Quantifying Risky Prospects with Expected Utility

1. The Decision Rule: Maximizing Expected Utility

When faced with uncertainty, a simple approach might be to choose the investment with the highest expected monetary payoff. However, as we saw last lesson, a risk-averse person cares about more than just the average outcome; they are sensitive to the downside.

The Expected Utility Theorem formalizes the decision-making process for a rational, risk-averse individual. It states that when choosing among risky prospects, an individual will choose the option that yields the highest expected utility.

For a lottery or gamble $L$ that has a set of possible final wealth outcomes $(W_1,W_2,...,W_n)$, each occurring with a known probability $(p_1,p_2,...,p_n)$, the expected utility is calculated as the probability-weighted average of the utility of each outcome:

The decision rule is simple: given a choice between two lotteries, $A$ and $B$, you calculate $E[U(A)]$ and $E[U(B)]$. You choose A if $E[U(A)]>E[U(B)]$, and B if the reverse is true.

This framework is built upon the axioms of completeness and transitivity we discussed previously, plus one more: the Independence Axiom. In essence, this axiom states that if you prefer investment A over B, your preference shouldn't change if you mix both A and B with some other irrelevant third option. The following video provides a concise overview of this principle and the core idea of maximizing expected utility.

2. A Numerical Example: To Bet or Not to Bet?

Let's make this concrete with an example. Suppose you are an expected utility maximizer with an initial wealth of $300 and a logarithmic utility function, $U(W)=\ln (W)$. You are offered a bet with the following terms:

• A 60% probability of winning $150.
• A 40% probability of losing $100.

Should you accept this bet?

To answer this, we compare the expected utility of taking the bet with the utility of refusing it (i.e., keeping your current wealth).

Step 1: Calculate the utility of NOT taking the bet.
If you refuse the bet, your wealth remains $300. The utility of this certain outcome is:

Step 2: Calculate the potential outcomes and their utilities if you take the bet.

• Win: Your final wealth would be $W_{win}=300+150=450$. The utility of this outcome is $U(450)=\ln (450)\approx 6.109$.
• Lose: Your final wealth would be $W_{lose}=300-100=200$. The utility of this outcome is $U(200)=\ln (200)\approx 5.298$.

Step 3: Calculate the expected utility of taking the bet.
Using the expected utility formula, with $p_{win}=0.6$ and $p_{lose}=0.4$:

Step 4: Compare and decide.

• Utility of not betting: $U(\text{No Bet})\approx 5.704$
• Expected utility of betting: $E[U(\text{Bet})]\approx 5.784$

Since $E[U(\text{Bet})]>U(\text{No Bet})$, a rational investor with this utility function and wealth would accept the bet.

Notice that the expected monetary value of the bet is positive: $(0.6\times 150)+(0.4\times -100)=90-40=\$50$. A risk-neutral person would obviously take this bet. Our calculation shows that for a risk-averse person with a log utility function, the bet is still attractive enough to overcome the risk.

This type of problem is very common. The following resource provides more examples for you to work through.

3. A Critical Distinction: $E[U(W)]$ vs. $U(E[W])$

A common and serious mistake is to confuse the expected utility of wealth with the utility of expected wealth.

• Correct: Expected Utility of Wealth, $E[U(W)]=\sum p_{i}U(W_i)$. We just calculated this. You find the utility of each outcome first, then take the expectation.
• Incorrect: Utility of Expected Wealth, $U(E[W])$. This involves calculating the expected monetary outcome first, and then finding its utility.

For our example bet:
The expected wealth is $E[W]=0.6\times 450+0.4\times 200=270+80=350$.
The utility of this expected wealth is $U(E[W])=U(350)=\ln (350)\approx 5.858$.

Notice that for this risk-averse investor, $E[U(W)]\approx 5.784$ and $U(E[W])\approx 5.858$. They are not the same. The decision must always be based on $E[U(W)]$.

The following video provides another worked example and explicitly highlights this common error.

4. The Graphical View: Jensen's Inequality

The inequality $E[U(W)]\le U(E[W])$ is not a coincidence. It is a mathematical property known as Jensen's Inequality, which holds for any concave function. Since risk aversion implies a concave utility function, this inequality is the graphical signature of a risk-averse individual.

Let's visualize this with the utility curve for a risk-averse investor.

Imagine a gamble between a low wealth outcome (like 10 on the chart) and a high wealth outcome (like 20).

1. Expected Wealth, $E[W]$: If the probabilities were 50/50, the expected wealth would be the midpoint, 15.
2. Utility of Expected Wealth, $U(E[W])$: Find 15 on the wealth axis, go up to the curve, and read the utility on the vertical axis. This corresponds to point A, with a utility of 55.
3. Expected Utility, $E[U(W)]$: Find the utilities of the two outcomes (utility at 10 is 45; utility at 20 is 65). The expected utility is the probability-weighted average of these values. For 50/50 odds, it would be the midpoint of the straight line connecting the two points on the curve: $(0.5\times 45)+(0.5\times 65)=55$.
   Wait, in this specific diagram, the expected utility equals the utility of the expected outcome. This is only because the points A, B, and the initial point are chosen to lie on a straight line segment of the curve for illustration. In general, for a strictly concave curve, the straight line connecting two points will always lie below the curve itself.

Let's reconsider with points (10, 45) and (30, 75).

• The expected wealth (for 50/50 odds) is $(10+30)/2=20$.
• The utility of this expected wealth is $U(20)=65$.
• The expected utility is the average of the endpoint utilities: $(45+75)/2=60$.

Here we see the classic result for a risk-averse person: $E[U(W)]=60<U(E[W])=65$. This gap represents the individual's aversion to risk. They prefer the utility of the average outcome for certain over the average utility of the actual risky outcomes.

Conclusion

In this lesson, we have put the utility function to work as a decision-making tool. We've established a clear, numerical method for ranking risky choices, moving beyond simple intuition.

Here are the key takeaways:

• The Expected Utility Theorem states that rational individuals choose the option that maximizes their expected utility.
• The expected utility of a gamble is the probability-weighted average of the utilities of all possible wealth outcomes: $E[U(W)]=\sum p_{i}U(W_i)$.
• It is crucial to distinguish between the expected utility of wealth ($E[U(W)]$) and the utility of expected wealth ($U(E[W])$). Decisions are always based on the former.
• For a risk-averse individual, Jensen's Inequality tells us that $E[U(W)]\le U(E[W])$, visually demonstrating their preference for a sure thing over a gamble with the same expected value.

Now that we can calculate the expected utility of a gamble, our next step is to quantify exactly how much that gamble is worth to an investor. In the next lesson, we will calculate the certainty equivalent—the guaranteed sum of money that provides the same satisfaction as the risky prospect—and the risk premium, which is the "price" of bearing risk.