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Explain risk aversion and what indifference curves show

CFA® Level I · Portfolio Management · Portfolio Risk and Return: Part I · PM.1.d

The idea

Risk aversion means that for a given expected return an investor prefers less uncertainty, and will accept a lower expected return to obtain it. Utility is commonly written as expected return less a risk penalty proportional to variance and to a risk aversion coefficient, which produces indifference curves sloping upward in mean-standard-deviation space: more risk must be paid for with more return. A more risk-averse investor has steeper curves. The optimal portfolio is where the highest attainable indifference curve touches the efficient frontier, which is why two rational investors facing identical opportunities choose differently.

Formulas

Utility of a risky portfolio

U = E(R) − ½ × A × σ²

E(R)
the expected return of the portfolio
A
the investor's risk aversion coefficient, higher for the more averse
σ²
the variance of the portfolio's return

Risk is subtracted at a rate the investor sets. A risk-neutral investor has A = 0 and ranks portfolios by return alone; a risk seeker has a negative A. It is a ranking, so the number itself means nothing on its own.

A practice question

From the question bank, with the reason each wrong option is wrong.

What does it mean for an investor to be risk averse?

  1. They will not hold risky assetsWhy not: Risk-averse investors hold risky assets routinely — they simply demand to be paid for doing so.
  2. They require additional expected return to accept additional risk — correct
  3. They are indifferent between certain and uncertain outcomes of equal meanWhy not: That is risk neutrality. The averse investor prefers the certain outcome.

Risk aversion is about the price of risk, not its refusal. A risk-averse investor holds plenty of risk provided it is compensated.

A harder one

How does a higher risk aversion coefficient change an investor's indifference curves?

  1. It makes them steeper, requiring more return per unit of risk — correct
  2. It makes them flatterWhy not: Flatter curves mean less extra return is demanded per unit of risk, which is lower aversion.
  3. It makes them horizontalWhy not: A horizontal curve would mean risk is accepted with no compensation at all — risk neutrality in the limit.

A steeper curve means a larger return premium is demanded for the same increase in standard deviation, which shifts the chosen portfolio toward the low-risk end of the frontier.

The hardest tier

Two investors face the identical efficient frontier yet hold different portfolios. Is one of them necessarily wrong?

  1. Yes — only one portfolio on the frontier is optimalWhy not: The frontier is the same for both; which point on it is optimal depends on the investor's own preferences.
  2. Yes, unless one has superior informationWhy not: Different information would give different frontiers. Here the frontier is shared and only the risk aversion differs.
  3. No, their risk aversion differs — correct

The optimum is where each investor's own indifference curve is tangent to the frontier. The frontier describes what is available and preferences decide where on it to sit, so different risk aversion produces different tangency points from the same opportunity set.

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