Risk

Conditional Value at Risk

A tail risk metric, Conditional Value at Risk (CVaR) quantifies the scale of expected losses once the Value at Risk (VaR) breakpoint has been breached.

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.

Math Corner: 

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

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

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.

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.

Maximum Drawdown

A risk metric indicating capital preservation, the maximum drawdown measures the peak-to-trough loss of an investment.

How Is it Useful?

Maximum drawdown offers investors a worst case scenario. Maximum drawdown tells the investor how much would have been lost if an investor bought at the absolute peak value of an investment, rode it all the way down, and sold at rock-bottom.

Math Corner: 

The calculation of maximum drawdown looks at all subperiods of the time period in question and calculates the compound return of the manager over each subperiod. The maximum drawdown is the lowest value of all these compound returns.

Downside Deviation

Downside deviation is a risk statistic measuring volatility. It is a variation of standard deviation that focuses only upon the “bad” volatility.

How Is it Useful?

Downside deviation addresses a shortcoming of standard deviation, which makes no distinction between the “good” or upside deviations, and the “bad” or downside deviations. Both upside and downside deviations have an equal influence on the calculation of standard deviation. Downside deviation seeks to remedy this by ignoring all of the “good” observations and by instead focusing on the “bad” returns.

Math Corner: 

The most important variable in the equation for downside deviation is the definition for what counts as being a “bad” observation. Denoted as “c” below, only the returns less than “c” are included in the calculation for downside deviation. Frequently used values for “c” are the risk-free rate, a hard-target value like 0%, or the mean return of the return series itself.

Kurtosis

Kurtosis identifies where the volatility risk came from in a distribution of returns. Kurtosis improves one’s understanding of volatility risk.
PDF version: 

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.

Math Corner: 

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.

Up/Down Capture

Up capture measures the percentage of market gains captured by a manager when markets are up. Down capture measures the percentage of market losses endured by a manager when markets are down.

How Is it Useful?

It’s said that the two elements that drive the market are “greed” and “fear.” The up capture and down capture ratios are a useful way of separating the two so they can be analyzed independently. Also, up capture and down capture address the shortcomings of beta, which fails to distinguish between up and down markets.

Math Corner: 

For up capture, the first step identifies all the periods in which the market was up. For those up-market periods the returns for both the manager and the benchmark are geometrically compounded and then annualized. Finally, a ratio between the two is calculated.

The down capture process is the same, but for down market periods.

It is possible for the manager to have a negative return in a period when the market is up. By the same token, it is possible for the manager to have a positive return during a period when the market is down. Obviously, the latter is preferred to the former.

It is also worth noting that up capture and down capture values can differ significantly if the underlying periods used in the calculation are monthly or quarterly. For example, assume that over the course of a quarter the monthly returns were -1.2%, +5.2%, and -0.8%. Compounded, the quarterly return was +3.1%. Using monthly data would result in one up period and two down periods. However, using quarterly data in the calculation of the capture ratios results in one up period and no down periods.

Beta

Beta measures the sensitivity of the manager to movements in an underlying benchmark.
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How Is it Useful?

Beta answers the question, “When markets go up or down, does the manager typically go up or down more than the market or less than the market?” This is what is meant by market sensitivity. Beta is also used to quantify market risk, sometimes known as “systematic risk”.

Math Corner: 

The numerator of beta measures how the manager return moves relative to the benchmark movements. The denominator scales the results of the numerator so that the point of reference of beta is 1.0.

R-squared

R-squared represents the “goodness of fit” of a manager to its benchmark. R-squared is the percentage of variation in a manager’s returns explained by the benchmark’s returns.
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How Is it Useful?

R-squared is used primarily as a cross-check on the appropriateness of the benchmark. Many other statistics such as alpha, beta, information ratio, and up/down capture use a passive benchmark as a reference point. If the R-squared of a manager to the benchmark is too low, the usefulness of all these other benchmark-relative metrics diminishes.

Math Corner: 

R-squared literally takes the correlation of a manager versus a benchmark and squares it. Squaring correlation removes the directional aspect of correlation. It is impossible to have a negative R-squared. This is intentional, as the sole point of R-squared is to determine what percentage, from 0% to 100%, of the variation of a manager’s return is explained by the benchmark.

Tracking Error

Also known as the standard deviation of excess returns, tracking error measures how consistently a manager outperforms or underperforms the benchmark.

How Is it Useful?

Tracking error measures the consistency of excess returns. It is created by taking the difference between the manager return and the benchmark return every month or quarter and then calculating how volatile that difference is. Tracking error is also useful in determining just how “active” a manager’s strategy is. The lower the tracking error, the closer the manager follows the benchmark. The higher the tracking error, the more the manager deviates from the benchmark.

Math Corner: 

Calculating tracking error is a three-step process. First, an excess return series is created by calculating the periodic differences between the manager and the benchmark. Next, the mean of that excess return series is calculated. Finally, the dispersion of individual observations from the mean excess return is calculated.

Standard Deviation

Standard deviation measures how closely returns track their long-term average. Standard deviation measures volatility risk.

How Is it Useful?

Despite being the oldest way of looking at risk, standard deviation remains applicable. Highly volatile investments are hard for some people to stomach. Also, for those investors who are prone to taking the worst action at the worst time (e.g. chasing returns, or buying high and selling low) highly volatile investments offer many opportunities to make mistakes.

Math Corner: 

Standard deviation is a well-known statistical tool used across many industries in order to determine just how representative the mean value of an overall set of data is. The process of squaring the differences is used to remove negative values. Otherwise the positive and negative values would net out to zero.

 
 

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