory: How Should You Choose?

Now we get to the big one. Expected Utility Theory, or EUT. This is the framework economists have used since the 1940s to model how rational agents make decisions when the future is uncertain.

The story goes back to von Neumann and Morgenstern. They laid down a small set of axioms about how a rational person’s preferences should work. If you accept these axioms, then a mathematical result follows: there exists a utility function that represents your preferences, and you should maximize its expected value.

The axioms themselves sound reasonable enough:

Completeness. You can always compare two options. You either prefer A, prefer B, or you’re indifferent. No “I don’t know” allowed.

Transitivity. If you prefer A over B and B over C, then you prefer A over C. No circular preferences.

Continuity. If A is better than B and B is better than C, there’s some mix of A and C that’s exactly as good as B.

Independence. If you’re indifferent between A and B, mixing each of them with the same third option C shouldn’t change your mind.

Dominance. If A is better than B, then a mix with more of A should be better than a mix with less of A.

These seem almost obvious. But “almost obvious” is doing a lot of work here. The independence axiom, for example, is the one that gets challenged the most. Real people violate it all the time. Look up the Allais paradox if you want to see how.

Utility Functions and Risk Aversion

Once you accept the axioms, you get a utility function. It takes a payoff and turns it into a number representing how much you “value” that payoff. The key insight is that for most people, the relationship between money and happiness isn’t linear.

Getting your first $10,000 feels amazing. Going from $990,000 to $1,000,000? Nice, but not life-changing. This diminishing marginal utility is what makes people risk-averse.

The book introduces the Arrow-Pratt measure of absolute risk aversion. It’s a formula that uses the second derivative of your utility function to measure how risk-averse you are. If the measure is positive, you’re risk-averse. Zero means risk-neutral. Negative means you’re risk-loving (hello, casino regulars).

Three common utility functions show up:

• log(x) models a risk-averse person. The classic choice.
• x (linear) models someone who’s risk-neutral. They only care about expected value.
• x squared models a risk-loving person. They actually prefer gambles.

In the numerical example, Hilpisch takes a risk-averse agent with square-root utility and two portfolios that cost the same to set up. One portfolio is heavier on the risky stock (75/25 split). The other leans toward bonds (25/75). Even though they cost the same, the risk-averse agent prefers the stock-heavy portfolio because it delivers slightly higher expected utility.

Then comes the optimization. Given a fixed budget of 10, what’s the best split between stock and bond? The math says about 61% in the risky stock and 39% in the bond. That’s the portfolio that maximizes expected utility for this particular agent with this particular utility function.

Where Theory Meets Reality

Here’s my honest take on this section. The math is clean. The logic is tight. And if you’re studying finance or building pricing models, you absolutely need to understand this stuff. It’s the foundation everything else sits on.

But the assumptions are heavy. Real people don’t have neatly defined utility functions. They don’t always rank preferences consistently. They panic-sell in crashes and FOMO-buy at peaks. Behavioral finance has documented dozens of ways humans deviate from these axioms.

Hilpisch seems aware of this. The chapter is titled “Normative Finance” for a reason. These aren’t descriptions of how people actually behave. They’re descriptions of how a perfectly logical agent should behave. The gap between “should” and “does” is where a lot of the interesting problems in modern finance live.

And that gap is exactly where AI and machine learning enter the picture later in the book. If humans aren’t rational in the way these models assume, maybe algorithms can find the patterns that normative theory misses.

Previous: Superintelligence: Forms, Paths, and the Control Problem Next: Portfolio Theory, CAPM, and Arbitrage Pricing Part of the series: Artificial Intelligence in Finance Book by Yves Hilpisch | O’Reilly 2020 | ISBN: 978-1-492-05543-3