How does Student-t VaR handle fat tails compared to normal VaR?

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AcadiFi · Accepted Answer

The Student-t distribution has a parameter nu (degrees of freedom) that governs tail thickness: lower nu means fatter tails. For nu above 30 it resembles the normal; at nu = 5 it produces tails noticeably heavier, and for nu near 3 the kurtosis is undefined but tail quantiles are dramatically larger. Student-t VaR uses the t-quantile t_{nu, 1-c} in place of the normal z-score, and the standard deviation must be rescaled because a raw Student-t with nu degrees of freedom has variance nu/(nu-2). A practical implementation fits nu via maximum likelihood to portfolio returns, scales the residuals, and plugs the t-quantile into the VaR formula. Example: at 99% confidence, a normal VaR uses 2.326; at nu = 5 the t-quantile is 3.365, a 45% heavier tail at the same volatility. For a $50M book with daily vol 1.4%, switching from normal to t(5) raises 99% VaR from $1.63M to $2.35M. Limitations: Student-t is symmetric, so it does not capture skew; real loss distributions often show a heavier left tail. Extensions like skewed-t, Generalized Hyperbolic, or EVT-POT address this. For FRM questions, know the t-quantile direction (larger than z), the variance scaling adjustment, and that the fit must be to standardized residuals, not raw returns.

How does Student-t VaR handle fat tails compared to normal VaR?

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