A PROBLEM WITH VaR
Whilst there has been much debate on the strengths and weaknesses of different ways to estimate value at risk (VaR), there has been relatively little debate on the inherent weaknesses of VaR itself as a risk measure. Unfortunately for VaR proponents, recent work by Philippe Artzner, Freddy Delbaen, Jean-Marc Eber and David Heath has demonstrated that VaR fails to satisfy a basic property--subadditivity--that we should expect of any sensible risk measure. Subadditivity means that aggregating individual risks does not increase overall risk. A risk measure *(.) is therefore subadditive if the measured risk of a combined position is always no more than the sum of the measured risks of the positions considered on their own, i.e.:
where A and B are two different positions. If risks are subadditive, adding individual risks together will always give us an estimate of the combined risk that will either be correct or, if incorrect, biased on the conservative side.
The importance of subadditivity can be appreciated from considering the problems that can arise if our risk measures are not subadditive:
* Adding risks together will give us an under-estimate of combined risk--our true risks will be greater than they appear to be. This should make risk managers uncomfortable, and also makes the determination of capital requirements problematic. After all, if we are going to have biased estimates of capital requirements, we would prefer them to be biased on the conservative side, to be safe.
* A firm might be tempted to break itself up to reduce its capital requirements, since the sum of the capital requirements of the smaller units would be less than the capital requirement of the firm as a whole. This can create major problems for portfolio managers, derivative dealers and holding companies, because their portfolios may be riskier than they appear.
* Traders on an organized exchange would be tempted to break up their accounts, with separate accounts for separate risks, in order to reduce their margin requirements. This would be a matter of concern for the exchange because the margin requirements on the separate accounts would no longer cover the combined risks.
Clearly, subadditivity is a highly desirable property for any risk measure.
The non-subadditivity of VaR can be demonstrated by counter examples. To take a simple one, suppose we have two short positions in out-of-the-money binary options. The specific details are shown in Table 1. Each of our options has a 4% probability of a payout of -GBP100 and a 96% probability of a payout of zero, and the underlying variables of the two options are independent of each other. If we take the VaR at the 95% confidence level, then each of our positions has a VaR of 0. However, if we combine the two positions, the probability of a zero payout falls to less than 95%, and so the VaR of the combined position is positive (and, in this case, equal to GBP100). The VaR of the combined position is therefore greater than the sum of the VaRs of the individual positions, so the VaR is not subadditive.
AN ALTERNATIVE EXPECTED TAIL LOSS
There is, however, an alternative risk measure that is subadditive. This alternative is expected tail loss (ETL)--the loss we would expect in a 'tail event' where loss exceeds VaR. If E[.] is the expectations operator and L is the loss expressed as a positive amount, ETL is:
The relationship between VaR and ETL is illustrated in Figure 1. VaR is the cut-off point separating the tail of the P/L distribution from the rest of it, and can be regarded as the maximum loss if a tail event does not occur. By contrast, ETL is the expected loss if a tail event does occur, and is therefore bigger than VaR.
Of course, VaR is more familiar than ETL, and VaRs are often easier to estimate because VaR formulas are generally simpler than ETL ones. In some cases, ETL formulas can be very difficult to derive and we need to estimate ETLs by numerical methods. However, these are fairly trivial issues. Familiarity is hardly a decisive or long-lasting advantage, and the greater ease of VaR estimation is increasingly irrelevant in the face of ongoing improvements in computing power. More importantly, if we use ETL as our preferred risk measure, we can be confident of not running into the problems that can arise with non-subadditive risk measures, like VaR. ETL also has the attractive property that it tells us something about what happens in tail events--it tells us the loss we should expect when such events occur--whereas VaR tells us next to nothing about tail events. So why use VaR, when there is a clearly superior alternative?
This week's Learning Curve was written byKevin Dowd, professor of economics at theUniversity of Sheffield.