Student-t Parametric VaR with Degrees-of-Freedom Scaling
Summary
The document corrects a proposed method for estimating a lower-tail portfolio return threshold using a Student-t distribution. It explains that the standard deviation of observed returns should not be divided by the square root of the sample size when estimating VaR; that adjustment is associated with uncertainty in a sample mean, rather than the return dispersion used for the portfolio threshold.
A Student-t model requires a degrees-of-freedom parameter, which determines its tail shape and variance. Because a standard Student-t variable has variance greater than one when the degrees of freedom exceed two, the answer rescales its quantile so the modeled returns retain the estimated standard deviation. The resulting lower-tail level combines the estimated mean, this scale-adjusted dispersion, and the chosen t quantile. The treatment is a compact parametric prescription; it does not explain how to estimate degrees of freedom, validate the distributional assumption, or handle changing portfolio risk.
Key ideas
- A VaR threshold uses return standard deviation directly rather than dividing it by the square root of the sample size.
- The Student-t degrees of freedom parameter controls the distribution’s tail behavior.
- For degrees of freedom above two, the standard Student-t variance exceeds one.
- Scale the t variable so the modeled returns have the estimated standard deviation.
- The method depends on estimating degrees of freedom and assumes a parametric return distribution.
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# Parametric VaR with Student-t distribution
# Parametric VaR with Student-t distribution
Im using VaR to estimate parametric VaR. I have been able to do this using a Normal Distribution, however I want to also do this using a Student t-distribution and I'm unsure how to implement that in Matlab.
I have a dataset of portfolio values, I have log returns and returns as well as mean and standard deviation. The only suggested method I was able to find elsewhere was the following:
$\mu - t_\alpha * (sd/\sqrt{n})$
Where $\mu$ is the mean, sd is the standard deviation, t is the t stat at the alpha level and n is the number of returns. This would give the lower level of the alpha confidence interval.
Can anyone confirm if this is the correct or incorrect method to implement parametric VaR using a t-distribution. If it is incorrect, how would I be able to implement it?
Thanks
## Answer by Richi Wa (score 3)
https://quant.stackexchange.com/a/19522
You got some things wrong:
- You don't have to devide sd by $\sqrt{n}$, the division is already part of the definition of $sd$.
- The $t$ distribution has a parameter $\nu$, the degrees of freedom.
- The variance of a standard $t$ distributed random variable $T$ is $$ VAR(T) = \frac{\nu}{(\nu-2)}. $$ Thus you have to define $$\sigma = sd * \sqrt{\frac{(\nu-2)}{\nu}}$$ and a random variable $$ X = \sigma T. $$ Then you will have that $$ VAR(X) = VAR(T \sigma) = \sigma^2 VAR(T) = sd^2 \frac{(\nu-2)}{\nu} \times \frac{\nu}{(\nu-2)} = sd^2. $$
For VaR you estimate $\nu$ and $sd$ and look at $$ \mu - \sigma t_{\alpha}, $$ where $\sigma$ is defined above and $t_{\alpha}$ is the quantile of a t-distribution with $\nu$ degrees of freedom.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.