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Measuring More Than Money
Quantitative Analysis · Chapter 6: Bayesian Revision and UtilitySection 6.5 · lesson 30 of 5512 min · free with a free account
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In this lesson
Applying utility theory to complex business choices.
- Read a risk attitude off the shape of a utility function
- Define certainty equivalent, risk premium and expected utility
- Elicit utilities with the Standard Gamble Method
- Rank alternatives by expected utility instead of EMV
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Bayesian Revision and Utility · Section 6.5
Measuring More Than Money
Expected monetary value looks like the complete answer to a decision problem. It is not. EMV quietly assumes one thing about the decision-maker. It assumes they are perfectly risk-neutral. That a gamble is worth exactly its average payoff, no matter how wild the spread. Real managers do not behave that way, and they are not being foolish. They will give up expected value to avoid an outcome the firm cannot survive. Utility theory is the formal machinery for that trade-off. In this section you will build a utility function, and read risk attitude off its shape. You will define certainty equivalents and risk premiums, and compute expected utility. Then you will watch expected utility reverse an EMV ranking. Why it reverses is the whole point.
The Hidden Assumption
What EMV Quietly Takes for Granted
EMV maximisation is a powerful decision rule. But it rests on one critical assumption. The assumption is that the decision-maker is perfectly risk-neutral. Risk-neutral means indifferent between a sure amount, and any gamble with the same average. In practice, managers and firms are often risk-averse. They willingly sacrifice some expected value to reduce the chance of a catastrophic outcome. Notice the word sacrifice. It is deliberate. A risk-averse firm knows it is giving up expected dollars, and pays that price anyway. Utility theory gives us the formal framework to model those preferences. And to say which alternative a rational but risk-averse decision-maker should choose. Rational is the key word there. Risk aversion is a preference, not a mistake.
Risk Attitudes
The Shape of the Curve Is the Attitude
A utility function maps every possible monetary outcome to a single number. That number is called utility, and it is measured in utils. Utils are not dollars. They are a scale of satisfaction. The shape of the function is where the risk attitude actually lives. A concave function rises, but flattens as money increases. Each extra dollar adds less utility than the one before it. That is diminishing marginal utility, and it is the signature of risk aversion. For a risk-averse agent, the certainty equivalent sits below the expected monetary value. A linear function is the risk-neutral case. Every dollar is worth the same. There, maximising utility and maximising EMV are the same operation. A convex function is risk-seeking, with increasing marginal utility, and a certainty equivalent above the EMV.
Pricing The Attitude
Certainty Equivalent and Risk Premium
Two quantities describe how a risk-averse agent values a risky prospect. The first is the certainty equivalent. It is the guaranteed dollar amount that makes the decision-maker indifferent. Indifferent between taking that sure amount, and playing the risky gamble. For a risk-averse agent, the certainty equivalent is below the expected monetary value. The second quantity is the risk premium. Risk premium equals expected monetary value, minus certainty equivalent. Read it as a price. It is the most the decision-maker will pay to avoid the risk entirely. That is the same logic that makes an insurance premium worth paying. The third quantity is expected utility itself, the sum of probability times the utility of each payoff. And the rule is simple. Choose the highest expected utility, not necessarily the highest EMV.
Eliciting Utilities
Constructing U(x): The Standard Gamble
Before you can compute expected utility, you need the utility function. It does not arrive from nowhere. It has to be elicited from the decision-maker. The most widely used approach is the standard gamble method. It derives utility values by exploiting indifference comparisons. Step one, anchor the extremes. Assign zero utility to the worst outcome, and one hundred to the best. Step two, pose the indifference question for each intermediate payoff. What probability of the best outcome makes you indifferent to taking that amount for certain? Step three, record the answer. At indifference, utility equals one hundred times that probability. Step four, repeat across the payoff range to trace the full curve. Step five, fit a smooth function to those points, and apply it to every alternative. The anchors of zero and one hundred are chosen for convenience, and they do not change the ranking.
Worked Example
Two Capital Projects, One Risk-Averse Firm
Now the worked example. This is the heart of the section. A risk-averse firm must choose between two capital investment projects. Its utility function has already been estimated. Utility of x equals one hundred, times the square root of x divided by six hundred thousand. That is a concave square-root form, consistent with risk aversion. Six hundred thousand is the best payoff on the table, so it anchors the scale at one hundred. Project A has three outcomes. A fifty per cent chance of three hundred thousand dollars, thirty per cent of two hundred thousand, twenty per cent of fifty thousand. Project B is wider on both ends. Forty per cent of six hundred thousand dollars, and thirty per cent of two hundred thousand. Then thirty per cent of a loss of one hundred and fifty thousand dollars. That loss is the number to watch. Project A has no losing state at all.
Money-Only Baseline
The EMV Ranking Favours Project B
Take the money-only view first, so we have a baseline to compare against. Expected monetary value is the sum, over all states, of probability times payoff. For Project A, the three terms are one hundred and fifty thousand, sixty thousand, and ten thousand. That totals two hundred and twenty thousand dollars. For Project B, the good state contributes two hundred and forty thousand. The moderate state adds sixty thousand. And the poor state subtracts forty-five thousand, because that payoff is negative. Project B's expected monetary value is two hundred and fifty-five thousand dollars. So on money alone, Project B wins by thirty-five thousand dollars. A risk-neutral firm stops right here and picks B. But look at what the average hides. EMV ignores variance, and expected utility captures it.
Payoffs Into Utils
Applying the Utility Function
The next step converts every distinct payoff into utils. Divide the payoff by six hundred thousand, take the square root, then multiply by one hundred. Six hundred thousand dollars gives a ratio of one, and a utility of one hundred. Three hundred thousand gives a ratio of point five zero zero zero. Its square root is point seven zero seven one, so the utility is seventy point seven one. Stop on that for a second. Half the money, but seventy-one per cent of the utility. That gap is diminishing marginal utility, made concrete. Two hundred thousand converts to fifty-seven point seven four utils. Fifty thousand converts to twenty-eight point eight seven utils. The negative payoff needs a ruling from the analyst. A loss of one hundred and fifty thousand lies below the worst reference outcome, so utility zero is assigned. That floor is a modelling choice, and it is worth stating out loud.
Expected Utility
Weighting Utils by Probability
Now weight those utilities by their probabilities. For Project A, point five zero times seventy point seven one gives thirty-five point three five five. Point three zero times fifty-seven point seven four gives seventeen point three two two. Point two zero times twenty-eight point eight seven gives five point seven seven four. Add the three, and Project A has an expected utility of fifty-eight point four five utils. For Project B, point four zero times one hundred gives forty. The moderate state again contributes seventeen point three two two. And the poor state contributes nothing at all, because its utility is zero. Project B's expected utility is fifty-seven point three two utils. The margin is one point one three utils. Narrow, but decisive. The ranking has flipped. On money B led, and on utility A leads.
Decision
Choose A, and Know Why
Put the two rankings side by side. On expected monetary value, Project B beats Project A. On expected utility, Project A beats Project B. The risk-averse firm should select Project A. Even though its expected monetary value is thirty-five thousand dollars lower. So why does the ranking reverse? Because of the possible loss of one hundred and fifty thousand dollars in Project B. That outcome carries a severe utility penalty, and the penalty outweighs the higher monetary gain. Look at where the damage happens. Project B's best state is capped at one hundred utils, the very top of the scale. Its poor state is floored at zero, and carries a thirty per cent weight. Concavity punishes the downside harder than it rewards the upside.
When It Matters
When Utility Analysis Changes the Decision
Utility analysis takes real effort, so when is it worth doing? First, on high-stakes, irreversible decisions. When a bad outcome could threaten organisational survival, the consequences are asymmetric. Then variance matters as much as the mean. EMV ignores variance. Expected utility captures it. Second, whenever the dispersion of returns is very large. Even with a clear EMV ranking, a wide standard deviation should send you to expected utility. It accounts for downside exposure that the average quietly conceals. Third, and this one surprises people, decisions taken under desperation. A firm in financial distress may become risk-seeking. It accepts a low-probability, high-payoff gamble, because the baseline, bankruptcy, is already catastrophic. Utility theory models that reversal explicitly, using a convex function.
Key Takeaways
Formulae and What to Carry Forward
Four formulae carry this section. Expected utility is the sum, over all states, of probability times the utility of the payoff. Choose the alternative that maximises it. Risk premium is expected monetary value minus certainty equivalent. It is positive for a risk-averse agent, zero for a risk-neutral one, and negative for a risk-seeker. The standard gamble formula gives the utility of a payoff at the indifference point. And expected monetary value stays the reference case. Maximise it only if you are risk-neutral. Four things to carry forward. The shape of the utility function is the risk attitude. Concave, linear, or convex. The certainty equivalent and the risk premium put a dollar price on that attitude. And the standard gamble is how those utility values get elicited, anchored at zero and one hundred. And for a risk-averse decision-maker, expected utility can reverse the EMV ranking, especially when the downside is severe.
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