historical VaR without computing volatilities and correlations

[karabouchi!]

In the RiskMetrics Monitor (2nd quarter, 1997, page 19), there is an article titled: "A general approach to calculating VaR without volatilities and correlations". (This article is available on the RiskMetrics site, http://www.riskmetrics.com/publications/journals.html, you may have to sign up to access these articles).



According to my understanding, here is the process they have layed out on page 22 to perform Monte Carlo simulations using historical data and without computing a covariance matrix: (Note, I am applying their process to calculate the VaR for a single security, just for the purpose of illustration.)



1) Calculate historical returns over say 151 days. Subtract the mean from this time series. Call this R. (Note R has length 150.)

2) Generate a vector of independent N(0,1) random variables of length 150. Call this e.

3) Now compute the inner product, transpose(e)*R.



This results in one Monte Carlo simulation. Now repeat this process say 999 times and you've generated 1000 simulations. Pick the 5% quantile to get a 95% VaR over a one day horizon. Done. The problem with this is that the VaR results are huge.



If you consider the fact that the linear combination of n random variables X~N(0,1) , with coefficients c1,...,cn:



Z = c_1x_1 + c_2x_2+ ... + c_nx_n



has variance equal to c1^2 + c2^2 + ... + cn^2. ie, Z~N(0,c1^2 + ... + cn^2). This explains why the VaR numbers are very large.



I would very much appreciate any assistance towards helping me understand where the short circuit between my ears is happening! Where am I missing the point?

Cheers...

[Corey]

Article access here for interested parties...



I didn't read it, but from you interpretation, it has a funky smell to me



1) If X ~ iid N(0, 1), you have that 68.2% of the randomly generated values will be between [-1,1], which seems to indicate to me that you would be shrinking the historic returns. Am I missing something here? Intuitively, it seems like this would reduce the overall variance in the return distributions for your MC simulations...



2) To do step (3) (aggregate across time), your returns must be compounding, not linear (see this paper). Are you applying the appropriate transformations?



3) Z is the 150 day return (assuming those are compounding returns). Therefore, when you generate your 1000 simulations, they are for returns over 150 days (since you are adding up the returns) -- so you are really computing 150 day VaR, not 1 day VaRs...





Those are just three quick thoughts

[karabouchi!]

Corey,

Thank you for your feedback.

Concerning (1), it turns out that the variance of a linear combination of N(0,1) random variables is equal to the sum of the squares of the coefficients. So for example if x1, x2 and x3 are N(0,1) and Z = x1 + x2 +2x3, then Z~N(0,6).

What I find interesting about points (2) and (3) is that this leads to the conclusion that the longer the time series, the higher the variance of transpose(e)*R, which doesn't make much sense in terms of computing VaR. Clearly I'm doing something wrong. I will look into using compound returns rather than linear.

Thanks again!

[silverside]

after a few seconds reading of your post, the obvious drawback is that of limited historical data.



150 days is basically half a year, so how much relevant historical data are you going to have? 10 years or 20 data points? seems a bit iffy to me.



like corey said, since I haven't read the original article I may be missing the point.

[FatChoi]

It looks like a normalisation issue. Equations [2] and [10] should be equivalent but equation 10 omits division by T.

[karabouchi!]

FatChoi I think you hit the nail on the head.

I came to the same conclusion overnight.

The question is, was that normalization constant omitted on purpose? Or is it just a typo?

Wink