Statistical Distributions for Financial Asset Prices and Returns
Welcome to one of the foundational modules in Quantitative Methods! Financial markets are full of uncertainty. When you invest in a stock, bond, or fund, you cannot know with certainty what the future price or return will be. To quantify this uncertainty, financial analysts rely on statistical distributions.
Don't worry if probability and statistics feel intimidating at first. We will break every concept down into small, digestible parts, use intuitive analogies, and walk through real-world applications step by step. According to the CFA Level I syllabus, this module centers on four core pillars:
• Unconditional expected values for mean, variance, and covariance
• Principal moments of key distributions used in finance
• Conditional expectations, variances, and covariances
• Bayesian updating (updating probabilities when new information arrives)
Cross-reference note: Building portfolios using these parameters is explored in detail in The Return and Risk of a Financial Portfolio, while sampling and hypothesis testing are covered in Estimation and Hypothesis Testing.
---1. Unconditional Expected Value, Variance, and Covariance
Unconditional Expected Value (Mean)
The unconditional expected value of a random variable is simply its weighted average outcome, where each possible outcome is weighted by its probability of occurrence, before any new or conditioning information is observed.
Think of it like calculating your grade in a course: each assignment has a weight (probability), and your overall score is the sum of each grade multiplied by its weight.
Mathematically, for a discrete random variable \(X\) with \(n\) possible outcomes \(x_i\), each having probability \(P(X = x_i)\):
\(E(X) = \sum_{i=1}^{n} P(X = x_i) x_i\)
Properties of Expected Value:
• Constant: The expected value of a constant \(c\) is just the constant itself: \(E(c) = c\).
• Scalar multiplication: If you multiply a random variable by a constant \(c\), you multiply its expected value: \(E(c X) = c E(X)\).
• Additivity: The expected value of a sum of random variables is the sum of their individual expected values: \(E(X + Y) = E(X) + E(Y)\). This holds whether \(X\) and \(Y\) are independent or not!
Unconditional Variance
While the expected value tells you the central anchor of a distribution, variance measures how widely the outcomes spread out around that mean. It represents the average squared deviation from the expected value:
\(\sigma^2(X) = Var(X) = E\left([X - E(X)]^2\right) = \sum_{i=1}^{n} P(X = x_i) [x_i - E(X)]^2\)
The standard deviation, \(\sigma(X)\), is the positive square root of variance: \(\sigma(X) = \sqrt{Var(X)}\). In finance, standard deviation is the primary measure of total risk or volatility.
Properties of Variance:
• The variance of a constant is zero: \(Var(c) = 0\) (a constant never varies).
• Adding a constant shifts the center but doesn't change the dispersion: \(Var(X + c) = Var(X)\).
• Multiplying by a constant scales the variance by the square of that constant: \(Var(c X) = c^2 Var(X)\).
Unconditional Covariance and Correlation
Covariance measures the extent to which two random variables move together. If asset \(X\) tends to be above its mean whenever asset \(Y\) is above its mean, their covariance is positive. If one tends to rise when the other falls, their covariance is negative.
\(Cov(X, Y) = \sigma_{XY} = E\left([X - E(X)][Y - E(Y)]\right) = \sum_{i=1}^{n} P(x_i, y_i) [x_i - E(X)][y_i - E(Y)]\)
An alternative computational formulation is:
\(Cov(X, Y) = E(XY) - E(X)E(Y)\)
Note that the covariance of a variable with itself is simply its variance: \(Cov(X, X) = Var(X)\).
Because covariance depends on the units in which variables are measured, it is standardized into the correlation coefficient, denoted \(\rho(X, Y)\) or \(Corr(X, Y)\):
\(\rho(X, Y) = \frac{Cov(X, Y)}{\sigma(X) \sigma(Y)}\)
Correlation is strictly bounded: \(-1 \le \rho(X, Y) \le +1\). A value of \(+1\) means perfect positive linear co-movement, \(-1\) means perfect inverse linear co-movement, and \(0\) means no linear relationship.
Key Takeaway: Expected value is the probability-weighted average outcome, variance captures dispersion around that center, and covariance measures the direction and strength of linear co-movement between two assets.
---2. Principal Moments of Key Financial Distributions
In statistics, moments describe the shape of a probability distribution. The first four central moments are:
1. First moment (Mean / Location): Expected value \(E(X)\).
2. Second central moment (Variance / Dispersion): \(\sigma^2\).
3. Third standardized moment (Skewness): Degree of asymmetry. A symmetric distribution has skewness of 0. Positive (right) skew has a long right tail; negative (left) skew has a long left tail.
4. Fourth standardized moment (Kurtosis): Peakedness and fatness of tails. A normal distribution has a kurtosis of 3. Excess kurtosis is defined as \(\text{Kurtosis} - 3\). Distributions with excess kurtosis greater than 0 are called leptokurtic (fat-tailed), which is common in asset returns.
A. Discrete Uniform Distribution
A random variable that takes on a finite number of equally likely outcomes follows a discrete uniform distribution. A simple example is rolling a fair six-sided die, where each outcome \(x \in \{1, 2, 3, 4, 5, 6\}\) has probability \(P(X = x) = \frac{1}{6}\).
B. Continuous Uniform Distribution
A continuous random variable has equally likely outcomes over a continuous bounded range \([a, b]\). The probability density function is constant: \(f(x) = \frac{1}{b - a}\) for \(a \le x \le b\).
Principal Moments:
• Mean: \(E(X) = \frac{a + b}{2}\)
• Variance: \(Var(X) = \frac{(b - a)^2}{12}\)
• Skewness: 0 (the distribution is perfectly symmetric around its midpoint).
C. Binomial Distribution
The binomial distribution models the number of "successes" \(x\) in \(n\) independent Bernoulli trials, where each trial has only two possible outcomes (success or failure) and the probability of success \(p\) is constant across trials.
The probability of observing exactly \(x\) successes in \(n\) trials is:
\(P(X = x) = \binom{n}{x} p^x (1 - p)^{n - x} = \frac{n!}{x!(n - x)!} p^x (1 - p)^{n - x}\)
Principal Moments:
• Mean: \(E(X) = n p\)
• Variance: \(Var(X) = n p (1 - p)\)
• Standard deviation: \(\sigma = \sqrt{n p (1 - p)}\)
Exam Tip: In financial modeling, a binomial tree is widely applied to model upward and downward movements of asset prices over discrete time steps.
D. Normal Distribution
The normal distribution is the cornerstone of classical financial theory. It is a continuous, symmetric, bell-shaped distribution completely defined by just two parameters: its mean \(\mu\) and variance \(\sigma^2\), written \(X \sim N(\mu, \sigma^2)\).
Key Properties and Principal Moments:
• Mean, median, and mode: All equal to \(\mu\).
• Skewness: 0 (completely symmetric).
• Kurtosis: Exactly 3 (excess kurtosis is 0).
• Linear combinations: A linear combination of independent normal random variables is itself normally distributed.
Standard Normal Distribution (\(Z\)):
Any normal variable \(X \sim N(\mu, \sigma^2)\) can be standardized to have a mean of 0 and standard deviation of 1 by calculating the \(Z\)-score:
\(Z = \frac{X - \mu}{\sigma}\)
Empirical Rule (Interval Probabilities):
• Approximately 68% of observations fall within \(\mu \pm 1\sigma\)
• Approximately 95% of observations fall within \(\mu \pm 1.96\sigma\) (roughly \(\pm 2\sigma\))
• Approximately 99% of observations fall within \(\mu \pm 2.58\sigma\) (roughly \(\pm 3\sigma\))
E. Lognormal Distribution
A random variable \(Y\) follows a lognormal distribution if its natural logarithm, \(\ln(Y)\), is normally distributed: \(\ln(Y) \sim N(\mu, \sigma^2)\).
Why is the lognormal distribution essential for asset prices?
• A normal distribution allows negative values, but standard financial asset prices (like stocks) have limited liability and cannot fall below zero.
• The lognormal distribution is bounded below by zero (\(Y > 0\)), making it ideal for modeling asset prices.
• While continuously compounded asset returns can be negative and are often approximated by a normal distribution, the resulting asset prices follow a lognormal distribution.
• Skewness: The lognormal distribution is positively skewed (skewed to the right) with a long upper tail, reflecting that prices can grow unbounded but cannot drop below zero.
Key Takeaway: Normal distributions model symmetric phenomena with zero excess kurtosis (often used for returns). Lognormal distributions are bounded at zero and right-skewed, making them well suited for asset prices.
---3. Conditional Expectations, Variances, and Covariances
In finance, you constantly receive new information: economic reports, interest rate decisions, or corporate earnings. When new information arrives, your expectations change. This leads to conditional moments.
Conditional Expectation
The conditional expectation of a random variable \(X\) given that another event or variable \(S\) has occurred is denoted \(E(X \mid S)\). It is calculated by taking the weighted average of \(X\) using conditional probabilities.
The Total Probability Rule for Expected Value:
If a set of events \(S_1, S_2, \dots, S_k\) forms a mutually exclusive and exhaustive set of economic states (for example: Recession, Normal, Expansion), the unconditional expected value is the weighted average of the conditional expectations across all states:
\(E(X) = \sum_{j=1}^{k} P(S_j) E(X \mid S_j)\)
Step-by-Step Example:
Suppose you are estimating the return of Stock A next year under two economic states:
• State 1 (Boom): Probability \(P(S_1) = 0.40\), Conditional expected return \(E(R_A \mid S_1) = 20\%\)
• State 2 (Bust): Probability \(P(S_2) = 0.60\), Conditional expected return \(E(R_A \mid S_2) = 5\%\)
Applying the total probability rule:
\(E(R_A) = (0.40)(20\%) + (0.60)(5\%) = 8\% + 3\% = 11\%\)
Conditional Variance
The conditional variance, \(Var(X \mid S)\), measures the uncertainty or dispersion of \(X\) given that event \(S\) has occurred:
\(Var(X \mid S) = E\left([X - E(X \mid S)]^2 \mid S\right)\)
The total unconditional variance reflects both the expected conditional variance within states and the variance across conditional expectations between states.
Conditional Covariance
Similarly, the conditional covariance between two assets \(X\) and \(Y\) given state \(S\) measures how they move together within that specific state:
\(Cov(X, Y \mid S) = E\left([X - E(X \mid S)][Y - E(Y \mid S)] \mid S\right)\)
Did you know? In periods of severe market distress, conditional covariances and correlations between risky assets often rise significantly. Diversification benefits can diminish precisely when investors need them most!
Key Takeaway: Unconditional moments reflect expectations before any event occurs; conditional moments adjust expectations once a specific state or piece of information is known.
---4. Bayesian Updating
The Logic of Bayes' Formula
Investment analysis is an ongoing learning process. You start with an initial belief about a state or asset performance, observe new data, and update your belief. Bayes' formula provides the formal mathematical rule for updating probabilities.
Terminology to Master:
• Prior Probability \(P(A)\): The baseline probability of event \(A\) before observing new information.
• Likelihood / Conditional Probability \(P(B \mid A)\): The probability of observing new evidence \(B\) given that event \(A\) is true.
• Unconditional Probability \(P(B)\): The total probability of observing evidence \(B\) across all possible states.
• Posterior Probability \(P(A \mid B)\): The updated probability of event \(A\) given that evidence \(B\) has occurred.
Bayes' Formula Formulation
\(P(A \mid B) = \frac{P(B \mid A)}{P(B)} P(A)\)
Using the total probability rule to expand the denominator \(P(B)\) across mutually exclusive states \(A\) and \(A^C\) (not \(A\)):
\(P(A \mid B) = \frac{P(B \mid A) P(A)}{P(B \mid A) P(A) + P(B \mid A^C) P(A^C)}\)
Step-by-Step Bayes' Calculation
Scenario: An equity research department evaluates earnings forecasts.
• An analyst classifies a company as having either "High Earnings Growth" (\(H\)) or "Low Earnings Growth" (\(L\)).
• Historically, \(30\%\) of companies achieve high growth: \(P(H) = 0.30\), so \(P(L) = 0.70\).
• A leading indicator signal produces a positive reading (\(+\)):
- If a company is high growth, the signal is positive with probability \(P(+ \mid H) = 0.80\).
- If a company is low growth, the signal gives a false positive with probability \(P(+ \mid L) = 0.20\).
• Question: If the signal is positive (\(+\)), what is the updated (posterior) probability that the company has high growth, \(P(H \mid +)\)?
Step 1: Compute the joint probabilities for the numerator and denominator components:
• High growth and positive signal: \(P(+ \mid H) P(H) = 0.80 \times 0.30 = 0.24\)
• Low growth and positive signal: \(P(+ \mid L) P(L) = 0.20 \times 0.70 = 0.14\)
Step 2: Calculate the total unconditional probability of a positive signal, \(P(+)\):
\(P(+) = 0.24 + 0.14 = 0.38\)
Step 3: Calculate the posterior probability \(P(H \mid +)\):
\(P(H \mid +) = \frac{0.24}{0.38} \approx 0.6316 \text{ or } 63.16\%\)
Interpretation: Observing the positive indicator shifted our belief from the prior probability of \(30\%\) up to a posterior probability of roughly \(63.2\%\).
Common Pitfalls to Avoid
• Confusing \(P(B \mid A)\) with \(P(A \mid B)\): The probability of having a positive signal given high growth (\(80\%\)) is very different from the probability of high growth given a positive signal (\(63.2\%\)).
• Forgetting the denominator expansion: Always ensure the denominator includes all ways the conditioning event could have occurred.
Key Takeaway: Bayes' formula systematically updates prior beliefs to posterior probabilities as new empirical evidence or signals emerge.
---Module Summary and Quick Review
• Expected Value \(E(X)\): The probability-weighted average outcome across all possible events.
• Variance \(\sigma^2\) and Covariance \(\sigma_{XY}\): Variance quantifies dispersion from the mean; covariance measures linear co-movement between two assets.
• Normal Distribution: Fully defined by \(\mu\) and \(\sigma^2\), symmetric (skewness = 0), kurtosis = 3. Models returns well, but allows negative values.
• Lognormal Distribution: Bounded below by zero and right-skewed. Well suited for modeling asset prices.
• Total Probability Rule: Connects conditional expectations to unconditional expected value: \(E(X) = \sum P(S_j) E(X \mid S_j)\).
• Bayesian Updating: Converts prior probability \(P(A)\) into posterior probability \(P(A \mid B)\) using conditional likelihoods.