Statistical Distributions for Financial Asset Prices and Returns

A portfolio manager gets fresh earnings data and has to update return expectations on the spot. The math behind that update, expected values, distribution moments, and Bayesian revision, is exactly what this reading drills.

To calculate the unconditional expected value, treat every scenario as one probability-weighted sum, ignoring new information. You interpret and evaluate it as the mean you would expect before any new signal arrives.

$$E(X)=\sum _{i=1}^{n}P(x_i)\cdot x_i$$

Variance measures squared deviation from the mean. Covariance measures the joint movement of two variables.

$${\sigma }^2(X)=E[(X-E(X){)}^2]\text{Cov}(X,Y)=E[(X-E(X))(Y-E(Y))]$$

KEY: Covariance units = units of X times units of Y, so magnitude alone is uninterpretable. Divide by σ(X)·σ(Y) to get correlation, which is unit-free and bounded by [-1, +1].

Every distribution is characterized by four moments. Each adds a piece of the shape.

Skewness. Positive skew has a long right tail (occasional large gains). Negative skew has a long left tail (occasional crashes). Equity returns typically show negative skew.

Common mistakes

• Confusing P(E|H) with P(H|E). Likelihood is not posterior. Trap: stating "80% of buy signals beat the market" when the data only says 80% of market-beaters get a buy signal.
• Treating covariance magnitude as comparable. Covariance scales with the units of both variables. Trap: ranking diversification benefit by covariance instead of correlation.
• Equating unconditional variance with the weighted average of conditional variances. That formula omits spread of the conditional means. Trap: forgetting the between-scenario variance term.

Bottom line

• Unconditional $E(X)=\sum P(x_i)\cdot x_i$ uses prior probabilities. Conditional E(X|A) re-weights outcomes by P(scenario|A)
• Four moments: mean (location), variance (spread), skewness (asymmetry), kurtosis (tail weight). Normal has skewness = 0, excess kurtosis = 0
• Lognormal models prices (cannot go negative, right-skewed). Normal models continuously compounded returns
• Covariance sign matters; magnitude depends on units. Correlation rescales to [-1, +1]

Exam shortcut

For Bayes, draw the 2×2 tree of (H, not H) × (E, not E) before reaching for the formula. Numerator is one cell; denominator is the column total. For distribution choice: prices use Lognormal; log returns use Normal; small-sample mean tests use Student's t; variance tests use chi-square or F. For moments, remember MVSK in order: Mean, Variance, Skewness, Kurtosis.

The full lesson (about 1,901 words, 13 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

Learning objectives

• statistical distributions for financial asset prices and returns

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