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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!

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    $\begingroup$ Can you please describe WML more detail? Thanks. $\endgroup$ – emcor Sep 5 '14 at 14:17
  • $\begingroup$ How do you determine winners and loosers exante? $\endgroup$ – emcor Sep 5 '14 at 17:00
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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 $-50\%$.
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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.

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  • $\begingroup$ Do you mean that the $n$ stocks are iid with one $\sigma$? This is rather unlikely. $\endgroup$ – Ric Sep 8 '14 at 13:39
  • $\begingroup$ @Richard You have $n$ stocks, but $1/n$ iid weighted, so the portfolio variance is just the sum: $n1/n\sigma=\sigma$. $\endgroup$ – emcor Sep 8 '14 at 13:41
  • $\begingroup$ no - only if they are uncorrelated - and usually each stock has it's own $\sigma_i$. For random variables $A,B$ and real numbers $a,b$ it holds that $VAR(a*A+b*B) = a^2VAR(A) + 2 a b COVAR(A,B) + b^2 VAR(B)$. $\endgroup$ – Ric Sep 8 '14 at 13:44
  • $\begingroup$ @Richard iid, means they are uncorrelated identically distributed. $\endgroup$ – emcor Sep 8 '14 at 13:54
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    $\begingroup$ I know but if you form a portfolio how can you assume that the stocks are uncorrelated? And why should all the variances be the same? Stocks are usually correlated and you have riskiers ones and less risky ones, if you form pairs (winner minus looser) then both volatility and correlation between the stocks will determine your risk. $\endgroup$ – Ric Sep 8 '14 at 13:57

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