Skip to content
DP
Risk Foundationsintermediate

The Normal Distribution and Z-Scores

Much of parametric risk measurement assumes returns are approximately normal, described fully by a mean and a standard deviation. Standardization converts any value into a z-score, the number of standard deviations from the mean, which lets a single standard normal table answer probability questions for any normal variable. The left-tail z-scores that recur in risk work are about -1.28 at the 90% level, -1.65 at 95%, and -2.33 at 99%, the multiples that turn a volatility into a Value at Risk. The same standard-normal cumulative function reappears across finance, for example as the N(d1) and N(d2) terms in the Black-Scholes-Merton option price.

Why it matters

The normal curve is the workhorse because two numbers, its centre and its width, pin down the whole shape. A z-score just asks "how many standard deviations out is this?", which puts every normal variable on one common ruler. The magic numbers to memorize are the tail z-scores: about 1.65 standard deviations covers 95% of outcomes on one side and 2.33 covers 99%. Those are exactly the dials a parametric VaR turns. The catch, picked up later, is that real returns have fatter tails than the bell curve admits.

Formulas

Standardization (z-score)
z=x−μσz = \frac{x - \mu}{\sigma}
Subtract the mean μ\mu and divide by the standard deviation σ\sigma to get the z-score, the distance from the mean in standard-deviation units. The result follows a standard normal, mean 0 and variance 1.
Left-tail critical z-values
z0.90=−1.28,z0.95=−1.65,z0.99=−2.33z_{0.90} = -1.28, \quad z_{0.95} = -1.65, \quad z_{0.99} = -2.33
These are the one-tailed z-scores cutting off the worst 10%, 5%, and 1% of a standard normal. They become the confidence multipliers in parametric VaR.

Worked examples

Scenario

Daily returns are normal with mean 0 and standard deviation 2%. What return marks the worst 5% of days (the 95% one-tailed cutoff)?

Solution

Use x=μ+zσx = \mu + z\sigma with the 95% one-tailed z of −1.65-1.65: x=0+(−1.65)(0.02)=−0.033x = 0 + (-1.65)(0.02) = -0.033, a -3.3% return. So on the worst 5% of days the loss is at least 3.3% of the position. This is precisely the parametric VaR calculation, computed here on the return rather than the dollar amount.

Scenario

A portfolio gains 4% on a day when the mean is 1% and the standard deviation is 1.5%. How unusual is that?

Solution

The z-score is z=(0.04−0.01)/0.015=2.0z = (0.04 - 0.01)/0.015 = 2.0, two standard deviations above the mean. Under normality only about 2.3% of days exceed a +2 z-score, so it is a notably good day. The same standardization, run on the loss side, is what feeds tail probabilities into VaR.

Common mistakes

  • ✗A z-score changes the underlying probabilities. Standardization only rescales the axis; it relabels a value as a distance from the mean without altering any probability.
  • ✗The 95% VaR uses a z of -1.96. The -1.96 value is the two-tailed 95% critical value; VaR is one-tailed and uses -1.65 for 95% and -2.33 for 99%.
  • ✗Real asset returns are exactly normal. Empirical returns are leptokurtic with fatter tails and often skew, so the normal model understates the probability of extreme losses, a limitation addressed later.
  • ✗A higher confidence level always means a more accurate VaR. A higher level reaches further into the tail, where the normal assumption is least reliable, so the estimate can become less trustworthy, not more.

Revision bullets

  • •Normal distribution: fully described by mean and standard deviation
  • •Z-score = (x - mean) / sd: distance from the mean in sd units
  • •One-tailed tail z: -1.28 (90%), -1.65 (95%), -2.33 (99%)
  • •These z-values become the VaR confidence multipliers
  • •Same N(.) appears as N(d1), N(d2) in Black-Scholes-Merton

Quick check

The one-tailed z-score used for a 99% confidence parametric VaR is approximately

A value has a z-score of -2. Under the standard normal, this means it is

Connected topics

Market RiskVaR DefinitionParametric VaRFat Tails

Sources

  1. Jorion (2007), Ch. 2-5
    Jorion, P. Value at Risk: The New Benchmark for Managing Financial Risk. 3rd ed. McGraw-Hill, 2007.
    Reviews the normal distribution and the confidence multipliers that convert volatility into VaR.
  2. Hull (2018), Ch. 11-12
    Hull, J. C. Risk Management and Financial Institutions. 5th ed. Wiley, 2018.
    Connects the normal distribution and standardization to VaR and to the N(d) terms in option pricing.
How to cite this page
Dr. Phil's Quant Lab. (2026). The Normal Distribution and Z-Scores. Derivatives Atlas. https://phucnguyenvan.com/concept/frm-normal-distribution-var-prep
← Back to the atlasSee in the network →