Random variables — returns are random variables

The abstraction: random variables

A random variable is a function that assigns a number to each outcome of a random process. Two examples:

• Discrete: the number of heads in 10 coin flips. Possible values: 0, 1, ..., 10.
• Continuous: tomorrow's log return on SPY. Possible values: any real number (mostly small ones).

The capital convention is to use $X$ for the random variable itself and lowercase $x$ for a specific realization. So:

• $X$ = "tomorrow's SPY return" (the abstract concept, not yet known)
• $x = 0.0034$ (a specific realization, observed after tomorrow)

This distinction matters: a strategy is built around the distribution of $X$ (many possible futures), not the value of any single $x$.

Two flavors

Discrete

Takes one of countably many values, each with its own probability:

$P(X = x_i) = p_i, \quad \sum_i p_i = 1$

Example: in a hypothetical "up day or down day" model:

• $P(X = \text{up}) = 0.53$
• $P(X = \text{down}) = 0.47$

That's a complete distribution.

Continuous

Takes any value in an interval. The probability of any single point is zero (this surprises people the first time). Instead, probability is described by a density function $f(x)$:

$P(a \le X \le b) = \int_a^b f(x)\,dx$

For continuous random variables, we ask "what's the probability $X$ is between $a$ and $b$" — not "what's the probability $X$ equals exactly 1.7%."

Daily stock returns are usually modeled as continuous.

Why daily returns are well-modeled this way

Daily log-returns of liquid US equities tend to look approximately like draws from a distribution centered slightly above zero with some width. The exact distribution isn't perfectly normal (we'll come back to fat tails in A1-03), but the abstraction — that returns are draws from a distribution we can characterize — is the foundational move that lets us build:

• Risk metrics (variance of returns)
• Portfolio theory (mean and covariance of asset returns)
• Strategy evaluation (is the realized return distribution consistent with the hypothesis?)
• Option pricing (the entire Black-Scholes machinery)

Without random variables, none of this exists.

The cumulative distribution function (CDF)

A useful object for any random variable is the CDF:

$F(x) = P(X \le x)$

For SPY daily returns: historically $F(-0.02) \approx 0.025$, meaning "about 2.5% of days have returns at or below −2%." The CDF lets you read off any "what fraction of days have returns below X?" question directly.

A related concept: quantiles. The 5th percentile of daily returns is the value $x$ such that $F(x) = 0.05$. We'll use these constantly when we discuss Value-at-Risk and tail risk.

Explain it back

Without scrolling up, write 2–3 sentences explaining why we treat tomorrow's stock return as a random variable rather than just "an unknown number." Compare to the lesson when you're done.