[R-SIG-Finance] [R-sig-finance] VaR

[Brian G. Peterson]

Mark is definitely on the right track here.  Bastian has also pointed 
out the major paper that introduced coherent risk measures.

First, I need to say that there are many definitions of Value at Risk, 
using different underlying assumptions to calculate the risk.  Jorion's 
book "Value at Risk" is probably the most often cited collection of 
these definitions.

Now, that bit aside, VaR, generally speaking, is defined as the  minimum 
loss level at a certain confidence level. So, for 95% confidence, the 
minimum amount you could expect to lose.  With this definition on daily 
data, 95% VaR thus describes that the losses should be below this level 
on 19 out of 20 trading days.  The problem comes in the 20th day, or the 
day that exceeds your 95% threshold.  On that day, your "tail loss" is 
defined by the process beyond the VaR, it is outside of the VaR 
estimating process.

Many common VaR estimators for single instruments use some variation of 
a normal distribution (Gaussian) assumption.  This means, as a result, 
that for a single instrument the VaR is defined by the Gaussian 
distribution quantile with the mean and variance of the observed 
series.   In a portfolio context, the VaR using these distributional 
assumptions will be defined by the joint distribution of the moments, 
the mean of the portfolio and the covariance of the instruments.   To 
Mark's point, you can end up with a portfolio variance that is larger 
than the sum of the variance of the individual instruments.

Your original understanding of sub-additive is incorrect as well.  A 
sub-additive measure will add up precisely to the portfolio measure. 
(neither more nor less). See Artzner's papers.

Several other people responding to this post have pointed out that 
getting the distributional assumption right is the important bit.  This 
is exactly correct, as what you are trying to figure out is the risk to 
a real portfolio.  Most real instruments are do not have normally 
distributed returns, so you need to account for that non-normality.

Portfolio measures of VaR and ES  are either marginal, which shows what 
happens when one instrument is removed, or component. Component measures 
of VaR should show the contribution to VaR/ES of individual instruments 
to the risk of the whole portfolio.  Here subadditivity is required if 
you wish to rearrange your aggregation criteria (e.g. to re-examine the 
portfolio by currencies, instrument types, or styles you want the 
portfolio risk measure to stay the same, but allow you to look at the 
component risks of these various slices).  In other cases, subadditivity 
doesn't really gain you anything, for instance if you are trying to 
determine the risk of loss of an individual position to calculate 
stop-loss or exit criteria.

PerformanceAnalytics contains several different methods for calculating 
VaR and CVaR/ES, including their portfolio and component variants.  Many 
of the measures we present are indeed sub-additive, if you're looking 
for a subadditive measure.

Regards,

    - Brian

-- 
Brian G. Peterson
http://www.braverock.com/brian/
IM: bgpbraverock
Ph: +1 773-459-4973


markleeds at verizon.net wrote:
>  Hi  Christofer: I don't know if the analogy is allowed but this can 
> happen with regular statistical
> variance so maybe it can happen with Value at Risk also ? if you have 
> a covariance matrix
> of 2 assets with portfolio weights w_1 and w_2 and the 2 assets have 
> positive covariance, then the resulting variance of the portfolio will 
> be greater than the sum of the individual variances of the two assets 
> with weights w_1 and w_2. ( w_1*v_1 + w_2*v_2 ).
>
> now I have no idea if the result for statistical variance holds for 
> Value at Risk ( i don't know the definition of Value at Risk ) but, if 
> it does, then that's probably the answer. Hopefully someone else will 
> tell us if the analogy is allowed ?
>
>
>
> On Tue, Mar 3, 2009 at  6:20 AM, Bogaso wrote:
>
>> I frequently hear Value at risk i.e. VaR is not a coherent risk measure
>> because, sum of VaR for two individual assets may be LOWER than VaR of
>> portfolio consists of that two aseets i.e. VaR may not be sub-additive.
>> However when I calculate VaR for general assets like Equity, 
>> commodity etc,
>> I see that VaR is actually sub-addtive i.e. portfolio VaR is always less
>> than sum of individuals, which is reported as "diversification 
>> benefit". Can
>> anyone give me a particular example why VaR is not sub-additive?
>>
>> Thanks