The basic concept
The normal distribution is a symmetric bell-shaped model determined by its mean μ (mu) and positive standard deviation σ (sigma). The mean locates the centre. The standard deviation measures spread. We write X ~ N(μ, σ²), with variance as the second parameter.
Its support extends over all real numbers. This can suit an approximation for small changes, but it creates problems for variables that must remain positive or bounded.
Normal distribution at a glance
| Property | Meaning |
|---|---|
| Mean, median and mode | All equal μ |
| Standard deviation | σ > 0 |
| Variance | σ² |
| Skewness | 0 |
| Kurtosis | 3, or excess kurtosis 0 |
| Within one, two and three standard deviations | Approximately 68.27%, 95.45% and 99.73% |
Here exp means the exponential function and π (pi) is approximately 3.14159. The probability density function (PDF) must be integrated to obtain probability.
Standard normal distribution and the z-score
A z-score measures distance from the mean in standard deviation units. Standardisation changes units and scale. It does not make a skewed or heavy-tailed variable normally distributed.
Worked example: a one-day return model
Assume a portfolio’s one-day return is normal with mean 0.05% and standard deviation 1.20%. The figures are illustrative and exclude fees.
The 1st-percentile return is 0.0005 + (−2.32635 × 0.012) = −2.7416%. On a ₹10 crore portfolio, this corresponds to a one-day 99% Value at Risk (VaR) of approximately ₹27.42 lakh, using a linear return-to-loss conversion and the model assumptions.
Excel illustration
| Task | Formula |
|---|---|
| Probability return is at most −2% | =NORM.DIST(-0.02,0.0005,0.012,TRUE) |
| 1st-percentile return | =NORM.INV(0.01,0.0005,0.012) |
| Density at the mean | =NORM.DIST(0.0005,0.0005,0.012,FALSE) |
| Standard normal probability | =NORM.S.DIST(-1.708333333,TRUE) |
Use decimal returns consistently: 1.20% is 0.012. A density output is not a probability.
Why normality needs checking
Financial returns may show extreme observations, skewness and changing volatility. A normal model can understate tail losses if the data have heavier tails. Examine histograms, quantile comparisons, stability over time and out-of-sample performance before relying on it.
The central limit theorem concerns suitably normalised sums under specific conditions. It does not establish that every individual return, price or loss is normal.
Check your understanding
Is a 95% central interval the same as the 95th percentile?
No. The central 95% interval is approximately μ ± 1.96σ. The one-sided 95th percentile is approximately μ + 1.645σ.
Can a normally distributed return be below −100%?
Yes, the mathematical model allows it. This is a support mismatch for an unleveraged asset’s simple return and matters when volatility or the horizon is large.
Interactive distribution laboratory
Change the parameters and threshold, then switch between cumulative probability and density or mass. Values are numerical approximations for learning.
Horizontal axis: the variable’s value. Vertical axis: density, probability mass, or cumulative probability as labelled. Shading highlights values at or below the threshold. The plot shows a finite viewing window.
Sources and further reading
- National Institute of Standards and Technology: Gallery of Distributions
- Distribution definition and properties
- Microsoft: Statistical functions reference
All numerical finance and project examples are hypothetical teaching illustrations. Probability assumptions require validation before practical use.