Statistical Distributions for Financial Asset Prices and Returns

An analyst models next year's return as a set of scenarios, then new data arrives and the scenario odds shift. This reading is the toolkit for both halves: describing a distribution before the news, and updating it after.

Suppose a stock returns −10% in a recession, 8% in a normal year and 20% in an expansion, with probabilities 25%, 50% and 25%. You cannot know which year you will get. The expected value is the average you would earn if you could replay the year many times: weight each outcome by its probability and add.

$$E(X) = \sum_{i=1}^{n} P(x_i)\, x_i$$

This is the unconditional expected value, because it uses the probabilities before you learn anything new. For the stock, it is 0.25(−10) + 0.50(8) + 0.25(20) = 6.5%. Interpret it as a long-run average, not a forecast of any one year: the stock never actually returns 6.5% in any scenario.

The mean says where outcomes center. Variance says how far they spread. Take each outcome's deviation from the mean, square it so misses on both sides count, and weight by...

Common mistakes

• Averaging only the conditional variances. In Example 3, 0.75 × 32.0 = 24.0 is the trap for the unconditional variance. The correct figure adds the 90.75 spread of the conditional means, for 114.75.
• Forgetting to rescale conditional probabilities. Conditioning on "no recession" and leaving the weights at 0.50 and 0.25 gives 0.50(8) + 0.25(20) = 9.0%. Trap: the conditional mean is 12.0%, because the weights must be rescaled to 2/3 and 1/3.
• Flipping the conditional in Bayes. In Example 4, answering 80% (the chance a winner gets a buy signal) is the trap. The chance a buy signal marks a winner is 57.8%.

Bottom line

• Expected value = Σ probability × outcome. Variance = Σ probability × squared deviation; standard deviation is its square root.
• Covariance = Σ probability × product of the two deviations. Correlation = covariance / $(\sigma_X \sigma_Y)$, always between −1 and +1.
• Moments: mean, variance, skewness (third, standardized), kurtosis (fourth, standardized). Normal: skewness 0, kurtosis 3.
• Binomial: mean np, variance np(1 − p). Uniform: mean (a + b)/2, variance (b − a)²/12. Chi-square: mean k, variance 2k.

Exam shortcut

For Bayes, draw a two-branch tree before touching the formula: the numerator is one branch, and the denominator is every branch that produces the signal. For any conditional question, check first that your updated probabilities sum to 1. When a question gives a continuously compounded return distribution and asks for an expected price, the answer choice without the σ²/2 adjustment is the median, and it is the trap.

The full lesson (about 4,346 words, 29 min read) adds 11 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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