Is a price-to-earnings ratio of 25 high? It depends on what that ratio usually is. A z-score answers exactly that question: how far is today's value from normal, measured in standard deviations?
Our Wealth Principles series uses z-scores to put very different indicators on one scale. This tutorial shows how, and the three mistakes that turn a good tool into a bad bet.
Quick check. If returns were normally distributed, how often would a day more than 3 standard deviations from the average happen?
About 0.27% of days, roughly once every 1.5 years. In real markets it happens far more often. The answer to why is below.
The formula
z = (x − μ) ÷ σ
- x is today's value
- μ is the historical average
- σ is the historical standard deviation
A z-score of +2 means "two standard deviations above average". A z-score of −1.5 means one and a half below. Because every indicator is converted to the same scale, you can compare a valuation ratio, a sentiment survey and an interest rate side by side.
What ±1, ±2 and ±3 mean (if the data is normal)
| Distance from average | Share of observations outside it (both tails) |
|---|---|
| beyond ±1σ | ≈ 31.7% |
| beyond ±2σ | ≈ 4.6% |
| beyond ±3σ | ≈ 0.27% |
These figures apply only if the data really follows a normal distribution. Standardising data does not make it normal.
Mistake 1: assuming the bell curve
In 1963 Benoit Mandelbrot showed that price changes have fat tails: extreme moves happen far more often than the normal distribution predicts. Rama Cont later summarised these "stylized facts" of asset returns: heavy tails, volatility that comes in clusters, and little predictability in the direction of daily returns.
A real example: on 19 October 1987 the S&P 500 fell about 20% in one day. With typical daily volatility around 1%, that is a move of roughly 20 standard deviations. Under a normal distribution it should essentially never happen in the lifetime of the universe. It happened on a Monday.
Lesson: read a z-score of 3 as "very unusual", not as "0.27% chance".
Mistake 2: the sign is a convention, not a law
For a plain z-score, positive means above average. Whether "above average" is cheap or expensive depends on the indicator:
- A high dividend yield usually means prices are low relative to dividends.
- A high price-to-earnings ratio usually means prices are high relative to earnings.
Many systems flip the sign so that positive always means "cheap". That is fine, as long as you write the convention down. Mixing the two is a classic way to buy exactly when you meant to sell.
Mistake 3: a threshold is a rule, not a fact
"Buy when the z-score is below −1.8" is a rule someone chose. It is not a statistical law. The right threshold depends on the indicator, the period and what you are trying to do. Test it, and expect it to work less well than it did in the past.
Where valuation z-scores do help
John Campbell and Robert Shiller showed that when valuation ratios such as price-to-earnings are far above their historical average, subsequent long-horizon returns have tended to be lower. The key words are long-horizon: valuation says little about the next month and more about the next ten years. A high valuation z-score is a reason to lower your expectations, not a timing signal.
The principle
A z-score measures "unusual", not "probable". Check whether the data is stable over time, state your sign convention, treat thresholds as rules you must test, and expect fat tails.
Your checklist
- Pick one indicator you follow and compute its z-score over 10+ years.
- Plot the history first: if the average drifts, a single long-run mean misleads.
- Write down whether positive means cheap or expensive in your system.
- Count how often |z| > 3 actually occurred compared with the 0.27% the bell curve predicts.
Sources
- NIST/SEMATECH e-Handbook of Statistical Methods: Normal distribution. Link
- Mandelbrot, B. (1963). The Variation of Certain Speculative Prices. Journal of Business, 36(4), 394–419. Link
- Cont, R. (2001). Empirical Properties of Asset Returns: Stylized Facts and Statistical Issues. Quantitative Finance, 1(2), 223–236. Link
- Campbell, J. Y. & Shiller, R. J. (1998). Valuation Ratios and the Long-Run Stock Market Outlook. Journal of Portfolio Management, 24(2), 11–26. Link
Educational content, not financial advice. Nothing here is a buy or sell signal for any asset. Investing carries the risk of loss.