Value-At-Risk, The Good And The Bad

There is a nice article at the Financial Times giving an introduction to Value-at-Risk (VaR), along with a discussion of the good and bad aspects of using VaR for risk management. A few illustrative examples are also given. It is worth a read for those without much math background, or those who would like a quick and simple introduction and explanation of VaR.

In general, VaR calculations find the maximum loss that is not exceeded for a defined probability over a given period of time. Sound confusing already? Let us put it another way. As an example, one VaR calculation might find that we are "95% confident that we will not lose more than $1 million over the next month." In other words, the most we expect to lose this month is $1 million, and we are 95% confident that losses will not exceed this figure. Past normally distributed return and volatility values of our asset or portfolio allow us to make such a calculation.

While the calculation is useful and intuitive, it is not without its problems. Worth noting from the article is how VaR is backward looking, such that if the distribution of volatility and stock returns change, the values given by the VaR calculation will over- or under-estimate the risk. To have more confidence in your calculations, it is important to make sure you are using past data that is similar to the data in the current time frame you are concerned with.

VaR is also not designed in its basic form to deal with what are increasingly being called “black swans,” made popular by Nassim Nicholas Taleb in his book of the same name - The Black Swan: The Impact of the Highly Improbable. In essence, a black swan is a hard to predict event that is rare and beyond the current level of expectations, but when it occurs, it has the ability to not only be relatively unique, but also carry a large impact. The events of 9-11, or the recent subprime credit events leading to the problems at Bear Stearns and elsewhere are such events. These events are difficult to predict, and not often seen (like a black swan), but nonetheless have lasting effects. [For what it is worth, both of Taleb's books related to the subject - The Black Swan and Fooled by Randomness - are worth your time].

Another problem with VaR that is often discussed concerns herd mentality. During an event, like a market meltdown, if everyone moves in the same direction and performs the same task (such as panic selling), they are essentially moving to the same location on the normal distribution. This in and of itself will change the curve. What before looked like an extreme event is now much more probable. As such, your level of risk and exposure will also increase.

Personally, while teaching VaR and performing my own calculations for finance organizations, I find that helping them understand the math and basic concepts is relatively easy. This is especially true for graduate students in finance, or those with a basic background in statistics, such as engineers and computer scientists, among others. What becomes difficult to teach and put into practice is understanding the proper distribution of a complex portfolio, especially one that includes non-linear derivative securities. Furthermore, getting a proper historical data set to model the distribution and calculate VaR can be difficult. Linear approximation and non-linear quadratic models have been developed and are often used, but they are also difficult to formulate, or at times require unrealistic generalization and/or assumptions.

Of course, sometimes the math itself gets misused, or is misunderstood. The Financial Times article gives an example offered by David Einhorn, a New York hedge fund manager. For instance, assume that "... you are offered odds of 127 to one on $100 that when you toss a coin, heads will not come up seven times in a row. The chance that you will win is 99.2 per cent. So you can say with 99 per cent confidence that you have no value at risk. Using a VaR model, a bank could hold no capital to guard against a loss on this bet. But in fact there is a 0.8 per cent chance (not an unimaginable black swan) that they will lose $12,700."

In other words, it is unlikely that you will incur a loss, therefore regulators will allow you to hold less capital. Of course, if you do happen to fall into the tail of the distribution, your loss would be significant, so significant that it could take down your company. When derivatives are used, and the non-linearity of options, complex swaps, CDOs, etc., are considered, the tail can become not only long, but also have a bump in it. This has the effect of further magnifying the potential loss and value that is at risk. In fact, the tail can be designed to be even less likely to occur (narrow and farther out, but having a taller bump), making the company look even less risky, but if the event does occur ........ , well, you get the picture. Unfortunately, this type of risk is hard to model, and even harder to understand. It also allows for abuse since it creates incentives to take excessive risk, albeit more remote. No doubt we are seeing the effect of this in the credit markets and elsewhere as we speak.