FINM7008 Chap.4 Utility, Risk Aversion and Indifference Curves
Utility, Risk Aversion and Indifference Curves
Choosing when dominance fails
When two portfolios differ in both mean and risk, a preference model is needed.
This chapter introduces utility theory, expected utility and the mean-variance utility score, which rewards expected return and penalises variance in proportion to the risk-aversion coefficient A.
Curves and attitudes
It compares risk-averse, risk-neutral and risk-loving investors, draws indifference curves, and finds the level of risk aversion at which an investor switches between a risky portfolio and bills.
How the tutorials use it
The tutorial questions for this topic ask you to find the level of risk aversion at which a risky portfolio and bills tie, to draw indifference curves for given utility levels, and to score a table of bill and index mixes for investors with different coefficients.
All three use the same score, so fluency with it pays off several times.
What this chapter covers
- 01
Dominance and its limits
- 02
Utility theory axioms
- 03
Expected utility
- 04
The mean-variance utility score
- 05
Risk-averse, neutral and loving investors
- 06
Indifference curves
- 07
Typical values of risk aversion
Worked example · free
Does an investor with A = 3 prefer portfolio X to bills?
- 1Utility of X: 0.11 − 0.5 × 3 × 0.202 = 0.11 − 0.06 = 0.05.
- 1Utility of bills: 0.04, since their variance is zero, so X is preferred.
- 1X stays preferred while 0.11 − 0.02A > 0.04, that is A < 3.5.
Key terms
- Expected Utility
- The probability-weighted average of the utility of each possible outcome.
- Risk Neutrality
- Indifference to risk, so that only expected value matters and A is zero.
- Certainty Equivalent
- The sure amount that gives the same utility as a risky prospect.
- Non-Satiation
- The assumption that more of a good is always preferred to less.
- Concave Utility
- A utility function whose slope falls as wealth rises, the shape that describes risk aversion.
Utility, Risk Aversion and Indifference Curves FAQ
What does the coefficient A measure?
A measures how strongly an investor dislikes variance. A positive A marks a risk-averse investor, zero a risk-neutral one and a negative value a risk lover. Typical estimates lie between 1 and 5, with 2 or 3 the usual benchmark.
Why can one investor's indifference curves never cross?
Each curve joins portfolios with one utility level. If two curves crossed, the portfolio at the crossing point would have two different utility levels at once, which is impossible for a single investor.
Why is a risk-neutral investor's indifference curve flat?
With A equal to zero, the utility score is just the expected return, so risk neither adds nor subtracts. Every portfolio with the same mean gives the same utility, which draws a horizontal line in mean and standard deviation space.
Why is the mean-variance score only an approximation?
It follows from expected utility if the portfolio's returns are normal and the investor's risk attitude does not change with wealth. Outside those conditions it is a convenient summary rather than an exact measure, which is why the lecture notes that U really stands for expected utility.
Can a rational investor ever prefer a gamble to a certain amount?
Yes, in special cases. The lecture's example is someone who needs $1.5 million for life-saving surgery: a coin flip for $2 million can be worth more to them than a certain $1.1 million, because only the larger sum meets the need.
What are the assumptions of utility theory?
A person can rank any two options, prefers more of a good to less, and ranks consistently, so that preferring A to B and B to C implies preferring A to C. The person then chooses the option with the highest utility.
Exam move
Compute a utility score for three portfolios with two different values of A and watch the ranking flip. Then sketch an indifference curve by tabulating expected returns for six standard deviations. Last, explain in words why a steeper indifference curve means a more risk-averse investor, using the extra return demanded for each extra unit of risk.
Note the decimal convention on your sheet: returns and standard deviations enter the score as decimals, so a 25% risk becomes 0.0625 once squared.
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