## Reference Materials

## Zephyr K-Ratio

## 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.

However, there is no concrete dividing line above which one can say the ratio is good or bad. One must compare the Zephyr K-ratio of a manager to the Zephyr K-ratios of its peers or of an index to get a feel for whether or not it is relatively better or worse than the alternatives.

## What Are the Limitations?

The Zephyr K-ratio quantifies wealth creation and consistency using methods more commonly found in pure statistical analysis. The mathematical techniques behind the Zephyr K-ratio are well established and frequently used in the world of statistics. However it tends to be difficult to easily explain the inner workings of the calculation.

## What Does the Graph Show Me?

The blue line that zigs and zags is the cumulative return of a manager, while the red line is a best-fit straight line superimposed over the actual data series. The slope of that best-fit line is the rate of appreciation of wealth and is the numerator of the Zephyr K-ratio. The steeper the slope, the greater the appreciation of wealth. The investor would hope the straight line is sloped upwards as steeply as possible.

Risk is defined by how much the actual blue line of returns strays from the idealized, straight and narrow red line of consistent wealth creation. The statistical term for this is the standard error of the slope, and it is the denominator of the Zephyr K-ratio. The larger this number, the more inconsistent the wealth creation.

## What Are Typical Values?

The values typically seen for Zephyr K-ratios will vary widely depending upon two main factors: the asset class and the time frame being analyzed. Looking at asset classes over long time horizons, we see that investment grade fixed income investments exhibited the highest Zephyr K-ratios. This is driven more by the consistency of the wealth creation rather than the overall, absolute return. Those asset classes that are more momentum-driven, like emerging markets, tend to have lower Zephyr K-ratios. While emerging markets has offered long-term wealth creation, the fact that the returns are so “feast or famine” leads to lower Zephyr K-ratios.

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