Methodology — Value at Risk & Conditional VaR

Transparency is a feature. This page explains what Value at Risk and Conditional VaR measure and why we compute them — in plain terms. The precise formulas and numeric conventions are documented in our internal methodology; the summary here is deliberate, not an omission.

These are tail-risk measures: they summarize how bad a bad period could look. We work in signed returns, where a loss is a negative number, so a five-percent loss is written as a negative five percent. Everything is expressed at a chosen confidence level — for example the ninety-five percent level focuses on the worst one-in-twenty periods. By construction, at the same confidence level Conditional VaR is always at least as negative as Value at Risk, because it looks deeper into the tail. Both figures are reported on whatever single-period return series is supplied and are not annualized.

Historical VaR

Historical VaR reads the loss threshold straight off the realized track record: it is the return level such that only the chosen tail fraction of past periods came out worse. It makes no assumption about the shape of the distribution — it simply lets the actual history speak. It needs a non-empty return series to have any distribution to draw from, and the confidence level must sit strictly between zero and one.

Gaussian VaR

Gaussian VaR is the parametric counterpart: it assumes returns follow a normal (bell-curve) distribution and derives the tail threshold from just the mean and the standard deviation of returns. A larger standard deviation deepens the estimated loss, and a higher confidence level pushes the figure further into the tail. Because it leans on the normal assumption, it can understate risk when real returns have fatter tails than a bell curve.

Cornish-Fisher VaR

Cornish-Fisher VaR keeps the parametric approach but corrects the tail for skewness and excess kurtosis — that is, for asymmetry and fat tails that the plain normal model ignores. When the return distribution has a fatter, heavier left tail, this adjustment pushes the loss estimate further into the red than Gaussian VaR would. If there is no skew and no excess kurtosis, it collapses back to exactly the Gaussian result, so it is a strict refinement rather than a different animal.

Historical CVaR (Expected Shortfall)

Historical CVaR, also called Expected Shortfall, answers "if we do land in the tail, how bad is it on average?" It averages the actual returns that fell at or beyond the Historical VaR threshold, so it captures the depth of losses that VaR alone only bounds. It is always at least as negative as the historical VaR at the same confidence, and like historical VaR it needs a non-empty series and a valid confidence level.

Gaussian CVaR (Expected Shortfall)

Gaussian CVaR is the parametric version of Expected Shortfall: under the normal assumption there is a closed-form expression for the average loss in the tail, again built from just the mean and standard deviation. It is always at least as negative as the Gaussian VaR at the same confidence, and it inherits the same caveat — if real returns are fatter-tailed than a bell curve, it can be optimistic about how deep the tail really runs.