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CFA Level I Exam · The Return and Risk of a Financial Portfolio

Risk-Averse Investors and Utility Theory Explained for CFA Level 1

Updated 7 October 2026 · Fact-checked

A risk-averse investor prefers less risk for the same expected return and needs extra return to accept more risk. Utility theory scores portfolios with U = E(r) − 0.5 × A × σ², where A is risk aversion. You compute U for each portfolio and choose the highest score.

Understand Risk-Averse Investors and Utility Theory

Start with a simple choice. Two portfolios offer the same expected return, but one is riskier. Most investors pick the safer one. This is risk aversion. A risk-averse investor will take more risk only if the expected return rises enough to pay for it.

There are three attitudes. A risk-averse investor dislikes risk and demands compensation for it. A risk-neutral investor cares only about expected return and ignores risk. A risk-seeking investor likes risk and would accept a lower expected return for a gamble. Level I theory assumes most investors are risk averse.

To compare portfolios, we turn expected return and risk into one number called utility. The common form is U = E(r) − 0.5 × A × σ². Here E(r) is expected return, σ² is variance, and A is the risk aversion coefficient. A higher A means you penalize risk more. A risk-neutral investor has A = 0, so utility equals expected return. A risk-seeking investor has a negative A. Return and variance must be in the same units, both as decimals or both as percentages.

An indifference curve joins all risk-return combinations that give the same utility. It is drawn with standard deviation on the horizontal axis and expected return on the vertical axis. For a risk-averse investor the curve slopes upward and is convex. It slopes up because more risk needs more return to keep utility equal. A more risk-averse investor has a steeper curve. Curves higher and to the left give higher utility, and indifference curves for one investor never cross.

The optimal portfolio is where the investor reaches the highest indifference curve that still touches the set of available portfolios, such as the capital allocation line. In practice, you compute utility for each candidate and pick the largest. A portfolio with a utility below the risk-free rate is worse than holding the risk-free asset.

Key formulas to remember

Utility of a portfolio
U = E(r) − 0.5 × A × σ²
Use the same units for E(r) and σ². A is the risk aversion coefficient. Higher U is better.
Risk-neutral investor
A = 0, so U = E(r)
Risk is ignored. Choose the highest expected return.
Risk-averse investor
A > 0
Variance lowers utility. A larger A means a stronger penalty and a steeper indifference curve.
Risk-seeking investor
A < 0
Variance raises utility. The indifference curve slopes downward.
Certainty equivalent
Utility U equals the certainty-equivalent return of a risky portfolio
The risk-free return that gives the same utility. A risky portfolio beats the risk-free asset only if U > risk-free rate.
Indifference curve properties
Risk-averse: upward sloping, convex, non-crossing; higher and to the left = higher utility
Steeper curve means more risk averse.

How to solve Risk-Averse Investors and Utility Theory questions

Use this method for any question on risk attitude, utility scores or optimal portfolio choice.

  1. 1Identify the investor type from the wording: dislikes risk, ignores risk, or likes risk. This tells you the sign of A.
  2. 2Write the formula U = E(r) − 0.5 × A × σ².
  3. 3Convert standard deviation to variance by squaring it. Use decimals for both return and variance.
  4. 4Compute U for each portfolio, or for the risk-free asset if it is offered.
  5. 5Pick the portfolio with the highest U. If U is below the risk-free rate, the risk-free asset wins.
  6. 6For graph questions, remember that curves higher and to the left are better, and a steeper curve means higher risk aversion.
  7. 7Check that your answer matches one of the three options and that the investor type makes sense.

Quickest way: Score and compare in decimals

When to use it: When the question gives several portfolios with E(r) and σ and asks which one a risk-averse investor prefers.

  1. Square each σ as a decimal, for example 20% becomes 0.20² = 0.04.
  2. Compute 0.5 × A × variance for each portfolio and subtract it from E(r).
  3. Compare the scores. Do not bother with exact decimals once one score is clearly higher.
  4. Eliminate any option that ignores risk or contradicts the investor type, such as choosing the highest return for a risk-averse investor with a large A.
  5. On the BA II Plus, key 0.5 × A × 0.04 then subtract from E(r) using the chain calculation.

Common mistakes in Risk-Averse Investors and Utility Theory

  • Using standard deviation instead of variance in the utility formula

    Questions list σ, and students plug it straight in.

    Fix: Always square σ first. Write σ² on your scratch line before computing.

  • Mixing percentages and decimals

    E(r) is entered as 12 but variance as 0.04.

    Fix: Convert everything to decimals, or everything to percentages with variance as 400 for σ = 20%. Decimals are safer.

  • Thinking risk-averse means the investor will never take risk

    The label sounds absolute.

    Fix: A risk-averse investor accepts risk when the extra expected return compensates for it. The question is how much.

  • Reading indifference curves wrongly

    Students confuse the direction of preference on the graph.

    Fix: Higher and to the left is better: more return, less risk. Curves for one investor never cross.

  • Choosing a risky portfolio whose utility is below the risk-free rate

    Students compare risky portfolios only with one another.

    Fix: Always compare U with the risk-free rate, which has zero variance and U equal to its rate.

  • Assuming a higher A means a flatter curve

    Students guess the link between A and slope.

    Fix: Higher A penalizes risk more, so more return is needed per unit of risk. The curve is steeper.

Worked examples

Example 1

An investor has a risk aversion coefficient A = 4. Portfolio X has E(r) = 10% and σ = 15%. Portfolio Y has E(r) = 13% and σ = 25%. Portfolio Z has E(r) = 7% and σ = 5%. Which portfolio gives the highest utility? (A) Portfolio Z (B) Portfolio X (C) Portfolio Y

Show the solution
  1. Use U = E(r) − 0.5 × A × σ², with 0.5 × 4 = 2.
  2. Portfolio X: σ² = 0.15² = 0.0225. Penalty = 2 × 0.0225 = 0.045. U = 0.10 − 0.045 = 0.055, or 5.5%.
  3. Portfolio Y: σ² = 0.25² = 0.0625. Penalty = 2 × 0.0625 = 0.125. U = 0.13 − 0.125 = 0.005, or 0.5%.
  4. Portfolio Z: σ² = 0.05² = 0.0025. Penalty = 2 × 0.0025 = 0.005. U = 0.07 − 0.005 = 0.065, or 6.5%.
  5. Compare: 6.5% > 5.5% > 0.5%. Z is highest.

Answer: (A) Portfolio Z, with utility of 6.5%.

Example 2

A risk-averse investor with A = 3 can invest in a risky portfolio with E(r) = 9% and σ = 20%, or in a risk-free asset earning 4%. Which statement is correct? (A) The investor prefers the risk-free asset because the risky utility is 3.0% (B) The investor is indifferent because both give the same utility (C) The investor prefers the risky portfolio because its utility is 9.0%

Show the solution
  1. Compute the risky portfolio utility: U = 0.09 − 0.5 × 3 × 0.20².
  2. σ² = 0.04. Penalty = 1.5 × 0.04 = 0.06.
  3. U = 0.09 − 0.06 = 0.03, or 3.0%.
  4. The risk-free asset has no variance, so U = 4%.
  5. Since 4% > 3%, the investor prefers the risk-free asset.

Answer: (A) The investor prefers the risk-free asset because the risky utility is 3.0%.

Exam tips

  • Questions are three-option MCQs, so compute utility for each portfolio only until you can rule out two options. Often one portfolio is clearly dominated.
  • Watch units. If the stem gives σ as 20%, square 0.20, not 20.
  • Always check the risk-free rate as a benchmark when it is available.
  • For conceptual items, link A to slope: higher A means steeper, more convex indifference curves.
  • Do not spend more than about 90 seconds. This is a short calculation, so avoid over-checking.

Practice questions from The Return and Risk of a Financial Portfolio

Risk-Averse Investors and 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-Averse Investors and Utility Theory: frequently asked questions

What is the difference between risk averse, risk neutral and risk seeking?

A risk-averse investor needs extra return to accept more risk. A risk-neutral investor ignores risk and looks only at expected return. A risk-seeking investor likes risk and may accept a lower expected return for a gamble.

What does the risk aversion coefficient A mean?

A measures how heavily the investor penalizes variance in the utility formula. A larger A means a stronger dislike of risk. A = 0 is risk neutral and A < 0 is risk seeking.

Why are indifference curves convex for risk-averse investors?

As risk rises, the investor needs more and more extra return for each additional unit of risk. This makes the curve rise at an increasing rate, which is convex. The curve is upward sloping because more risk needs more return.

How is the optimal portfolio chosen using utility?

Compute utility for each available portfolio and pick the highest. Graphically, the optimal portfolio is where the highest reachable indifference curve touches the set of available portfolios, such as the capital allocation line.