understanding Value-at-Risk correclty

Question

The are several types how to calculate the VaR. I am focussing on the method of calculation the VaR in percentage.

$VaR=I*z*std*\sqrt{t}$

This gives the VaR in €.

I have the z-value, the daily standard deviation std, the holding period t, the investment I. Now let's assume the holding period is one year, that ist 258 trading days for which I have the daily standard deviation. So I multiply the whole thing by root(258). My results are reasonable.

Now to get the percentage of VaR as rate of the Investment, I will devide by I, which then will be cancelled out.

Now, I increase the holding period by, say 10 years. t becomes 2580. VaR in percentage becomes extremely large. Of course the Investment grows over time. Since it gets cancelled out, I can't grow it.

The VaR should be the rate of the average investment amount, right?

So I am struggling to implement the rate of VaR for a dynamic investment over a longer time horizon. Can anybody help me please or am I completely mistaken by the usage of VaR?

Answer 1

Scaling volatility (standard deviation) is not the best option while calculating long term VaR. This has been discussed extensively in this post. See this page for the paper by Diebold et al. (1996).

Keep in mind that long term volatility is believed to mean revert to its long term average. So if an investment is currently in high volatility regime, then VaR calculated based on scaling this volatility is going to be misleading.

What you can do here is to change the sampling frequency to weekly/monthly to somewhat remedy the scaling issue. Then you can also incorporate your expected return on your Investment $I$ for the holding period into the VaR equation:

$$VaR = I\times \mu + I\times z \times \sigma \times \sqrt{t}$$

Answer 2

Let's $\Phi$ represent the standard normal CDF, and q the required var quantile (e.g., 95%) so your $z=\Phi^{-1}\left(q\right)$.

Now assume the return x is normally distributed with annualised mean $\mu$ and annualised standard deviation $\sigma$. By the way you can annualise your daily volatility by scaling it by $\sqrt{258}$ because we are in simple normal distribution world, and you are assuming 258 days in a year. So we can write the quantile as follows:

$\Phi \left(\frac{x_q-\mu t}{\sigma \sqrt{t}}\right)=q$

We can rearrange,

$x_q=\mu t+{\sigma \sqrt{t}}\Phi^{-1} \left(q\right)$

So it is similar to your formula but with a drift. But please note t here is measured in years, and $\mu$ and $\sigma$ are annualised.

For smaller holding periods such as 1 day, you can ignore the mean/drift, but this becomes significant for longer holding periods. So as you increase the holding period, the process drift (upward or downward depending on the sign) and the variance grows. If you assume zero drift, the VaR will grow with the holding period. Which is intuitive because holding a stock, which is risky asset, for 10 years can make you a lot richer (or poorer!). But if you think the variance is growing too fast than could be considered realistic, then you can consider alternative specifications for the return process. In the interest rate world, an alternative mean reverting specification is more common, so the variance grows with horizon but at a deceasing rate. The simplest examples is the Vasicek model and is based on the Ornstein Uhlenback process, but I can see your question relates to simple settings so won’t go there.

There is also a chance you might be referring to the holding period as the length of the observation windows that you used to estimate the daily volatility. If that's the case, then just changing the observation window shall not by itself increase the VaR because we are annualising the volatility (the annual variance will be 258 times the daily variance under the above assumptions, irrespective of whether you estimate it using 258 days period or 2580 days period).