Parametric/Analytical VaR

Question

Suppose I want to calculate VaR for a known distribution with mean $\mu$, variance $\sigma^2$ and $\alpha$-quantile as, $VaR_{\alpha}$ = $\mu + \sigma q_{\alpha}$.

For a Gaussian distribution it is clear that $q_{\alpha}=z_{\alpha}$ where $z_{\alpha}$ is the $\alpha$-quantile of a standard normal distribution with $\mu,\sigma$ being the moments.

Now I have two questions

1. For returns with student-t distribution, there can be two ways to calculate VaR. First, use moment definition of student-t and standard t quantile. This gives an unbiased estimate for VaR. Second, use moment definition of student-t and quantile from standard normal distribution. This is a biased estimate for VaR. How does one understand which VaR is used most non-normal definitions of VaR are loosely defined in that sense.

2. For parametric distributions which are defined using location and scale such as Azzalini's skew-t, where mean and standard deviation are different from location and scale there can be three definitions of VaR. First, use moment definition of skew-t and standard skew-t quantile. This gives an biased estimate for VaR. Second, use moment definition of skew-t and quantile from standard normal distribution. This is a biased estimate for VaR. Third, define VaR using location, scale and standard skew-t quantile. This gives an unbiased estimate. How does one name and distinguish the three cases to avoid any ambiguity.

In general the challenge in defining VaR is which moments to use and which quantile should be used. Are there any references that elaborate on using parametric VaR for non-normal distributions

Answer 1

For any continuous distribution we can define $$VaR_{\alpha}=-F^{-1}(1-\alpha)$$ where $F^{-1}$ is the inverse of the CDF. Now suppose that you have a distribution which comes from a location-scale family with location parameter $\mu$ and scale parameter $\sigma$ then $$F^{-1}(1-\alpha)=\mu+\sigma \phi^{-1}(1-\alpha)$$ where $\phi^{-1}$ is the inverse CDF of the distribution with $\mu=0$ and $\sigma=1$. Thus we have $$VaR_{\alpha}=-(\mu+\sigma \phi^{-1}(1-\alpha))$$ for any location-scale distribution. In particular for the student-t $$VaR_{\alpha}=-(\mu+\sigma \;t_v^{-1}(1-\alpha))$$ where $t_v^{-1}$ is the inverse CDF of the standard student-t with $v$ degrees of freedom.

Answer 2

The quantile used is a choice of the user (e.g. 99%, 95%) the moments to be used would be dependent on the distribution as a distribution can be parametrized by it's moments.

To answer your first question: Value at Risk is always defined for any distribution as it's the probability returns will not exceed some threshold (e.g. 95% of the time losses will not exceed X).

For your second one, simply state which distribution you are sampling from, and calculate quantiles based on those distributions (e.g. the qbeta/qnorm/qwhatever in R). See http://www.inside-r.org/packages/cran/PerformanceAnalytics/docs/VaR for some various ways to calculate VaR in R (it's very simple!).

For your general problem:

Wouldn't the simple solution to be to either:

a) Choose a distribution of choice, estimate required moments and calculate the VaR

b) Computationally fit returns using a variety of distributions and choose the one with the best fit and then calculate VaR

c) Simulate a data generating process and run N trials and calculate VaR using that

Answer 3

Let's start with one observation: Take a random variable of the form $X=\mu + \sigma Z$ for some real $\mu$ and $\sigma>0$ then $$ P[X \le x] = P[X-\mu \le x - \mu] = P[\frac{X-\mu}{\sigma} \le \frac{x-\mu}{\sigma}] = P[Z \le z], $$ where $z = \frac{x-\mu}{\sigma}$. In the case of location scale families the distribution of $Z$ is a special case of the distribution of $X$ and we can write quantiles as $$ q^X_\alpha = \mu + \sigma q^Z_{\alpha}. $$

Thus the answer is:

1. Why should one use a biased VaR estimate? Or just a a wrong one? It is not biases it is just wrong to insert the quantile of the normal distribution. As you say you have to choose $\sigma$ right. if $\hat{\sigma}$ is the sd of your sample then due to the fact that the variance of a $t$ distribution is $\frac{n}{n-2}$ you set $\sigma := \hat{\sigma} \sqrt{\frac{n-2}{n}}$.
2. Looking at Azzalini's skew-t they define a distribution by $$ Y = \xi + \omega X, $$ then if you know the quantile of $X$ then you can calculate the quantile of $Y$ and it will be $\xi + \omega \sigma q^Z_{\alpha}$ for the reasons explained above. Everyhing else is biased or simply wrong.