Expected Value: Definition, Formula, and Examples

What is expected value?

Expected value (EV), also called the expectation or mean, is the probability-weighted average of all possible outcomes of a random variable. It represents the long-run average value you would expect if an experiment were repeated many times. EV quantifies the center of a probability distribution but does not describe the variability or risk around that center.

Key points:
* EV is a probability-weighted average, not a guaranteed outcome.
* For continuous variables, EV is calculated using integrals; for discrete variables, it is a sum.
* By the law of large numbers, the sample average converges to the EV as the number of repetitions grows.

Formula

For a discrete random variable X with possible values Xi and probabilities P(Xi):

EV = Σ P(Xi) × Xi

Where:
* Xi = a possible value of X
* P(Xi) = probability that X equals Xi

For continuous variables, EV = ∫ x f(x) dx, where f(x) is the probability density function.

Interpreting EV

• EV describes the long-term average outcome, not the outcome of a single trial.
• EV accounts for probabilities; an outcome with low probability but a large payoff can substantially influence EV.
• EV does not capture risk, variance, or investors’ utility; two investments can have the same EV but very different risk profiles.

Examples

Die roll (discrete uniform distribution)
A fair six-sided die has outcomes 1–6, each with probability 1/6:

EV = (1/6)(1) + (1/6)(2) + (1/6)(3) + (1/6)(4) + (1/6)(5) + (1/6)(6) = 3.5

Investment scenario (simple two-outcome example)
An investment has:
* 60% chance to gain $10,000 → 0.6 × 10,000 = $6,000
* 40% chance to lose $5,000 → 0.4 × (−5,000) = −$2,000

EV = $6,000 + (−$2,000) = $4,000

This $4,000 is the expected return; it does not indicate the variability or risk of the investment.

EV in finance and portfolio construction

• Scenario analysis: EV is used with probability-estimated scenarios to compare likely outcomes for investments.
• Asset comparison: Calculate and compare EVs across assets (stocks, bonds, ETFs) to help select holdings, but always consider risk and correlations.
• Portfolio optimization: Modern portfolio theory uses expected returns (means) together with variances/covariances (risk) to find efficient portfolios via mean–variance optimization.
• Rebalancing/selection: Investors can use EV to evaluate whether to replace underperforming assets with higher-EV alternatives.

Valuation vs. statistical EV
* Dividend-paying stocks: Valuation models (e.g., discounted cash flow or dividend discount models such as the Gordon Growth Model) estimate the intrinsic value as the net present value of expected future dividends. This valuation approach differs from the statistical EV concept but serves a similar purpose in estimating worth.
* Non-dividend stocks: Analysts may use multiples (e.g., P/E ratios) versus peers to estimate value—again a valuation method distinct from the probability-weighted EV used in statistics.

Limitations and cautions

• Requires reliable probabilities: EV is only as accurate as the probability estimates used.
• Ignores risk preference: EV does not reflect risk aversion or utility; two options with the same EV may be preferred differently by risk-averse investors.
• Doesn’t capture extreme outcomes: EV can understate tail risk or the impact of rare catastrophic events.
• Single-trial relevance: EV is most meaningful over many repetitions; for single, non-repeatable decisions, utility-based analyses or risk-adjusted metrics may be more appropriate.

Key takeaways

• Expected value is the probability-weighted average of all possible outcomes and indicates a long-term average, not a guaranteed result.
• EV is useful for comparing investments and informing portfolio construction but must be considered alongside risk, variance, and investors’ objectives.
• Valuation methods for stocks (dividend discount models, multiples) are conceptually related to estimating expected future returns but are distinct from the statistical EV calculation.
• Always assess the reliability of probabilities and complement EV analysis with risk measures and scenario testing.

Bottom line

Expected value is a foundational concept for decision-making under uncertainty. It helps quantify average outcomes and compare alternatives, but effective use requires careful probability estimates and attention to risk, variance, and investor preferences.