Risk preferences, mean–variance utility and indifference curves
Mean–variance utility scores an investment as its expected return minus a penalty for risk: U = E[r] − ½Aσ². The coefficient A measures risk aversion: positive for risk-averse investors, zero for risk-neutral, negative for risk-seeking. The investor picks the highest score. An indifference curve joins equal scores.
Before this: The minimum-variance portfolio and the efficient frontier
On the exam
- Explain the selection of an optimal portfolio, given an investor’s risk aversion and the capital allocation line, and how this extends to the market portfolio and the capital market line
Explain how risk aversion drives portfolio choice. Expect a utility calculation and a ranking of several investments for different values of A (including negative utilities and a negative A), the utility of a risk-free asset, the three risk attitudes, and the shape and slope of indifference curves.
How deeply to learn each part
- Know the idea
- Understand it in general. It frames other topics, but you will not calculate with it.
- Use it
- Understand it well enough to solve a question with it. Recognize when it applies.
- Memorize
- Learn it by heart. The exam will not give it to you.
You know this one. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still do.
You tested strong here. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still know it.
What does portfolio theory assume?
The assumptions of portfolio theory are four simple ones:
- Investors are rational and risk-averse.
- They plan for a single period.
- There are no taxes or transaction costs.
- A risk-free asset exists that everyone can lend and borrow at the same rate, in any amount.
The next lesson relies on the last assumption. The limits of all four close that lesson.
Three attitudes to risk
Take a choice between INR 9,000 for sure and a coin toss that pays INR 0 or INR 20,000. The expected value of the toss is INR 10,000.
- A risk-averse investor takes the sure INR 9,000, even though the gamble’s expected value is higher. This investor aims to minimize risk for a given return.
- A risk-neutral investor looks only at expected value, so takes the gamble (10,000 is more than 9,000). Risk does not matter. This is typical when the stake is a tiny part of wealth.
- A risk-seeking investor prefers the gamble even at a lower expected value, as with lottery tickets and casinos.
Real investors are, with few exceptions, risk-averse. Institutions that place very many small, independent trades can be modeled as risk-neutral: by the law of large numbers, the variability averages out and the expected result becomes close to certain. That idea underpins arbitrage reasoning.
- Risk-averse Memorize A > 0; takes the sure thing; convex indifference curves
- Risk-neutral Memorize A = 0; U = E[r]; flat curves
- Risk-seeking Memorize A < 0; risk adds utility; concave curves
How do you score an investment?
Utility rises with expected return and falls with variance, more steeply for a more risk-averse investor. The utility function turns that into a single score.
Here E[r] and σ are in decimals, σ² is the variance, and A is the coefficient of risk aversion . A is seldom below 1, and studies have documented values between 2 and 3.
This is one example of mean–variance utility : any utility that depends only on expected return and risk, U = U(E[r_P], σ_P). The formula above is the standard special case. Other forms exist, such as logarithmic and power utility, and classical economics defines utility over wealth instead.
A risk-free asset has σ = 0, so its utility equals its return for every investor, whatever A.
For a risk-seeking investor with A = −2, the scores are X 0.0564, Y 0.1224 and Z 0.2200. Risk now adds to utility.
An investment has an expected return of 9.00% and a standard deviation of 18.00%. For an investor whose coefficient of risk aversion is 2.5, its utility is closest to:
0.09 − 0.5 × 2.5 × 0.18² = 0.0495, so 4.95%
Utility, U4.95%
- 0.5×2.5×0.18x²=the risk penalty
- +|-+0.09=the utility as a decimal
x² square · +|- change sign
- 0.09ENTER0.5ENTER2.5×0.18ENTER××−the utility as a decimal
ENTER enter (on the HP, puts the number on the stack)
Try this 0 of 3
- Find an A at which the bill beats every fund
- Make Fund Z win with a positive A
- Set A below 0 and look at the curves
- Indifference curves
Fund X: risk 8.0%, return 5.0%; Fund Y: risk 18.0%, return 9.0%; Fund Z: risk 30.0%, return 13.0%; Risk-free bill: risk 0.0%, return 3.0%. With risk aversion 2.5, Fund Y gives the highest utility (4.95%).
Show working
U(Fund X) = 0.05 − 0.5 × 2.5 × 0.08² = 0.042 (4.20%)
U(Fund Y) = 0.09 − 0.5 × 2.5 × 0.18² = 0.0495 (4.95%)
U(Fund Z) = 0.13 − 0.5 × 2.5 × 0.3² = 0.0175 (1.75%)
U(Risk-free bill) = 0.03 − 0.5 × 2.5 × 0² = 0.03 (3.00%)
Highest utility: Fund Y. Utility uses decimals: 0.5 × A × σ² with σ = 0.18, not 18.
Utility uses decimals and the variance: σ = 18% is 0.18, so σ² = 0.0324. Multiply by 0.5 × A = 1.25 and subtract from E[r]. The bill has σ = 0, so its utility is its 3% return for every investor.
Calculator, Fund Y at A = 2.5. TI BA II Plus: 0.18 [x²] [×] 0.5 [×] 2.5 [=] // 0.0405, the risk penalty; [+|-] [+] 0.09 [=] // 0.0495. HP 12C: 0.09 [ENTER] 0.18 [ENTER] [×] 0.5 [×] 2.5 [×] [−] // 0.0495.
Negative utilities are allowed. Rank them; the highest (least negative) wins.
Risk tolerance and three kinds of risk aversion
Risk tolerance is the amount of risk an investor will accept to reach a goal. It moves opposite to risk aversion, and it changes with age, health, wealth, family circumstances and the economic climate.
Risk aversion itself comes in three kinds:
- Absolute risk aversion is how much utility changes for each extra unit of risk, with wealth held fixed.
- Relative risk aversion is the share of wealth the investor is prepared to put at risk.
- Varying risk aversion is aversion that changes as wealth changes.
- Absolute Know the idea utility change per unit of risk, wealth fixed
- Relative Know the idea share of wealth put at risk
- Varying Know the idea changes as wealth changes
The sign of A sorts the three attitudes:
| Attitude | A | Utility | Prefers, at a given return |
|---|---|---|---|
| Risk-seeking | A < 0 | E[r] − ½Aσ² (risk adds) | more risk |
| Risk-neutral | A = 0 | E[r] | indifferent to risk |
| Risk-averse | A > 0 | E[r] − ½Aσ² (risk subtracts) | less risk |
How do indifference curves show risk preferences?
An indifference curve joins every risk and return combination that gives one investor the same utility. Higher curves, up and to the left, mean more utility. Solving the utility formula for E[r] gives the curve.
The slope of the curve is the marginal rate of substitution between risk and return: the extra return the investor needs for a bit more risk. A steeper curve means a more risk-averse investor. Risk-averse investors (concave utility of wealth) have convex, upward-sloping curves. Risk-neutral investors have flat, horizontal curves. Risk-seeking investors (convex utility of wealth) have concave curves.
The next lesson places these curves on the capital allocation line to find each investor’s best portfolio.
Memorize this
MemorizeUtility in one line
| A | Attitude | Indifference curves |
|---|---|---|
| A > 0 | risk-averse | upward, convex; steeper = more averse |
| A = 0 | risk-neutral | flat; U = E[r] |
| A < 0 | risk-seeking | concave |
Memory hook U = E[r] − ½Aσ²: half, A, VARIANCE, all in decimals. The bill scores its own yield.
Know it cold Pick the highest U, even if all are negative. A is seldom below 1; studies put it between 2 and 3.
- Risk-averse Memorize A > 0; takes the sure thing; convex indifference curves
- Risk-neutral Memorize A = 0; U = E[r]; flat curves
- Risk-seeking Memorize A < 0; risk adds utility; concave curves
- Absolute Know the idea utility change per unit of risk, wealth fixed
- Relative Know the idea share of wealth put at risk
- Varying Know the idea changes as wealth changes
Quick example
E[r] = 8%, σ = 15%, A = 2. 0.08 − 0.5 × 2 × 0.0225 = 0.0575 (5.75%). Use decimals, with σ² the variance.
- 0.5×2×0.15x²= 0.0225
- +|-+0.08= 0.0575
x² square · +|- change sign
- 0.08ENTER0.5ENTER2×0.15ENTER××− 0.0575
ENTER enter (on the HP, puts the number on the stack)
Quick example
A = 4 and U = 0.03. At σ = 20% the curve sits at 0.03 + 2 × 0.04 = 11.0%. A higher σ needs a higher E[r] to keep U unchanged.
- 2×0.2x²+0.03= 0.11
x² square
- 2ENTER0.2ENTER××0.03+ 0.11
ENTER enter (on the HP, puts the number on the stack)
How it is asked
-
Rank by utility
A = 3: P 7%/12%, Q 11%/22%, R 16%/35%.
U: P 0.0484, Q 0.0374, R −0.0238 → P. Trap: R (highest return, ignores risk).
-
Negative A
Same menu, A = −4.
Risk adds utility: R wins.
-
Certainty equivalent
Fund 12%, σ 28%, A = 2. Lowest risk-free rate the investor would take instead?
0.12 − 0.0784 = 4.16%. Trap: −3.68% (no ½).
-
Curve shape
More risk-averse investor's indifference curves?
Steeper.
Exam traps
- Dropping the ½ (this doubles the penalty).
- Using σ instead of σ², or percentages instead of decimals (20 instead of 0.20).
- Thinking negative utilities rule an investment out. You still pick the highest.
- Giving a risk-free asset a utility other than its return.
- Saying a more risk-averse investor has flatter indifference curves.
Checkpoint
3 exam-style questions. About 90 seconds each.
1A risk-neutral investor choosing among several investments will most likely pick the one with the:
2An investment has an expected return of 14% and a standard deviation of 24%. For an investor whose coefficient of risk aversion is 3, its utility is closest to:
3An investor with a coefficient of risk aversion of 2 can buy a fund with an expected return of 11% and a standard deviation of 25%. The lowest risk-free return the investor would accept instead of the fund is closest to:
Key takeaways
- U = E[r] − ½Aσ², in decimals, with σ² the variance.
- A > 0 risk-averse, A = 0 risk-neutral, A < 0 risk-seeking. A is seldom below 1, often 2 to 3.
- Pick the highest utility, even when all are negative. A risk-free asset's utility is its return.
- Indifference curves join equal utility. Steeper means more risk-averse, and convex for risk-averse investors.
Quick example
E[r] = 8%, σ = 15%, A = 2. 0.08 − 0.5 × 2 × 0.0225 = 0.0575 (5.75%). Use decimals, with σ² the variance.
- 0.5×2×0.15x²= 0.0225
- +|-+0.08= 0.0575
x² square · +|- change sign
- 0.08ENTER0.5ENTER2×0.15ENTER××− 0.0575
ENTER enter (on the HP, puts the number on the stack)
Quick example
A = 4 and U = 0.03. At σ = 20% the curve sits at 0.03 + 2 × 0.04 = 11.0%. A higher σ needs a higher E[r] to keep U unchanged.
- 2×0.2x²+0.03= 0.11
x² square
- 2ENTER0.2ENTER××0.03+ 0.11
ENTER enter (on the HP, puts the number on the stack)
Questions about this lesson?
Where this comes back
- From the capital allocation line to the capital market line and the CAPM Portfolio Return and Risk
- Risk aversion, utility and indifference curves Portfolio Risk and Return: Part I
Questions candidates ask
What is the utility formula in CFA Level 1?
U = E[r] − ½Aσ², with returns and variance in decimals. A fund with E[r] 9% and σ 18% scores 0.09 − 0.5 × 2.5 × 0.0324 = 0.0495 for A = 2.5. The investor prefers the highest score.
What does the risk-aversion coefficient A mean?
How strongly the investor dislikes variance. A > 0 is risk-averse, A = 0 risk-neutral (only expected return counts) and A < 0 risk-seeking. Research puts typical values between 2 and 3, and A is seldom below 1.
Why are indifference curves convex for risk-averse investors?
Each extra unit of risk needs a larger extra return to keep utility unchanged, because the penalty grows with the square of σ. So the curve gets steeper as risk rises.
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