Student’s t VaR When Fitted Degrees of Freedom Are Below Two
Summary
The post describes fitting several distributions to daily log returns of a minimum-variance portfolio containing four cryptocurrencies, then estimating value at risk and expected shortfall. A fitted Student’s t distribution has 1.73 degrees of freedom. The author’s stated VaR expression includes a square-root normalization involving the degrees of freedom minus two, which becomes invalid at this fitted value, while the author reports being able to calculate expected shortfall.
The author asks why a fit might produce fewer than two degrees of freedom and in what circumstances this occurs, contrasting the result with prior stock-portfolio analyses. The post provides no answer, figures, or calculation details that would resolve the issue. It highlights a modeling and parameterization question: a standard deviation based VaR expression may not apply when the fitted t distribution lacks finite variance, so the assumed scale and distributional formula require scrutiny.
Key ideas
- A cryptocurrency portfolio’s return series is fitted to several candidate distributions, including Student’s t.
- The fitted t degrees of freedom are reported as 1.73, below the threshold required for finite variance.
- The VaR expression shown uses a variance-based scale adjustment that becomes invalid at that fitted parameter.
- The author reports an expected-shortfall calculation but provides no supporting derivation or figures.
- The document poses the modeling question without supplying a resolution.
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Full text
# 36327
# Trouble computing the VaR for Student's t-distribution for a minimum-variance portfolio composed of four cryptocurrencies (BTC, ETH, LTC, and XMR)
I have modelled the time-series of daily log-returns from August 2015 to October 2017 of a minimum-variance portfolio composed of four cryptocurrencies (BTC, ETH, LTC, XMR) by fitting the data to four different distributions: the Cauchy distribution, the Normal distribution, the exponential distribution, and Student's t distribution. I've then subsequently yielded the value-at-risk (VaR) and expected shortfall (ES) for all of the aforementioned distributions except for the Cauchy one, as this one has an undefined mean and variance. However, I've particularly run into a problem when computing the VaR for Student's t distribution. This is because when fitting my data I obtain a degrees of freedom parameter of $$ \nu=1.73 $$ and since the VaR formula for Student's t distribution is defined as $$ \sqrt{\frac{\nu-2}{\nu}}t_{\nu}^{-1}(1-\alpha)\sigma-\mu $$ where $t_{\nu}^{-1}(1-\alpha)$ is the quantile function of the Student's t distribution, $\alpha$ is a probability level such that $0<\alpha<1$, $\sigma$ is the standard deviation parameter, and $\mu$ is the mean (or location) parameter, this then obviously implies that $$ \nu-2<0 $$ which means that the square root "does not exist" in this context.
Interestingly enough, I can nevertheless yield the expected shortfall for Student's t distribution, since its closed-form expression doesn't involve taking the square root of any number. This is all illustrated in the figures below
Therefore, does anybody have any idea as to why I obtain a degrees of freedom parameter $\nu<2$, and in which specific circumstances this tends to occur? Typically the opposite result should be obtained, and as it can be seen above, the distribution is somewhat heavy-tailed. In the past I have modelled and have performed stress testing (i.e. computing the parametric VaR and ES) on a minimum-variance portfolio of stocks, and I have never encountered this issue of yielding $\nu<2$. Any input would be highly appreciated. Thank you very much in advance.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.