MATH2070 Chap.8 Utility and Risk Aversion
Utility and Risk Aversion
Utility theory explains choices that expected monetary return cannot. Expected utility averages utility across possible final wealth outcomes. Increasing concave utility represents risk aversion and makes the certainty equivalent fall below expected wealth. Their difference is the risk premium. Absolute and relative risk-aversion measures compare curvature after scaling by marginal utility.
Domain conditions matter, and utility numbers cannot be compared across investors as if they were money. Final wealth must be formed before applying the utility function, because initial resources can change the significance of the same payoff. Expected utility and utility of expected wealth are different operations, with Jensen's inequality explaining their order under concavity.
The certainty equivalent is obtained by applying the inverse utility function after averaging utilities. Positive affine transformations preserve one investor's rankings, but raw utility levels have no direct interpersonal or monetary interpretation. Beliefs enter through probabilities, while preferences enter through curvature.
What this chapter covers
- 01
Expected return principle
- 02
St Petersburg paradox
- 03
Expected utility
- 04
Concave utility
- 05
Certainty equivalents
- 06
Risk premiums
- 07
Risk-aversion measures
Worked example · free
Value a two-state gamble
- 1Compute expected final wealth.
- 1Average utility across states.
- 1Invert utility for the certainty equivalent.
- 1Subtract to find the risk premium.
Key terms
- Expected utility
- The probability-weighted average utility of possible final wealth outcomes.
- Certainty equivalent
- Certain wealth producing the same utility as a risky prospect.
- Risk premium
- Expected wealth minus certainty-equivalent wealth for a risky prospect.
- Risk aversion
- Preference curvature under which utility is increasing and concave in wealth.
- Final wealth
- The investor's initial wealth adjusted by the payoff in a particular state. Utility is normally applied to this state-dependent amount, subject to the function's domain, before utilities are averaged.
- Affine transformation
- A change from one utility representation to a positive constant plus a positive multiple of it. Such a transformation preserves expected-utility rankings even though the displayed utility numbers change. A negative multiplier would reverse preferences, while a nonlinear transformation need not preserve expected-utility choices.
Utility and Risk Aversion FAQ
Why can a risk-averse investor reject a fair gamble?
Concave utility makes the utility loss from downside wealth larger than the utility gain from an equal upside change. Expected monetary gain can therefore be zero while expected utility falls.
How is the certainty equivalent calculated?
Compute expected utility across states and then apply the inverse utility function. Do not take utility of expected wealth, because concavity makes that a different and usually larger value.
Can utility values be compared between two people?
Not as cardinal welfare or money. Positive affine transformations preserve one person's rankings, and different people may use different functions and wealth levels. Compare each person's prospects within a consistent utility model.
Exam move
Draw a pipeline for each risky prospect: initial wealth and payoffs produce final-wealth states; final wealth enters utility; probabilities average those utilities; inverse utility produces the certainty equivalent. Keep expected monetary wealth in a separate calculation. This order prevents the common substitution of utility of expected wealth for expected utility.
Check the utility domain state by state before taking logarithms, roots or powers. Predict the ranking before calculating. Increasing utility preserves preference for more wealth in a certain comparison, while concavity makes a risk-averse investor prefer smoothing. For a concave function, expect the certainty equivalent to lie below expected wealth and the risk premium to be non-negative.
If the numbers contradict that prediction, inspect the probability weights, wealth baseline and inverse-function step rather than forcing the interpretation. Practise curvature classification from derivatives and from graphs. Identify increasing concave, linear and convex cases, then calculate absolute and relative risk-aversion measures with their domains.
Use positive affine transformations to confirm that rankings remain unchanged, but do not compare raw utility levels across different investors. Include pairs of investors with different wealth or curvature so that you can explain why the same gamble may receive different certainty equivalents. For exam-style work, show the state values, expected utility equation, inversion and monetary comparison on separate lines.
Report certainty equivalent in currency and risk premium as expected wealth minus that amount. Add one sentence linking the sign to risk attitude. Keep an error log for final wealth, domain, averaging order, inverse utility and interpretation. Revisit calculations with altered probabilities and ask which part changes: beliefs affect the weights, while preferences remain in the utility curve.
This separation prepares the transition from individual risky choices to portfolio selection.
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