Term Glossary

Metrics, Models & Concepts

Value-at-Risk


Value-at-risk (VaR) is a category of risk metrics that describe probabilistically the market risk of asset holding. In its most general form, the Value at Risk measures the potential loss in value of a risky asset or portfolio over a defined period for a given confidence interval.

For example, if the VaR on an asset is 100 million USD at a one-week, 95% confidence level, there is a only a 5% chance that the value of the asset will drop more than 100 USD million over any given week.

Delta-Normal VaR

The delta-normal method assumes that asset returns are normally distributed. \begin{equation} VaR_{\alpha} = \sigma z_{\alpha} \end{equation}

\(\sigma\)
standard deviation of returns (square root of variance)
\(\alpha\)
confidence level, \(\alpha \in (0, 1)\)
\(z_{\alpha}\)
\(\alpha\)-quantile of standard Gaussian cumulative distribution function

Modified VaR

Return distributions (particularly for intraday returns) often have asymmetry (skew) or/and fat tails (kurtosis). Modified Value-at-Risk accomodates deviations from normality using Cornish-Fisher expansion around z-core with return skewness and kurtosis estimates: \begin{equation} MVaR_{\alpha} = \sigma z_{CF\alpha} \end{equation} \begin{equation} z_{CF\alpha} = z_{\alpha} + \frac{({z_{\alpha}}^2-1)S}{6} + \frac{(z_{\alpha}^3 - 3z_{\alpha})(K - 3)}{24} - \frac{(2z_{\alpha}^3 - 5z_{\alpha})S^2}{36} \end{equation}

\(\sigma\)
standard deviation of returns (square root of variance)
\(S\)
return skewness
\(K\)
return kurtosis
\(\alpha\)
confidence level, \(\alpha \in (0, 1)\)
\(z_{\alpha}\)
\(\alpha\) - quantile of standard Gaussian cumulative distribution function


Function Reference
portfolio_VaR, position_VaR