The Distributions Finance Runs On

In the last lesson you met the machinery of a random variable: a PDF that says how densely outcomes cluster, a CDF that accumulates probability up to a point, and an expectation that pins down the average. Lovely abstractions — but finance doesn’t run on abstractions. It runs on a short list of named distributions that show up again and again: the normal for returns, the lognormal for prices, and the fat-tailed cousins that take over the moment markets stop behaving.

This lesson is the bestiary. Learn these few shapes and you’ll recognize the assumptions hiding inside option pricers, risk models, and that “two-sigma move” your PM just muttered. Get them wrong — assume a bell curve where a fat tail lurks — and you’ll be the one explaining why a “once in a million years” loss happened twice this decade.

The normal distribution: the bell everyone defaults to

Analogy. Picture every analyst’s height in a giant firm marked on a wall. Most cluster near average; a few are unusually short or tall; nobody is 30 feet tall. Plot the pile-up and you get the famous bell curve — dense in the middle, thinning symmetrically toward both edges. The normal distribution is that shape made precise.

Definition. A normal (Gaussian) distribution is fully described by just two numbers: its mean $\mu$ (where the peak sits — the center of mass) and its standard deviation $\sigma$ (how wide the bell is — the typical distance from the mean). Its PDF is

You rarely touch that formula by hand. What you do use constantly is its signature 68–95–99.7 rule: for any normal distribution,

Worked example. A stock has a monthly return that’s roughly normal with mean $\mu = 1\%$ and standard deviation $\sigma = 5\%$. Then:

• About 68% of months land between $1\% - 5\% = -4\%$ and $1\% + 5\% = 6\%$.
• About 95% land between $1\% - 10\% = -9\%$ and $1\% + 10\% = 11\%$ (two sigmas out).
• Only about 0.3% of months — roughly 1 in 300 — should fall outside $\pm 3\sigma$, i.e. below $-14\%$ or above $16\%$.

The same two-curve picture below makes the spread visible: both bells share an average return, but the wider one scatters its outcomes much farther from that center. That width is exactly $\sigma$.

Why it’s the default. The normal is the workhorse for three reasons. First, the Central Limit Theorem: add up many small independent shocks (the thousands of trades that move a price each day) and their sum tends toward a normal, almost regardless of the individual shocks’ shapes. Second, it’s mathematically frictionless — sums of normals are normal, and the whole thing is pinned down by $\mu$ and $\sigma$ alone, which makes portfolio math tractable. Third, convention and tooling: VaR, the Sharpe ratio, and option-pricing models all start from a Gaussian assumption, so it’s the lingua franca.

Standardizing: the z-score

Analogy. Two students score 80 and 80 on different exams. Identical? Not until you know each class’s average and spread. Converting a raw score into “how many standard deviations above or below average” puts them on one common ruler. That common ruler is the z-score, and it does the same job for returns measured in wildly different units.

Definition. The z-score of a value $x$ from a distribution with mean $\mu$ and standard deviation $\sigma$ is

It answers one question: how many standard deviations is $x$ from the mean? Subtracting $\mu$ recenters the distribution to zero; dividing by $\sigma$ rescales it to unit width. The result is the standard normal — a normal distribution with $\mu = 0$ and $\sigma = 1$ — and crucially, every normal becomes the same standard normal after this transform. Compute one CDF table once, reuse it forever.

Worked example. Two funds have a $-3\%$ month.

Same headline loss, completely different stories. For Steady-Eddie a $-3\%$ month is a 2-sigma event — out near the 95% edge, genuinely surprising. For Wild-Child it’s a $-0.5$-sigma yawn, well inside an ordinary month. The z-score is what lets you say so.

The lognormal distribution: why PRICES, not returns, get this one

Analogy. Returns can swing either way around zero — up 5%, down 5%, symmetric. But a price has a hard floor: a share can fall to $0 and stop, yet it can in principle climb without limit. That lopsidedness — a wall on the downside, open sky on the upside — is exactly what the lognormal distribution captures.

Definition. A variable is lognormally distributed if its logarithm is normally distributed. The link runs through compounding: if continuously-compounded (log) returns are normal, then price is the exponential of those returns, and the exponential of a normal is lognormal. Two consequences fall straight out:

1. Prices can’t go negative. The exponential function never dips below zero, so a lognormal variable lives strictly on the positive side — matching reality, where a stock price is never negative.
2. It’s right-skewed. Exponentiating stretches the upper tail and compresses the lower one, so the distribution leans left with a long tail trailing off to the right. The mean sits above the median, dragged up by those rare large gains.

This is why the same asset is modeled two ways at once: its returns are taken as (roughly) normal and symmetric, while its price — the compounded result of those returns — comes out lognormal and skewed. Black–Scholes option pricing is built on exactly this: log returns normal, terminal price lognormal.

Worked example. A stock at $100 has a continuously-compounded annual log return that’s normal with mean $0$ and standard deviation $\sigma = 0.2$. Over one year, the log return $r$ is normal, and the price ends at $S = 100\,e^{r}$.

• A typical outcome ($r = 0$) gives $100 \times e^{0}$, landing the price at $100 — the median.
• A good year ($r = +0.2$, one sigma up) gives $100 \times e^{0.2}$, about $122.
• A bad year ($r = -0.2$, one sigma down) gives $100 \times e^{-0.2}$, about $82.

Notice the asymmetry: the up-move adds about $22 while the matching down-move subtracts only about $18. Equal-and-opposite log returns produce unequal price moves — the upside runs a little hotter. That gap is the right-skew, and it’s why the lognormal’s mean (about $102 here) sits above its median ($100).

Fat tails: when the bell underpredicts the extremes

Analogy. The normal distribution is an optimist: it assumes the world calms down quickly as you move away from average, so anything past $\pm 4\sigma$ is essentially impossible. Real markets are pessimists. They serve up crashes, gaps, and limit-down days far more often than that. The distribution of real returns has fat tails (the technical term is leptokurtosis — excess kurtosis): more mass piled in the extremes than a Gaussian permits, often with a taller, skinnier peak in the middle to compensate.

Definition. Kurtosis measures tail-heaviness. The normal has a kurtosis of exactly $3$ (or “excess kurtosis” of $0$). A distribution with excess kurtosis $> 0$ — leptokurtic — has fatter tails and a sharper peak; real daily equity returns routinely show excess kurtosis well above zero. The practical translation: events the normal labels “once in millennia” actually arrive every few years.

A standard fix is to swap the Gaussian for the Student-t distribution, which is bell-shaped but heavier-tailed. It has a single dial, the degrees of freedom $\nu$ (nu): low $\nu$ means very fat tails, and as $\nu \to \infty$ the Student-t converges back to the normal. Slide the dial below and watch how a 4-sigma move goes from a freak event to a frequent visitor.

Worked example. Under a normal distribution, a daily move beyond $-4\sigma$ has probability about $0.003\%$ — roughly once every 30,000 trading days, or about once every 125 years. Yet across long market histories, $-4\sigma$ (and far worse) days show up every handful of years. October 19, 1987 — Black Monday — was a daily move of roughly $-20\sigma$ on the prevailing volatility. Under a strict normal model that’s not “rare”; it’s so absurdly improbable it shouldn’t happen in the lifetime of the universe. It happened on a Monday. That single contradiction is the whole case for fat tails.

Skewness: why equity returns lean the wrong way

Analogy. Imagine the staircase out of a crowded stadium versus the panic of a fire drill. People file out calmly over many minutes but stampede out in seconds. Stock markets have the same rhythm: they grind up slowly and crash down fast. That asymmetry — many small gains punctuated by occasional violent drops — gives equity returns a negative skew.

Definition. Skewness measures the lopsidedness of a distribution. A negative (left) skew means a long tail stretching into the losses: most outcomes are small positives, but the rare big moves are disproportionately to the downside. (Positive skew is the mirror image — a long right tail of big gains, the shape of, say, lottery tickets or venture bets.) For a left-skewed distribution the mean sits below the median: those occasional crashes drag the average down even though a typical day is mildly positive.

Worked example. Suppose a year of monthly equity returns reads: eleven months of $+1\%$ and one month of $-15\%$ (a crash).

• The median month is $+1\%$ — the calm, typical experience.
• The mean month is $\dfrac{11(1\%) + (-15\%)}{12} = \dfrac{11\% - 15\%}{12} = \dfrac{-4\%}{12} \approx -0.33\%$.

The mean is below the median, pulled under water by the single ugly month. That’s negative skew in one line of arithmetic — and it’s why “the market was up most months” can still coexist with “the fund lost money this year.”

The big pitfall: normality underprices the tail

Stack the previous two sections together and you get finance’s most expensive modeling mistake. Real returns are fat-tailed and negatively skewed. The normal distribution is thin-tailed and symmetric. So whenever you model returns as normal, you make two errors that point the same dangerous direction:

1. Fat tails mean extreme losses are bigger and more frequent than the bell allows — you under-count how often disaster strikes.
2. Negative skew means the worst surprises cluster on the downside — and a symmetric model splits its (already too-thin) tail evenly, under-counting how bad the left tail gets.

Both errors understate downside risk. A VaR or capital model built on a clean Gaussian will look prudent in calm markets and then be blindsided, repeatedly, by losses it rated as near-impossible. This isn’t a hypothetical: the 2008 crisis was littered with “25-standard-deviation events, several days in a row” — a phrase that, under a true normal, describes something that cannot happen even once in the age of the cosmos, let alone on consecutive Tuesdays. The model wasn’t unlucky; it was wrong about the shape.

Putting it together

Six shapes, one ladder. The normal is the symmetric bell that two numbers — $\mu$ and $\sigma$ — fully describe, with its memorable 68–95–99.7 rule and a built-in claim that up and down surprises are equally likely. The z-score collapses every normal onto one standard ruler, so a single CDF table serves them all. Prices get the lognormal because they compound, can’t go negative, and skew right. And then reality intrudes: returns have fat tails (leptokurtosis, tamed by the Student-t) and negative skew, both of which mean a Gaussian quietly underprices the crash. Know which shape applies where, and you’ll read a risk model’s assumptions as clearly as its outputs.

Next we put these shapes to work: with returns, expectations, and a notion of spread in hand, the following lessons turn distributions into the core risk-and-return statistics — variance, covariance, correlation, and the ratios desks actually trade on.