Back-testing Value at Risk with a WML investment strategy

Question

I'm currently taking a course in Financial Econometrics and there is a question in the lecture notes regarding back-testing of VaR which I'm have difficulty with.

First of all the procedure for back-testing of VaR using a rolling scheme is described as follows:

assume we have returns data $r_{t}, t=1,...,T$ for some large enough $T$. We take the first 60 observations and calculate their sample mean $\mu_{1-60}$ and variance $\sigma_{1-60}^{2}$ and then calculate: $$\mbox{VaR}_{1-60}\left(\alpha\right)=-\mu_{1-60}+\Phi^{-1}\left(\alpha\right)\cdot\sigma_{1-60}$$ Where $\alpha$ is our confidence parameter (usually $\alpha=0.05$) and $\Phi^{-1}$ is the inverse standard normal CDF. If $r_{61}<-\mbox{VaR}_{1-60}\left(\alpha\right)$ we mark $1$ and otherwise $0$. We proceed doing the same thing for observations $2...62$ compared to $63$ and so on and in the end we count the number of times in which the result was $1$. while under the null hypothesis we expect that the proportion of times we got $1$ to be at most $\alpha$. In particular it's important to notice we had an assumption here that the returns are all normally distributed, $r_{t}\sim N\left(\mu_{t},\sigma_{t}^{2}\right)$.

Now for the actual question, suppose $r_{t}, t=1,...,T$ are the returns using a WML investment strategy (momentum trading strategy). Is the procedure described above suitable for testing the VaR in such a case and if not why?

Usually the phrasing of the question would hint towards there being some sort of problem with using the procedure in such a case but I don't see why that would be the case... Help would very much be appreciated!

Answer 1

What you could do is to apply the methods of portfolio risk analysis. If you buy $n$ stocks with percentages $w_i,i=1,\ldots,n$ then your portfolio return is $r = \sum_{i=1}^n w_i r_i$.

Dealing with investment strategies I would not include an expected profit in the VaR calculation and put $\mu=0$ for this reason.

To calculate the volatility of your portfolio you can do the following:

• calculate the covariance matrix of your assets on the past $N$ (e.g. 60) days, $\Sigma$
• caclulate portfolio ex-ante volatility by $\sigma = \sqrt{w \Sigma w^T}$.

You ca plug this $\sigma$ into your formula and proceed. This is the basis set-up, assumptions about the dependence of the assets or the distributions of asset returns can improve the risk analysis.

IMPORTANT addon:

• the formulae above are valid for negative weigths too. All you have to do is to determine a cash basis from which you calculate the weights. Say you have $50 000$ cash, buy a stock for $25 000$ and sell one for $25 000$ then you have $100\%$ cash and weights of $+50\%$ and $-50\%$.

Answer 2

If the returns are $N(\mu,\Sigma)$ distributed, then $WML\sim N(0,\sigma)$, because the equally-weighted $\mu$'s cancel while $\Sigma=\sqrt{w \Sigma w'}$ with $w=\{1/n...1/n\}$.

So your new VaR becomes:

$$\mbox{VaR}\left(\alpha\right)_{WML}=\Phi^{-1}\left(\alpha\right)\cdot\sigma$$

Your sampling formula from above remains still valid though, just with $(0,\sigma)$ parameters.