Volatility

Sortino Ratio

A variation of the Sharpe ratio, the Sortino ratio is a return-versus-risk trade-off metric that uses downside deviation as its measure of risk.

How Is it Useful?

The Sortino ratio addresses a shortcoming of using standard deviation as a measure of risk in a return-versus-risk trade-off ratio. Standard deviation punishes a manager equally for “good” risk and “bad” risk. Downside deviation adjusts for this by only counting the “bad” risk and ignoring “good” observations in a return series.

Math Corner: 

The below calculation for the Sortino ratio is not complicated, as it is simply a variation of the Sharpe ratio. It is up to the user to define what the breakpoint is for minimum acceptable return (MAR) in the calculation of downside risk. Frequently used values for MAR are the risk-free rate or a hard-target value like 0%.

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.

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.

Sharpe Ratio

The most famous return-versus-risk measurement, the Sharpe ratio represents the added value over the risk-free rate per unit of volatility risk.

How Is it Useful?

The Sharpe ratio simplifies the options facing the investor by separating investments into one of two choices: the risk-free rate or anything else. Thus, the Sharpe ratio allows investors to compare very different investments by the same criteria. Anything that isn’t the risk-free investment can be compared against any other investment. The Sharpe ratio allows for apples-to-oranges comparisons.

Math Corner: 

First proposed by William Sharpe in his landmark 1966 paper “Mutual Fund Performance,” the original version of the Sharpe ratio was known as the reward-to-variability ratio. Sharpe revised the formula in 1994 to acknowledge that the risk-free rate used as the reference point is variable, not a constant.

Zephyr K-Ratio

A return-versus-risk statistic, the Zephyr K-ratio measures the rate at which wealth is created and the consistency of the path of wealth creation.
PDF version: 

How Is it Useful?

The Zephyr K-ratio answers two questions many investors care about most: “At what rate did I grow my wealth?” and “Was that growth of wealth consistent?”

What Is a Good Number?

With the Zephyr K-ratio, a high numerator indicates a high rate of wealth creation. A low denominator indicates consistency in that rate of appreciation. Roll those two goals together and you would hope to see a high Zephyr K-ratio.

Math Corner: 

The original variant of what would eventually become the Zephyr K-ratio was proposed by Lars Kestner in 1996 using well-established statistical theories. The summary formula comparing the rate of wealth appreciation against the consistency of wealth appreciation is:

To calculate the Zephyr K-ratio, one should first replace the dates on the horizontal axis of the portfolio’s cumulative return graph with consecutive integers starting at 0. With these integers as independent x-values and the corresponding cumulative return values as dependent y-values, one can now calculate the slope of the regression line, the numerator of the Zephyr K-ratio, by the well-known formula

The standard error of the slope, the denominator of the Zephyr K-ratio, can
be calculated from the x- and y-values by the formula

 
 

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