## Tail

## Conditional Value at Risk

## How Is it Useful?

Conditional Value at Risk attempts to address some of the shortcomings of Value at Risk (VaR). VaR is defined as a breakpoint that is breached only under extreme conditions. However, VaR does not describe what happens beyond that breakpoint. CVaR does. It is the average of the returns that fall beyond the VaR cut-off. CVaR is a more pessimistic measure of tail risk than VaR.

The very name, Conditional Value at Risk, indicates how it is calculated. CVaR values are conditional to the calculation of VaR itself. Therefore, all of the decisions that go into the calculation of VaR will also impact CVaR. The shape of the distribution, the cut-off level used, the periodicity of the data, and assumptions about volatility stochasticity all set the value known as VaR.

Once the VaR has been established, calculating CVaR is trivial. It is simply the average of those values that fall beyond the VaR:

Where p(x)dx term is the probability density of getting a return with value “x” and “c” is the cut-off point along the distribution curve where one sets the VaR breakpoint.

## Value at Risk

## 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.

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.

## Upside / Downside Omega

## How Is it Useful?

Upside omega and downside omega are simply the numerator and denominator of the omega ratio separated into individual parts. While the omega ratio is useful for quantifying the trade-off between upside gains and downside losses, sometimes the details get lost because both the good and the bad are rolled into one number. By breaking omega into its constituent parts, one can focus only on the return or only on the risk element.

The upside omega is an integral, defined by the minimum acceptable return (MAR) on one axis and the count and scale of observations above the MAR at the upper bound. Downside omega is the same, but counting the observations below the MAR.

## Kurtosis

## How Is it Useful?

Kurtosis tells us where the risk exists. On a monthto- month basis, does the investment typically display a moderate amount of risk? Or does the investment appear to have little risk until the risk suddenly hits all at once? Kurtosis tells us whether the risk is spread evenly through the distribution of returns or if it tends to be concentrated in tail events.

Kurtosis is also known as the fourth moment of the distribution, used in conjunction with mean, standard deviation, and skewness to understand the shape of a distribution of returns. In its base case, kurtosis has a neutral value of 3.0. The calculation is frequently modified by the second term in the equation below, which scales kurtosis so that the baseline, neutral value is 0.0.

## Skewness

## How Is it Useful?

One way of thinking about skewness is that it compares the length of the two “tails” of the distribution. Another way of thinking of skewness is that it measures whether or not the distribution of returns is symmetrical around the mean. The two are related, because if the distribution is impacted more by negative outliers than positive outliers (or vice versa) the distribution will no longer be symmetrical. Therefore, skewness tells us how outlier events impact the shape of the distribution.

Skewness is also known as the third moment of the distribution. By cubing the differences of the individual observations away from the mean, positive or negative values are possible, which indicate the tilt of the distribution. The process of cubing exacerbates the deviations from the mean, which is why skewness is used for measuring tail risk.

## Omega

## How Is it Useful?

Omega represents one useful way of understanding tail risk, the impact that extreme observations have on an overall set of numbers. If the returns of a manager are close to the minimum acceptable return (MAR) breakpoint, they don’t strongly affect omega. However, if many returns lie above or below the MAR, or if the returns are extreme, those returns will impact the value of omega significantly.

Omega was first proposed by Con Keating and William Shadwick in their 2002 paper “A Universal Performance Measure”. Omega is the ratio of two integrals: the area above the minimum acceptable return (MAR) in the numerator and the area below the MAR as the denominator. Omega captures all four moments of the distribution (return, standard deviation, skewness, and kurtosis) in a single measure.