Lesson 5: Expected Value and Variance: Measuring Uncertain Outcomes
1. Expected value: the probability-weighted average
Suppose an uncertain payoff X takes values x1, …, xk with probabilities p1, …, pk (summing to 1). Its expected value (mean) is:
Expected value is the long-run average if the gamble were repeated many times under identical conditions. It is not the most likely outcome, and it may be a value that never actually occurs.
2. Variance and standard deviation: how spread out are outcomes?
Variance is the probability-weighted average squared distance from the mean:
Standard deviation is the square root of variance: SD(X) = √Var(X). It is reported in the same units as X (for returns, percentage points), which is why finance usually quotes standard deviation rather than variance. Variance is in squared units — for returns, “percent-squared” — and is harder to interpret directly, but it is the quantity that combines across assets in Lesson 6.
Worked example 1 — an investment with three outcomes
Possible one-year returns: +20% with probability 0.30, +5% with probability 0.50, −10% with probability 0.20.
Expected value: E = 0.30 × 20 + 0.50 × 5 + 0.20 × (−10) = 6 + 2.5 − 2 = 6.5%.
Variance: Var = 0.30 × (20 − 6.5)2 + 0.50 × (5 − 6.5)2 + 0.20 × (−10 − 6.5)2
= 0.30 × 182.25 + 0.50 × 2.25 + 0.20 × 272.25 = 54.675 + 1.125 + 54.45 = 110.25 (percent-squared).
Standard deviation: SD = √110.25 = 10.5 percentage points. So the investment is summarised as “6.5% expected return, 10.5% standard deviation”.
Worked example 2 — same idea, a second distribution
Outcomes: −10 with probability 0.25, 0 with probability 0.50, +20 with probability 0.25 (units are percentage points of return).
Expected value: E = 0.25 × (−10) + 0.50 × 0 + 0.25 × 20 = −2.5 + 0 + 5 = 2.5.
Variance: Var = 0.25 × (−12.5)2 + 0.50 × (−2.5)2 + 0.25 × (17.5)2 = 0.25 × 156.25 + 0.50 × 6.25 + 0.25 × 306.25 = 39.0625 + 3.125 + 76.5625 = 118.75.
Standard deviation: SD = √118.75 ≈ 10.90 percentage points.
3. Why expected value alone is not enough
Worked example 3 — identical expected value, different risk
Option A pays 10 with certainty: E = 10, Var = 0, SD = 0.
Option B pays 0 with probability 0.5 and 20 with probability 0.5: E = 0.5 × 0 + 0.5 × 20 = 10, but Var = 0.5 × (0 − 10)2 + 0.5 × (20 − 10)2 = 50 + 50 = 100, SD = 10.
Both options have expected value 10, yet B is plainly riskier. A decision-maker who cares only about expected value is indifferent between them; most real decision-makers are not — which is why finance reports both numbers together, as a return–risk pair.
4. Notes and cautions
- Population vs sample. The formulas above describe a known probability distribution. When variance is estimated from observed data, textbooks distinguish the population variance (dividing by n) from the sample variance (dividing by n − 1); the latter corrects a small-sample bias. This course uses known-distribution formulas throughout, so the distinction does not change any number here.
- Variance treats upside and downside alike. A surprisingly good outcome raises variance just as a bad one does. Investors usually dislike downside more than upside, which is one reason variance is a useful but incomplete risk measure.
- Probabilities are assumptions. Every number above depends on probabilities that were stipulated for teaching. For a real asset, probabilities must be estimated from data or models, and the estimates themselves are uncertain.
Key terms
- Expected value
- Probability-weighted average of possible outcomes.
- Variance
- Probability-weighted average squared deviation from the mean.
- Standard deviation
- Square root of variance; in the same units as the outcome.
References & further study
- Standard introductory probability and statistics textbooks for expected value, variance and standard deviation.
- Standard introductory corporate-finance and investments textbooks (Brealey / Myers / Allen; Bodie / Kane / Marcus; or equivalents) for the conceptual framework used in this course.
- Standard introductory probability and statistics textbooks for expected value, variance, covariance and correlation.
- MIT OpenCourseWare (generic entry point for free course materials in mathematics, probability and finance-related topics): ocw.mit.edu — use the site search for the relevant topic; no specific course is prescribed here.
📝 Quiz
1. Outcomes +20% (prob. 0.30), +5% (0.50), −10% (0.20). The expected value is:
Show answer
b) E = 0.30×20 + 0.50×5 + 0.20×(−10) = 6 + 2.5 − 2 = 6.5%. (10.5% is the standard deviation.)
2. For that same distribution, the variance and standard deviation are:
Show answer
a) Var = 0.30×182.25 + 0.50×2.25 + 0.20×272.25 = 110.25; √110.25 = 10.5.
3. Two options both have expected value 10: A pays 10 certainly, B pays 0 or 20 with equal probability. Which is true?
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b) Expected value alone hides risk. B's variance is 100 (SD 10); A's variance is 0.