K-Ratio: Performance Metric by Lars Kestner

Developed by derivatives trader and statistician Lars Kestner, the K-ratio is a performance metric that tries to address the problem of how returns and consistency of returns are analyzed. Although the K-ratio has been around since 1996 — with some modifications though — do you really know what the K-ratio is and how it is used?

The K-ratio is a statistical metric that is used to measure the growth of return and the consistency of that growth over a specified period. It measures return versus risk by analyzing how steady a security’s, portfolio’s, or manager’s returns are over time using the value-added monthly index (VAMI).

Now, we’ve added the K–ratio (also known as the Zephyr k–ratio) to our discussion. In this post, we will discuss the following:

• What the K-ratio is
• How to calculate it
• What the ratio signifies
• How to use the K-Ratio

What is K-ratio?

The K–ratio is a return vs. risk ratio that measures the growth of return and the consistency of that growth over a certain period. It takes into account both the returns themselves and the order of those returns. The calculation is performed by plotting a linear trend for the return data and estimating the slope/variability (standard error of the slope) of the data. Hence, the K–ratio not only measures the return of a security over time but also examines the consistency of that return over that period.

The data for the ratio is derived from a value-added monthly index (VAMI). VAMI is a metric that uses linear regression to track the progress of a $1,000 initial investment in the security being analyzed. The K-Ratio’s calculation requires creating a time series of data that is being analyzed. Because it takes the return trend into account, the ratio is a good tool to measure the performance of equities.

The k-ratio is included in most software packages, like Amibroker and Tradestation, for example.

How to calculate the K-ratio

The K-ratio is calculated by running a linear regression on the logarithmic cumulative return of a Value-Added Monthly Index (VAMI) curve. Here is how we do it:

• We generate a scatter plot of log[VAMI] verses (Number of periods).
• Then, the return per period (days, weeks, months, whatever) is measured by the Slope. For example, we may use the monthly returns of XOM stock over 10 years.
• Finally, the deviation of points from the regression line is measured by the Standard Error. This Error is a measure of “risk” associated with the stock.

The ratio: Slope / Standard Error may be interpreted as Return / Risk, just like in the Sharpe Ratio.

That would give:

K-ratio = (Slope of logVAMI regression line) / (Standard Regression Error)

Or

Where there are n return periods in the monthly return data, the formula would be given as:

K–Ratio (Kestner) = (Slope of logVAMI regression line) / n (Standard Error of the Slope)

The Significance of the K-ratio

Since investors are often interested in returns and consistency, the K–ratio is a return vs. risk ratio that was created to measure both. Calculated by plotting a linear trend for the return data and estimating the slope/variability (standard error of the slope) of the data, the slope measures the return, while the standard error of the slope represents the risk.

Thus, the K-ratio will increase when the slope increases (cumulative P&L increasing faster) and will decrease when there are outsized gains or losses (increasing inconsistency). A higher k-ratio implies a better performance both in positive returns and consistency.

What is a good K-ratio? We consider a K-ratio value greater than 2.0 as good.

How to use the K-ratio

K–ratio tests the consistency of an equity return over time, but it isn’t created to be a unique measure. You should use it in combination with other trading strategy and system performance metrics to measure the viability of an investment.

You can use the K-ratio to compare cumulative returns for different equities over a given time so as to determine the ones that are worthy of your hard-earned money. But it is not used for stocks alone; you can use the ratio to analyze other assets, fund managers, and trading strategies.

The history of K-ratio

The K-ratio was created by Lars Kestner in 1996. In a book called Quantitative Trading Strategies, he introduced the K-ratio as an alternative to the Sharpe Ratio. It goes something like this:

Slope / Standard Error, where the slope component accounts for return, and the error component accounts for risk.

The formula has undergone two modifications over the years: one in 2003 and another in 2013. In one of his papers, Kestner made this comment:

I introduced the K-ratio in 1996 as a reward to risk measurement to compliment the popular Sharpe ratio. The K-ratio is calculated by fitting a linear trend series to cumulative returns and estimating the slope and variability of the slope. Over the years there have been comments on adjustments factors needed to account for a varying number of return observations and return periodicity. In this paper, I show that the correct adjustments to the raw K-ratio include dividing by the number of return observations and multiplying by the square root of expected observations in a calendar year.

What is K-ratio? Conclusion

When you are looking at trading performance metrics, like the K-ratio, please make sure you understand what you are measuring. What is K-ratio? What does it measure? There is no plain right or wrong in trading and speculation, and thus you need to both understand and comprehend what you are measuring.

FAQ:

How is the K-ratio calculated?

The K-ratio is calculated by running a linear regression on the logarithmic cumulative return of a Value-Added Monthly Index (VAMI) curve. The formula is expressed as the slope of the logVAMI regression line divided by the standard error of the slope.

What does the K-ratio signify?

The K-ratio measures the return versus risk by analyzing the steadiness of a security’s, portfolio’s, or manager’s returns over time. The slope represents the return, while the standard error of the slope indicates the risk. A higher K-ratio implies better performance in both positive returns and consistency.

How can the K-ratio be used?

The K-ratio tests the consistency of an equity return over time. It can be used to compare cumulative returns for different equities, assess the viability of investments, and analyze other assets, fund managers, and trading strategies. However, it is recommended to use the K-ratio in conjunction with other performance metrics.