ETF5952 · Quantitative Methods for Risk Analysis
Utility Theory and Decision-Making Under Risk
In ETF5952 Quantitative Methods for Risk Analysis at Monash University, this chapter is the bridge from describing risk to acting on it: how a rational investor turns a whole distribution of returns into a single ranked choice. It develops expected utility, Jensen's inequality, the equivalence between a concave utility function and risk aversion, the certainty equivalent and risk premium, and the Arrow-Pratt coefficient of absolute risk aversion. It supports Learning Outcome 2 (how risk affects investment decisions) and sits in Week 2 of the quantitative risk-management sequence.
What this chapter covers
- 011. Expected utility E[U(X)] = Σ pᵢ U(xᵢ): score a gamble by the average utility of its outcomes, then pick the highest
- 022. Utility is ordinal: a positive-affine rescaling Ũ = aU + b (a > 0) leaves the ranking unchanged, and a sure payoff y has E[U(y)] = U(y)
- 033. Jensen's inequality: U concave ⇒ U(E[X]) ≥ E[U(X)]; U convex ⇒ U(E[X]) ≤ E[U(X)]
- 044. Risk attitude ⇔ curvature: risk-averse = concave = U''< 0; risk-neutral = linear = U''= 0; risk-loving = convex = U''> 0
- 055. Certainty equivalent x_CE: the sure wealth with the same utility as the gamble, U(x_CE) = E[U(X)]
- 066. Risk premium π = E[X] − x_CE: positive for a risk-averse investor (x_CE < E[X]), zero if risk-neutral, negative if risk-loving
- 077. Arrow-Pratt absolute risk aversion A(w) = − U''(w) / U'(w) (the minus sign makes it positive for a risk-averse investor)
- 088. Small-risk approximation π ≈ ½ A(w) σ², and DARA vs CARA (whether A(w) falls or stays flat as wealth rises)
- 099. Decision rule: a risk-averse investor maximises expected utility, not expected wealth
Certainty equivalent and risk premium with square-root utility
- +1Expected wealth: E[X] = 0.5(900) + 0.5(100) = 450 + 50 = 500 (wealth units).
- +1Expected utility: U(900) = √900 = 30 and U(100) = √100 = 10, so E[U(X)] = 0.5(30) + 0.5(10) = 20.
- +1Certainty equivalent: solve U(x_CE) = √x_CE = 20, giving x_CE = 20² = 400. A guaranteed 400 is exactly as good as the gamble.
- +1Risk premium: π = E[X] − x_CE = 500 − 400 = 100. The investor would give up as much as 100 of expected wealth to shed the risk.
- +1Direction check: x_CE = 400 < E[X] = 500 and π = 100 > 0 confirm risk aversion, consistent with the concave √ utility (Jensen: U(E[X]) = √500 = 22.36 > 20 = E[U(X)]).
- +1Arrow-Pratt: U'(w) = ½ w^(−1/2) and U''(w) = −¼ w^(−3/2), so A(w) = − U''(w) / U'(w) = 1 / (2w). At w = x_CE = 400, A(400) = 1/800 = 0.00125 > 0; because A(w) falls as w rises, the investor exhibits decreasing absolute risk aversion (DARA).
Key terms
- Expected utility
- The rule that a rational investor ranks a risky payoff X by the probability-weighted average of the utility of its outcomes, E[U(X)] = Σ pᵢ U(xᵢ), and chooses the option with the highest value. Utility is ordinal, so only comparisons matter, not the level; a positive-affine rescaling Ũ = aU + b (a > 0) gives the same ranking.
- Jensen's inequality
- For a concave utility function, the utility of the expected outcome is at least the expected utility: U(E[X]) ≥ E[U(X)]; for a convex function the inequality reverses. This gap is exactly why a risk-averse investor prefers a sure E[X] to the gamble X with the same mean.
- Risk aversion (concave utility)
- An investor is risk-averse when E[U(X)] < U(E[X]), which holds precisely when the utility function is concave (U''< 0) with diminishing marginal utility. Then the pain of a loss exceeds the pleasure of an equal-sized gain, so a fair bet has negative expected utility.
- Certainty equivalent
- The guaranteed wealth x_CE that leaves the investor exactly as well off as taking the gamble, defined by U(x_CE) = E[U(X)], i.e. x_CE = U⁻¹(E[U(X)]). For a risk-averse investor x_CE is strictly below the expected wealth E[X].
- Risk premium
- The amount of expected wealth an investor will forgo to avoid the risk: π = E[X] − x_CE. It is positive for a risk-averse investor, zero for a risk-neutral one, and negative for a risk-loving one. It is the microfoundation of ‘expected return = risk-free rate + risk premium’.
- Arrow-Pratt absolute risk aversion
- A local measure of risk aversion at wealth w, A(w) = − U''(w) / U'(w). The minus sign makes A(w) positive for a risk-averse investor (U''< 0, U'> 0); a larger A(w) means more risk aversion and, for a small gamble, a larger risk premium π ≈ ½ A(w) σ².
- Risk-neutral vs risk-loving
- A risk-neutral investor has linear utility (U''= 0), so E[U(X)] = U(E[X]), x_CE = E[X] and π = 0, and ranks choices purely on expected wealth. A risk-loving investor has convex utility (U''> 0), so E[U(X)] > U(E[X]) and π < 0.
Utility Theory and Decision-Making Under Risk FAQ
Why do investors maximise expected utility instead of expected wealth?
Because expected wealth ignores attitude to risk. A risk-averse investor feels a loss more than an equal gain (diminishing marginal utility), so a fair gamble can have positive expected wealth yet negative expected utility. Ranking by E[U(X)] captures that trade-off: it is why such an investor can rationally prefer a sure 102 to a risky project with an expected 104. Rank on utility, then read off the certainty equivalent and risk premium to see how much the risk costs.
What is the difference between the certainty equivalent and the risk premium?
They are two views of the same thing. The certainty equivalent x_CE is the sure wealth that matches the gamble's expected utility, U(x_CE) = E[U(X)]. The risk premium π = E[X] − x_CE is the shortfall of that certainty equivalent below the expected wealth — the amount of expected wealth the investor sacrifices to be rid of the risk. For a risk-averse investor x_CE < E[X], so π > 0.
Can AI help me with utility theory and decision-making under risk in ETF5952?
Yes — as a study aid, not an answer service. Sia can explain expected utility, Jensen's inequality and Arrow-Pratt step by step, generate extra practice gambles with worked solutions, and check your reasoning on a certainty-equivalent or risk-premium calculation. Use it to understand the method and the sign conventions; it will not sit your closed-book exam or guarantee a mark. Always confirm assessment details and the exam date on Moodle.
Exam move
Master one template and reuse it everywhere: for any gamble, write E[X], then E[U(X)] = Σ pᵢ U(xᵢ), invert the utility to get the certainty equivalent from U(x_CE) = E[U(X)], and finish with π = E[X] − x_CE. Memorise the attitude-curvature map (concave = risk-averse = U''< 0 = x_CE < E[X] = π > 0) and the Arrow-Pratt definition A(w) = −U''/U', taking special care to keep the minus sign so the coefficient is positive for a risk-averse investor. Practise the square-root and log utilities until the inversions are automatic, and always add a one-line direction check because that is where the interpretation marks live. The final examination is worth 40% of ETF5952 and is a closed-book, centrally scheduled, invigilated paper in the ~June 2027 Semester-1 exam period — confirm the exact date and time on Moodle. The duration is not published in the unit materials, so plan by spending time in proportion to the marks rather than a fixed clock. Because Assignment 2 is set in the style of the final exam, rehearsing these setup-then-method-then-compute steps is the most efficient preparation.
Working through Utility Theory and Decision-Making Under Risk in ETF5952? Sia is AskSia’s AI Finance tutor — ask any ETF5952 Utility Theory and Decision-Making Under Risk question and get a clear, step-by-step explanation grounded in how ETF5952 is taught and assessed. Read this chapter free, then take your hardest questions to Sia.
