Module 4 · Distributions Core

Use the normal distribution and z-scores, and explain why real returns have fatter tails.

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0:00 Black Monday 0:19 The bell curve 0:45 Z-scores 1:23 Time narrows the curve 1:59 Lognormal and binomial 2:23 Fat tails 3:02 What fat tails mean for you 3:20 Your mission

What a distribution is

A description of every possible outcome and its probability. A model of returns is a choice of distribution.

The normal distribution

The symmetric bell curve, fully described by its mean and standard deviation.

Z-scores

A z-score converts any value into standard deviations from the mean: z = (value minus mean) / standard deviation. With a monthly mean of 1% and standard deviation of 4%, a return of minus 7% has z = (minus 7 minus 1) / 4 = minus 2. Under a normal distribution the chance of a result that bad or worse is about 2.3%.

Useful cut-offs

The lognormal distribution

If log returns are normal, prices are lognormal: they cannot fall below zero and are skewed to the right. This is the assumption inside the Black-Scholes model (RS 303).

The binomial distribution

Counts successes in n independent trials, each with probability p. If a stock market rises in 70% of years independently, the chance of five up years in a row is 0.7^5 = 16.8%. The binomial tree for options is built from this.

Where the normal curve fails

Real returns have fat tails and, for equities, negative skew.

• On October 19, 1987, US stocks fell more than 20% in a day. With daily volatility of about 1%, the normal model treats that as a move of some 20 standard deviations, an event it says should never happen in the life of the universe.
• Daily moves beyond 4 standard deviations occur many times more often than the model predicts.
• Calm periods and turbulent periods cluster (Module 9), so the true distribution changes over time.

Consequences

Models that assume normality understate the probability and size of extreme losses. Options far from the current price are priced with higher implied volatility because the market knows this. Risk estimates such as value at risk should be treated as a floor on what can go wrong in bad times, not a ceiling.

The Student’s t distribution

Bell-shaped with fatter tails, used when the sample is small and the standard deviation is estimated from the data. With fewer than about 30 observations, use t critical values instead of normal ones; they are larger.

More worked examples

From probabilities to cut-offs

NORM.S.DIST(z, TRUE) gives the probability below z; NORM.S.INV(p) does the reverse, giving the z-score for a probability. NORM.S.INV(0.05) = -1.645, the cut-off used in 95% value at risk (Module 10).

Mean and median under a lognormal model

If log returns are normal, the median outcome grows at the mean log return, while the arithmetic mean grows faster, by half the variance. Over long horizons most investors end up below the mean outcome, because a few very good paths pull the mean up. Projections that show 'average' wealth overstate what a typical investor will see.

Fat-tailed alternatives

A Student's t distribution with few degrees of freedom (often 3 to 5 for daily returns) fits stock returns much better than the normal. Under a t distribution with 4 degrees of freedom, a move beyond 5 standard deviations is thousands of times more likely than under the normal curve. Risk managers use t distributions, historical simulation or explicit stress scenarios to capture this.

Jumps

Some price moves are discontinuous: an earnings surprise, a takeover, a default. A model with occasional jumps on top of normal day-to-day noise explains why short-dated options on single stocks around earnings carry so much implied volatility (RS 303).

Checking normality

Compare the histogram with a normal curve of the same mean and standard deviation, compute skewness and excess kurtosis, and plot a quantile-quantile (Q-Q) chart: the actual quantiles against the normal quantiles. Points bending away from the straight line at the ends reveal fat tails.

Put it into practice