Significance level

Statisticians rarely say a range holds true all the time. They pick a confidence level instead — say, 95% — and accept a small chance of being wrong.

Significance level is that small chance. It is 1 minus the confidence level. At 95% confidence, the significance level is 5%.

Because the normal distribution is symmetric, that 5% usually splits evenly. 2.5% of the time the true value is too high. 2.5% of the time it is too low. A portfolio manager uses the significance level to pick the right z-score — the cut-off point used to work out Value at Risk.

The workbook's own example (section 17.4.2.1): at an approximate 95% confidence interval, 1 − 95% = 5% is the significance level. That 5% splits into 2.5% on each side of the distribution, because a standard normal distribution is symmetric around its mean.

For a two-sided range, the relevant z-value at 5% significance is 1.96. But Value at Risk only cares about the downside, so the entire significance level is pushed onto the negative side. The workbook gives three worked z-scores this way: 1.645 for 95% VaR (5% significance), 1.282 for 90% VaR (10% significance), and 2.33 for 99% VaR (1% significance).

The workbook's standard-normal table lets a reader go either direction — look up the probability for a given z-score, or look up the z-score for a given probability.

Following the workbook's own worked case. A portfolio manager wants to be 99% confident about a risk estimate. The significance level is 1 − 99% = 1%.

For a two-sided estimate, that 1% would split into 0.5% on each side. But because the manager only cares about the loss side, the whole 1% is assigned to the left tail, giving a one-tailed z-score of 2.33 — the value the workbook uses in its 99% VaR example on a $100 million portfolio with a 15% expected annual return and 20% annual standard deviation.

Contrast this with a 90% confidence choice on the same portfolio: significance level 10%, one-tailed z-score 1.282 — a smaller cushion, because the manager has accepted a bigger chance of being wrong.

Chapter 17 (Risk), section 17.4.2.1 (Parametric Method), derives significance level from confidence interval and uses it to justify the z-scores in the Value at Risk formula. Expect a direct question converting a stated confidence level into a significance level, or asking for the correct one-tailed z-score at a given significance level.

• Significance level = 1 − confidence level. A 90% confidence level is a 10% significance level, not 90%.
• A two-sided z-value (1.96 at 5% significance) is not the same as VaR's one-tailed z-value (1.645 at the same 5%) — VaR always uses the one-tailed figure because it only cares about the loss side.
• Higher confidence means a lower significance level and a bigger z-score — 99% confidence (1% significance) uses the largest z-score, 2.33, of the workbook's three examples, not the smallest.