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Explain risk aversion and what indifference curves show

CFA® Level I · Portfolio Construction · 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.

Being risk averse does not mean avoiding risk altogether. It means that, given two investments with the same expected return, you prefer the less risky one, so you need extra expected return as payment for taking extra risk. Rule: higher risk requires higher expected return. A risk-averse investor happily holds risky assets as long as the pay-off is enough. An investor who does not care about risk at all, and is indifferent between certain and uncertain outcomes with the same average, is called risk neutral. There is no formula.

A harder one

An investor chooses among risky portfolios only, with no risk-free asset. Where does the investor's optimal portfolio sit?

  1. At the global minimum-variance portfolio, the point of lowest riskWhy not: Optimal only for an investor of essentially infinite risk aversion.
  2. Where the highest reachable indifference curve touches the frontier - correct
  3. At the highest-return point on the frontier, where return is maximisedWhy not: Optimal only for an investor indifferent to risk. It ignores the risk penalty.

Two things decide the choice. The efficient frontier shows the best portfolios available, with the most return for each level of risk. Indifference curves show what the investor prefers. The best choice is the point where the highest attainable indifference curve just touches the frontier. A very risk-averse investor ends near the lowest-risk portfolio, the global minimum-variance portfolio. A less risk-averse investor ends higher up the frontier. The highest-return point suits only someone who ignores risk.

The hardest tier

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

  1. Yes, since only one portfolio on the frontier is optimalWhy not: The frontier shows all efficient choices. Which one is best depends on the investor's risk aversion, so different investors can choose different ones.
  2. No, since their forecasts of returns must differWhy not: Identical frontiers mean the same inputs. The differing choices come from differing risk aversion.
  3. No, since their risk aversion differs - correct

The efficient frontier is the set of best portfolios, those with the most return for each level of risk. It shows what is available, and it is the same for both investors. Which point on it is best depends on personal taste for risk. Each investor picks the point where their own indifference curve just touches the frontier. A more cautious investor picks a lower-risk point, and a bolder one picks a higher-risk point. Neither is wrong. There is no formula; the frontier is what is possible, and preferences decide the choice.

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