Statistical Distributions for Financial Asset Prices and Returns

Forecasts in finance are statements about probability: what returns are possible, how likely each is, and how the picture changes when new information arrives. This module shows how to compute expected values and risk from a probability distribution, how the binomial, normal, lognormal, t, chi-square and F distributions are used, how to work with conditional expectations and scenario trees, and how Bayes' formula updates a view in the light of new evidence.

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What you will learn

1. Calculate and interpret unconditional expected values, variances, standard deviations and covariances of returns from a probability distribution.

Expected value E(R) = sum of P(s) x R_s over all scenarios.
Variance = sum of P(s) x [R_s - E(R)]^2; no n - 1 adjustment when using probabilities.
Covariance = sum of P(s) x (deviation of A) x (deviation of B); correlation = covariance / (sigma_A x sigma_B).
Unconditional moments average across all scenarios.
Extreme scenarios have a large effect on variance because deviations are squared.

2. Describe the binomial and normal distributions, calculate probabilities and moments under each, and apply the normal distribution to shortfall risk.

Binomial: P(X = x) = C(n, x) p^x (1 - p)^(n - x); mean np, variance np(1 - p).
Normal: symmetric, described by mean and standard deviation; 68% within 1 sigma, 95% within 1.96 sigma, 99% within 2.58 sigma.
Standardize with z = (X - mu) / sigma to use the standard normal table.
Safety-first ratio = [E(R_P) - R_L] / sigma_P; choose the portfolio with the highest ratio.
Normal probabilities understate extreme losses when returns are fat-tailed.

3. Describe the lognormal, Student's t, chi-square and F distributions, including their shapes, parameters and uses in finance, and calculate the mean of a lognormal variable.

If log returns are normal, prices are lognormal: non-negative and right-skewed.
Lognormal mean = e^(mu + sigma^2 / 2) > median = e^mu.
Student's t: symmetric, fatter tails than normal, converges to normal as df rises; used when the variance is estimated.
Chi-square: non-negative, right-skewed, mean k, variance 2k; used for variance tests and independence tests.
F: ratio of two scaled chi-squares, two df parameters; used to compare variances and in regression ANOVA.

4. Calculate and interpret conditional expected values and variances, and use the total probability rule to combine scenario forecasts.

E(R | S) is the expected return given that scenario S occurs.
Total probability rule: E(R) = sum of E(R | S_i) P(S_i) over mutually exclusive, exhaustive scenarios.
Conditional variance uses conditional probabilities and the conditional mean.
Multiply probabilities along each branch of a tree to get joint probabilities.
Unconditional variance exceeds the average conditional variance when the conditional means differ.

5. Use Bayes' formula to update probabilities and investment views when new information arrives.

Posterior P(A | I) = P(I | A) x P(A) / P(I).
P(I) comes from the total probability rule over A and not-A.
Information more likely under A than in general raises the probability of A.
Ignoring the base rate (prior) leads to overreacting to signals about rare events.
Updating is sequential: each posterior becomes the next prior.

Module at a glance

Expected value, variance and covariance from a distribution are probability-weighted, with no n - 1 adjustment.
Binomial: P(x) = C(n, x) p^x (1 - p)^(n - x), mean np, variance np(1 - p).
Normal: 68% / 95% / 99% within 1 / 1.96 / 2.58 standard deviations; standardize with z = (X - mu) / sigma.
Choose the portfolio with the highest safety-first ratio to minimize shortfall probability.
Prices are lognormal if log returns are normal; mean = e^(mu + sigma^2 / 2) exceeds the median.
t: symmetric, fat-tailed; chi-square and F: non-negative and right-skewed.
Total probability rule: E(R) = sum of E(R | S_i) P(S_i).
Bayes: posterior = likelihood x prior / P(information).

This free module has 5 lessons, 13 practice questions with full explanations.

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