Common Probability Distributions in Finance › The Normal Distribution

The normal distribution, also called the Gaussian distribution, is a common baseline in financial modeling. One mathematical reason is the classical Central Limit Theorem: standardized sums of independent, identically distributed random variables with finite, strictly positive variance converge in distribution to a normal law. Thinking of a return as the aggregate effect of many small shocks can motivate a normal approximation, but market shocks need not be independent or identically distributed, so this is a modeling heuristic rather than a theorem about returns.

To understand what it means for a random variable to follow a normal distribution, we must first grasp the concept of a probability density function. Unlike discrete random variables that take on specific values with certain probabilities, a continuous random variable like one following a normal distribution can take any value along the real line. Probability over a region is the integral of the density over that region; density at one point is not itself a probability. A higher density at a point means that sufficiently small, equal-width neighborhoods around that point have greater probability than neighborhoods around a lower-density point.

A nondegenerate normal variable $X$ has mean $\mu$ and variance $\sigma^2>0$, written $X \sim N(\mu, \sigma^2)$, with probability density function:

Each component serves a specific purpose. Let's examine each component:

• $x$: the value at which we evaluate the density
• $\mu$: the mean (center of the distribution)
• $\sigma$: the standard deviation (controls the spread)
• $\frac{1}{\sigma\sqrt{2\pi}}$: a normalization constant ensuring the density integrates to 1
• $(x-\mu)^2/(2\sigma^2)$: one half of the squared standardized distance from $x$ to the mean

The heart of the normal distribution lies in the exponential term. The quantity $(x-\mu)^2$ measures the squared distance between the point $x$ and the center of the distribution $\mu$. Dividing by $2\sigma^2$ scales this distance relative to the spread of the distribution. A value two units away from the mean is much more unusual when $\sigma = 0.5$ than when $\sigma = 10$, because the standardized distance is larger. The negative sign in the exponent ensures that the function decreases as we move away from the mean, and the exponential function turns this squared distance into a probability density that decays smoothly and symmetrically.

The exponential of a negative quadratic creates the characteristic bell shape: values near the mean have high probability density, while values far from the mean have exponentially decreasing density. For a standard normal variable, the two-sided probability beyond four standard deviations is about $6.3 \times 10^{-5}$, and beyond five it is about $5.7 \times 10^{-7}$. Whether those benchmark probabilities adequately describe a financial return series is an empirical question tied to the asset, horizon, and sample. The quadratic term in the exponent is the key: because squaring grows faster than linear growth, the tails of the normal distribution thin out quickly.

The normalization constant $\frac{1}{\sigma\sqrt{2\pi}}$ deserves special attention. This factor ensures that when we integrate the density function over all possible values from negative infinity to positive infinity, we obtain exactly 1. This is the total probability. The $\sqrt{2\pi}$ arises from the Gaussian integral, one of the most famous results in calculus, while the $\sigma$ in the denominator accounts for how wider distributions (larger $\sigma$) must have lower peak heights to maintain the same total area under the curve.

Key parameters:

• Mean ($\mu$): The center of the distribution and the expected value of $X$. In finance, this represents the expected return.
• Variance ($\sigma^2$): Measures the spread of the distribution. The square root, $\sigma$, is the standard deviation, which in finance we call volatility.
• Skewness: Zero for the normal distribution, meaning it is perfectly symmetric around the mean.
• Kurtosis: Equal to 3 for the normal distribution (or excess kurtosis of 0). Kurtosis is the fourth standardized moment; by itself, it does not uniquely determine how probability is divided among the center, shoulders, and tails.

The symmetry of the normal distribution is both its elegance and its limitation. Symmetry is a modeling assumption: it assigns equal probabilities to equal-sized deviations above and below the mean. Some financial return series exhibit asymmetry. This possibility is one reason to treat the normal distribution as a baseline rather than a universal model of returns.

The standard normal distribution is the special case where $\mu = 0$ and $\sigma^2 = 1$. Any normal variable with $\sigma>0$ can be standardized:

For $\sigma>0$, this process turns a normal random variable into the canonical standard normal form. The operation $\frac{X - \mu}{\sigma}$ first centers the distribution by subtracting the mean (so the new center is at zero), then scales it by dividing by the standard deviation (so the new spread equals one). This step preserves the essential shape of the distribution while placing it in a standard coordinate system.

This standardization is central because it allows us to use a single table of probabilities for any normal distribution. The cumulative distribution function of the standard normal, denoted $\Phi(z)$, gives the probability that a standard normal variable is less than $z$. Before computers, statisticians relied on printed tables of $\Phi(z)$ values, and the standardization formula allowed them to answer questions about any normal distribution using this single reference. Even today, the standard normal is the universal reference point: when we speak of a "3-sigma event," we mean an outcome more than three standard deviations from the mean, regardless of what the actual mean and standard deviation are.

One of the most useful properties of the normal distribution is how probability mass is concentrated around the mean:

• Approximately 68% of observations fall within one standard deviation of the mean
• Approximately 95% fall within two standard deviations
• Approximately 99.7% fall within three standard deviations

These percentages emerge directly from integrating the normal density function, but their approximate values (68, 95, and 99.7) are worth committing to memory because they provide instant intuition for assessing how unusual any particular observation is. If someone tells you that an event was a "two-sigma move," you immediately know that such events occur roughly 5% of the time, or about one trading day in twenty.

This rule provides quick mental estimates for risk. If daily returns are normal with a standard deviation of 1%, the probability of a deviation from their mean larger than 3 percentage points is about 0.27%. For independent daily draws, the expected waiting time is about 370 trading days, or roughly 1.5 trading years at 252 days per year. An absolute return larger than 3% has that same probability only when the mean is zero. These are model-based estimates; a different return distribution can produce very different frequencies.

The normal distribution appears throughout quantitative finance:

Portfolio theoryLink Copied

Portfolio theory often summarizes portfolios by expected return and variance. Normality is sufficient for those two moments to determine the full distribution of portfolio returns, but the mean-variance framework itself does not require normally distributed returns. When candidate portfolio returns are normal, expected utility for a fixed utility function can be evaluated from the mean and variance because those parameters determine the distribution, provided that expected utility exists and is finite.

Value at Risk (VaR)Link Copied

Parametric normal VaR assumes normally distributed returns. Under that model, the signed VaR loss quantile at confidence level $0<\alpha<1$ is:

This formula translates a tail quantile into a loss estimate. For normally distributed returns with mean $\mu$ and standard deviation $\sigma>0$, we seek the threshold such that returns fall below it with probability $1-\alpha$, or $100(1-\alpha)\%$. The standard normal quantile $z_\alpha$ determines the lower-return threshold $\mu-z_\alpha\sigma$.

Each component of this formula plays a specific role:

• $\text{VaR}_\alpha$: the signed loss quantile at confidence level $\alpha$; under a continuous model, losses exceed this threshold with probability $1-\alpha$
• $\mu$: the expected return over the time horizon
• $\sigma$: the volatility (standard deviation of returns) over the time horizon
• $z_\alpha$: the $\alpha$-quantile of the standard normal distribution (e.g., $z_{0.95} \approx 1.645$ for 95% confidence)

The formula works because under normality, the return quantile at probability $1-\alpha$ is exactly $\mu-z_\alpha\sigma$. For example, the 95% VaR uses $z_{0.95}\approx1.645$, and losses exceed the threshold with probability 5%. Taking the negative of this return quantile yields the signed loss quantile $z_\alpha\sigma-\mu$. It is positive when $z_\alpha\sigma>\mu$, zero at equality, and negative when the modeled lower-tail return quantile is still a gain. A reporting policy that disallows negative VaR can use $\max(0,z_\alpha\sigma-\mu)$, but that floor is a reporting convention rather than the raw quantile.