Economic Modelling · Mean-variance portfolio theory

Risk Aversion and Utility Assumptions in Mean-Variance Theory

Q: What are the main assumptions of mean-variance portfolio theory?

Investors are non-satiated and risk averse, and they judge portfolios by expected return and variance over a single period. Markets are assumed to have known means, variances and covariances, and investors are price takers.

Q: What is the difference between a risk-averse and a risk-neutral investor?

A risk-averse investor has concave utility and needs extra expected return to accept extra risk. A risk-neutral investor has linear utility and cares only about expected value, ignoring variance.

Q: Why does quadratic utility lead to the mean-variance criterion?

Expected quadratic utility contains only E(W) and E(W²). Since E(W²) = Var(W) + (E(W))², expected utility depends only on mean and variance.

Q: What is wrong with quadratic utility?

It has a satiation point where marginal utility becomes negative, so it violates non-satiation beyond that wealth. It also implies absolute risk aversion that increases with wealth, which is unrealistic.

Updated 11 October 2026 · Fact-checked

Mean-variance theory assumes investors prefer more wealth to less (non-satiation), dislike risk (risk aversion), and judge portfolios only by expected return and variance. This holds exactly if utility is quadratic or returns are normal. To answer questions, state the assumption, link it to utility, then apply the criterion.

Understand Risk Aversion and Utility Assumptions

Mean-variance portfolio theory assumes you choose between portfolios using only two numbers: the expected return and the variance of return. Everything else about the return distribution is ignored. The assumptions explain why this is reasonable and where it fails.

The first assumption is non-satiation: you always prefer more wealth to less. In utility terms, U'(w) > 0, so utility rises with wealth. For two portfolios with the same variance, you pick the higher expected return.

The second is risk aversion: you prefer a certain amount to a gamble with the same expected value. In utility terms, U''(w) < 0, so utility is concave. For two portfolios with the same expected return, you pick the lower variance. A risk-neutral investor has linear utility (U'' = 0) and cares only about expected value. A risk-seeking investor has convex utility (U'' > 0).

Together these give the mean-variance criterion: portfolio A is preferred to B if E(A) ≥ E(B) and Var(A) ≤ Var(B), with at least one strict inequality. Portfolios that no other portfolio dominates in this way are efficient. Where one has higher return and higher risk, the choice depends on the investor's degree of risk aversion.

Why can two moments be enough? Expected utility depends on the whole distribution. It depends only on mean and variance in two cases: utility is quadratic, or returns are jointly normal (or more generally, elliptically distributed). Quadratic utility has a flaw: it has a satiation point, beyond which marginal utility turns negative, so it breaks non-satiation. It also implies increasing absolute risk aversion. Normal returns avoid the utility problem but allow negative wealth and ignore skewness and fat tails.

The theory also assumes a single period, investors act as price takers, and the mean, variances and covariances are known and agreed.

Key rules to remember

Non-satiation: U'(w) > 0; Marginal utility is positive: more wealth is always preferred.
Risk aversion: U''(w) < 0; Concave utility. Risk neutral: U'' = 0. Risk seeking: U'' > 0.
Jensen's inequality for risk aversion: E[U(W)] < U(E[W]); For strictly concave U and non-degenerate W. The investor prefers the certain mean to the gamble.
Mean-variance dominance: A preferred to B if E(A) ≥ E(B) and Var(A) ≤ Var(B), with at least one strict; Applies to risk-averse, non-satiated investors using the mean-variance criterion.
Quadratic utility: U(w) = w − (b/2)w², b > 0, valid for w < 1/b; Gives E[U] = E(W) − (b/2)[Var(W) + (E(W))²], so only mean and variance matter.
Absolute risk aversion: A(w) = −U''(w) ÷ U'(w); Higher A means more risk averse. For quadratic utility A(w) = b ÷ (1 − bw), which increases with w.

How to solve Risk Aversion and Utility Assumptions questions

Use this method for any question on assumptions, utility and the mean-variance criterion.

1Identify what is asked: an assumption, a classification of an investor, a utility calculation, or a preference between portfolios.
2Write the utility function and find U'(w) and U''(w).
3Use the signs: U' > 0 means non-satiation; U'' < 0 risk averse, = 0 risk neutral, > 0 risk seeking.
4If asked about the mean-variance link, expand E[U(W)] and show it depends only on E(W) and Var(W) (quadratic utility), or state the normal returns condition.
5For portfolio comparison, check dominance on mean and variance. If neither dominates, say the choice depends on risk aversion.
6State the limits: valid range of w for quadratic utility, increasing absolute risk aversion, and ignoring skewness or fat tails.
7Give a final one-line conclusion in the words of the question.

Quickest way: Sign test on U' and U''

When to use it: Multiple-choice questions that ask you to classify an investor or check whether a utility function meets the assumptions.

Differentiate twice.
Check U' > 0 over the stated wealth range.
Check the sign of U'': negative means risk averse.
For quadratic utility, remember the range w < 1/b, where U' stays positive.
Eliminate options that confuse risk neutral (U'' = 0) with risk averse.

Common mistakes in Risk Aversion and Utility Assumptions

Saying a risk-averse investor will never take risk.
Risk aversion sounds like risk avoidance.
Fix: A risk-averse investor accepts extra risk only if compensated by extra expected return. Risk neutral means no compensation is needed.
Stating that quadratic utility satisfies non-satiation at all wealth levels.
Students check U'' < 0 and forget to check U'.
Fix: U'(w) = 1 − bw is positive only for w < 1/b. Beyond that, more wealth reduces utility.
Claiming mean-variance works for any utility function.
The criterion is used so widely that its conditions get forgotten.
Fix: State that it is exact only for quadratic utility or normally distributed returns, and approximate otherwise.
Confusing a risk-neutral investor with a risk-averse one.
Both prefer higher expected return.
Fix: Risk neutral has linear utility and ignores variance. Risk averse has concave utility and penalises variance.
Applying mean-variance dominance when one portfolio has both higher return and higher variance.
Students assume the higher return always wins.
Fix: Dominance needs better or equal on both measures. Otherwise both may be efficient and the choice depends on preferences.

Worked examples

Example 1

An investor has utility U(w) = w − 0.001w², with w in ₹ thousands, valid for w < 500. Show that the investor is non-satiated and risk averse for w < 500, and find E[U(W)] for a portfolio with E(W) = 100 and Var(W) = 400.

Show the solution

U'(w) = 1 − 0.002w. For w < 500, 0.002w < 1, so U' > 0. The investor is non-satiated.
U''(w) = −0.002 < 0, so the investor is risk averse.
Here b/2 = 0.001, so E[U(W)] = E(W) − 0.001 E(W²).
E(W²) = Var(W) + (E(W))² = 400 + 10,000 = 10,400.
E[U(W)] = 100 − 0.001 × 10,400 = 100 − 10.4 = 89.6.

Answer: U' > 0 and U'' < 0 for w < 500. E[U(W)] = 89.6. It depends only on the mean and variance.

Example 2

Portfolio A has expected return 8% and standard deviation 10%. Portfolio B has expected return 8% and standard deviation 14%. Portfolio C has expected return 11% and standard deviation 16%. For a risk-averse, non-satiated investor using the mean-variance criterion, which portfolios can be ruled out?

Show the solution

Compare A and B: same expected return, A has lower variance. A dominates B, so B is ruled out.
Compare A and C: C has higher return but also higher risk. Neither dominates.
Compare B and C: C has higher return and higher risk, so B and C do not dominate each other on this basis, but B is already ruled out by A.
So A and C remain.

Answer: B is ruled out because A dominates it. A and C are both efficient; the choice between them depends on the investor's degree of risk aversion.

Exam tips

Always link each assumption to a sign: non-satiation to U' > 0, risk aversion to U'' < 0.
When asked to discuss mean-variance, give the two conditions that justify it and at least two limitations.
For quadratic utility, always state the valid wealth range.
In portfolio comparisons, say explicitly when neither portfolio dominates.

Practice questions from Mean-variance portfolio theory

Risk Aversion and Utility Assumptions: frequently asked questions

What are the main assumptions of mean-variance portfolio theory?

Investors are non-satiated and risk averse, and they judge portfolios by expected return and variance over a single period. Markets are assumed to have known means, variances and covariances, and investors are price takers.

What is the difference between a risk-averse and a risk-neutral investor?

A risk-averse investor has concave utility and needs extra expected return to accept extra risk. A risk-neutral investor has linear utility and cares only about expected value, ignoring variance.

Why does quadratic utility lead to the mean-variance criterion?

Expected quadratic utility contains only E(W) and E(W²). Since E(W²) = Var(W) + (E(W))², expected utility depends only on mean and variance.

What is wrong with quadratic utility?

It has a satiation point where marginal utility becomes negative, so it violates non-satiation beyond that wealth. It also implies absolute risk aversion that increases with wealth, which is unrealistic.