## Value at Risk

A tail risk metric, Value at Risk (VaR) quantifies the amount of expected loss under rare-but-extreme market conditions. StatFacts_Value_at_Risk.pdf

## How Is it Useful?

Markets experience losses, and occasionally those losses are extreme. Investors should be financially and mentally prepared to deal with the outcomes of these rare, but traumatic, events. Value at Risk describes how much is typically lost in a day, month, or quarter when markets are at their worst.

## What Is a Good Number?

Since VaR is a risk metric measuring loss, the smaller the VaR, the better. Ideally the VaR would be 0.0%, but no investment carries zero risk. Therefore, it is up to the investor to ask what level of occasional loss would be acceptable to bear.

## What Are the Limitations?

It is important to remember that VaR does not represent the maximum amount one can possibly lose. After all, the most one could potentially lose is 100% of an investment. VaR is not an absolute number. Rather VaR represents a breakpoint that is exceeded only under extreme conditions

In addition, VaR does not take into account the upside potential of an investment. Oftentimes those asset classes or investments with the highest potential for loss also offer the greatest potential for gain.

## What Does the Graph Show Me?

The graph below illustrates an idealized distribution curve of returns. Most of the time an investment’s returns occur near the center or “peak” of the distribution. When markets are doing very well, the returns will fall to the far right of the curve. However, at other times the returns will fall to the left or far-left of the distribution. Value at Risk is calculated by placing a cut-off point on this part of the curve.

There are many different ways to calculate Value at Risk. Two major inputs to the calculation are the shape of the distribution and where to place the cutoff point. In the real world, most return distributions are not as smooth as the one shown below. Different mathematical solutions accommodate the shape of real world data. Finally, the analyst must determine the appropriate location of the VaR cut-off point. Usually, cut-offs are set between 95% and 99%. ## What Are Typical Values?

Context is all-important when analyzing Value at Risk. One must take the asset class and the underlying time period into consideration when evaluating an investment’s VaR. Below are VaRs calculated at a 95% cut-off level. Not surprisingly, equities have had steeper VaRs than fixed income, with emerging market stocks being the worst. In addition, the VaRs tended to be much worse in the 2000s. While the 1980s and 1990s are remembered as bull markets, the 2000s started and ended with two significant bear markets.  Math Corner:

The general equation for Value at Risk can be stated as: Calculating Value at Risk requires different assumptions about the variables in the above equation. For example, “c” represents the cut-off point along the distribution curve where one sets the VaR breakpoint. Values typically fall between 95% to 99%.

The p(x)dx term is the probability density of getting a return with value “x”. It addresses the shape of the distribution of returns. StyleADVISOR provides two options in defining the distribution. The first is to use a non-parametric distribution, where the historical data is assumed to be representative of all possible outcomes. While trivial to calculate, it requires a large amount of data in order to be considered robust.

The second option is to use a Cornish-Fisher distribution, which assumes the distribution is close to the classic, normal distribution but does have some amount of skewness and kurtosis. Cornish-Fisher presents a better alternative with smaller data sets. However, it does not work well if the data has large degrees of skewness or kurtosis.

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