Economic Modelling · Measures of investment risk

Risk and Investor Utility Theory: Risk Aversion Explained

Q: What is the difference between absolute and relative risk aversion?

Absolute risk aversion A(w) = −U''(w) ÷ U'(w) describes attitude to a fixed rupee risk at wealth w. Relative risk aversion R(w) = w × A(w) describes attitude to a risk that is a fixed proportion of wealth. Constant R means the investor keeps the same proportion of wealth in risky assets as wealth grows.

Q: How do I calculate the risk aversion coefficient from a utility function?

Differentiate twice to get U'(w) and U''(w). Then compute A(w) = −U''(w) ÷ U'(w). Multiply by w if you need R(w).

Q: Why is a risk-averse utility function concave?

Concavity means each extra rupee adds less utility than the last. So losing a rupee hurts more than gaining one helps. By Jensen's inequality, the expected utility of a gamble is then less than the utility of its expected wealth.

Q: What is the certainty equivalent?

It is the certain amount of wealth that gives the same utility as a risky gamble. You find it by setting U(CE) = E[U(W)] and solving for CE. For a risk-averse investor it is below expected wealth.

Updated 11 October 2026 · Fact-checked

Utility theory says investors choose the option with the highest expected utility, E[U(W)], not the highest expected wealth. A risk-averse investor has a concave utility function, U''(w) < 0. You measure risk aversion with A(w) = −U''(w) ÷ U'(w) and R(w) = w × A(w). Solve by differentiating the utility function.

Understand Risk and Investor Utility Theory

Investors do not just want more money. They also dislike uncertainty. Utility is a number that measures how much satisfaction an investor gets from a level of wealth. Utility theory lets you turn that preference into a calculation.

The expected utility rule says the investor compares gambles by E[U(W)], the probability-weighted average of the utility of each outcome. Compare this with expected wealth, E[W]. The two can rank options differently, and the difference is where risk attitude shows up.

The usual assumptions in the IAI course are: non-satiation (more wealth is preferred to less, so U'(w) > 0) and risk aversion (U''(w) < 0, so U is concave). A risk-averse investor prefers a certain amount equal to E[W] to a gamble with the same expected wealth. This follows from Jensen's inequality: for concave U, E[U(W)] ≤ U(E[W]). A risk-neutral investor has U'' = 0 and only cares about E[W]. A risk-seeking investor has U'' > 0.

To measure how risk averse someone is, you cannot use U'' alone, because U can be rescaled (a positive linear transformation, aU + b with a > 0) without changing choices. So you scale U'' by U'. This gives the absolute risk aversion A(w) = −U''(w) ÷ U'(w), which relates to the rupee amount at risk. Relative risk aversion R(w) = w × A(w) relates to the proportion of wealth at risk.

Common utility functions are exponential, U(w) = −e^(−aw), with constant absolute risk aversion a; power (isoelastic), U(w) = w^(1−γ) ÷ (1−γ), with constant relative risk aversion γ; logarithmic, U(w) = ln w, with R = 1; and quadratic, which has increasing absolute risk aversion and is only valid over a range where U' > 0. Typically, investors are assumed to have decreasing absolute risk aversion: they hold more rupees in risky assets as they get wealthier.

Key rules to remember

Expected utility: E[U(W)] = Σ p_i × U(w_i); For a continuous outcome use the integral of U(w) f(w) dw. Choose the option with the higher value.
Non-satiation: U'(w) > 0; More wealth is always preferred.
Risk aversion, neutrality, seeking: U''(w) < 0, = 0, > 0; Concave, linear, convex respectively.
Jensen's inequality: E[U(W)] ≤ U(E[W]) for concave U; A risk-averse investor prefers the certain expected wealth to the gamble.
Absolute risk aversion: A(w) = −U''(w) ÷ U'(w); Unchanged by a positive linear transformation of U.
Relative risk aversion: R(w) = w × A(w) = −w U''(w) ÷ U'(w); Measures aversion to proportional wealth risk.
Certainty equivalent: U(CE) = E[U(W)]; CE is the certain wealth giving the same utility as the gamble. Solve by inverting U.
Exponential utility: U(w) = −e^(−aw); A(w) = a; R(w) = a w; Constant absolute, increasing relative risk aversion.
Power utility: U(w) = w^(1−γ) ÷ (1−γ), γ > 0, γ ≠ 1; A(w) = γ ÷ w; R(w) = γ; Constant relative, decreasing absolute risk aversion.
Logarithmic utility: U(w) = ln w; A(w) = 1 ÷ w; R(w) = 1; Special case of power utility with γ = 1.
Quadratic utility: U(w) = w − b w²; A(w) = 2b ÷ (1 − 2b w); b > 0, valid only for w < 1 ÷ (2b). Absolute risk aversion increases with wealth.

How to solve Risk and Investor Utility Theory questions

Use this order for any question on utility, risk aversion or choice between investments.

1Write down U(w) and check the range of w over which it applies.
2Differentiate: find U'(w) and U''(w). Check the signs to classify the investor.
3If asked for risk aversion, compute A(w) = −U''(w) ÷ U'(w), then R(w) = w × A(w) if needed.
4Look at how A and R change with w to say whether aversion is increasing, constant or decreasing.
5For a choice between investments, list each outcome with its probability and compute E[U(W)] for each option.
6Pick the option with the highest expected utility. Compare with expected wealth if the question asks.
7For a certainty equivalent, set U(CE) = E[U(W)] and invert U. State the risk premium as E[W] − CE.
8State the conclusion in words, linking it back to the investor's attitude to risk.

Quickest way: Shortcut: recognise the utility family

When to use it: When the question gives a standard utility function and asks for A(w) or R(w), or asks you to classify the investor.

Exponential U = −e^(−aw): A = a (constant), R = a w (increasing).
Power U = w^(1−γ) ÷ (1−γ): R = γ (constant), A = γ ÷ w (decreasing).
Log U = ln w: A = 1 ÷ w, R = 1.
Quadratic: A increases with w, so it is unrealistic for large wealth.
Still show U' and U'' in a written answer, since marks go for method.

Common mistakes in Risk and Investor Utility Theory

Choosing the option with the highest expected wealth.
It feels natural to maximise the average payoff.
Fix: Compute E[U(W)] for each option. A risk-averse investor can prefer a lower-mean, safer option.
Defining A(w) as −U''(w) without dividing by U'(w).
Students remember that U'' measures curvature and stop there.
Fix: Always use A(w) = −U''(w) ÷ U'(w). Dividing removes the effect of rescaling U.
Forgetting to multiply by w when finding relative risk aversion.
Absolute and relative measures have similar names.
Fix: Write R(w) = w × A(w). Check: power utility must give a constant R = γ.
Taking E[U(W)] as U(E[W]).
Treating U as a linear function.
Fix: Apply U to each outcome first, then take the weighted average. They are equal only when U is linear.
Reporting the expected utility as the certainty equivalent.
Both come from the same calculation.
Fix: Expected utility is a utility value. Invert U to get the certainty equivalent in rupees.
Using quadratic utility outside its valid range.
The formula looks fine for any w.
Fix: State that U' > 0 needs w < 1 ÷ (2b). Beyond that, more wealth reduces utility.

Worked examples

Example 1

An investor has utility U(w) = ln w. She can hold a certain wealth of ₹1,00,000, or a gamble giving ₹50,000 or ₹2,00,000 with equal probability. Which does she prefer? Find the certainty equivalent of the gamble.

Show the solution

Expected wealth of the gamble = 0.5 × 50,000 + 0.5 × 2,00,000 = ₹1,25,000, which is above ₹1,00,000.
Expected utility of the gamble = 0.5 ln 50,000 + 0.5 ln 2,00,000 = 0.5 ln(50,000 × 2,00,000) = 0.5 ln(10,00,00,00,000) = 0.5 ln(10^10) = 5 ln 10.
5 ln 10 = 5 × 2.302585 = 11.5129.
Utility of certain ₹1,00,000 = ln(10^5) = 5 ln 10 = 11.5129.
The two expected utilities are equal, so she is indifferent.
Certainty equivalent: ln(CE) = 5 ln 10, so CE = ₹1,00,000.
Risk premium = 1,25,000 − 1,00,000 = ₹25,000.

Answer: She is indifferent between the two. The certainty equivalent of the gamble is ₹1,00,000, and the risk premium is ₹25,000.

Example 2

An investor has utility U(w) = −e^(−0.002w), where w is wealth in ₹ thousands. Find A(w) and R(w), and state how each changes with wealth.

Show the solution

U'(w) = 0.002 e^(−0.002w), which is positive, so non-satiation holds.
U''(w) = −0.002² e^(−0.002w) = −0.000004 e^(−0.002w), which is negative, so the investor is risk averse.
A(w) = −U''(w) ÷ U'(w) = 0.000004 ÷ 0.002 = 0.002.
R(w) = w × A(w) = 0.002w.
A(w) does not depend on w, so absolute risk aversion is constant.
R(w) rises in proportion to w, so relative risk aversion is increasing.

Answer: A(w) = 0.002 (constant). R(w) = 0.002w (increasing in wealth). The investor is risk averse with constant absolute risk aversion.

Exam tips

Show U' and U'' every time. Marks go for the derivatives and signs even if the final number slips.
State what each risk aversion measure means in words: A relates to rupee amounts at risk, R to proportions of wealth.
Be ready to discuss the realism of each utility function, for example quadratic increasing A and log utility R = 1.
In certainty equivalent questions, invert U carefully and give the risk premium as E[W] − CE with units.
Link utility to portfolio choice: a more risk-averse investor chooses a point on the efficient frontier with lower risk.

Practice questions from Measures of investment risk

Risk and Investor Utility Theory in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Risk and Investor Utility Theory: frequently asked questions

What is the difference between absolute and relative risk aversion?

Absolute risk aversion A(w) = −U''(w) ÷ U'(w) describes attitude to a fixed rupee risk at wealth w. Relative risk aversion R(w) = w × A(w) describes attitude to a risk that is a fixed proportion of wealth. Constant R means the investor keeps the same proportion of wealth in risky assets as wealth grows.

How do I calculate the risk aversion coefficient from a utility function?

Differentiate twice to get U'(w) and U''(w). Then compute A(w) = −U''(w) ÷ U'(w). Multiply by w if you need R(w).

Why is a risk-averse utility function concave?

Concavity means each extra rupee adds less utility than the last. So losing a rupee hurts more than gaining one helps. By Jensen's inequality, the expected utility of a gamble is then less than the utility of its expected wealth.

What is the certainty equivalent?

It is the certain amount of wealth that gives the same utility as a risky gamble. You find it by setting U(CE) = E[U(W)] and solving for CE. For a risk-averse investor it is below expected wealth.