Z-score

Different variables come in different units and different scales. To compare them fairly, statisticians convert each one into a common yardstick: how many standard deviations it sits away from its own average.

That converted number is the z-score. A standardised, or 'normalised', variable has a mean of 0 and a standard deviation of 1. A z-score tells you exactly where a particular value falls on that common scale.

In portfolio risk work, the z-score is what turns a chosen confidence level, such as 95%, into an actual cut-off point for calculating Value at Risk.

The workbook's formula (footnote 84, section 17.4.2.1): a standardised value of any variable is (X − Mean X) ÷ Standard Deviation of X. Such standardised values are called 'normal deviates' or 'Z-Scores.'

The workbook ties specific z-scores to specific confidence levels for VaR, using the one-tailed figure because VaR only cares about losses: 1.645 for 95% VaR, 1.282 for 90% VaR, and 2.33 for 99% VaR. It separately notes that, for a standard normal distribution, approximately 68%, 95% and 99% of observations lie within 1, 2 and 3 standard deviations of the mean respectively — the more familiar two-sided z-scores of roughly 1, 2 and 3.

Z-score = (X − Mean of X) ÷ Standard deviation of X

One-tailed VaR z-scores: 1.282 (90%), 1.645 (95%), 2.33 (99%)
Two-sided ranges (approx.): 1 SD ≈ 68%, 2 SD ≈ 95%, 3 SD ≈ 99%

Following the workbook's own VaR application. A portfolio has an expected annual return of 15% and annual standard deviation of 20%, on a $100 million base — the workbook's own figures.

For a 95% VaR, the one-tailed z-score is 1.645. The daily return is 15% ÷ 250 = 0.06% and the daily standard deviation is 20% ÷ √250 ≈ 1.26%. Plugging into the VaR formula: 0.06% − (1.645 × 1.26%) ≈ −2.01%, so the 95% one-day VaR is about 2.01% of $100 million, or roughly $2.01 million.

Swap the z-score to 2.33 for 99% confidence instead, and the same portfolio's estimated one-day loss rises to about $2.88 million — a bigger z-score demanding a bigger cushion for the extra confidence.

Chapter 17 (Risk), section 17.4.2.1 (Parametric Method) and its footnote on standardised values, define the z-score and its role in the Value at Risk formula. Expect a question asking for the correct one-tailed z-score at a stated confidence level, or one applying the z-score formula to a given VaR calculation.

• In VaR work, z-score is always the one-tailed figure (1.645 at 95%, not 1.96) — 1.96 is the two-sided value for a symmetric confidence range, and is the classic swap error here.
• Mean 0, standard deviation 1 describe the standardised distribution, not the original variable's own mean and standard deviation.
• 'Normal deviate' and 'z-score' are the same thing in the workbook's own terminology — do not treat them as different concepts if a question uses either name.
• A bigger z-score means a bigger, more conservative VaR estimate — because it corresponds to a higher confidence level, not a lower one.