Risk aversion and the portfolio
The more you dislike risk, the less of the risky asset you hold. Move the risk-aversion slider and watch the chosen point slide along the risk–return line.
Example numbers — returns, swings and risk aversion are for illustration. This is not investment advice.
The more you dislike risk (higher A), the less of the risky asset you hold at the same reward. With no extra reward you hold none; with enough reward even a risk-averse person holds some.
Risk–return line (solid 0–100%, dashed = borrowing) Indifference curve (higher is better) Chosen mix
| Price of risk (slope) | 0.325 | (8.5 − 2) ÷ 20 |
|---|---|---|
| Calculated w* | 0.812 | (0.085 − 0.02) ÷ (2 × 0.2²) — model assumption |
| Expected return E_p | 7.28% | 2 + 0.8125 × (8.5 − 2) |
| Swing σ_p | 16.25% | 0.8125 × 20 |
Model assumption: the share comes from mean–variance utility U = E − (A/2)σ². Only the direction (more risk aversion, less of the risky asset) is in the study guide; the formula is a learning supplement.
Not in this calculation
- Several risky assets — co-movement (correlation) and diversification are a different calculation
- Borrowing to buy more or short selling — clipped to 0–100% by default
- Taxes and fees
- Anyone’s real risk aversion — the slider is an assumption
- That expected returns and swings stay the same
- Investment advice
Using the example values
At a glance — risk and choice
A choice is not settled by the average alone; how much someone dislikes the spread changes the pick. Open a box for its one-line meaning and the study-guide page.
1Measuring risk
Describing risk
When outcomes are spread by probability, measure the average (expected value) and the spread (variance, standard deviation) separately.
Guide p. 21
Expected utility
The probability-weighted sum of satisfaction (utility) from each outcome. Choices follow this, not the money average.
Guide p. 21–22
2Attitude
Risk-averse
Takes the sure thing at the same average. Concave utility, diminishing marginal utility.
Guide p. 22
Risk-neutral
Indifferent between a sure amount and a gamble with the same average. Linear utility, constant marginal utility.
Guide p. 22
Risk-loving
Takes the gamble at the same average. Convex utility, increasing marginal utility.
Guide p. 22
3The price
Risk premium
The most a risk-averse person would pay to avoid the risk — a price paid, not a reward received.
Guide p. 22
Tool ▶ risk premiumCertainty equivalent (supplement)
The sure amount as good as the gamble; expected value minus the risk premium. Not defined in the study guide — a supplement.
Outside the guide
Tool ▶ risk premium
4Reducing risk
Insurance
Hands the risk to someone else. With fair insurance (premium = expected loss), a risk-averse person buys full cover.
Guide p. 22
Diversification
Spreading resources over activities that are not closely related, to reduce the spread.
Guide p. 22
Information
Better prediction reduces risk. The value of complete information is the gap between expected values with and without it.
Guide p. 22
5Allocating assets
Portfolio
Splitting wealth between a risky asset (uncertain outcome) and a risk-free one.
Guide p. 22–23
Price of risk
The extra expected return asked for one more unit of risk — the slope of the risk–return line.
Guide p. 22–23
Tool ▶ risk aversion and the portfolio
6Choice
Optimal choice
Where the risk–return (budget) line touches an indifference curve. The more risk-averse, the more of the risk-free asset.
Guide p. 23
Tool ▶ risk aversion and the portfolio
Show 11 links
- Describing risk → Expected utility — Outcomes and probabilities feed a weighted average of utility
- Expected utility → Risk-averse · Risk-neutral · Risk-loving — The shape of the utility function sets the attitude
- Risk-averse → Risk premium — Only the risk-averse pay to avoid risk
- Describing risk → Risk premium — The expected value is the benchmark
- Risk-averse → Insurance · Diversification · Information — The risk-averse try to reduce risk
- Describing risk → Diversification — Lowering the spread is what diversification is for
- Diversification → Portfolio — The concrete form of spreading assets
- Portfolio → Price of risk — The slope of the line is the price of risk
- Portfolio · Price of risk → Optimal choice — Choose on the budget line
- Risk-averse → Optimal choice — More risk aversion moves the choice toward the risk-free asset
- Risk premium → Certainty equivalent (supplement) — Expected value − risk premium (supplement)
Solid boxes are within the study guide (1.2.2, pp. 21–23); the dashed box is a supplement not defined there. Numbers are worked in the tools.
Check next
Embed this tool
Paste the code below; the source line and link come with it (CC BY 4.0). Numbers are examples.
Preview