# [R-SIG-Finance] VaR again

Brian G. Peterson brian at braverock.com
Wed Apr 29 14:56:06 CEST 2009

```megh wrote:
> Hi all Gurus, I have a problem to quantify the riskiness of a typical
> position wherein this position is in some foreign country. Let me be more
> specific on my problem.
>
> Say I am a British investor and taken a position in NYSE, say in ATT (AT&T).
> Therefore apart from the risk due to fluctuation in stock quote of that, I
> am exposed of additional risk due to fluctuation in USD/GBP exchange rate. I
> intend to calculate the VaR of this position in GBP. Here I used monte carlo
> simulation approach to find that, which is as follows, please see the R code
> :
>
> # calculation of risk on an unit position
> att <- 25.67
> # last traded price of AT&T in USD
> usdgbp <- 0.68366
> # last quote for USDGBP
> vcv <- matrix(c(5.33727E-05, 2.56709E-05, 2.56709E-05,0.000176556), 2)
>
> # VCV matrix for AT&T and USDGBP
>
> # We simulate 1-day ahead stock price and ex. rate assumig a Bi-variate
> normal dist. with above VCV structure
> library(mnormt)
> simu <- exp(rmnorm(10000, c(log(25.67),log(0.68366)), vcv))
> simu.pos.val <- apply(simu, 1, function(X) X[1]*X[2])                 #
> Simulated value of my position in USD
> abs(quantile(simu.pos.val, 0.05) - att*usdgbp)               # VaR (in GBP)
> in terms of Maximum possible loss
>
> Upto this point I am OK. However next thing automatically comes is that what
> is contribution of Stock and Ex. rate in total risk, i.e. Dissecting the
> risk. Can anyone please guide me how to do this for n-asset portfolio (of
> this kind) under MCS framework?
>
> Thanks
>

My first comment is that your Monte Carlo simulation makes the
assumption that the simulation is normally distributed.  Currency
markets are almost all much fatter tailed than that.  You might want to
consider a different VaR method that takes the fat tails into account.
Many people believe that because Monte Carlo VaR methods are
"non-parametric" that there are no distributional assumptions built in,
but a quick glance at your code should make it obvious that this is not
true, and how dangerous it is.

Next, If you want to separate the two components, and do it for a large
portfolio that may be mixed in different currencies  and instrument
types, you should be calculating VaR in percentages, using returns.
Then you calculate the market risk to the portfolio/instrument and the
currency risk to the portfolio/instrument separately. You can always
turn the aggregate number back to dollars/pounds/whatever if you
need/want to.  To get a univariate VaR number, you can aggregate the
positions by weight and calculate a univariate return for the portfolio
in a given currency, from which VaR may be taken (but see below on
portfolio VaR).

I think it is rather important to look at the actual observed (or
estimated) variance, skewness, and kurtosis of each instrument in
working out the VaR for that instrument, via simulation or any other
method.  Then, you can separately work out the currency risk to your
portfolio once you aggregate the risks of each position in a given currency.

Finally, if you are doing this in a portfolio context, the co-moments of
the instruments have a massive effect on the total portfolio risk.
There is a large literature on "Portfolio VaR" or "Component VaR" that
discusses this and its solution. Component VaR is subadditive, so you
can dissect the risk as you ask for above.

Regards,

- Brian

--
Brian G. Peterson
http://braverock.com/brian/
Ph: 773-459-4973
IM: bgpbraverock

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