Expectation, Variance & the Moments

A probability distribution is a whole shape — an infinite cloud of possible returns, each with its own likelihood. That’s far too much to carry around in your head, so finance squeezes the cloud into a handful of summary numbers called moments. The first tells you where the distribution sits (its average), the second how wide it is (its spread), the third how lopsided it is, and the fourth how heavy its tails are. Master these four and you can describe any return series — a stock, a fund, a strategy — in four numbers that a risk manager will actually understand.

This lesson builds them from the ground up, straight from raw return numbers, and connects every one to the bell curves and fat tails you met last lesson.

Expectation: the probability-weighted average

Analogy. Imagine a weighted roulette wheel where each outcome is a possible return and the size of its slot is its probability. Spin it a million times, average the results, and you converge on the expectation — the long-run center of gravity of the distribution. It is not the most likely single outcome; it’s the balance point of the whole wheel.

Definition. For a discrete random variable $X$ taking values $x_i$ with probabilities $p_i$, the expectation (or mean) is

Each outcome is weighted by how often it happens. In finance $\mathbb{E}[X]$ is the expected return — the average return you’d earn if the same odds repeated forever.

Worked example. A trade has three possible one-year outcomes:

Add the last column: $7.5\% + 4.0\% - 5.0\% = 6.5\%$. So $\mathbb{E}[X] = 6.5\%$. Notice the expected return (6.5%) isn’t any of the three outcomes — it’s the weighted balance point, pulled down from the 8% base case by the fat 25% chance of a −20% bust.

Linearity of expectation

The single most useful property of expectation: it passes straight through sums and scalings, no matter what. For any constants $a$ and $b$,

Worked example. Suppose you lever the trade above $2\times$ and the broker skims a flat 1% fee, so your net return is $Y = 2X - 1\%$. Then

You never had to rebuild the scenario table — linearity let you scale the answer directly. This is why expected returns of a portfolio are just the weighted average of the components’ expected returns: expectation glides through the sum.

Variance and standard deviation: the spread

Analogy. Two archers can have the same average — both centered on the bullseye — yet one peppers the whole target while the other clusters tight. Variance measures that scatter: the average squared distance of each outcome from the mean. Its square root, the standard deviation, brings the units back to plain returns and is what finance calls volatility.

Definition. With mean $\mu = \mathbb{E}[X]$,

We square the deviations for two reasons: it stops positive and negative gaps from cancelling, and it punishes big misses far more than small ones. The square root then undoes the squaring so $\sigma$ is back in percent.

Worked example. A stock’s five monthly returns are:

First the mean: $(4 - 2 + 6 - 4 + 6)/5 = 10/5 = +2\%$.

Now the squared deviations from that 2% mean:

Sum of squares $= 4 + 16 + 16 + 36 + 16 = 88$. Treating these five as the whole population, divide by $n = 5$: $\mathrm{Var} = 88/5 = 17.6$ (in %²). The volatility is $\sigma = \sqrt{17.6} \approx 4.2\%$ per month. The bigger the typical swing from 2%, the bigger that number.

The wider bell below has the larger $\sigma$ — same average return, more risk:

Population vs sample: the n−1 correction

Analogy. When you estimate spread from a sample rather than the full population, you’ve already spent some of the data computing the sample mean — and the data always hugs its own mean a little too snugly. Dividing by $n$ would therefore underestimate the true spread. The fix is to divide by $n - 1$ instead of $n$: you’ve “used up” one degree of freedom pinning down the mean, so you have only $n - 1$ independent pieces of spread information left.

Definition. The two variance formulas differ only in the denominator:

The $n - 1$ version (called Bessel’s correction) is the unbiased estimator of the true variance — on average across many samples it lands on the right answer rather than systematically low.

Worked example. Reuse the five returns above. The sum of squared deviations was 88.

• As a population ($\div\,5$): $\sigma^2 = 88/5 = 17.6$, so $\sigma \approx 4.20\%$.
• As a sample ($\div\,4$): $s^2 = 88/4 = 22.0$, so $s \approx 4.69\%$.

The sample number is larger — exactly the upward nudge Bessel’s correction provides to undo the built-in underestimate. With only five data points the gap is sizable; with hundreds of daily returns, $n$ and $n - 1$ barely differ.

Annualizing volatility: the √252 rule

Analogy. Volatility, like risk over time, grows like a random walk, not a straight march. For independent returns, variance adds across periods — so over $T$ periods the variance is $T$ times the one-period variance, and the standard deviation is $\sqrt{T}$ times the one-period $\sigma$. To turn a daily volatility into an annual one, you scale by the square root of the number of trading days in a year (about 252).

The rule. With roughly 252 trading days per year,

Note $\sqrt{252} \approx 15.87$, a number worth memorizing: annual vol is daily vol times roughly 16.

Worked example. A stock has a daily volatility of $1\%$. Then

Going the other way, an index quoted at $20\%$ annual vol has a daily vol of $20\% / \sqrt{252} \approx 1.26\%$. And to annualize from monthly data you’d use $\sqrt{12}$ instead, because there are twelve months in a year — the rule generalizes to $\sqrt{(\text{periods per year})}$.

Skewness: the third moment (lopsidedness)

Analogy. A symmetric bell is a balanced see-saw. Skewness measures which way it tips. A long tail stretching to the right (rare big gains) is positive skew; a long tail to the left (rare big losses) is negative skew. The mean gets dragged toward the long tail, away from the bulk of typical outcomes.

Definition. Skewness is the standardized third moment:

Cubing keeps the sign of each deviation (unlike squaring), so the formula registers which side carries the heavier, more extreme outcomes. Zero skew means perfectly symmetric, like the normal distribution.

What the sign means for returns. This is where skew earns its keep:

• Negative skew — frequent small gains, occasional brutal crashes. Most equity indices, credit, and short-volatility strategies live here. Investors dislike it: the rare event is a disaster.
• Positive skew — frequent small losses, occasional huge wins. Lottery tickets, far out-of-the-money call options, and trend-following live here.

Kurtosis and excess kurtosis: the fourth moment (fat tails)

Analogy. Two distributions can share the same mean, the same variance, and the same symmetry, yet one still serves up far more extreme outcomes than the other. Kurtosis measures that — how much of the action lives in the tails versus the shoulders. High kurtosis means a sharper peak and fatter tails: lots of tiny moves punctuated by rare giant ones.

Definition. Kurtosis is the standardized fourth moment:

The fourth power weights extreme deviations enormously — a $4\sigma$ event contributes $4^4 = 256$ times as much as a $1\sigma$ one — so kurtosis is dominated by the tails. The normal distribution has a kurtosis of exactly 3, so it’s convenient to subtract that benchmark:

Reading the sign. Excess kurtosis is the fat-tail meter:

• Excess kurtosis > 0 (leptokurtic) — fatter tails than the normal. Extreme moves are more frequent and larger than a bell curve predicts. Real financial returns live here, often with excess kurtosis well above zero.
• Excess kurtosis = 0 — exactly normal-tailed (mesokurtic).
• Excess kurtosis < 0 (platykurtic) — thinner tails, more uniform-looking outcomes.

This is the spaced-recall payoff: the lognormal prices and fat tails from last lesson aren’t a separate topic — they’re just a distribution’s third and fourth moments talking. The chart above lets you widen σ; below, drag the same intuition onto where the mass sits.

Putting it together

Any return distribution, however messy, can be summarized by four moments: the expectation (where it’s centered, and it glides through sums via linearity), the variance / volatility (how wide it spreads, scaled across time by √t), the skewness (which way it leans), and the kurtosis (how fat its tails are). Use the $n - 1$ sample correction when you’re estimating from data, and remember that real returns are negatively skewed and fat-tailed — exactly the shape a tidy normal model refuses to admit.

Next up — covariance, correlation and regression — we stop describing one return series in isolation and ask how two of them move together: the covariance that drives diversification, the correlation that standardizes it onto a −1-to-+1 scale, and the regression line that turns a cloud of points into an estimate of beta.