Expected value and risk premium
Two bets can share an average and still feel different. Enter the sure amount you would take instead. The gap to the average is the risk premium.
Example numbers — the sure amount differs from person to person. This tool only works out the average and the risk premium from your inputs.
Takes a sure amount below the average — the gap is the price paid to avoid the risk.
| Expected value | 104 | 0.5 × 64 + 0.5 × 144 |
|---|---|---|
| Standard deviation | 40 | √(0.5 × (64 − 104)² + 0.5 × (144 − 104)²) |
| Risk premium | 4 | 104 − 100 |
Not in this calculation
- Bets with three or more outcomes — only two are worked here
- Anyone’s actual utility function — it differs from person to person
- Feeling losses more than gains — loss aversion is a different theory
| RP > 0 | Risk-averse | Takes a sure amount below the average — the gap is the price paid to avoid the risk. |
|---|---|---|
| RP = 0 | Risk-neutral | Treats the bet and a sure amount equal to the average as the same. |
| RP < 0 | Risk-seeking | Needs more than the average to give up the bet — the swing itself is worth something. |
Using the example values
In pictures
A pays 90 or 110; B pays 0 or 200. Every outcome has a one-half chance, so both average 100 — but B swings far more. Bar height is probability.
A bet paying 0 or 200 with equal chances. Its expected value is 100, but for someone whose utility curves like √W the bet is worth the same as a sure 50 (½√0 + ½√200 ≈ 7.07, and 7.07² = 50). That 50 is the certainty equivalent; the gap of 50 to the expected value is the risk premium (orange line).
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.
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